ON THE GEOMETRY OF RATE INDEPENDENT DROPLETEVOLUTIONWILLIAM M FELDMAN, INWON C KIM, AND NORBERT POŽÁR
Abstract. We consider a toy model of rate independent droplet motion ona surface with contact angle hysteresis based on the one-phase Bernoulli freeboundary problem. Taking advantage of two notions of weak solutions, energybased and comparison-principle-based, we study the dynamic contact angle ofmoving contact lines and the geometry of de-pinning. We show that these two
notions of solutions coincide in a star-shaped setting, where we show (almost)
optimal regularity of the contact line and the convergence of a minimizing
movements scheme. In a general setting, the notions differ essentially in how
they handle jumps, but both are shown to satisfy a weak motion law.

Contents
1. Introduction
2. Motivating examples
3. Regularity theory of Bernoulli obstacle problems
4. Local laws: weak solutions, regularity, and comparison principle
5. Basic space and time regularity properties of energy solutions
6. Limit of the minimizing movement scheme
7. The dynamic slope condition for energy solutions
Appendix A. Technical results
References

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63

1. Introduction
In this work we consider a toy model, based on the one-phase Bernoulli freeboundary problem, for the rate independent motion of droplets on a solid surface
with the effects of contact angle hysteresis. We study two related notions of weaksolution which handle time jump discontinuities differently: an energetic notion anda notion based on comparison principle and a free boundary obstacle problem. We
study the geometry of the contact line and the dynamic contact angle of moving
contact lines for these solutions. Our first main result is in a star-shaped setting.
We show equivalence between the typical minimizing movements energy solutions
and the obstacle solutions based on a novel comparison principle. We also show1
the optimal C 1, 2 − regularity of the contact line. Our second main result is in a
general setting. We show that both energy solutions and obstacle solutions satisfy
a weak motion law essentially reducing the difference of these two evolutions to
Key words and phrases. Rate-independent evolution, free boundaries, contact angle hysteresis,
free boundary regularity.
12

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

their jump laws. To our knowledge these are the first rigorous results establishingthese properties for a general class of weak solutions to a rate independent modelof droplet evolution.
It is well understood in the physics and engineering literature that capillary
droplets on many solid surfaces experience a phenomenon known as contact angle
hysteresis [18]. Rather than a single contact angle specified by the material properties, as would be predicted by the classical Young’s law in capillarity theory [32],there is a range of stable contact angles. Consequently droplets can “stick” to thesolid surface; the contact line (the curve separating the wet and dry regions) does
not move under small applied forces although the free surface (the liquid-air interface) does move. The origin and appropriate modeling of contact angle hysteresisis the subject of much debate in the engineering literature [18, 32].
Similar phenomena occur in the context of two phase flow in a porous medium,
called capillary pressure hysteresis in this context [16, 34, 37]. Typically the flow
velocity in each phase is determined by Darcy’s law and the phase interface moves
with velocity proportional to a pressure differential. However, when capillarity
forces at the interface, where individual grains of the medium are only partiallywetted, are of the same order as the pressure differential, pinning can occur. The
origin of this pinning is exactly the microscopic surface roughness of the matrix
medium and the associated contact angle hysteresis. The phase interface only
advances or recedes when the pressure differential exceeds a certain threshold.
The present paper is on the qualitative features of a macroscopic model for thesetypes of interface pinning phenomena. Our model is inspired by the one proposed
by DeSimone, Grunewald, and Otto [17], with theory developed by Alberti and
DeSimone [1].
Instead of the capillarity energy with slowly varying volume constraint, as considered in [1, 17], we study a quasi-static and rate independent evolution associatedwith the Alt-Caffarelli one-phase energy functional under a slowly varying Dirichlet
forcing. The mathematical reason for considering the one-phase problem insteadof the capillarity problem is to simplify various technical aspects of the problem,in particular giving us access to the more developed regularity theory of the onephase free boundary problem. Specifically, the work of Chang-Lara and Savin [12]on Bernoulli obstacle problems and its extension by Ferreri and Velichkov [22] willplay a major role. We emphasize that, as it concerns capillarity, the models weconsider in this work are toy models. Nonetheless, as is typically the purpose of toy
models, we expect the mathematical philosophies developed here to be relevant to
the true capillarity model.
Our contribution in this work is in connecting several energetic and geometricnotions of weak solution, to study the pointwise qualitative description of those solutions: (1) global energetic solutions (which we just refer to as “energy solutions”),(2) obstacle evolution solutions, and (3) motion law solutions, see Section 1.1 below
for definitions. Each of these notions of weak solution has different advantages, andonly by connecting the concepts can we obtain a more complete understanding of
the model.
Energy solutions can be defined in a quite general setting, but are diﬀicult towork with in terms of geometric properties and their jump law can be un-physical.
Among the general energy solutions, minimizing movements solutions generated
by a time incremental scheme behave especially well. Obstacle solutions on the

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

3uF{u > 0}u = u(x)
gas

U∁
u=F

liquidU∁{u > 0} ∂{u > 0} ∩ U
solid

x

∂{u > 0} ∩ U

Figure 1. Side view (left) and the top view (right) of the setup
for the one-phase free boundary problem.
other hand are easier to work with in terms of geometry, and have a desirable jumplaw, jumping “as late and as little as possible”. However the obstacle notion only
makes sense in the important special case of piecewise time monotone forcing andit is diﬀicult to derive energetic properties. These two evolutions have different
jump laws, but conceptually are linked by the idea that their motion at non-jumptimes is closely related. Both evolutions, at least formally, satisfy the dynamic
slope condition which governs the continuous motion but only partially specifies
the jumps.
In what follows we we introduce the notion of solutions that we will consider inthe paper, and then state our main results.
1.1. One-phase free boundary problem. Consider a connected domain U in
Rd with compact complement. For u : U → [0, ∞) consider the Alt–Caffarelli
one-phase free boundary energy functionalˆ
J (u) =|∇u|2 + 1{u>0} dx,(1.1)
U

where 1{u>0} is the indicator function of the set Ω(u) := {u > 0}.
In the context of capillarity modelling the droplet’s free surface (liquid–gas) is
given by the graph of u on U , and the positivity set Ω(u) is the wetted area:
the contact between the liquid and solid phase; see Figure 1. The triple junction
or contact line (liquid-gas-solid)
is located at the free boundary ∂Ω(u) ∩ U . The´
Dirichlet energy U |∇u|2 dx comes from a linearization of the surface area.
We recall that the Euler-Lagrange equation associated with the energy functional
J is the one-phase Bernoulli free boundary problem(∆u = 0in Ω(u) ∩ U(1.2)
2|∇u| = 1 on ∂Ω(u) ∩ U.
We are interested in rate independent evolutions associated with the energyfunctional J , driven by a Dirichlet boundary data F (t) : [0, T ] → (0, ∞) acting as
the external forcing. We augment the energy functional (1.1) by adding an energy
dissipation in a form of a dissipation distance: for any pair of sets Ω0 and Ω1 in Rd

4W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

we defineDiss(Ω0 , Ω1 ) = µ+ |Ω1 \ Ω0 | + µ− |Ω0 \ Ω1 |.
This non-symmetric distance measures the energy dissipated by the motion of thefree interface under (monotone) motion of the positive phase from state Ω0 to stateΩ1 . The coeﬀicients µ+ > 0 and µ− ∈ (0, 1) can be viewed as the friction forces perunit length of the free interface, respectively for advancing and receding regimes.
In what follows we will abuse notation and also write Diss(u, v) = Diss(Ω(u), Ω(v))
for the dissipation distance between the positive phase of two profiles u and v.
In the simple model we assume that free interface can move only if the “force”|∇u|2 −1 per unit length on the interface coming from the first variation of potential
energy (1.1) can overcome the static friction force µ+ or µ− , depending whether
the contact line advances or recedes. Furthermore, the scale at which the contactline moves is much faster than the scale on which we observe the state of thesystem, and therefore at each time t the interface is assumed to be in an equilibrium
configuration and cannot move. This state u(t) can be characterized as a local
minimizer of the augmented energy functional(1.3)

E(u, u′ ) := J (u′ ) + Diss(u, u′ ),that expresses the total of the potential energy J (u′ ) in an alternative state u′ andthe energy dissipated by the friction forces on the contact line required to reach thealternative state.
The formal first variation of the augmented energy function E(u, ·) results in thepinned one-phase free boundary problem(∆u = 0in {u > 0} ∩ U,
(1.4)
1 − µ− ≤ |∇u|2 ≤ 1 + µ+ on ∂{u > 0} ∩ U.
Now the gradient is allowed to take a range of values on the interface, this is the
manifestation of contact angle hysteresis in this model as “gradient hysteresis”.
The reason why the state might change with t is that it is subject to the forcing
by Dirichlet boundary conditionu(t) = F (t)on ∂Uat each t ≥ 0. Since we can observe the state only in an equilibrium, the evolution
is rate-independent in the sense that the path does not depend on how fast Fchanges, that is, the time variable can be monotonically reparametrized yielding anequivalent evolution.
Varying F (t) pulls up (or down) the profile u(t) but the free boundary remains
pinned as long as the gradient at the free boundary is within the pinning intervalin (1.4). Once the gradient saturates one of the endpoints in the pinning condition(1.4) somewhere, the interface needs to move. And, indeed, the free boundary only
advances or recedes when the gradient saturates the corresponding endpoint of thepinning interval. This heuristic suggests the dynamic slope condition(1.5)|∇u(t)|2 = 1 ± µ±if

± Vn (Ω(u(t)), x) > 0

on ∂Ω(u(t)) ∩ U

where Vn (Ω(u(t)), x) is the outward normal velocity of Ω(u(t)) at x ∈ ∂Ω(u(t)).
This condition is analogous to the dynamic contact angle condition which is studied
in the physics of capillarity.

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

5

1.2. Rate independent evolution rules. Now we proceed to detailed descriptions of the (1) energy solutions, (2) obstacle solutions, and (3) motion law solutions.
Energetic weak solutions. The view of the evolution as a continuum of stablestates of the energy functional E motivates the first notion of solutions of the rateindependent evolution of the Bernoulli functional. However, the local minimality
is strengthened to a global one, as is common in the theory of rate-independent
systems [1, 30]. Additionally, an evolution between any two states at two times
along the solution must be energetically allowed: the dissipated energy required toreach the second state from the first cannot be larger than the difference of potential
energies of the two states together with the work done by the external forcing.
Definition 1.1. A measurable u : [0, T ] → H 1 (U ) is a energy solution (E) of thequasistatic evolution problem driven by Dirichlet forcing F if the following hold:(1) (Forcing) For all t ∈ [0, T ]u(t) = F (t) on ∂U(2) (Global stability) The solution u(t) ∈ H 1 (U ) and satisfies for all t ∈ [0, T ]:J (u(t)) ≤ J (u′ ) + Diss(u(t), u′ )for all u′ ∈ u(t) + H01 (U ).

(3) (Energy dissipation inequality) For every 0 ≤ t0 ≤ t1 ≤ T it holdsˆ t1
2Ḟ (t)P (t) dt ≥ Diss(u(t0 ), u(t1 )).
(1.6)
J (u(t0 )) − J (u(t1 )) +Here P (t) = P (u(t)) =

t0

´
∂U

∂u(t)
∂n dS is an associated pressure.

In some works, for example [1], energy solutions are required to satisfy an energy dissipation balance condition instead of the inequality we use. That notion isequivalent to ours, see Remark 5.8 later for more details.
The Euler-Lagrange equations for the global stability condition (2) is the pinned
one-phase problem (1.4). Taking the time derivative in the energy dissipation balance, see (5.4) and (7.1) below for details of the computation, shows that smoothenergy solutions satisfy the dynamic slope condition (1.5). Justifying this formal
computation for general energy solutions is quite diﬀicult and is the content ofTheorem 1.6 below.
Minimizing movements scheme. The most typical way to construct an energy solution (E) is based on a time-incremental or minimizing movements scheme. Consider
the time-discrete approximation scheme, w) : w ∈ F (kδ) + H01 (U ) .
ukδ ∈ argmin J (w) + Diss(uk−1(1.7)δ
Using piecewise constant interpolation define, for all t ∈ [0, T ],(1.8)

uδ (t) := ukδ and Fδ (t) = F (kδ) if t ∈ [kδ, (k + 1)δ).

The time incremental scheme does turn out to produce energy solutions via a compactness idea introduced by Mainik and Mielke [27].
Definition 1.2. Say that u : [0, T ] → H 1 (U ) measurable is a minimizing movements energy solution if there is uδ (t) solving the scheme (1.7) and a subsequenceδk → 0 so that uδk (t) → u(t) in L2 (U ) for every t ∈ [0, T ].

6

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

Note that, despite the terminology, it definitely requires proof to show thatminimizing movements energy solutions actually solve (E). Our results for minimizing movements solutions are stronger than for general energy solutions of (E),see Theorem 1.5.
Geometric weak solutions: obstacle solution. Next we introduce an, apparently, quite different notion of rate independent evolution associated with the
one-phase problem which we call the obstacle solution. We will see that this notion
is physically quite natural, but it is diﬀicult to classify the evolution in terms ofenergetics, see Section 1.4 for discussion of possibly related energetic notions.
This notion of solution does rely on a certain monotonicity hypothesis on theforcing. More precisely, we assume that the forcing F : [0, T ] → (0, ∞) is Lipschitz
regular and only changes monotonicity on a finite set Z ⊂ [0, T ], that is, for any(s, t) ⊂ [0, T ] \ Z the forcing F is either strictly increasing or strictly decreasing on[s, t].
Definition 1.3. We say that u : [0, T ] × U → [0, ∞) is a obstacle solution (O) in
U driven by F on ∂U if(1) (Initial data) u(0) is a viscosity solution of the local stability conditions(1.4).
(2) (Dirichlet forcing) For all t ∈ [0, T ](1.9)u(t) = F (t) on ∂U.

(3) (Obstacle condition) For every (s, t) ∩ Z = ∅, so that F is monotone on[s, t], u(t) is the minimal supersolution of (1.4) and (1.9) above u(s) when
F is increasing on [s, t] (resp. maximal subsolution below u(s) when F isdecreasing).
The notions of minimal supersolution and maximal subsolution above are in
the viscosity solutions / Perron’s method sense. The stability condition on theinitial data is for convenience, otherwise the solution would jump at the initial time
resulting in a “replacement” initial data arising from a Bernoulli obstacle problem.
Geometric weak solutions: dynamic slope condition. Although the obstacle
solution (O) and the energetic solution (E) have different jump laws, our philosophy
is that they should “only” differ due to the jump laws. The commonality of these
two will be expressed in the notion of motion law solution. The motion law solution
property imposes a local stability condition and a dynamic condition on the slope.
In this model the local evolution law can be naturally interpreted in the framework of viscosity solutions (i.e. solutions based on the comparison principle). This
motivates the notion of motion-law viscosity solution.
Definition 1.4. u : [0, T ] → C(U ) is a motion-law viscosity solution (M) if(1) (Forcing) For all t ∈ [0, T ]u(t) = F (t) on ∂U.
(2) (Local stability condition) For all t ∈ [0, T ] the function u(t) ∈ C(U ) is acontinuous viscosity solution of (1.4) in the semicontinuous envelope sense.
(3) (Dynamic slope condition) u is a viscosity solution of (1.5) on [0, T ] × U in
the semicontinuous envelope sense.

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

7

We recall viscosity solutions in Section 4, and in particular the precise meaningof the local stability condition in Definition 4.11 and the dynamic slope conditionis given in Definition 4.15.
1.3. Statements of main results. Our first collection of results is on the uniqueness and regularity properties of the three evolutions in the case of strongly starshaped data. Star-shapedness is a natural topological hypothesis in the context
of free boundary problems which often allows for uniqueness and regularity andprevents topological changes. In our context star-shapedness also rules out jumps
and so we can show a significant (but not total) overlap of the three notions ofsolution considered in this work.
Theorem 1.5 (see Theorem 4.17 and Theorem 6.6). Suppose that Rd \ U andΩ0 are strongly star-shaped and bounded, Ω0 is C 1,α , and F : [0, T ] → (0, ∞) is
Lipschitz and only changes monotonicity finitely many times. Let u be the uniqueobstacle solution (O) on [0, T ] with a strongly-star-shaped initial data u(0), andu(0) satisfies (1.4). Then(i) The profile u(t) is strongly star-shaped for each time, and it is the uniquemotion law viscosity solution (M) with initial data u(0).
(ii) The profile u(t) and positivity set Ω(u(t)) have the regularity Ct0,1 Cx ∩1,min{ 1 ,α}−

2
L∞
.
t Cx
(iii) The unique (M) solution u(t) is also the unique minimizing movement
energy solution with initial data u(0). Moreover the solutions of the discretetime minimizing scheme converge uniformly to u(t) with a uniform rate that
only depends on F , µ± and d.

Besides the regularity theory developed in Section 3, the novel comparison principle for solutions of (M), Proposition 4.26, also plays a central role in this theorem. The convergence result for minimizing movements solutions part (iii) follows
a similar idea to Chambolle’s proof of convergence of the Almgren-Taylor-Wang /Luckhaus-Sturzenhecker schemes for mean curvature flow [2, 11, 26]. We establishthat the discrete-time minimizing movements satisfy (M) and we use the aforementioned comparison principle.
The star-shapedness is a key hypothesis in Theorem 1.5. Perhaps most importantly, the star-shaped geometry is used to ensure that there is no jump in the
obstacle solution (O). This is crucial to obtain part (iii), since the energy solutions and obstacle solutions have different jump laws, see Section 2.2. The occurrence of jumps is also exactly related to non-uniqueness of the exterior obstacle
Bernoulli problem and the (M) problem. The local cone monotonicity implied bystar-shapedness is also used to obtain the regularity result in (ii). Cone monotonicity allows us to show, in Section 3, that all blow-ups at the free boundary
are half-plane solutions. Then we can invoke the “flat means smooth” results of[12, 22] for Bernoulli obstacle problems. All these properties are consistent with theidea that the energy landscape on the star-shaped sets has a convexity property, it
would be interesting to make this precise.
Next we address the general case without the topological restrictions of thestar-shaped setting. In Section 4 we show that all non-degenerate (O) solutions,regardless of geometry, satisfy (M), see Lemma 4.20. But whether general energy
solutions are (M) is unclear: since jumps are now possible, (O) and (E) are typicallydifferent evolutions. Nonetheless we still show that energy solutions satisfy (M)8W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

in some sense. More precisely, our second main result says that general energy
solutions still satisfy the dynamic slope condition in a geometric measure theoretic
sense
Theorem 1.6 (see Proposition 5.16 and Theorem 7.1). Suppose u is an energy
solution on [0, T ]. Then(i) (Basic regularity properties) The states u(t) are uniformly Lipschitz andnon-degenerate and Hd−1 (∂Ω(t)) is uniformly bounded in time. Also t 7→Ω(u(t)) is in BV ([0, T ]; L1 (R)) and u(t) has left and right limits in uniform
metric at every time, denoted uℓ (t) and ur (t).
(ii) (Upper and lower envelopes) The upper and lower semicontinuous envelopesof u, called u∗ and u∗ , are themselves energy solutions and actually u∗ (t) =max{uℓ (t), ur (t)} and u∗ (t) = min{uℓ (t), ur (t)}.
(iii) (Dynamic slope condition a.e.) For all t ∈ [0, T ] the function u(t) satisfies
the stability condition (1.4) and satisfies (in terms of u∗ and u∗ ) the dynamic slope condition (1.5) at Hd−1 almost every point of its free boundary∂Ω(u(t)) ∩ U .
The regularity properties in (i) follow from relatively standard energy arguments,
we are recalling them here for context. The precise description of the semicontinuous envelopes (ii) is a somewhat unusual result in the context of literature on
rate-independent motion. Our analysis of the energetic properties of the upper andlower envelopes in (ii), and see Proposition 5.16 below, is more typical of the theoryof discontinuous viscosity solutions [13] and plays an important role in the proof ofthe dynamic slope condition.
1.4. Discussion. In order to contextualize our results we now explain further
about the advantages and disadvantages of the different notions of solution andtheir interrelations.
Discussion of the energy solution notion. Energy solutions have the advantage of“baking in” the energy dissipation balance and this notion of solution generalizes
better to the more physically relevant case of volume constrained evolution. There
may also be advantages of the variational framework in studying questions related
to homogenization.
On the other hand it is diﬀicult to derive geometric properties of the evolution
purely from the energetic structure. We are not aware of any purely energetic
method to obtain higher regularity of the free boundary. Even seemingly simple
properties are diﬀicult to prove just from energetics, for example it is not clear
whether arbitrary (E) solutions must respect time monotonicity of the Dirichlet
forcing.
The central diﬀiculty, in general, with energy solutions is to find a suitableglobally defined dissipation distance. In the isotropic case that we consider in thispaper there is a simple and natural dissipation distance. Such a quantity is notalways available. For example, it is not clear how to define any global dissipation
distance in the case of anisotropic media arising from periodic homogenization[9, 19, 21, 25]. Furthermore, even with a mathematically elegant choice of global
dissipation distance, as we have, the energy evolution handles jump discontinuities
in a “non-physical” way. This is a well-known and common shortcoming of the,so-called, global energetic solutions in the dissipative evolution framework [28].

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

9One of main proposals, within the energetic framework, for improving the “locality” of the jumps is the notion of balanced viscosity solution [14, 28, 29, 31] (different from the comparison viscosity solutions we consider below) and see the book[30, Chapter 3.8.2]. This notion is based on a modification of the time incremental
minimization scheme (1.7), adding an L2 -type distance term in the incremental
minimization problem which strongly penalizes jumps. The balanced viscosity approach leads to a notion of energetic solution which jumps “as late and as little aspossible”.
In the context of the literature on rate independent motions our work is novelin that we can take advantage of certain monotonicity and comparison structuresin the problem in order to (1) give a simple non-energetic characterization of thesolutions which jump “as late and as little as possible”, (2) display the optimal
regularity of energy solutions induced by the dynamic slope condition.
Discussion of the obstacle solution notion. The obstacle solution (O) notion is well
suited to study the regularity theory and qualitative geometric features of the freeboundary.
In particular using this notion, in Section 3 and Section 4, we can study theshape of the free boundary as it de-pins. We will use regularity results for Bernoulli
obstacle problems due to Chang-Lara and Savin [12] and very recently extended
to the obstacle from below case by Ferreri and Velichkov [22]. The regularity at
de-pinning turns out to be governed, at first order in the flat asymptotic expansion,1
by a thin obstacle problem. The thin obstacle problem is well known to have C 1, 2
1
optimal regularity and we inherit C 1, 2 − regularity for obstacle solutions in thestar-shaped case, see Theorem 1.5 below.
The other advantage of the obstacle solution, even outside of the star-shaped
case, is that it handles time jump discontinuities well. Specifically, the obstacle
solution jumps “as late and as little as possible”, which is regarded as a more
physical jump condition the same as the notion of balanced viscosity solution, seefor example [28] or [30, Chapter 3.8.2]. The obstacle solution dissipates the “right”
amount of energy on its jumps, but does not obviously yield an energetic notion ofjump dissipation. It would be interesting to study the possible connection of thebalanced viscosity notion with our obstacle solution (O), but we do not addressthat in this work.
Finally we mention that the problem (O) has an easy uniqueness property fromits definition, although establishing local conditions equivalent to the minimality /maximality properties in (O) is not so easy in general, see Section 4.
Discussion of the motion law solution notion. The concept of the motion law solution property, which we can partially but not fully justify in this work, is that the
motion law viscosity solution property contains all the information in (E) or (O)except for the jump law. We do not have a way to make this concept fully precise,but our results are in support of this notion.
The motion-law viscosity solution property, philosophically speaking, contains
significantly less information than either the notion of energy solution or the notionof obstacle solution. The (M) property only imposes a very mild condition on thejumps via (1.5). Specifically that the dynamic slope is saturated at any point thathas jumped strictly inwards or outwards. This property is indeed satisfied by both
energy solutions and obstacle solutions.

10

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

Despite this philosophy it is still highly nontrivial to actually prove the (M)property, especially for general energy solutions. In Lemma 4.20 we show that (O)solutions do satisfy (M). And our second main result, Theorem 1.6, shows thatthe dynamic slope condition of (M) at least holds in the surface measure almost
everywhere sense.
1.5. Acknowledgments. W. Feldman was partially supported by the NSF grantDMS-2009286. I. Kim was partially supported by the NSF grant DMS-2153254. N.
Pozar was partially supported by JSPS KAKENHI Kiban C Grant No. 23K03212.
1.6. Notations and conventions. We list several notations and conventions which
will be in force through the paper.
▷ We call a constant universal if it only depends on d and µ+ > 0, µ− ∈ (0, 1),
a constant is called dimensional if it only depends on the dimension.
▷ We will refer to universal constants by C ≥ 1 and 0 < c ≤ 1 and allow such
constants to change from line to line of the computation.
▷ Denote a ∨ b = max{a, b) and a ∧ b = min(a, b), respectively, for the maximum and minimum of two real numbers a and b.
▷ We often abuse notation and write Ω(t) instead of Ω(u(t)) etc.
▷ u∗ and u∗ denote the upper-semicontinuous envelope and the lower-semicontinuous
envelope of u, respectively, see (4.4). We write USC and LSC respectively
as shorthand for upper semicontinuous and lower semicontinuous.
▷ F + H01 (U ) refers to the space of functions in H 1 (U ) with trace F on ∂U .
2. Motivating examples
In order to introduce the problem and motivate the phenomena we will study inthe paper we present several examples with analytical computations and numericalsimulations.
2.1. Numerical simulations. The obstacle solutions (O) can be relatively easily
approximated by a large time limit (τ → ∞) of a dynamic contact angle problem:
Suppose that F is increasing on [s, t]. Given u(s), u(t) can be found as the limitτ → ∞ of the unique, monotone solution of the free boundary problem−∆w(τ ) = 0,
in {w(τ ) > 0} ∩ U,




1/2

Vn = max(|∇w| − (1 + µ+ ) , 0) on ∂{w(τ ) > 0} ∩ U,
w(τ ) = F (t),on ∂U,
{w(0) > 0} = {u(s) > 0}.
Analogously, for F decreasing on [s, t] we replace the velocity law by Vn = min(|∇w|−(1 − µ− )1/2 , 0).
A numerical solution can be found by adapting the level set method introducedin [23], stopping at τ when |Vn | < ε for some small parameter ε > 0. Plots in
Figure 2 were produced this way.
2.2. Jump conditions for (global) energetic vs obstacle solutions. Next
we consider an example where the jump time for the energy solutions is different
from the jump time for the obstacle evolution. Heuristically speaking the obstacle
evolution solutions jump as late and as little as possible, while the (global) energy
solutions jump whenever it becomes energetically favorable.

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

11

Consider a domain U which is the complement of two disjoint closed disks andinitial data given by two disjoint annuli. Then under increasing F (t) the (O)solution will consist of two disjoint annuli until the value of the forcing when theboundaries of the two annuli meet at a single point. As F continues increasing pastthat critical value the solution will need to jump outwards to a new state. Thissituation is depicted in the simulations in Figure 2.
The energy solution with the same data and forcing must jump before the valueof the forcing when the two annuli touch. One way to see this is that the blow-up
at the touching point of the annuli is a two plane solution of the formv(x) = (1 + µ+ )1/2 |x · e|.
However this blow-up does not satisfy the global stability condition in (E), since
the harmonic replacement in any open region has the same positivity set but lower
Dirichlet energy. Since any blow-up of a globally stable state is also globally stable
we conclude that no energy solution can coincide with the obstacle solution all theway to the jump time.
2.3. Convexity is not preserved under the obstacle evolution. Consider the
Bernoulli free boundary problem in the complement of a convex obstacle K−∆u = 0 in {u > 0} \ K, with |∇u| = 1 on ∂{u > 0} \ K.
It is known, see Henrot and Shahgholian [24], that the (unique) compactly supported solution of this problem has convex super-level sets. In particular if one
considers the obstacle solution (O) without pinning µ± = 0, then the solution u(t)of the obstacle evolution (which depends only on the current value F (t) due to thelack of hysteresis) is convex at all times.
It is natural to ask whether convexity is still preserved under (O) in the caseof nontrivial pinning interval µ± > 0. In fact it is not. We give a simulation of acounterexample in Figure 2 and present a sketched proof here.
Consider the case of K = B1 , F (0) = 1, and an initial region Ω0 which is a
“stadium” type initial data which is a large portion of the strip −a < x2 < acapped off by two circles of radius a centered at (±b, 0). Here 0 < a − 1  1and b  1. The initial region Ω0 and K uniquely determine the initial profile u0 ,which does have convex super-level sets due to [10]. Fix the pinning interval by the
relationsµ+ := max(|∇u0 |2 − 1) and µ− := max(1 − |∇u0 |2 ).
∂Ω0

∂Ω0

By symmetry considerations the first maximum is achieved at (0, ±a) and the second at (±(b + a), 0). We can guarantee that both maxima are positive by choosing
the a close to 1 and b large. Now increasing F (t) slightly above 1 the free boundaryneeds to immediately move outwards near (0, ±a) since the slope there is alreadysaturated at |∇u0 (0, a)|2 = 1 + µ+ , but since this is exactly the maximum valueof the slope the outwards movement will only be in a small neighborhood of those
two points. This motion must produce nonconvexity because the domain Ω(t) will
have outward normal e2 both at some point (0, a + δ(t)) and at (b, a).
This argument could be justified rigorously using Theorem 1.5.
2.4. Hysteresis. In order to understand the macroscopic effect of the hysteresis
inherent in the slope pinning condition we consider a simple explicitly solvable

12

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

Figure 2. Plots of boundaries of obstacle solution (O) simulations. Solid curves represent ∂Ω(t) ∩ U plotted for evenly spaced
values of F (t), with the initial shape dashed. ∂U is given by adotted curve. Top left: Disconnected annuli initial data, jump discontinuity on touching. Top right: Receding situation (decreasing
F (t)) with initial data given by the last step of the top left image.
Note that the jump occurs at a different configuration, as late as
possible. Bottom left: Different radius annuli, free boundary peelsfrom the larger annulus after the jump. Bottom right: Stadium
type initial data, convexity is not preserved.
radially symmetric example in dimension d = 2. See Figure 3 which will be further
explained below.
Denote the family of radially symmetric solutions Vλ,F of ∆Vλ,F = 0 in Ω(Vλ,F ) \ B1 ,Vλ,F = F on ∂B1 ,|∇Vλ,F | = λ on ∂Ω(Vλ,F ) \ B1 .
These can be explicitly computed as
Vλ,F (x) = F

1−

log |x|log ζ(λ−1 F )+where ζ : (0, ∞) → (1, ∞) strictly monotone increasing is the inverse of R 7→ R log Ri.e.
ζ(s) log ζ(s) = s for s > 0.
Note that the radius of the support of Vλ,F is R = ζ(λ−1 F ).

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

u

13

u

F1

F1

1
σ F1

σF0
F0
R0

R1

F

γ+

F0

|x|

R0

γ−
F1

1
σ F1
σF0

1
σ F1
σF0

R

|x|

F

F1

F0

R1

F0

R0 R1

t
T

Figure 3. Hysteresis loop for radial solutions. Top left: profiles ofadvancing solution starting from (R0 , F0 ) ∈ γ− . Top right: profilesof receding solution starting at (R1 , F1 ) ∈ γ+ . Bottom left: hysteresis loop diagram in (R, F )-plane. Bottom right: plot of F (t).
Parameters used to generate top and bottom set of pictures do
not exactly match in order to better display the respective graphs.
1+µ+ 1/2
The factor σ is defined to be 1−µ.
−
Now let’s consider the radially symmetric solution of (M) in U = R2 \ B1 withsome forcing F (t) and
u0 = V(1−µ− )1/2 ,F0 .
The solution must be of the formu(t) = Vλ(t),F (t) with λ(t)2 ∈ [1 − µ− , 1 + µ+ ]and the dynamic slope condition becomesdd± R(t) = ± ζ(λ(t)−1 F (t)) > 0 implies λ(t) = 1 ± µ± .
dt
dt
In other wordsd± R(t) > 0 implies R(t) = ζ((1 ± µ± )1/2 F (t)).
dt
or viewing matters in the (R, F ) plane the state of the system is always in theregion


−1/2
−1/2
S = (R, F ) ∈ [1, ∞) × (0, ∞) : ζ((1 + µ+ )F ) ≤ R ≤ ζ((1 − µ− )F)

14

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

and R can only increase/decrease while on the respective boundary curves
γ± = {(R, F ) : R = ζ((1 ± µ± )−1 F )}.
3. Regularity theory of Bernoulli obstacle problems
In this section we consider a pair of obstacle problems for the Bernoulli freeboundary problem, one with an obstacle from above and the other with an obstacle
from below. The two problems are similar but not exactly symmetric, as we willsee below in the analysis. The regularity theory of the problem with obstacle fromabove has been developed by Chang-Lara and Savin [12], and as we were finishingpreparing this paper their theory has been extended to the obstacle from below
case by Ferreri and Velichkov [22]. We present the problems, recall the flat implies
smooth regularity results from the above-mentioned works, and then show how toachieve the initial flatness and full regularity under a cone monotonicity hypothesis
which is appropriate for our work.
3.1. Bernoulli obstacle problems. Let U be an open region, the domain. We saythat u is a solution / supersolution / subsolution of the (unconstrained) Bernoulli
free boundary problem in U if(
∆u = 0in {u > 0} ∩ U(3.1)|∇u| = 1 on ∂{u > 0} ∩ U.
Let the obstacle O be another open region with C 1,α boundary.
Definition 3.1. A function u ∈ C(U ) is a solution of the Bernoulli problem in U
with obstacle O from below if∆u = 0in {u > 0} ∩ Uu > 0in O ∩ U
(3.2)

|∇u|
=
1
on (∂{u > 0} \ O)|∇u| ≤ 1 on Λ := (∂{u > 0} ∩ ∂O)
Definition 3.2. A function u ∈ C(U ) is a solution of the Bernoulli problem in U
with obstacle O from above if∆u = 0in {u > 0} ∩ Uu = 0in U \ O
(3.3)

|∇u|
=
1
on
(∂{u > 0} ∩ O)|∇u| ≥ 1 on Λ := (∂{u > 0} ∩ ∂O).
See Figure 4 for depictions of obstacle solutions.
3.2. Additional hypotheses. The obstacle below (resp. above) problems do notprovide any a-priori bound on the slope at the free boundary from below (resp.
above). Given a regular domain O and a specific boundary data on ∂U one could
establish such bounds on the interior. However it is more convenient for us to justlist these additional bounds as hypotheses which will be in force for some (but notall) of the statements below. Let 0 < κ < 1 and consider the hypothesis: in the
obstacle from below case(3.4)|∇u| ≥ κ in the viscosity sense on ∂{u > 0} ∩ U,

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION|∇u| = 1{u = 0}O|∇u| ≥ 1

15|∇u| ≤ 1O{u = 0}|∇u| = 1

Figure 4. Left: Obstacle from above, slope is larger than 1 everywhere and saturates where free boundary bends into O. Right:
Obstacle from below, slope is smaller than 1 everywhere and saturates where free boundary bends away from O.
and in the obstacle from above case(3.5)|∇u| ≤ κ−1 in the viscosity sense on ∂{u > 0} ∩ U.

3.3. Existence and typical examples. An indicative example of u solving a
Bernoulli problem with obstacle from below (3.2) is the Perron’s method obstacle
minimal supersolution(3.6)
u(x) := inf{v(x) : v is a supersolution of (3.1), v = g on ∂U , and v > 0 in O ∩ U }.
Here g ∈ C(∂U ) is some boundary condition. In order for the minimal supersolution
to also be positive on O we need to put some additional condition on g. Noticethat every supersolution in (3.6) is above w, the solution of the Dirichlet problem∆w = 0 in O ∩ U, w = 0 on ∂O ∩ U, and w = g on ∂U ∩ O.
So if w > 0 in O then u will be positive in O as well. In particular it would suﬀice
to assume thatg > 0 on O ∩ ∂U.
The obstacle solution property follows from the typical Perron’s method arguments.
Local upward perturbations are possible everywhere in U , so u is a supersolution
everywhere, while local downward perturbations are possible only away from ∂{u >0} ∩ O limiting the subsolution property.
An indicative example of u solving a Bernoulli problem with obstacle from above(3.3) is the obstacle maximal subsolution(3.7)
u(x) := sup{v(x) : v a subsolution of (3.1) in U , v = g on ∂U , and {v > 0} ⊂ O}.
For consistency the positivity set of g on ∂U should be contained in O ∩ ∂U . As
before Perron’s method applies, arbitrary downward perturbations are possible so u
is a subsolution everywhere, but local upward perturbations are only allowed awayfrom ∂{u > 0} ∩ O.
Remark 3.3. As in [12] it is also possible to construct solutions to (3.3) by minimal
supersolution Perron’s method, or by energy minimization. However, it is the
maximal subsolutions that we will encounter in this work. This is something we16

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

need to be careful with, since non-degeneracy at the free boundary is a more delicate
issue for maximal subsolutions.
3.4. Flat implies C 1, 2 − and regularity of cone monotone solutions. First
let us recall the flat implies C 1,1− regularity away from the obstacle. This result isoriginally due to Caffarelli [7,8]. A more recent alternative proof by De Silva [15] has
motivated the techniques used to study the obstacle problems we are considering..
1

Theorem 3.4. (flat implies C 1,1− for unconstrained Bernoulli [7, 8]) Let u solves(3.1) in B1 and 0 ∈ ∂{u > 0}. For any β ∈ (0, 1) there is ε0 > 0 and C ≥ 1
universal so that if(x · e − ε)+ ≤ u(x) ≤ (x · e + ε)+ in B1
for some ε ≤ ε0 , then for all 0 < r < 1 and some |∇u(0)| = 1(x · ∇u(0) − Cεr1+β )+ ≤ u(x) ≤ (x · ∇u(0) + Cεr1+β )+ in Br .
Next we recall a similar flat implies smooth result on the contact set between thesolution and the obstacle. Consider the contact set of the free boundary with the
obstacle Λ := ∂{u > 0} ∩ ∂O. Let us denote ∂ ′ Λ to be the boundary of Λ relative
to ∂O.
Theorem 3.5. (flat implies C 1, 2 − at the contact set [12, 22]) Let u solves either(3.3) or (3.2) in B1 and 0 ∈ ∂ ′ Λ, O a C 1,α obstacle, and e the inward normal to
O at 0. For any 0 < β < min{α, 12 } there is ε0 > 0 and C ≥ 1 universal so that if
1(x · e − ε)+ ≤ u(x) ≤ (x · e + ε)+ in B1
for some ε ≤ ε0 , then for all 0 < r < 1(x · e − Cεr1+β )+ ≤ u(x) ≤ (x · e + Cεr1+β )+ in Br .
In both the obstacle above and obstacle below cases the proof is based on establishing an asymptotic expansion for flat solutions of the formu(x) ∼ (x · e + εw(x) + o(ε))+
where w is a solution of a certain Signorini or thin obstacle problem and ε is the1
flatness in B1 as in the hypothesis of Theorem 3.5. The C 1, 2 optimal regularity for1
the Signorini problem is the reason for the C 1, 2 − regularity which appears here,
and is likely (almost) optimal. The idea of the argument, which is based on acompactness principle using a special Harnack inequality for flat solutions, goesback to the work of De Silva [15].
It is important for us to obtain a full regularity result without the flatness assumption. Of course some assumption is still necessary, singular solutions exist
in higher dimensions, and fitting with the strongly star-shaped geometry we study
later in the paper we will consider obstacle solutions satisfying a cone monotonicity
condition. For the statement define(3.8)

Cone(e, θ) := {e′ : e′ · e ≥ 1 − θ} for e ∈ S d−1 and θ ∈ (0, 1).

Theorem 3.6. Suppose O is a bounded set with C 1,α boundary, and that u solves
either (3.3) and (3.5) or (3.2) and (3.4) in B1 . Assume also that there are e ∈ S d−1
and 1 ≥ c0 > 0 so that u is monotone increasing in the directions of the coneCone(e, c0 ).

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

17

Then for any 0 < β < min{α, 12 } the positivity set {u > 0} is a C 1,β domain in
B1/2 and u ∈ C 1,β ({u > 0} ∩ B1/2 ). The C 1,β norms depend only on α, κ, c0 , d,and the C 1,α norm of ∂O.
1

Remark 3.7. Chang-Lara and Savin [12] actually prove optimal C 1, 2 regularity.
This matches the regularity of the thin obstacle / Signorini problem, which appears
as the first order term in the asymptotic expansion for ε-flat solutions in the flatness
parameter ε. Since this optimal regularity requires significantly more work, and it
is not necessary for our purposes here, we will not address that question for the
obstacle from below problem.
We need to establish the initial flatness in order to apply Theorem 3.4 and/or
Theorem 3.5 [12]. This is a bit different in our case from the way that initial flatness
is established in [12] (minimal supersolutions) or [22] (one-sided energy minimizers).
The first difference is in the obstacle from below case. This issue appears inLemma 3.8, where we establish the initial flatness of u at points of ∂ ′ Λ based on
the existence of a non-tangential slope. In the case of the obstacle from above (3.3)
the positive set of u lies inside of O which has smooth boundary. This means thata non-tangential gradient of u at the free boundary point in contact with ∂O can
describe the leading-order behavior of u near the point in the entire positive set.
This is no longer true with (3.2), where the positive set contains O. This motivatesthe cone monotonicity hypothesis on u in the obstacle from below case.
The other difference is in the obstacle from above case. Unlike [12], who consider
minimal supersolutions and energy minimizers, we are interested to study maximal
subsolutions. Non-degeneracy at the free boundary is not known in general for
maximal subsolutions. However, in the case when the free boundary is a Lipschitz
graph, in particular in the cone monotone case, non-degeneracy does hold for all
viscosity solutions including the maximal subsolution, see Lemma A.1 below. This
motivates the cone monotonicity hypothesis on u in the obstacle from above case.
In both cases the cone monotonicity hypothesis is probably overly strong, however it suﬀices for the applications in this paper. The possibility of a more generalregularity result is an open question.
We also remark that Lipschitz regularity is easier than non-degeneracy and justfollows from the viscosity supersolution property (3.5), see Lemma 5.5 below.
3.5. Initial free boundary flatness. In order to obtain the initial flatness wewill show the blow-up limit at a free boundary point on the contact set Λ exists
and is a half-planar supersolution of the unconstrained problem (3.1).
Lemma 3.8. Assume that u is a solution of (3.2) and (3.4) (or (3.3) and (3.5))
in B1 which is monotone with respect to the directions of a cone Cone(e, c0 ), thenu(rx)= (∇u(0) · x)+ locally uniformly in Rdrwith some gradient c(d, κ) < |∇u(0)| ≤ 1 (resp. 1 ≤ |∇u(0)| ≤ C(d, κ)). In
particular, for any ε > 0 there exists r0 > 0 suﬀiciently small (depending on u) sothat for any r ≤ r0
(3.9)

(3.10)

lim

r→0|∇u|(0)(xd − εr)+ ≤ u(x) ≤ |∇u|(0)(xd + εr)+ in Br .

Proof. We will just consider the obstacle from below (3.2), the obstacle above caseis in [12, Lemma 2.6]. The idea is that the interior ball condition furnished by the18

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

O0

xn = 0∂{u > 0}Figure 5. Asymptotic expansion in non-tangential cone plus
monotonicity also gives control in {xn ≤ 0}.
regular obstacle gives a non-tangential blow-up limit and then the cone monotonicity upgrades this to a local uniform limit on the whole space.
Note that u is harmonic in its positivity set, {u > 0} ⊃ O in B1 and O is a
half-space in B1 . Therefore 0 is an inner regular point so there is a slope ∇u(0),parallel to the inward normal ed to O at 0, such thatu(x) = (∇u(0) · x)+ + o(|x|) as x → 0 non-tangentially in O.
See [6, Lemma 11.17] for the proof, which does extend to the case of Hölder continuous coeﬀicients. By non-degeneracy, Lemma A.1, and the Lipschitz estimate,standard recalled in Lemma 5.5 below, |∇u(0)| is bounded from below away fromzero, and from above.
Here is where we need the cone monotonicity property. The non-tangential
information, by itself, does not give us any control in B1 \ O.
From the non-tangential limit, for any ε > 0, there is r0 > 0 such thatu(x) ≤ 2|∇u(0)|εr on {xd = εr} ∩ Br for all r ≤ r0 .
Since u is monotone increasing in the ed direction alsou(x) ≤ 2|∇u(0)|εr on {xd ≤ εr} ∩ Br for all r ≤ r0 .
See Figure 5.
Thus limr→0 u(rx)= 0 for xd ≤ 0 and we have upgraded the non-tangentialrlimit to local uniform convergence of the blow-up sequence (3.9). Then the uniform
stability of viscosity solutions implies that |∇u(0)| ≤ 1.
Using non-degeneracy again we can prove the convergence of the free boundaries
as well (3.10).
□
If we use Lemma 3.8 as stated then the initial flatness radius r0 would depend
on u is a non-universal way. In turn the C 1,β norm of the solution depends on theinitial flatness, and would also depend on u in a non-universal way. Next we show,
using the flat implies smooth results, that r0 is actually universal.
Lemma 3.9. Assume that u is a solution of (3.2) and (3.4) (or (3.3) and (3.5)) in
B1 which is monotone with respect to the directions of a cone Cone(e, c0 ). For any
ε > 0 there exists r0 > 0 suﬀiciently small, depending only on ε, c0 , κ, d and the
C 1,α property of O, so that for any r ≤ r0 there is κ ≤ q ≤ 1 (resp. 1 ≤ q ≤ κ−1 )so that(3.11)q(xd − εr)+ ≤ u(x) ≤ q(xd + εr)+ in Br .

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

19

Proof. We just consider the obstacle from below case, the obstacle from above caseis similar. The proof is by compactness. Suppose otherwise, then there is ε0 > 0,a sequence of uk solving (3.2) in B1 , a sequence of obstacles Ok all uniformly
C 1,α regular, and a sequence of radii rk → 0 so that (3.11) fails with r = rk andε = ε0 . By uniform Lipschitz regularity and non-degeneracy we can assume that, on
compact subsets of B1 , the uk converge uniformly to some u, the ∂{uk > 0} converge
in Hausdorff distance to ∂{u > 0}, and the Ok and ∂Ok also converge in Hausdorff
distance to another C 1,α domain O. Standard viscosity solution arguments showthat u solves (3.2) in B1 . This implies that u is in C 1,β ({u > 0} ∩ B1/2 ) so for any
ε > 0 there is r1 > 0 so that (3.11) holds for r ≤ r1 . Let ε1 > 0 suﬀiciently small
so that Theorem 3.4 and Theorem 3.5 hold, and then r1 > 0 small so that|∇u|(0)(xd − 21 ε1 r1 )+ ≤ u(x) ≤ |∇u|(0)(xd + 12 ε1 r1 )+ in Br1
which implies, for k large enough,|∇u|(0)(xd − ε1 r1 )+ ≤ uk (x) ≤ |∇u|(0)(xd + ε1 r1 )+ in Br1 .
Then, applying either Theorem 3.4 or Theorem 3.5 as the case may be, implies thatthat for all 0 < r ≤ r1|∇uk |(0)(xd − Cε1 r1+β )+ ≤ u(x) ≤ |∇uk |(0)(xd + Cε1 r1+β )+ in Br
which contradicts the hypothesis for large k.

□

4. Local laws: weak solutions, regularity, and comparison principle
In this section we study the viscosity solution notions (O) and (M), especiallyin the star-shaped setting. We establish regularity of (O) solutions using the results of Section 3. The central element of a viscosity solutions theory is always
the comparison principle. One of the main results of this section is a comparison
principle in the star-shaped setting, Proposition 4.26, between the (O) solution andan arbitrary (M) solution.
4.1. Definitions and main results. Let F = F (t) be a strictly positive C 1 function on the time interval [0, T ] with a discrete set of critical points, or more precisely,(4.1)

F ∈ C 1 ([0, T ])and Z := {t ∈ (0, T ) : F ′ (t) = 0} is discrete.

In other words, F only changes monotonicity at most finitely many times on any
bounded interval.
We start by introducing a notion of solution that states explicitly what equations
the obstacle solution (O) satisfies, see Lemma 4.18 below.
Definition 4.1. A function u : [0, T ) × U → [0, ∞) is an obstacle viscosity solution(OVS) if u(t) is continuous for each t ∈ [0, T ], satisfies u(t) = F (t) on ∂U , and thefollowing holds for any (s, t) ⊂ (0, T ) \ Z:(a) If F is monotone increasing on [s, t] then u(t) solves in the viscosity sense∆u(t) = 0in Ω(u(t)) ∩ Uu(t) > 0in Ω(u(s)) ∩ U
2|∇u(t)|≤1+µ+ on (∂Ω(u(t)) ∩ ∂Ω(u(s))) ∩ U
2|∇u(t)| = 1 + µ+ on (∂Ω(u(t)) \ Ω(u(s))) ∩ U

20

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

(b) If F is monotone decreasing on [s, t] then u(t) solves in the viscosity sense∆u(t) = 0in Ω(u(t)) ∩ Uu(t) = 0
in U \ Ω(u(s))
2|∇u(t)| = 1 − µ− on (∂Ω(u(t)) ∩ Ω(u(s))) ∩ U|∇u(t)|2 ≥ 1 − µ− on (∂Ω(u(t)) ∩ ∂Ω(u(s))) ∩ U.
Remark 4.2. Note in particular in case of (a) u is monotone increasing on [s, t],
and in case of (b) u is monotone decreasing in [s, t]. Given the condition on F , itfollows that u also only changes monotonicity finitely many times on any boundedinterval.
4.1.1. Slope condition in the comparison sense. The slope conditions are interpreted
in the comparison / viscosity sense which we make precise in two ways. First we
define a notion of sub and superdifferential which is suited to the problem. This isalso called the first order semi-jet J +,1 in [13].
{u>0}Definition 4.3. Given a non-negative continuous function u on U define, for eachx0 ∈ ∂Ω(u) ∩ Uno
D+ u(x0 ) = p ∈ Rd : u(x) ≤ p · (x − x0 ) + o(|x − x0 |) in Ω(u)Definition 4.4. Given a non-negative continuous function u on U define, for eachx0 ∈ ∂Ω(u) ∩ U
D− u(x0 ) = p ∈ Rd : u(x) ≥ p · (x − x0 ) − o(|x − x0 |)This leads to a comparison notion of slope conditions.
Definition 4.5. Suppose u : U → [0, ∞) is continuous and is subharmonic inΩ(u) ∩ U . Then u is a subsolution of|∇u|2 ≥ Q on ∂Ω(u) ∩ U
if, for every x ∈ ∂Ω(u) and every p ∈ D+ u(x),|p|2 ≥ Q.
Similarly, if u : U → [0, ∞) is continuous and is superharmonic in Ω(u) ∩ U , thenu is a supersolution of|∇u|2 ≤ Q on ∂Ω(u) ∩ U
if, for every x ∈ ∂Ω(u) and every p ∈ D− u(x),|p|2 ≤ Q.
In this section we will define the spatial slope conditions in terms of the sub
and superdifferential notions above. However, in later sections we prefer to use
the usual touching test functions notion which is more convenient to argue with inmany cases.
Definition 4.6. Suppose u : U → [0, ∞) is continuous and is subharmonic inΩ(u) ∩ U . Then u is a subsolution of|∇u|2 ≥ Q on ∂Ω(u) ∩ U
if, whenever φ is a smooth test function such that φ+ touches u from above atx ∈ ∂Ω(u) ∩ U with ∆φ(x) < 0,|∇φ(x)|2 ≥ Q.

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

21

Similarly, suppose u : U → [0, ∞) is continuous and is superharmonic in Ω(u) ∩ U .
Then u is a supersolution of|∇u|2 ≤ Q on ∂Ω(u) ∩ U
if, whenever φ is a smooth test function touching u from above at x ∈ ∂Ω(u) ∩ U
with ∆φ(x) > 0,|∇φ(x)|2 ≤ Q.
These two notions of viscosity solution are equivalent.
Lemma 4.7. Suppose that u : U → [0, ∞) is continuous and subharmonic (resp.
superharmonic) in Ω(u)∩U . Then u is a viscosity subsolution (resp. supersolution)in the sense of Definition 4.5 if and only if it is a viscosity subsolution (resp.
supersolution) in the sense of Definition 4.6.
We provide a proof sketch in the appendix for convenience, see Section A.4.
Remark 4.8. Note that this equivalence requires both solution properties to holdeverywhere. If φ touches u from above at x0 ∈ ∂Ω(u) ∩ U then ∇φ(x0 ) ∈ D+ u(x0 ).
However the reverse is not true, if p ∈ D+ u(x0 ) it does not necessarily guaranteethat u can be touched from above by a smooth test function at the same point with
∇φ(x0 ) ≈ p, rather one can find a nearby point where the touching occurs.
In Section 7 this distinction is important since we only prove a sub/superdifferential
notion Hd−1 -almost everywhere.
4.1.2. Dynamic slope condition in the comparison sense. In order to formulate
the dynamic slope condition (1.5) we need a weak pointwise sense of positive andnegative normal velocity. We will use a notion based on space-time “light cones” inplace of standard barrier functions, see Figure 6 for a sketch of the geometry. This
notion of viscosity solutions is stronger than the usual comparison definition of levelset velocity, since we test more free boundary points that have weaker space-time
regularity. Nonetheless, this stronger notion of normal velocity fits well with othernotion of solutions that we use.
Definition 4.9. Given a time varying family of domains Ω(t) such that χΩ(t) is
upper semi-continuous, and for given c, r0 > 0, we say that the outward normal
velocity at x0 ∈ ∂Ω(t0 ) is at least c at scale r0 and write
V r0 (t0 , x0 ) ≥ c
if the positive cone condition holds, namely(4.2){x : |x − x0 | ≤ c(t0 − t)} ⊂ Ω(t)∁ for t0 > t ≥ t0 − r0 .

DefineV + (t0 , x0 ) := 0 ∨ sup{c : ∃r0 > 0 so that V r0 (t0 , x0 ) ≥ c}.
We write V (t0 , x0 ) > 0 if V + (t0 , x0 ) > 0, i.e., there are c > 0 and r0 > 0 so that
V r0 (t0 , x0 ) ≥ c. If we need to clearly specify the domain we write V (·; Ω), V r0 (·; Ω),etc.
Definition 4.10. Given a time varying family of domains Ω(t) such that χΩ(t)is lower semi-continuous, and for given c, r0 > 0, we say that the inward normal
velocity at x0 ∈ ∂Ω(t0 ) is at least c at scale r0 and write
V r0 (t0 , x0 ) ≤ −c

22

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

tt = t0Ωt = t0Ωt|x − x0 | ≤ c(t0 − t)|x − x0 | ≤ c(t0 − t)Figure 6. Left: velocity c cone touches Ω(t) from the outsideat (t0 , x0 ), interpreted as Vn (t0 , x0 ) ≥ c. Right: velocity c cone
touches Ω(t) from the inside at (t0 , x0 ), interpreted as Vn (t0 , x0 ) ≤−c.
if the negative cone condition holds, namely(4.3){x : |x − x0 | ≤ c(t0 − t)} ⊂ Ω(t) for t0 > t ≥ t0 − r0 .

DefineV − (t0 , x0 ) := 0 ∧ inf{−c < 0 : ∃r0 > 0 so that V r0 (t0 , x0 ) ≤ −c}.
We write V (t0 , x0 ) < 0 if V − (t0 , x0 ) < 0, i.e., there are c > 0 and r0 > 0 sothat V r0 (t0 , x0 ) ≤ −c. If we need to clearly specify the domain we write V (·; Ω),V r0 (·; Ω), etc.
Finally, let us recall the semicontinuous envelopes u∗ and u∗ of a function u :[0, T ] × U → R:
(4.4)

u∗ (t, x) := lim sup u(s, y),(s,y)→(t,x)

u∗ (t, x) := lim inf u(s, y),(s,y)→(t,x)

where (s, y) is always assumed to be in [0, T ] × U . Recall that u∗ is USC and u∗ is
LSC.
Equipped with the notions of nonzero normal velocity and subdifferentials we can
now precisely define the meaning of the local stability condition and the dynamic
slope condition of (M) in Definition 1.4:
Definition 4.11. We say that a bounded map u : [0, T ] → C(U ) is a viscosity
solution of (1.4) on [0, T ] × U in the semicontinuous envelope sense if(a) u(t) satisfies (1.4) for every t ∈ [0, T ].
(b) If p ∈ D+ u∗ (t0 , x0 ) for some x0 ∈ ∂Ω(u∗ (t0 )) then |p|2 ≥ 1 − µ− .
Remark 4.12. Note that (1.4) and uniform boundedness imply that u(t) are uniformly Lipschitz continuous in compact subsets of U by Lemma 5.5. Therefore theupper and lower envelopes u∗ (t) and u∗ (t) are continuous in x for each t. It is also
standard that u∗ (t) and u∗ (t) are respectively subharmonic and superharmonic in
their positivity sets.
Remark 4.13. Note that if u : [0, T ] → C(U ) such that u(t) is non-degenerateuniformly in t then (b) in Definition 4.11 follow from (a). Indeed, say that u∗ (t) − ϕ
has a strict local maximum at x0 in Ω(u∗ (t)). By the uniform nondegeneracy we
can deduce that there exists a sequence tn → t and xn → x0 such that u(tn ) − ϕ
has a local maximum at xn in Ω(u(tn )), from which it follows that u∗ (t) satisfies(1.4).
Remark 4.14. A similar argument to Remark 4.13 can be done for u∗ but nondegeneracy is not necessary because the test functions touch from below in Rd

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

23

instead of in Ω(u(t)). So (a) actually directly implies the supersolution analogueto (b): if p ∈ D− u∗ (t0 , x0 ) for some x0 ∈ ∂Ω(u∗ (t0 )), then |p|2 ≤ 1 + µ+ .
Definition 4.15. We say that u : [0, T ] → C(U ) is a viscosity solution of (1.5) on[0, T ] × U in the semicontinuous envelope sense if(a) If V (t0 , x0 ; Ω(u∗ )) > 0 for some x0 ∈ ∂Ω(u∗ (t0 )), then any p ∈ D+ u∗ (t0 , x0 )satisfies |p|2 ≥ 1 + µ+ .
(b) If V (t0 , x0 ; Ω(u∗ )) < 0 for some x0 ∈ ∂Ω(u∗ (t0 )), then any p ∈ D− u∗ (t0 , x0 )satisfies |p|2 ≤ 1 − µ− . in U × (0, T ].
4.1.3. Star-shaped comparison principle / equivalence of solution notions. We willshow the equivalence of the notions in a strongly star-shaped setting.
Definition 4.16. In the following we say that a set is strongly star-shaped if itis star-shaped with respect to all x in a neighborhood of the origin. A function
u : U ⊃ Rd → R is strongly star-shaped if each of its superlevel sets {u > η} ∪ U ∁are strongly star-shaped.
The important property is that a strongly star-shaped set has a Lipschitz boundary. This can be seen by drawing the interior and exterior cones created by drawing
the rays from every star-shaped center though a point on the boundary, see for example [20, Figure 2].
The main result of this section is the following theorem that asserts that in thestrongly star-shaped setting all three notions of solutions that we introduced turnout to be equivalent.
Theorem 4.17. If Rd \ U and Ω0 are both bounded strongly star-shaped sets, Ω0
is a C 1,α domain for some α > 0, F satisfies (4.1) and u(0) ∈ C(U ) satisfies (1.4),
thenu is (O) solution ⇔ u is (OVS) solution ⇔ u is (M) solution.
Furthermore Ω(u(t)) are C 1,β domains uniformly in t ∈ [0, T ] for some β > 0.
Proof. To prove this theorem, we first show the easy equivalence of (O) and (OVS)
that follows from a standard viscosity solution argument and the uniqueness ofan obstacle Bernoulli problem in the star-shaped setting (Lemma 4.18). The factthat an obstacle solution (O) is also motion-law viscosity solution (M) is shown inLemma 4.20. Then we deduce the regularity in both space and time of (O)/(OVS)
solutions (Lemma 4.22, Corollary 4.25). Finally showing that (M) implies theother two notions (Corollary 4.27) follows a comparison-type viscosity argument
using sup- and inf-convolutions between a regular (M) solution and a general (M)solution (Proposition 4.26).
□
Lemma 4.18. The obstacle solution (O) is also an obstacle viscosity solution(OVS). Under the strong star-shapedness assumption of Theorem 4.17 this is theunique (OVS) solution.
Proof. (O) ⇒ (OVS) can be verified by a standard viscosity solution argument anddoes not require star-shapedness.
(OVS) ⇒ (O) on the other hand follows from the uniqueness of the obstacle
Bernoulli problem in the strongly star-shaped case: (O) always exists and by theprevious step, it is also (OVS), and hence by Theorem A.4 it must be the unique(OVS).
□

24

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

Remark 4.19. Let us point out that, while most of the analysis in this section
regards star-shaped initial data, for general initial data we still have a unique (O)solution by its definition, and this solution is also an (OVS) by the lemma above.
We do not know whether (OVS) would imply (M), or vice versa, for general initial
data. However formally both notions (M) and (OVS) come with the same dynamic
slope condition at space-time regular points of the free boundary.
4.2. Obstacle solutions are motion law solutions. Next we show that, in a
general setting without star-shaped assumption, obstacle solutions of (O) solve themotion law (M).
Lemma 4.20. If u : [0, T ] → C(U ) is an obstacle solution (O) on [0, T ] that is
uniformly non-degenerate at its free boundary, then u solves (M).
Proof. The state u(t) is harmonic in Ω(u(t)) by definition. By Remark 4.12 u∗ andu∗ are respectively sub and superharmonic in their positivity sets.
Now we aim to check the stability conditions in Definition 4.11(a). We first checkthe lower bound condition |∇u(t)|2 ≥ 1 − µ− in the viscosity sense on ∂Ω(t) ∩ U
by induction on the monotonicity intervals of F provided by assumption (4.1). We
know that the condition is satisfied at t = 0 by the compatibility of the initial data.
Suppose now that we know that u satisfies this condition up to s ∈ Z ∪ {0} andconsider t > s such that (s, t) ∩ Z = ∅.
• If F is decreasing on [s, t] then the lower bound on |∇u(t)|2 follows from
(OVS)-(b) immediately.
• Otherwise F is increasing on [s, t]. In this case u(t) are minimal supersolutions above u(s) for t0 ∈ (s, t] and so, by Lemma A.1, they are uniformly
non-degenerate. On (∂Ω(t) \ Ω(s)) ∩ U we get the same bound from (OVS)(a) since |∇u(t)|2 = 1 + µ+ ≥ 1 − µ− . Finally, on (∂Ω(t) ∩ ∂Ω(s)) ∩ U ,as u(t) ≥ u(s), any test function touching u(t) from above in {u(t) > 0} atx ∈ (∂Ω(t)∩∂Ω(s))∩U also touches u(s) and hence D+ u(t, x) ⊂ D+ u(s, x).
Hence u(t) is a solution |∇u(t)|2 ≥ 1 − µ− at x in the viscosity sense sinceu(s) is by the induction hypothesis.
By induction we can therefore extend the property to all t ∈ [0, T ] since Z isdiscrete. An analogous argument gives |∇u(t)|2 ≤ 1 + µ+ .
By the assumption of uniform non-degeneracy, Remark 4.13 yields Definition
4.11(b). We note that the assumption is actually needed only when F is decreasing.
When F is increasing, u(t) is a minimal supersolution above u(s) and hence nondegenerate by Lemma A.1.
We now check the remaining dynamic slope conditions. First consider monotone
increasing interval [s, t] of F and we aim to check the advancing condition Definition
4.15 part (a). Suppose the positive cone condition (4.2) holds for x0 ∈ ∂Ω(u∗ (t0 )) ∩U for some s < t0 ≤ t. By the positive cone condition and monotonicity there is∁a neighborhood Br (x0 ) ⊂ Ω(u(s)) , and thus we have |∇u|2 ≥ 1 + µ+ on ∂Ω(u)in (s, t] × Br (x0 ) by (OVS)-(a). If t0 = t then u∗ (t) = u(t) because u(t) is a local
maximum by increasing monotonicity to the left and decreasing monotonicity to theright, so we can reduce to t0 ∈ (s, t). Then, by uniform non-degeneracy of minimal
supersolutions Lemma A.1 and Remark 4.13, we can conclude that |∇u∗ |2 ≥ 1+ µ+on ∂Ω(u∗ ) in (s, t) × Br (x0 ), in particular it also holds at (t0 , x0 ).
Next we check that the receding condition Definition 4.15 part (b) holds formonotone decreasing interval [s, t] of F . The argument is mostly the same with a

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

25

subtle difference which we need to point out. Suppose the negative cone condition(4.3) holds for x0 ∈ ∂Ω(u∗ (t0 )) ∩ U for some s < t0 ≤ t. As before there is aneighborhood Br (x0 ) ⊂ Ω(u(s)), and thus we have |∇u|2 ≤ 1 − µ− on ∂Ω(u) in(s, t] × Br (x0 ) by (OVS)-(b). If t0 = t then u∗ (t) = u(t) so we reduce to the caset0 ∈ (s, t). Then by Remark 4.14, we can conclude that |∇u∗ | ≤ 1 − µ− on ∂Ω(u∗ )in (s, t] × Br (x0 ), in particular it also holds at (t0 , x0 ). Notice that we do not need
uniform non-degeneracy, which is not known in general for maximal subsolutions,because we are testing a supersolution condition.
□
4.3. Spatial regularity. We start with star-shapedness of (O).
Lemma 4.21. Suppose that u solves (O), K = Rd \ U and Ω0 = {u0 > 0} arestrongly star-shaped, u0 satisfies (1.4) and F satisfies (4.1). Then u(t) is strongly
star-shaped for every t ≥ 0 with respect to the same set of centers as K and Ω0 .
Proof. We show the star-shapedness only with respect to the origin assuming that
K and Ω0 are star-shaped with respect to the origin. The proof with respect toother centers can be done analogously by translation.
We will use induction on the monotonicity intervals of F . Suppose that t > 0
with (0, t) ∩ Z 6= ∅. Consider the case F (t) > F (0). First note that for λ < 1(min u(t, λx), F (0) x ∈ λ−1 U,
v(x) =F (0)x ∈ U \ λ−1 U,
is a supersolution of (1.4). Since {u(0) > 0} is star-shaped and {u(t) > 0} ⊃{u(0) > 0} also
λ−1 {u(t) > 0} ⊃ {u(0) > 0}.
Thus v is a supersolution of (1.4) above u(0) and sou(t, λ·) ≥ u(t)in λ−1 U

by the minimality of u(t) in the definition of (O). Since λ > 0 was arbitrary, u(t)is star-shaped (as are all of its super-level sets) with respect to 0.
In the case F (t) < F (0) we consider the dilation with λ > 1 and easily check thatu(t, λ·) ≤ u(0) in U and u(t, λ·) is a subsolution of (1.4) and hence u(t, λ·) ≤ u(t)in U by the maximality of u(t) in the definition of (O). In particular, u(t) is starshaped with respect to 0.
By translation we have star-shapedness at any point at which K and {u0 > 0}
are star-shaped. Finally by induction on the monotonicity intervals of F , we cancontinue the star-shapedness property up to any t > 0.
□
A solution of (O) is a solution of the Bernoulli obstacle problem on each monotonicity interval of F , where the obstacle is the solution at the beginning of theinterval. This allows us to use the regularity of the obstacle problem, Section 3,
only a finite number of times to establish the regularity of (O) at an arbitrary time.
Lemma 4.22. Suppose that u solves (O) with F satisfying (4.1), K = Rd \ U
has C 2 boundary, Ω0 has a C 1,α boundary, and both K and Ω0 are strongly starshaped. Then for any 0 < β < min{α, 21 } the domains {u(t) > 0} are C 1,β andu(t) ∈ C 1,β ({u(t) > 0} ∩ U ) uniformly in t ∈ [0, T ].
Proof. Due to the regularity of K = Rd \ U the function u(t) is in C 2 ({u(t) >0} ∩ ∂U ). We need to show the regularity at the free boundary of u(t).

26

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

By Lemma 4.21 the sets {u(t) > 0} are uniformly strongly star-shaped, andtherefore they are uniformly Lipschitz in [0, T ]. Let t > 0 be such that (0, t)∩Z = ∅.
Therefore by the regularity of the Bernoulli obstacle problem Theorem 3.6 with C 1,αobstacle {u0 > 0} and any 0 < β0 < min{α, 12 } the domain {u(t) > 0} is C 1,β0 andu(t) ∈ C 1,β0 ({u(t) > 0} ∩ U ).
Consider now t > 0 such that (0, t) ∩ Z = {s1 }. By the previous step {u(s1 ) >0} ∈ C 1,β0 . Applying the regularity of the Bernoulli obstacle problem with α = β0
for any 0 < β1 < β0 we find that u(t) has regularity C 1,β1 .
We can continue inductively on the monotonicity intervals of F up to t = T , justtaking a strictly decreasing sequence of min{α, 21 } > β0 > β1 > . . . > β#(0,T )∩Z =β.
□4.4. Time regularity. The regularity in space Lemma 4.22 of a star-shaped (O)solution yields in particular that u(t) is Lipschitz locally uniformly in time. Thisin particular implies (with the nondegeneracy of the solutions of the Bernoulli
problem) that its support moves in a Lipschitz way as well.
Lemma 4.23. Suppose that u solves (O), K = Rd \ U has C 2 boundary, and K
and Ω0 = {u0 > 0} are strongly star-shaped and bounded, and u0 satisfies (1.4).
Then
(4.5)

|u(t, x) − u(s, x)| ≤ CkF ′ k∞ |t − s| in U

where C = C(L, R, min F ) with L the Lipschitz constant of u in space and R :=maxx∈K∪Ω0 |x|.
Proof. Step 1. By the minimality of u(t), including u(0) by (1.4),
u(t, x) ≤ F (t) + (1 + µ+ )(R − |x|) ,t ∈ [0, T ], x ∈ U ,+since the right-hand side is a supersolution. Therefore(4.6)

x ∈ Ω(t)

⇒

|x| ≤

F (t)
+ R.
1 + µ+

Step 2. Fix (s, t) ⊂ R \ Z with F (s) < F (t). Let λ := F (s)/F (t) < 1. Define(
λ−1 u(s, λx), x ∈ λ−1 U
v(x) :=
F (t),
otherwise.
By the choice of λ, v is continuous and hence a supersolution on U with boundary
data F (t). By minimality, u(t) ≤ v.
F (s)+R). Recall also that u(s, λx) ≤Fix λx ∈ Ω(s)∩U . (4.6) implies |x| ≤ λ−1 ( 1+µ+F (s). We havev(x) = λ−1 u(s, λx) = u(s, λx) + (λ−1 − 1)u(s, λx)≤ u(s, x) + k∇u(s)k∞ (1 − λ)|x| + (λ−1 − 1)F (s)


F (s)
≤ u(s, x) + (λ−1 − 1) L(+ R) + F (s) .
1 + µ+
Since ut ≤ v we conclude that0 ≤ u(t, x) − u(s, x) ≤



L
R
+1+
1 + µ+F (s)(F (t) − F (s)).

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION27

For x ∈ U \ λ−1 U we note that dist(x, ∂U ) ≤ (λ−1 − 1)R = FR(s) (F (t) − F (s))
and therefore
0 ≤ u(t, x) − u(s, x) ≤ F (t) − F (s) + k∇u(s)k∞ dist(x, ∂U )


LR
≤ 1+
(F (t) − F (s)).
F (s)Step 3. If (s, t) ⊂ R \ Z with F (s) > F (t), we have λ = F (s)/F (t) > 1 andv ≤ F (t). Hence v is a subsolution and therefore by maximality u(t) ≥ v. We only
need to consider x ∈ Ω(s) ∩ U . This timev(x) = λ−1 u(s, λx) = u(s, λx) + (λ−1 − 1)u(s, λx)≥ u(s, x) − k∇u(s)k∞ (λ − 1)|x| − (1 − λ−1 )F (s)
F (s) − F (t)F (s)+R− F (t) + F (s).
≤ u(s, x) − L(
1 + µ+F (t)Hence by u(t) ≥ v

0 ≥ u(t, x) − u(s, x) ≥ L

R
F (s)
+1+
(1 + µ+ )F (t)F (t)(F (s) − F (t)).

For general (s, t), we recover the estimate (4.5) by triangle inequality.

□

Using the nondegeneracy of cone monotone solutions of the Bernoulli problem,
Lemma A.1, we can deduce Lipschitz regularity of Ω(u(t)) in time.
Corollary 4.24. Under the assumptions of Lemma 4.23, the sets Ω(u(t)) are locally
Lipschitz in time with respect to the Hausdorff distance.
Proof. Let us fix s 6= t. Let z ∈ ∂Ω(u(t)) be the point that maximizes the distancefrom ∂Ω(u(s)). Let us denote r := dist(z, ∂Ω(u(s))).
Suppose that z ∈/ Ω(u(s)). We have Br (z) ∩ Ω(u(s)) = ∅ and hence us = 0 on
Br (z). On the other hand, the free boundary is locally a Lipschitz graph and so
the non-degeneracy (note that u(t) satisfies (1.4) by Lemma 4.20) for the Bernoulli
problem in this setting Lemma A.1 yields supBr (z) u(t) ≥ c0 r. By the Lipschitz
continuity of u in time, we deduce
c0 r ≤ sup u(t) ≤ L̃|t − s|,Br (z)

where L̃ is the Lipschitz constant in (4.5). In particular, r ≤ cL̃0 |t − s|.
If z ∈ Ω(u(s)) then Br (z) ⊂ Ω(u(s)). Using the non-degeneracy, Harnack inequality and the Lipschitz regularity, we haver
CH c0 ≤ CH sup u(s) ≤ u(s, z) ≤ L̃|t − s|,2
Br/2 (z)which yields r ≤ C2HL̃c0 |t − s|.

□Finally we mention that the time regularity and space regularity can be interpolated to get time regularity of ∇u.
Corollary 4.25. Under the assumptions of Lemma 4.22 for any 0 < β < min{α, 21 }β1,βso by interpolation u ∈ Ct1+β Cx1 .
u ∈ Ct0,1 Cx ∩ L∞
t Cx
2

−

In particular, if α ≥ 21 then ∇u ∈ Ct3 Cx0 .

28

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

Proof. Since ∇u(t) are in C β (Ω(u(t))) uniformly in t we can use the uniform convergence of the difference quotients and the Lipschitz continuity in time of u(t)from Lemma 4.23
1
1(∇u(t, x) − ∇u(s, x)) · e = (u(t, x + he) − u(t, x)) + (u(s, x + he) − u(s, x)) + O(hβ )hh= O(h−1 |t − s|) + O(hβ ).
1choose h = |t − s| 1+β to match the orders of the two terms.

□

4.5. Comparison principle in star-shaped setting. Now we state a comparison principle that closes our discussion on equivalences of notions. We will utilize
this comparison principle later on in Section 6, when we discuss a specific energy
solution generated by minimizing movements.
Proposition 4.26. Suppose that w and u are respectively a subsolution and a LSC
supersolution (or a supersolution and an USC subsolution) of (M) in [0, T ] withstrongly star-shaped initial data, with the ordering w(0) ≤ u(0) (or w(0) ≥ u(0)).
Further suppose that w satisfies the following:(a) {w(t) > 0} is bounded and strongly star-shaped at each time, and its boundary is uniformly C 1,α in space and Lipschitz in time.
(b) There exists a finite partition {ti }i=ni=0 of [0, T ] such that w is monotone in[ti , ti+1 ].
Then w ≤ u (or w ≥ u).
As is typical with viscosity solutions, one of the main diﬀiculties in the comparison principle of weak solutions is the lack of regularity. We follow a typicalapproach based on sup / inf convolutions in order to regularize at the point where
the first touching occurs. The novel diﬀiculty has to do with the weak notion of
positive velocity and the diﬀiculties stemming from crossings which occur with zero
velocity. We can use sup / inf convolutions with decreasing radius to accelerate thefree boundary velocity, but a delicate argument relying on the regularity of the (O)solution is still needed to actually locate a positive velocity first touching point.
Let us mention that an (OVS) solution with star-shaped initial data satisfies (a)in Proposition 4.26 due to Lemma 4.22 and Lemma 4.23 and (b) due to Remark 4.2.
It is also an (M) by Lemma 4.20. Thus, from the comparison principle Proposition
4.26, we can derive the uniqueness:
Corollary 4.27. Any motion law solution (M) with a strongly star-shaped initial
data is the unique star-shaped obstacle solution (O) with the same initial data.
Proof of Proposition 4.26. Let us assume that w is a (M) subsolution with regularity properties, and u is a LSC (M) supersolution. We will show thatw(t, λ·) ≤ u(t, ·) for t > 0 and λ > 1,
which yields w ≤ u in the limit λ & 1. The proof for the other half of the statement
is parallel, arguing the opposite inequality with λ < 1.
Fix λ > 1. Define
W (t, x) := sup w(t, λy).
y∈B ρ(t) (x)−t

Here ρ(t) = ρ0 e , and ρ0 is chosen proportional to (λ − 1) so that W (0) < u(0)in {W (0) > 0}, which is possible by strong star-shapedness. The main motivation

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

29

for the sup-convolution is the effective reduction of the normal velocity of W byr′ (t) < 0 compared to w. Hence if W and u touch each other, either the normal
velocity of u is negative or w has positive normal velocity near the contact, andtherefore the dynamic slope condition can be used in either situation to arrive at acontradiction. Recall that W is subharmonic in its positive set.
We would like to show that W (t) < u(t) in {W (t) > 0} for all t ≥ 0. Define thefirst contact time
t0 = inf t > 0 : W (t) 6< u(t) in {W (t) > 0} .
Clearly t0 > 0 by lower semi-continuity of u in time. If t0 is finite, we want to geta contradiction. The only non-standard part here lies in the fact that the supportof u may jump in time, so let us discuss this point carefully. Since we assume
that W is star-shaped, W and its support continuously decrease as λ increases.
The support of {W (t0 ) > 0} indeed can be made arbitrarily close to B ρ(t0 ) (0), in
particular a subset of the fixed boundary U ∁ for u, if λ is suﬀiciently large. Thus if{W (·, t0 ) > 0} 6⊂ {u(·, t0 ) > 0} due to discontinuity in time in u at t0 , we increase λin the definition of W until {W (t0 ) > 0} precisely touches {u(t0 ) > 0} from inside,
so that there exists a point x0 ∈ ∂{W (t0 ) > 0} ∩ ∂{u(t0 ) > 0}. We point out that
ρ0 does not need to be changed even if we need to increase λ. Note that W ≤ u fort ≤ t0 for this larger choice of λ due to the definition of t0 and W (t0 ) − u(t0 ) being
subharmonic in {W (t0 ) > 0}. In particular u(t0 ) − W (t0 ) has a minimum at x0 .
By the definition of W there is a point y0 ∈ ∂{w(t0 , λ·) > 0} with y0 ∈ Bρ(t0 ) (x0 ),andW (t0 , x) ≥ w(t0 , λ(x − x0 + y0 )) =: ϕ(x).
As u(t0 ) − ϕ has a minimum at x0 , we deduce that
(4.7)

λ∇w(t0 , λy0 ) = ∇ϕ(x0 ) ∈ D− u(x0 ).

By our assumption (b), we know that w is either monotone increasing or decreasing in a closed time interval [t−1 , t0 ] with t−1 < t0 . If w is monotone decreasing in [t−1 , t0 ], then W has strictly decreasing support in [t−1 , t0 ], namely forc = −r′ (t0 ) > 0 we have
(4.8)

Bc(t0 −t) (x0 ) ⊂ {W (t) > 0} ⊂ {u(t) > 0} for t−1 ≤ t < t0 .

In particular the negative cone condition (4.3) is satisfied for u at (t0 , x0 ), and so as u
is a (M) supersolution we conclude by (4.7) that λ2 |∇w|2 (t0 , λy0 ) ≤ 1−µ− . However
as w is itself (M) subsolution, the stability condition yields λ2 |∇w|2 (t0 , λy0 ) ≥λ2 (1 − µ− ), a contradiction.
Now suppose w is monotone increasing in [t−1 , t0 ]. It is possible that in this caseu still satisfies the negative cone condition (4.3) for some c < 0 and r0 > 0, in whichcase one can still proceed as in the above case. So now suppose that (4.3) is nottrue for c := − 31 ρ′ (t0 ) > 0 and any r0 > 0. This means that along an increasingtime sequence tn converging to t0 , there is xn such thatxn ∈ {u(tn ) = 0} ∩ Bc(t0 −tn ) (x0 ).

30

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

(y0 , t0 )t = t0ρ(t0 ) + c(t − tn )Ω(wλ )(τn , yn )

xn
t

x0
ρ(t0 )
y0{W > 0}{wλ > 0}2c(t0 − tn ) c(t0 − tn )Figure 7. Left: spatial picture of the sup convolution balls attime t0 and tn overlayed, patterned ball is in the zero level set ofwλ at time tn . Right: space-time picture displaying the definitionof (τn , yn ).

Now we consider what this means in terms of w. Recall y0 above and consider
y such that |y − y0 | ≤ 2c(t0 − tn ). Then|y − xn | ≤ |y − y0 | + |y0 − x0 | + |x0 − xn |≤ 2c(t0 − tn ) + ρ(t0 ) + c(t0 − tn )≤ ρ(tn ),where the last inequality is by convexity of ρ(t). Thus, by definition of W , w(tn , λy) ≤W (tn , xn ) ≤ u(tn , xn ) = 0. Namely B 2c(t0 −tn ) (y0 ) ⊂ {w(tn , λ·) = 0}. See the left
image in Figure 7.
Now letτn := sup{s ∈ [tn , t0 ) : B c(t0 −s) (y0 ) ⊂ {w(s, λ·) = 0}}.
See the right image in Figure 7. Since {w(t) = 0} evolves continuously in time
(Corollary 4.24), we have τn > tn , and if τn < t0 then there is a contact pointyn ∈ ∂{w(τn ) > 0} ∩ ∂Bc(t0 −τn ) (y0 ). If the order is preserved up to t = t0 , i.e.
τn = t0 , then we set yn = y0 . From the definition of (yn , τn ) we see that positive
cone condition (4.2) holds for w(·, λ·) at (τn , yn ) and therefore for w at (tn , λyn )with λc. By (M) we have |∇w|2 (tn , λyn ) ≥ 1 + µ+ .
Since (τn , yn ) → (t0 , y0 ), by the gradient continuity in Corollary 4.25, we deduce
λ2 |∇w|2 (t0 , λy0 ) = lim λ2 |∇w|2 (τn , λyn ) ≥ λ2 (1 + µ+ ).
n→∞

But this contradicts the fact that u is a (M) supersolution and hence by (4.7) wemust have λ2 |∇w|2 (λy0 , t0 ) ≤ 1 + µ+ . Hence we conclude.
□t = tnON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

31

5. Basic space and time regularity properties of energy solutions
In this section we study energy solutions (E) establishing existence and various spatial and temporal regularity properties. We will work in a general setting,without the special geometry of star-shapedness which was used in the previous
section.
In Section 5.1 we recall several results from the literature on inwards and outwards minimizers of one-phase functionals. These results, in particular, also apply
to states satisfying the global stability condition of energy solutions (E).
In Section 5.2 we show BV time regularity of energy solutions using a typical
Grönwall argument via the energy dissipation inequality, see Lemma 5.7. With
this regularity we can also establish that the energy dissipation balance holds withequality, see Lemma 5.10.
Finally in Section 5.3 we establish the main temporal regularity result of thesection, Proposition 5.16. Abstract theory of bounded variation maps from [0, T ]into a metric space shows that u(t) has left and right limits uℓ (t) and ur (t) at every
time. Then we use the monotonicity structure of the problem to show that the lower
and upper-semicontinuous envelopes of an energy solution are exactly uℓ (t) ∧ ur (t)and uℓ (t) ∨ ur (t) and these are energy solutions as well. This is independently
interesting, elucidating the structure of the jump discontinuities, and it also plays
a key role in establishing the dynamic slope condition in Section 7.
5.1. Inwards and outwards minimality and spatial regularity. Let V be an
open region that contains Rd \ U . In this section we will discuss the regularity ofminimizers of the dissipation augmented energy, defined above in (1.3),
E(V, u) := J (u) + Diss(V, Ω(u)).
We will also abuse notation to write E(v, u) = E(Ω(v), u) when v is another nonnegative function on Rn .
Note that if u is an energy solution on [0, T ], from the global stability property
at each time u(t) is a minimizer for E(u(t), ·).
It is convenient to make a connection with the notions of inwards and outwards
minimality for the Bernoulli functional J . First of all we introduce the notationfor Q > 0 and an open region U ⊂ Rdˆ
JQ (u) :=|∇u|2 + Q1{u>0} dx.
U

We have written and will continue to write J = J1 abusing notation.
Next we introduce the notions of inwards and outwards minimizers. These notions have appeared in the literature before, for example see the book [36] for further
references.
Definition 5.1. u ∈ H 1 (U ) is an outward (resp. inward) minimizer of JQ (·; U ) if
(1) The set Ω(u) = {u > 0} is open and u is harmonic in Ω(u).
(2) For any v ∈ u + H01 (U ) with v ≥ u (resp. v ≤ u)
JQ (u) ≤ JQ (v).
Minimizers of E(V, ·) have a natural inward/outward J1±µ± minimality property.

32

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

Lemma 5.2. Let V be an open set that contains Rd \ U . Suppose that u is a global
minimizer ofJ (v) + Diss(V, Ω(v)) over v ∈ u + H01 (U ).
Then u is an outwards minimizer for J1+µ+ and an inwards minimizer for J1−µ− .
Furthermore if Br lies outside of V then u minimizes J1+µ+ in Br , and if Br ⊂ Vthen u minimizes J1−µ− in Br .
Proof. Suppose v ∈ u + H01 (U ) with v ≥ u. Then {v > 0} ⊃ {u > 0}, so(5.1)|V \ Ω(u)| ≥ |V \ Ω(v)| and |Ω(u) ∩ V | ≤ |Ω(v) ∩ V |.

We can apply (5.1) along with the minimality property of u to obtain
J1+µ+ (u) = J1 (u) + µ+ |Ω(u)|= J1 (u) + µ+ |Ω(u) \ V | + µ+ |Ω(u) ∩ V |= J1 (u) + Diss(V, Ω(u)) − µ− |V \ Ω(u)| + µ+ |Ω(u) ∩ V |≤ J1 (v) + Diss(V, Ω(v)) − µ− |V \ Ω(v)| + µ+ |Ω(v) ∩ V |= J1+µ+ (v).
Lastly note that if v = u outside of Br ⊂ Rd \ V then (5.1) holds with equalities.
Symmetrical computations give the (inwards) minimality property for J1−µ− .
□5.1.1. Viscosity solution properties of inward / outward minimizers. The notionsof inwards and outwards minimizers are reminiscent of viscosity sub and supersolutions. Indeed there is a direct correspondence between the two notions.
Lemma 5.3. Suppose that S > 0 and u is an inwards minimizer of JS in a domain
U . Then in the viscosity sense|∇u|2 ≥ Q on ∂{u > 0} ∩ U.
Similarly, if u is an outwards minimizer of JS then in the viscosity sense|∇u|2 ≤ Q on ∂{u > 0} ∩ U.
In general the implication cannot go the other way, there are viscosity solutions,i.e. stationary points of the energy, which are not energy minimizers.
It is well known that such inwards and outwards minimality properties imply
the corresponding viscosity solution conditions, see for example [36, Proposition
7.1]. We present a proof anyway since we will use similar, but more involved,
computations in the proof of Theorem 7.1.
Before proceeding with the proof let us write down a corollary of Lemma 5.2 andLemma 5.3 for energy solutions.
Corollary 5.4 (Stability implies slope in pinning interval). Suppose that u is anenergy solution on [0, T ], then for each time u(t) is a viscosity solution of
1 − µ− ≤ |∇u(t)|2 ≤ 1 + µ+on ∂{u(t) > 0} ∩ U.

Thus energy solutions satisfy the stability condition of (M).

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

33

Proof of Lemma 5.3. (Subsolution) Suppose that a smooth test function φ touches
u from above in Ω(u) strictly at x0 ∈ ∂Ω(u) with p := ∇φ(x0 ) 6= 0 and ∆φ(x0 ) < 0.
Let δ > 0 and consider the comparison functionuδ (x) := u(x) ∧ (φ(x) − δ)+ .
Since the touching is strict {u > 0} \ {uδ > 0} is contained in a ball of radius oδ (1)around x0 as δ → 0. In particular we can assume that δ is small enough so that φis superharmonic in {uδ < u}. By the inwards minimality of u
JQ (u) ≤ JQ (uδ ).
Now note that uδ extends to a superharmonic function in {u > 0} by(uδ (x)x ∈ Ω(uδ )ūδ (x) :=φ(x) − δ x ∈ Ω(u) \ Ω(uδ ).
So we can apply Lemma A.2, formula (A.2), and we findˆJQ (u) − JQ (uδ ) ≥Q − |∇φ(x)|2 dx{u>0}\{uδ >0}≥ [(Q − |p|2 ) − oδ (1)]|{u > 0} \ {uδ > 0}|
Combining the previous[(Q − |p|2 ) − oδ (1)]|{u > 0} \ {uδ > 0}| ≤ 0
so dividing through by |{u > 0} \ {uδ > 0}| > 0 and taking δ → 0 we findQ ≤ |p|2 .
(Supersolution) Suppose that a smooth test function φ touches u from below
strictly at x0 ∈ ∂Ω(u) with p := ∇φ(x0 ) 6= 0 and ∆φ(x0 ) > 0. Let δ > 0 andconsider the comparison functionuδ (x) = u(x) ∨ (φ(x) + δ).
Since the touching is strict {uδ > u} is contained in a ball of radius oδ (1) aroundx0 as δ → 0. In particular we can assume that δ is small enough so that ∆uδ ≥ 0.
By the outwards minimality property of u
JS (u) ≤ JS (uδ ).
Applying Lemma A.2 (note uδ not harmonic in {uδ > 0} but in general subharmonic) we findˆJS (u) − JS (uδ ) ≥|∇uδ |2 − Q dx
{uδ >0}\{u>0}ˆ
=|∇φ(x)|2 − Q dx
{uδ >0}\{u>0}≥ [(|p|2 − Q) − oδ (1)]|{uδ > 0} \ {u > 0}|
Combining the previous[(|p|2 − Q) − oδ (1)]|{uδ > 0} \ {u > 0}| ≤ 0
so dividing through by |{uδ > 0} \ {u > 0}| > 0 and taking δ → 0 we find|p|2 ≤ Q.
□

34

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

5.1.2. Regularity properties of inward / outward minimizers. Next we collect several
basic regularity results of JQ inwards / outwards minimizers.
Lemma 5.5.
(i) (Lipschitz estimate) There is C(d) ≥ 1 so that for any viscosity solution u of
∆u = 0 in Ω(u) ∩ B2 and |∇u|2 ≤ Q on ∂Ω(u) ∩ B2
we havepk∇ukL∞ (B1 ) ≤ C( Q + k∇ukL2 (B2 ) ).
In particular this holds for outward minimizers of JQ in B2 by Lemma 5.3.
(ii) (Non-degeneracy) There is c(d, Q) > 0 so that if u ∈ H 1 (U ) is an inward
minimizer of JQ in U , then
sup u ≥ cr

for any x ∈ ∂{u > 0} and Br (x) ⊂ U.

Br (x)(iii) (Density estimates) For Q1 ≤ Q2 , there is c(d, Q1 , Q2 ) > 0 so that if u is
an inward minimizer of JQ1 and an outward minimizer of JQ2 in U thenc≤|Ω(u) ∩ Br (x)|≤ 1 − c for any x ∈ ∂Ω(u) with Br (x) ⊂ U.
|Br |(iv) (Perimeter estimate) There is C(d) > 0 such that if u is an inward minimizer of JS in B2 thenpPer(Ω(u); B1 ) ≤ C Q(1 + k∇ukL2 (B2 ) ).
(v) (Hausdorff dimension) If u satisfies the conclusions of (iii) and perimeter
estimate (iv) in B2 thenHd−1 (∂Ω(u) ∩ B1 ) ≤ C(1 + k∇ukL2 (B2 ) )where C depends on the constants in (iii) and (iv).
Proof. See the books [6, 36] for presentations of the proofs and citations for their
original appearances in the literature, Lipschitz continuity of viscosity solutions isin [6, Lemma 11.19], non-degeneracy is in [36, Lemma 4.4], density estimates are in[36, Lemma 5.1], perimeter estimates are in [36, Lemma 5.6], Hausdorff dimension
estimates in [36, Lemma 5.9].
□
Remark 5.6. The C 1,β regularity of E(u, ·) minimizers is known due to the very
recent result of Ferreri and Velichkov [22] when {u > 0} is C 1,α .
5.2. Bounded variation regularity in t. BV regularity in time is typical forsolutions of (global) dissipative evolution problems [30]. Following the standard
argument, we establish the BV in time regularity using a Grönwall type argumentwith the energy dissipation inequality.
Lemma 5.7. Suppose u is an energy solution (E) on [0, T ]. Then1Ω(u(t)) ∈ BV([0, T ]; L1 (Rd ))andJ (u(t)), P (u(t)) ∈ BV([0, T ]; R),´ ∂u´´dS = F (t)−1 |∇u(t)|2 dx.
where recall the pressure is P (u(t)) = ∂Ω(t) |∇u|dS = ∂U ∂n
Before beginning the proofs we make a remark about the definition of energy
solution.

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

35

Remark 5.8. Energy solutions will also satisfy an energy dissipation balance condition. Define the total variation of the dissipation distance along a path Λ(t) offinite measure subsets of Rd
(5.2)
(N
)
X
N
Diss(Λ(·); [s, t]) := supDiss(Λ(tk−1 ), Λ(tk )) : (tk )k=0 partitions [s, t] .
k=1

Then let u(t) be an energy solution on [0, T ], i.e. satisfying the forcing, stability,and energy dissipation inequality conditions of (E). One can show immediately, byapplying (1.6) on each subinterval of an arbitrary partition of [t0 , t1 ], that
(5.3)

J (u(t0 )) − J (u(t1 )) +

ˆ t1

2Ḟ (t)P (t) dt ≥ Diss(u(·); [t0 , t1 ]).

t0

With more work (see Lemma 5.10 later) combining with the stability property one
can also show the identityˆ t1
2Ḟ (t)P (t) dt = Diss(u(·); [t0 , t1 ]).
(5.4)
J (u(t0 )) − J (u(t1 )) +t0

In some works, for example [1], the energy dissipation balance (5.4) is used in placeof the energy dissipation inequality (1.6) as part of the definition of energy solution.
This difference of definition is just a matter of preference at least in this problem.
Remark 5.9. It is quite natural, and important for applications, to consider general F (t, x) and allow for F to be only BV regular in time. However, this generality
does add serious complications which would significantly lengthen the presentation,and it is not so relevant to the goals of the present work.
Proof of Lemma 5.7. By the energy dissipation inequality (5.3)
(5.5)
ˆ t
J (u(0))−J (u(t))+
2Ḟ (s)P (u(s)) ds ≥ Diss(Ω(u(·)); [0, t]) ≥ µ+ ∧µ− [1Ω(·) ]BV([0,t];L1 )0

In the remainder of the proof we will denote the Dirichlet energyˆD(t) :=|∇u(t, x)|2 dx = F (t)P (u(t)).
Ω(u(t))

Note that D(t) ≤ J (u(t)) so in particularˆ tD(t) ≤ J (u(0)) +2Ḟ (s)P (u(s)) ds0ˆ t= J (u(0)) +20d(log F )(s)D(s) ds
ds

for all 0 ≤ t ≤ T so by Grönwall
D(t) ≤ J (u(0))

F (t)2.
F (0)2

36

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

Thus P (u(t)) ≤ F (0)−2 F (t), and from (5.5) we have


ˆ t
1
J (u(t)) + µ+ ∧ µ− [1Ω(·) ]BV([0,t];L1 ) ≤ J (u(0)) 1 +
2
Ḟ
(s)F
(s)
ds
F (0)2 0
F (t)2 − F (0)2
= J (u(0)) 1 +F (0)2F (t)2= J (u(0)).
F (0)2
To summarize,
J (u(t)) + µ+ ∧ µ− [1Ω(·) ]BV([0,t];L1 ) ≤ J (u(0))(5.6)

F (t)2.
F (0)2

In particular(5.7)[1Ω(·) ]BV([0,t];L1 ) ≤F (t)2
1
J (u(0)).
µ+ ∧ µ−F (0)2This gives the BV in time estimate of 1Ω(t) .
Now we turn to the BV estimate of the Dirichlet energy. Focusing on the otherterm on the left hand side in (5.6), and using the fact that J (u(t)) = D(t) +|Ω(u(t))|,F (t)2F (t)2
D(t) ≤D(0)+|Ω(u(0))| − |Ω(u(t))|
F (0)2F (0)2or rearranging thisD(t) − D(0) ≤

(5.8)

=

D(0)
1
(F (t)2 − F (0)2 ) +(F (t)2 − F (0)2 )|Ω(u(0))|
2F (0)F (0)2+ |Ω(u(t))∆Ω(u(0))|
J (u(0))(F (t)2 − F (0)2 ) + |Ω(u(t))∆Ω(u(0))|.
F (0)2Global stability implies a bound in the other direction


F (0)
J (u(0)) ≤ Ju(t) + µ+ ∨ µ− |Ω(u(0))∆Ω(u(t))|,F (t)which expands out asD(0) + |Ω(u(0))| ≤F (0)2
D(t) + |Ω(u(t))| + µ+ ∨ µ− |Ω(u(0))∆Ω(u(t))|,F (t)2

which we can use to findD(t)(5.9) D(0) − D(t) ≤(F (0)2 − F (t)2 ) + (1 + µ+ ∨ µ− )|Ω(u(0))∆Ω(u(t))|.
F (t)2Since 0 and t replaced by arbitrary t0 < t1 ∈ [0, T ], for any partition of [0, T ] wecan apply (5.9) and (5.8) together to find:XX
J (u(tj )) J (u(tj+1 ))max,
F (tj+1 )2 − F (tj )2|D(tj+1 ) − D(tj )| ≤F (tj )2
F (tj+1 )2
jj+ (1 + µ+ ∨ µ− )|Ω(u(tj+1 ))∆Ω(u(tj ))|.

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

37

Applying (5.6) again for F (t)−2 J (u(t)) ≤ F (0)−2 J (u(0)), we obtain thatX

|D(tj+1 ) − D(tj )| ≤jJ (u(0)) 2[F ]BV([0,T ]) + (1 + µ+ ∨ µ− )[1Ω(·) ]BV([0,T ];L1 ) .
F (0)2Thus we conclude that D(t) ∈ BV ([0, T ]) with the estimate, applying (5.7),
1 + µ+ ∨ µ−2
2 J (u(0))[D]BV ([0,T ]) ≤ [F ]BV([0,T ]) +F (t).
µ+ ∧ µ−F (0)2
Finally since J (u(t)) = D(t) + |Ω(u(t))| and both terms on the right are in
BV ([0, T ]) then so is J (u(t)). Similarly P (u(t)) = F (t)−1 D(t) is in BV ([0, T ])since F (t) is bounded from below.
□
From these regularity properties we can establish that the stability and energy
dissipation inequality properties of energy solutions force the energy dissipation
equality.
Lemma 5.10 (Energy dissipation equality). If u is an energy solution on [0, T ]then for all s ≤ t in [0, T ]ˆ t
J (u(s)) − J (u(t)) +2Ḟ (τ )P (u(τ )) dτ = Diss(Ω(u(·)); [s, t]).
s

Proof. We need to establish the upper bound to complement (1.6). This will follow
from the time regularity and the global stability property. We may take s = 0 andt = T . Let ε > 0. By the continuity of log F (t), we can choose a (finite) partition
0 = t0 < t1 < · · · < tN = T of [0, T ] such that
(5.10)

sup(tj+1 − tj ) ≤ ε andjF (t)≤ 1 + ε.
F
(tj+1 )
t∈[tj ,tj+1 ]supWe write the energy dissipation using a telescoping sum(5.11)

J (u(0)) − J (u(T )) =N
−1
X

[J (u(tj )) − J (u(tj+1 ))].

j=0

Applying the global stability condition for u(tj ),J (u(tj )) − J (u(tj+1 )) ≤ JF (tj )
F (tj+1 ) u(tj+1 )+ Diss(Ω(tj ), Ω(tj+1 )) − J (u(tj+1 ))F (t )2j= Diss(Ω(tj ), Ω(tj+1 )) + ( F (tj+1)2 − 1)D(u(tj+1 ))
{z
}
|

:=A

Now from the fact that D(u(t)) = F (t)P (u(t)),
A=

F (tj )2 − F (tj+1 )2
P (u(tj+1 )) =F (tj+1 )ˆ tj+1
2
tj

F (t)Ḟ (t)P (u(tj+1 )) dt.
F (tj+1 )38

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

From (5.10) we have
ˆ tj+1F (t)2A=Ḟ (t)P (u(tj+1 )) dt
F (tj+1 )tj
ˆ tj+1F (t)2≤Ḟ (t)P (u(t)) dt + 2(1 + ε)kḞ k∞ [P (u(·))]BV((tj ,tj+1 ]) (tj+1 − tj )F (tj+1 )tj(5.12)≤ (1 + ε)ˆ tj+1

2Ḟ (t)P (u(t)) dt + 2(1 + ε)εkḞ k∞ [P (u(·))]BV((tj ,tj+1 ])tj

Combining the previous estimates into (5.11) we find
ˆ T
N
−1
X
Diss(Ω(tj ), Ω(tj+1 ))J (u(0)) − J (u(T )) ≤ (1 + ε)2Ḟ (t)P (u(t)) dt +
0

+

N
−1
X

j=0

2(1 + ε)εkḞ k∞ [P (u(·))]BV((tj ,tj+1 ])

j=0

ˆ T

≤ Diss(Ω(u(·)); (0, T )) + (1 + ε)2Ḟ (t)P (u(t)) dt0+ 2(1 + ε)εkḞ k∞ [P (u(·))]BV((0,T ]) .
Since ε > 0 was arbitrary we get the result.
□5.3. Left and right limits and monotonicity of jumps. The BV time regularity of 1Ω(u(t)) allows us to establish further temporal left and right continuity
properties of the evolution at every time. Recall the dissipation augmented energyfunctional (1.3),
E(V, u) := J (u) + Diss(V, Ω(u)).
First we discuss a monotonicity property for minimizers of E.
5.3.1. No crossing property of E minimizers. The next lemma shows that minimizers of the dissipation distance augmented energy E, defined in (1.3), satisfy acertain ordering property.
Suppose that u1 and u2 are arbitrary H 1 (U ) functions. Note that
J (u1 ∨ u2 ) + J (u1 ∧ u2 ) = J (u1 ) + J (u2 )andDiss(V, Ω(u1 ∨ u2 )) + Diss(V, Ω(u1 ∧ u2 )) = Diss(V, Ω(u1 )) + Diss(V, Ω(u2 )).
Combining the previous two equations yields(5.13)

E(V, u1 ∨ u2 ) + E(V, u1 ∧ u2 ) = E(V, u1 ) + E(V, u2 ).

It is standard to derive from this additivity property that the pointwise minimum
and maximum of two minimizers with the same Dirichlet condition are energy
minimizers as well. Adding in the strong maximum principle for harmonic functions
we can show an ordering property of the collection of energy minimizers, again thisis a suﬀiciently standard idea that we do not have a particular original reference.

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

39

Lemma 5.11 (No crossing). Let V be an open set containing Rd \ U . If u1 and u2
both minimize E(V, ·) with respect to H01 (U ) perturbations and u1 − u2 ∈ H01 (U ).
Then u1 ∧ u2 and u1 ∨ u2 are also minimizers and u1 and u2 are ordered in eachconnected component of Ω(u1 ) ∪ Ω(u2 ) = Ω(u1 ∨ u2 ).
Remark 5.12. To be clear the ordering between u1 and u2 may differ in the
different connected components of Ω(u1 ) ∪ Ω(u2 ). For example, given two orderedminimizers the same scenario can be repeated far away with the reverse ordering
to create unordered minimizers.
Proof of Lemma 5.11. Recalling (5.13) we haveE(V, u1 ∨ u2 ) + E(V, u1 ∧ u2 ) = E(V, u1 ) + E(V, u2 ).
On the other hand, since u1 = u2 on ∂U in the trace sense, also u1 ∧ u2 , u1 ∨ u2 =u1 = u2 on ∂U in the trace sense. So, from the minimizer assumption on u1 andu2 , we conclude that all four terms in the above equality must be equal. Thus
u1 , u2 , u1 ∧ u2 , and u1 ∨ u2 are all minimizers of E(V, ·) over the admissible class.
In particular each one is harmonic in its positivity set. By unique continuation
for harmonic functions this means that u1 and u2 are ordered in each connectedcomponent of U ∩ {u1 ∨ u2 > 0}. Note that by harmonicity again if u1 6= u2 in
a given connected component of U ∩ {u1 ∨ u2 > 0} then the ordering is strict init.
□
5.3.2. Abstract left-right limits. Next we discuss some general facts about boundedvariation maps on intervals of the line R. Suppose (Y, dY ) is a complete metricspace.
Definition 5.13. Say f : [a, b] → Y is a bounded variation map if[f ]BV([a,b];Y ) :=

sup

N
X

a=t0 ≤t1 ≤···≤tN +1 =b j=0

d(f (tj+1 ), f (tj )) < +∞.

On open or half open intervals define[f ]BV([a,b);Y ) := lim[f ]BV([a,b′ ];Y ) ,′b ↗bor similar for (a, b] and (a, b). The limit exists by monotonicity.
Definition 5.14. Let f : [a, b] → Y . For each t ∈ [a, b], if the limits exist define
the left limit fℓ (t) at t ∈ (a, b]fℓ (t) := lim f (s)s→t−

and the right limit fr (t) at t ∈ [a, b)
fr (t) := lim f (s).
s→t+Lemma 5.15. Let (Y, d) a complete metric space and f a bounded variation map[a, b] → Y . Then f has left and right limits at each t ∈ (a, b] (resp. [a, b)) and f
has at most countably many jump discontinuities.
The proof is standard, but we give a sketch for convenience.

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W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

Proof. For the directional limits note that for any sequence sj % t ∈ (a, b]
X
dY (f (sj ), f (sj+1 )) ≤ [f ]BV([a,b];Y ) < +∞jand so f (sj ) is Cauchy in Y . The limits must agree on different approaching sequences by a typical interlacement argument with pairs of sequences. A symmetricargument produces the right limits.
Now, first arguing for any finite set T of times, and then taking the supremum
over all finite setsX
dY (fℓ (t), fr (t)) ≤ [f ]BV([a,b];Y ) .
t∈[a,b]

In particular fℓ (t) = fr (t) except for at most countably many times.

□

5.3.3. Characterization of the upper and lower semicontinuous envelopes for energy
solutions. Lemma 5.15 and Lemma 5.7 together yield the existence of left andright limits of Ω(u(t)), J (u(t)), and P (u(t)) at all times. Our next result says thatu(t) also has left and right limits in the uniform metric, even though we do notnecessarily establish that t → u(t) is a BV in time map into Cc (Rd ). The left andright limits of all the previous quantities are consistent, i.e. (P ◦ u)ℓ (t) = P (uℓ (t))
etc. Furthermore the left and right limits of u(t) satisfy certain global minimality
properties for the energy plus dissipation distance.
The special structure of the problem comes in when we prove certain monotonicity properties of all the jumps, namely properties (iii)-(iv) below. This allows us
to make a simple classification of the upper and lower semicontinuous envelopes ofenergy solutions, showing that they are energy solutions themselves. This simple
jump structure will be very useful in Section 7, where we prove a certain weak version of the dynamic slope condition, and make the following proposition a central
result of this section.
Semicontinuous envelopes are typically important in studying the geometricproperties of interface evolution problems. They are used, for instance, in thenotion of discontinuous viscosity solutions for free boundary problems, as seen in
Section 4. The fact that these envelopes are themselves energy solutions is extremely useful to us in Section 7.
In service of analyzing the properties of energy solutions at jump times andsimplifying repetitive notations let us introduce the set(5.14)

U (t) = {uℓ (t), u(t), ur (t), uℓ (t) ∧ ur (t), uℓ (t) ∨ ur (t)}which is a kind of multi-valued version of u(t) indexing the values taken by u(t)“near” time t as well as the upper and lower envelopes. The set U (t) comes with anatural partial order induced by the time variable. Since uℓ (t) is the limit from theleft of u(t) it is “before” all the other elements of U (t), and since ur (t) is the limit
from the right it is “after” all the other elements of u(t). Specifically we define thepartial order(5.15)uℓ (t) ⊴ u(t), uℓ (t) ∧ ur (t), uℓ (t) ∨ ur (t) ⊴ ur (t).

To be clear this partial ordering is purely related to the “time” of the elements ofU (t) and is not related to the spatial ordering between the different elements ofU (t).

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION41

The core step in the following proposition is part (ii). Essentially we show thatevery element of U (t) is a valid intermediate state for the evolution at time t. Morespecifically, the energy dissipation relation is still satisfied as long as the system
jumps from uℓ (t) to any element of U (t) and then to ur (t).
Proposition 5.16. Suppose u is an energy solution on [0, T ] then:(i) (Left and right limits exist and are consistent) At every t ∈ [0, T ], u(t),J (u(t)), P (u(t), and Ω(u(t)) have left and right limits and(5.16)

lims→ℓ/r t Ω(u(t)) = Ω(uℓ/r (t)), lims→ℓ/r t J (u(t)) = J (uℓ/r (t)),
and lims→ℓ/r t P (u(t)) = P (uℓ/r (t))(ii) (Jumps minimize energy plus dissipation) At every t ∈ [0, T ] for any v, w ∈U (t) (from (5.14)), with v ⊴ w (from (5.15))
w minimizes E(v, ·) over F (t) + H01 (U ).
Note that in particular this implies all v ∈ U (t) are globally stable sincev ⊴ v.
(iii) (Jumps are monotone per component) At every t ∈ [0, T ]uℓ (t) ∧ ur (t) ≤ u(t) ≤ uℓ (t) ∨ ur (t).
In particular at each t ∈ [0, T ] the upper semicontinuous and lower semicontinuous envelopes are u∗ (t) = ur (t) ∨ uℓ (t) and u∗ (t) = ur (t) ∧ uℓ (t).
(iv) (Re-definition at jumps) Suppose that v : [0, T ] × Rd → R satisfies theproperty:
for each t ∈ [0, T ] v(t) ∈ U (t) (from (5.14)).
Then v(t) is also an energy solution on [0, T ]. In particular the upper andlower semicontinuous envelopes u∗ (t) and u∗ (t) are also energy solutions.
Besides the independent interest in understanding the time regularity propertiesof energetic solutions, this result plays a key role in establishing the validity of thefree boundary motion law in Theorem 7.1.
Remark 5.17. Note that part (iii) does allow the possibility that the free boundaries ∂Ω(uℓ (t)) and ∂Ω(ur (t)) are distinct and monotonically ordered but touch insome nontrivial region. This is possible to occur: for example consider a small balland a large ball merging with a large pinning interval. This would allow a portionof the free boundary opposite the “contacting area” to stay fixed as depicted in thesimulation Figure 2. In other words, although strong maximum principle holds forthe solutions, it does not hold for the free boundaries.
Remark 5.18. Note that from Lemma 5.11 and (iii) in each connected component
V of U ∩ {u∗ (t) > 0} we have {u∗ (t, ·)|V , u∗ (t, ·)|V } = {ur (t)|V , uℓ (t)|V }.
Remark 5.19. Note that v(t) as in (iv) is always measurable since v(t) = u(t)except at countably many times.
Proof of Proposition 5.16. Part (i). By Lemma 5.7 the map
t 7→ (1Ω(u(t)) , J (u(t)), P (u(t))) is in BV ([0, T ]; L1 (Rd ) × R × R)
and so by Lemma 5.15 the map has left and right limits in L1 (Rd ) × R × R at eacht ∈ [0, T ].

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W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

Call Ωℓ (t0 ) to be the set obtained by the L1 left limit of Ω(u(t)) at t0 . Let
tn % t0 . By uniform Lipschitz continuity of u(tn ) there is a subsequence converging uniformly to some u∞ which satisfies u∞ = F (t0 ) on ∂U . By uniform
non-degeneracy of the u(t) we must have Ω(u∞ ) = Ωℓ (t0 ). This specifies that u∞
is the unique solution of(5.17)∆u∞ = 0 in Ωℓ (t0 ) ∩ U with u∞ = 0 on ∂Ωℓ (t0 ) and u∞ = F (t0 ) on ∂U.
Since the original sequence tn % t0 was arbitrary we find that u(t) → u∞ uniformly
as tn % t0 . The argument is the same for right limits.
Thus u(t) has left and right limits uℓ/r (t) in the uniform metric at each timet ∈ [0, T ]. Next we need to show the consistency properties (5.16). We alreadyshowed that lims→ℓ/r t Ω(u(t)) = Ω(uℓ/r (t)) using uniform non-degeneracy in theprevious paragraph. Recalling that P (u(t)) = F (t)−1 D(u(t)), to conclude we justneed to check the consistency of limits for the Dirichlet energy D(u(t))ˆˆ
lim|∇u(t)|2 dx =|∇uℓ/r (t)|2 dx.
(5.18)
s→ℓ/r tU

U

We claim that ∇u(s) converges locally uniformly on Ω(uℓ/r (t)) to ∇uℓ/r (t) as s % t(resp. s & t). Along with the uniform boundedness of ∇u(t) the dominatedconvergence theorem gives (5.18).
As usual we handle the left limit case and the right limit case is similar. Let
K be a compact subset of Ω(uℓ (t)). By the uniform convergence u(s) → uℓ (t) ass % t and by uniform non-degeneracy
inf u(s) ≥ cd(K, Ω(uℓ (t))∁ )K

and so by uniform Lipschitz continuityd(K, Ω(u(s))∁ ) ≥ cd(K, Ω(uℓ (t))∁ )for s < t suﬀiciently close to t. Thus the ∇u(s) are uniformly bounded and uniformly continuous on K for s < t suﬀiciently close to t. Typical arbitrary subsequence arguments show that ∇u(s) converges uniformly on K to ∇uℓ (t) as s % t.
Part (ii). We introduce the extraneous notation u0 (t) = u(t) in order to prove
minimizer properties for uℓ /u0 /ur at once.
First we check that each of uℓ/0/r (t) minimize their own respective E(Ω(uℓ/0/r (t)), ·)over F (t) + H01 (U ). For u0 (t) this is just the global stability property of an energy
solution. For uℓ (t), by the global stability property for any t− < t we knowF (t− )
v + Diss(u(t− ), v) for all v ∈ F (t) + H01 (U ).
J (u(t− )) ≤ J
F (t)Taking the limit as t− % t, using time continuity of F , and applying part (i)J (uℓ (t)) ≤ J (v) + Diss(uℓ (t), v) for all v ∈ F (t) + H01 (U ).
The proof that ur (t) minimizes E(ur (t), ·) is similar, applying global stability ofu(t+ ) and taking a limit t+ & t.
Next we check that u0/r minimize E(uℓ (t), ·) over F (t) + H01 (U ). Apply thedissipation relation with times t− ≤ t ≤ t+ˆ t+
2Ḟ (s)P (s) ds ≥ Diss(u(t− ), u(t+ )) + J (u(t+ )).
J (u(t− )) +t−

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

43Taking the limit as t− % t and t+ & t and using the left/right continuity of J (u(·))
and Ω(·) established in part (i)J (uℓ (t)) ≥ J (ur (t)) + Diss(uℓ (t), ur (t))
or in the case t+ = t and t− % t
J (uℓ (t)) ≥ J (u(t)) + Diss(uℓ (t), u(t))
Since we already established that uℓ (t) is always a minimizer of E(uℓ (t), ·) alsou0/r (t) must be minimizers. Finally in the case t− = t and t+ & t we obtain
similarly
J (u(t)) ≥ J (ur (t)) + Diss(u(t), ur (t))
which gives that ur (t) minimizes E(u(t), ·).
Lastly we consider uℓ (t)∧ur (t) and uℓ (t)∨ur (t). Since t is fixed for the remainderof this part of the proof we will write uℓ = uℓ (t) and ur = ur (t) to simplify
expressions.
First of all, since uℓ and ur minimize E(uℓ , ·) over F (t) + H01 (U ), so also, by(5.13), uℓ ∧ ur and uℓ ∨ ur both minimize E(uℓ , ·) over F (t) + H01 (U ).
Then for any v ∈ F (t) + H01 (U )E(uℓ , uℓ ∧ ur ) ≤ E(uℓ , v)orJ (uℓ ∧ ur ) ≤ J (v) + Diss(uℓ , v) − Diss(uℓ , uℓ ∧ ur )≤ J (v) + Diss(uℓ ∧ ur , v)
= E(uℓ ∧ ur , v)
by the dissipation distance triangle inequality Lemma A.3 for the last inequality.
Thus uℓ ∧ ur is globally stable. A similar argument applies to uℓ ∨ ur .
Finally we need to argue thatur minimizes E(uℓ ∧ ur , ·) and E(uℓ ∨ ur , ·) over F (t) + H01 (U ).
We just argue for E(uℓ ∧ ur , ·), the other case is similar,
E(uℓ ∧ ur , v) = E(uℓ , v) + Diss(uℓ ∧ ur , v) − Diss(uℓ , v)
≥ E(uℓ , ur ) + Diss(uℓ ∧ ur , v) − Diss(uℓ , v)
= J (ur ) + Diss(uℓ , ur ) + Diss(uℓ ∧ ur , v) − Diss(uℓ , v)
= J (ur ) + Diss(uℓ ∧ ur , v) + Diss(v, ur )+ [Diss(uℓ , ur ) − Diss(uℓ , v) − Diss(v, ur )]= J (ur ) + Diss(uℓ ∧ ur , ur )+ (µ− + µ+ )[|Ω(v) \ Ω((uℓ ∧ ur ) ∨ ur )| + |Ω((uℓ ∧ ur ) ∧ ur ) \ Ω(v)|]− (µ− + µ+ )[|Ω(v) \ Ω(uℓ ∨ ur )| + |Ω(uℓ ∧ ur ) \ Ω(v)|]= E(uℓ ∧ ur , ur ).
where we have used that ur minimizes E(uℓ , ·), then the sharp triangle inequality
Lemma A.3, and finally that (uℓ ∧ ur ) ∧ ur = uℓ ∧ ur and (uℓ ∧ ur ) ∨ ur = uℓ ∨ ur .
Part (iii). Since t is fixed in this part of the proof we write uℓ/0/r (t) = uℓ/0/r
dropping the t dependence. By global stability and part (ii) respectively we knowthat uℓ and ur both minimize E(uℓ , ·). Furthermore Lemma 5.11 yields that uℓ andur are ordered in each connected component of Ω(uℓ ∨ ur ).

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W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

Next we show that uℓ ∧ ur ≤ u ≤ uℓ ∨ ur . We use the multiple minimizations
in part (ii) to show an “equality in triangle inequality” which implies that u must
be in between uℓ and ur . Intuitively speaking an intermediate jump which crosses
either uℓ or ur would just incur extra dissipation cost to return to ur .
We compute using part (ii)J (ur ) + Diss(uℓ , ur ) = J (u) + Diss(uℓ , u)
= J (ur ) + Diss(u, ur ) + Diss(uℓ , u).
More specifically the first equality is since ur and u both minimize E(uℓ , ·), thesecond equality is since u and ur both minimize E(u, ·). Simplifying we find theequality(5.19)

Diss(uℓ , u) + Diss(u, ur ) − Diss(uℓ , ur ) = 0.

By the sharp triangle inequality Lemma A.3


Diss(uℓ , u)+Diss(u, ur )−Diss(uℓ , ur ) = (µ− +µ+ ) |Ω(uℓ ∧ur )\Ω(u)|+|Ω(u)\Ω(uℓ ∨ur )| .

Together with (5.19), above equality shows that Ω(uℓ ∧ ur ) ⊂ Ω(u) ⊂ Ω(uℓ ∨ ur ).
Since uℓ ∧ur , u, and uℓ ∨ur are harmonic in their respective positivity sets by Lemma
5.11, then the previous set ordering and maximum principle implies uℓ ∧ ur ≤ u ≤uℓ ∨ ur .
Part (iv). Since uℓ (t) = ur (t) = u(t) = F (t) on ∂U the boundary condition issatisfied by v(t). By (ii) any v(t) ∈ U (t) is globally stable.
Finally we need to check the dissipation relation (1.6) for v. We do this in
two steps, the dissipation on the open interval (t0 , t1 ) plus the dissipation at theendpoints.
Fix t0 < t1 and we claim that
ˆ t1
2Ḟ (s)P (s) ds ≥ Diss(ur (t0 ), uℓ (t1 ))J (ur (t0 )) − J (uℓ (t1 )) +(5.20)
t0

To prove this apply the dissipation relation for u(t) with sequences t0,k and t1,kwith t0,k & t0 and t1,k % t1
ˆ t1,kJ (u(t0,k )) − J (u(t1,k )) +2Ḟ (s)P (s) ds ≥ Diss(u(t0,k ), u(t1,k )).
t0,kSending k → ∞ and using the continuities from (i) shows (5.20).
Now consider the endpoint dissipations, by (ii),J (ur (t0 ))+Diss(v(t0 ), ur (t0 )) = J (v(t0 )) and J (v(t1 ))+Diss(uℓ (t1 ), v(t1 )) = J (uℓ (t1 ))since every v(t0 ) ⊴ ur (t0 ) for every v(t0 ) ∈ U (t0 ) and uℓ (t1 ) ⊴ v(t1 ) for everyv(t1 ) ∈ U (t1 ) in the temporal partial ordering (defined in (5.15)).
Adding these to (5.20) givesˆ t1
2Ḟ (s)P (s) ds
J (v(t0 )) − J (v(t1 )) +t0

≥ Diss(v(t0 ), ur (t0 )) + Diss(ur (t0 ), uℓ (t1 )) + Diss(uℓ (t1 ), v(t1 ))≥ Diss(v(t0 ), v(t1 ))by applying the triangle inequality Lemma A.3 in the last step. This completes theproof.
□

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

45

6. Limit of the minimizing movement scheme
In this section we will study the minimizing movements scheme introduced in(1.7). We recall that scheme here for convenience:ukδ ∈ argmin J (w) + Diss(uk−1, w) : w ∈ F (kδ) + H01 (U ) .
δThen interpolate discontinuously to define, for all t ∈ [0, T ],uδ (t) := ukδ and Fδ (t) = F (kδ) if t ∈ [kδ, (k + 1)δ).
We will analyze the behavior of the minimizing movements scheme as δ → 0.
First, in Section 6.1, we apply established ideas [1, 27] to show that, up to a
subsequence, the uδ converge pointwise in time to an energy solution (E). Our
argument will be quite similar to that in Alberti and DeSimone [1, Section 4] which
was itself inspired by Mainik and Mielke [27]. The conclusion of these arguments isthat, up to a subsequence, the uδ (t) and Ω(uδ (t)) converge pointwise everywhere intime in uniform / Hausdorff distance norm in space to a (global) energetic solution.
Then, in Section 6.2 below, we will show that the approximate solutions generated by this scheme will also satisfy the dynamic slope condition (M) in the viscosity
sense. In the star-shaped case we will be able to apply the comparison principle
proved in Section 4 to show convergence to the unique obstacle solution (O), see
Theorem 6.6. As a corollary we obtain that the obstacle solution (O) is an energy
solution.
These results will complete the proof of our first main theorem in the introduction, Theorem 1.5.
Remark 6.1. The very recent result of Ferreri and Velichkov [22] implies thatC 1,β regularity is propagated in the discrete scheme (1.7). However, at least witha naive application of their result, the constants would blow-up as δ → 0 since theregularity theorem needs to be applied O( 1δ ) times in order to get to a nontrivial
positive t in the limit. In the star-shaped case we go around this diﬀiculty due tothe equivalence with the (O) evolution. This equivalence allows us to apply theregularity theorem only finitely many times, at each monotonicity change, as inLemma 4.22.
6.1. Pointwise in time convergence to an energy solution. The goal of thissection is to analyze the limits of the minimizing movements scheme by energetic
methods. We start by proving uniform (in δ > 0) BV in time bounds for theminimizing movements scheme via a discrete Grönwall type argument similar toLemma 5.7. This establishes the necessary compactness to show that, up to a
subsequence, the uδ (t) and Ω(uδ (t)) converge at every time in uniform / Hausdorff
distance norm to a (global) energetic solution. The convergence at every time,
which follows from an application of Helly’s selection theorem, is a key detail toestablish the energy solution property.
Theorem 6.2. The uδ (t) generated by the minimizing movements scheme satisfy
the following:(i) The states uδ (t) are uniformly Lipschitz, χΩδ (t) and J (uδ ) are uniformly
bounded in BV ([0, T ]; L1 (U )) and BV ([0, T ]; R) respectively.
(ii) There is a subsequence δk → 0 and u : [0, T ] → F (t) + H01 (U ) so that forevery t ∈ [0, T ]kuδk (t) − u(t)kL∞ (U ) and dH (Ω(uδk (t)), Ω(u(t))) → 0 as δk → 0.

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W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

(iii) Any such subsequential limit u(t) is an energy solution.
Remark 6.3. By Theorem 6.2 any minimizing movements solution, Definition 1.2,
is indeed an energy solution of (E).
The proof will use many of the same ideas that were developed in Section 5 inthe continuous time case, so we will refer to the relevant arguments above in many
places to significantly shorten the presentation.
Proof. Part (i). A discrete version of the energy dissipation inequality will play acentral role in the proof. We claim that for each k ≥ 1
ˆ kδ
k−1
k
(6.1)
Diss(uδ , uδ ) ≤2(1 + gδ (t))Ḟ (t)Pδ (t) dt + J (uk−1) − J (ukδ )δ(k−1)δ

where the error term in the integrand has|gδ (t)| ≤ exp(k(log F )′ k∞ δ) − 1 → 0 uniformly as δ → 0,
´ ∂u[t/δ]δ
and we used the notation Pδ (t) := ∂U ∂ndS.
To prove this by using the minimizing scheme we build a comparable version ofF (kδ)
uk−1. Then we haveuk−1
to ukδ , by defining ũ := F ((k−1)δ)
δ
δ

(6.2)

Diss(uk−1, ukδ )+J (ukδ ) ≤ Diss(uk−1, ũ)+J (ũ) = J (ũ) = J (ũ)−J (uk−1)+J (uk−1
).
δ
δ
δ
δ
By integration by parts we can rewriteˆ2
Fk2 − Fk−1
∂uk−1
δ
dS
Fk−1
∂n
∂U
ˆ kδ
F (t)
Ḟ (t)Pδ (t) dt.
=
2
F
k−1
(k−1)δJ (ũ) − J (uk−1
)=
δ

(t)
− 1 for [t] = k − 1 which satisfies (6.2) by fundamental
Then define gδ (t) := FFk−1
theorem of calculus and the inequality 1 − e−k ≤ ek − 1 for k > 0. That completes
our proof of (6.1).
Next we sum up the one-step dissipation inequality (6.1) to get, for any 0 ≤ t0 ≤t1 ≤ T ,ˆ t1
2(1+gδ (t))Ḟ (t)Pδ (t)dt+J (uδ (t1 ))−J (uδ (t0 ))+Cδ
(6.3) Diss(uδ (t); [t0 , t1 ]) ≤t0

where Cδ accounts for the error from t0 and t1 not necessarily being integer multiples of δ and we are using the uniform Lipschitz bound of uδ (t) from Lemma
5.5.
From (6.3) we can derive uniform BV ([0, T ]; L1 (U ) × R × R) bounds on the map
t 7→ (χΩδ (t), Pδ (t), Jδ (t)) by the same arguments as in Lemma 5.7.
Part (ii). Due to the above compactness, Helly’s selection principle [1, Theorem
5.1] yields that, along a subsequence,(χΩδ (t), Pδ (t), Jδ (t)) → (χΩ(t) , P (t), J (t)) in L1 (U ) × R × R for every t ∈ [0, T ].
Then arguing as in the proof of part (i) of Proposition 5.16 we can also show thatuδ (t) → u(t) uniformly in U for each t ∈ [0, T ] and we can show the consistency ofthe limitsΩ(t) = Ω(u(t)), J (t) = J (u(t)), and P (t) = P (u(t)).

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

47

Furthermore the convergence of Ωδ (t) → Ω(t) can be upgraded to Hausdorff distance convergence using the uniform convergence of the uδ (t) → u(t) and theuniform non-degeneracy of the uδ (t).
Part (iii). Now let us show that u(t) is an energy solution. To show the stability
property, observe that for any v ∈ u(t) + H01 (U ) we can take k = [t/δ] and compareuδ (t) = ukδ with ṽ := FF(δk)(t) v(t), so that
Diss(uk−1, ukδ ) + J (ukδ ) ≤ Diss(uk−1, ṽ) + J (ṽ).
δδ
Note that, due to the possibility of a jump in the limit we don’t necessarily knowthat Diss(uk−1, ukδ ) → 0. So to get everything in terms of uδ (t) = ukδ we apply theδtriangle inequality Lemma A.3
J (uδ (t)) = J (ukδ ) ≤ Diss(uk−1, ṽ) − Diss(uk−1, ukδ ) + J (ṽ)δδ≤ Diss(ukδ , ṽ) + J (ṽ)= Diss(uδ (t), ṽ) + J (ṽ)
Now sending δ → 0 along the convergent subsequence yields the desired stability
inequality.
Last we show the energy dissipation inequality. For any 0 ≤ t0 < t1 ≤ T we canapply (6.3) to find
ˆ t1
2(1 + gδ (t))Ḟ (t)Pδ (t)dt + J (uδ (t1 )) − J (uδ (t0 )) + Cδ.
Diss(uδ (t0 ), uδ (t1 )) ≤t0

Using the consistency results from part (ii) above and sending δ → 0 gives thedissipation inequality for u. The convergence of the left hand side just follows fromthe L1 convergence of the indicator functions, and the convergence of the integral
is by dominated convergence theorem.
□
6.2. Comparison properties of the minimizing movements scheme. In thissection we discuss the comparison properties of the minimizing movement scheme
and its δ → 0 limit.
Lemma 6.4. The uδ (t) are viscosity solutions of (M) on [0, T ] in the sense of
Definition 1.4.
Remark 6.5. The uδ (t) are also viscosity solutions of (M) in the “standard” senseof touching from above and below by C 1 test functions in space-time. This notion,
unlike the notion in Definition 1.4 which defines positive velocity based on cones,
behaves well with respect to the limit δ → 0. One can apply the standard methodof upper and lower half-relaxed limits, of Barles and Perthame [3], to show that u∗
and u∗ are respectively sub and supersolutions of (M) in the standard sense.
We should remark that the upper and lower-half relaxed limits u∗ and u∗ are notnecessarily the same as the upper and lower semicontinuous envelopes u∗ (t) andu∗ (t) of the pointwise limit u(t) constructed in Theorem 6.2. It would be interesting
to know whether the pointwise in time (subsequential) limits u(t) are also viscosity
solutions of (M) in general either in the standard sense or in the cone sense of
Definition 1.4. Below in Theorem 6.6 we will show that it is so in the star-shaped
case.
Proof. We only check the subsolution condition at the free boundary since theremaining cases are standard or symmetrical.

48

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

Suppose that Ω(u∗δ (t)) has positive normal velocity Vn (t0 , x0 ) > 0 at some pointx0 ∈ ∂Ω(u∗δ (t0 )) in the sense of Definition 4.9, i.e.
{x : |x − x0 | ≤ c(t − t0 )} ⊂ Ω(u∗δ (t))∁ for t0 − r0 ≤ t < t0 .
Since uδ is constant on open intervals of the form (kδ, (k + 1)δ) this implies thatt0 = kδ for an integer k, that u∗δ (t0 ) = ukδ , and that u∗δ (t) = uk−1
for t0 − δ ≤ t < t0 .
δSince ukδ is a minimizer ofE(uk−1, v) = J (v) + Diss(uk−1, v) over v ∈ F (kδ) + H01 (U )δδ) we can apply Lemma 5.2 andand we have just established that x0 ∈ Rd \ Ω(uk−1δLemma 5.3 to conclude that|∇u(t0 , x0 )|2 ≥ 1 + µ+ in the viscosity sense.
□
Now we can apply the previous viscosity solution property in combination withthe comparison theorem in Section 4 to show that, in the star-shaped case, the limitof the minimizing movements scheme is the same as the solution of the obstacle
evolution (O). The convergence is in fact quantitative.
Theorem 6.6. Suppose that Ω0 is a C 1,α strongly star-shaped region, K = Rd \U iscompact and strongly star-shaped, F is positive Lipschitz continuous, and F changes
monotonicity at most finitely many times on [0, T ]. Call v(t) the obstacle solution(O) and uδ (t) to be a solution of the time incremental scheme (1.7) on [0, T ] × U .
Thenkuδ (t) − v(t)kL∞ (U ) ≤ C(µ± , d)(exp(k(log F )′ k∞ δ) − 1)kF k∞ → 0 as δ → 0.
Thus the viscosity solution of (O) is the unique minimizing movements solution of
the energetic evolution (Definition 1.2), in particular any minimizing movements
solution, Definition 1.2, is also a motion-law viscosity solution (M).
Let us clarify the conclusions of this statement, since several threads of the paper
have suddenly come together. We have shown the following:• In the star-shaped setting the obstacle solution (O) is an energy solution.
• In the star-shaped setting there is only one minimizing movements energy
solution (Definition 1.2) and it is also a motion-law viscosity solution (M).
• In the star-shaped setting all solutions of the discrete scheme converge
uniformly to the obstacle solution (O) with quantitative rate.
Proof. We prove the convergence by comparison principle Proposition 4.26 with v(t)∞ 1,βthe (O) solution which, by Corollary 4.25, is in Ct0,1 L∞
on [0, T ] × U .
x ∩ Lt Cx
Let M (δ) = supt suph∈[0,δ] F (t − h)/F (t), note that M (δ) ≤ exp(k(log F )′ k∞ δ).
Thenφδ (t, x) := M (δ)v(t, M (δ)−1 x)∞ 1,βis a Ct0,1 L∞
supersolution of (M) on [0, T ] with star-shaped level setsx ∩ Lt Cx
and so the comparison principle Proposition 4.26 implies that uδ ≤ φδ on [0, h] × U .
Note that we have chosen M (δ) ≥ 1 so that for x ∈ K since K star-shaped also
M (δ)−1 x ∈ K andφδ (t, x) = M (δ)v(t, M (δ)−1 x) = M (δ)F (t) ≥ sup F (t − h) ≥ Fδ (t)h∈[0,δ]ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

49

By uniform Lipschitz regularity of v(t)uδ (t, x) ≤ φδ (t, x)≤ M (δ)v(t, x) + M (δ)k∇v(t)k∞ (M (δ)−1 − 1)

sup

|x|

x∈Ω(v(t))≤ v(t, x) + C(M (δ) − 1)kF k∞
Note that Ω(v(t)) ⊂ BCF (t) (0) by uniform non-degeneracy of v(t) and the upperbound v(t) ≤ F (t).
This proves the upper bound, the lower bound is similar comparing with aninward dilation of size m(δ) = inf t inf h∈[0,δ] F (t − h)/F (t) instead.
Furthermore, applying Theorem 6.2, we also know that, along a subsequence,uδ (t) → u(t) pointwise in time and uniformly in space and u(t) is an energy solution.
By uniqueness of pointwise limits u ≡ v and so the solution of (O) is an energy
solution.
□
7. The dynamic slope condition for energy solutions
In this section we show that the energy dissipation balance law implies a weak
notion of the dynamic slope condition (1.5)|∇u|2 = 1 ± µ±if

± Vn (Ω(t), x) > 0.

The notion of weak solution is geometric measure theoretic, although there is someviscosity solution aspect as well since there is a subsolution and a supersolution
piece which are expressed in terms of sub and superdifferentials. This will complete
the proof of Theorem 1.6.
Note that in Section 6 we showed, at least in the star-shaped case, that all minimizing movements solutions are viscosity solutions of the motion law (M). However,this proof was based on the structure provided by the time incremental scheme.
Even in the star-shaped case, we do not know uniqueness of energy solutions orwhether all energy solutions can be generated by this scheme. Thus we must provesome notion of the dynamic slope condition (1.5) directly from the energy solution
property (E).
Recall by Proposition 5.16 part (iv) if u is an arbitrary energy solution on U ×[0, T ] then the envelopes u∗ and u∗ are also energy solutions and have a simple
representation in terms of the left and right limits at each time u∗ (t) = uℓ (t) ∧ ur (t)and u∗ (t) = uℓ (t) ∨ ur (t).
The main theorem of this section is that if u(t) is an energy solution then u∗ (t)and u∗ (t) are respectively (up to sets of surface measure zero) a motion law viscosity
subsolution and supersolution.
For the statement we will use the notions of positive and negative velocity
Vn± (t, x), based on inner and outer touching space-time cones given in Section
4, and the notions of sub and superdifferential. Specifically refer back to Definition
4.4, Definition 4.3, Definition 4.9, and Definition 4.10.
Theorem 7.1. Suppose that u is an upper semicontinuous energy solution (E) on
U × [0, T ]. Then for every t ∈ (0, T ] the set of points where the outward motion law
fails
Γ+ (u, t) := {x ∈ ∂Ω(t) : Vn (t, x) > 0, |p|2 < 1 + µ+ for some p ∈ D+ u(t, x)}
has Hd−1 measure zero.

50

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

Similarly, if u is an lower semicontinuous energy solution (E) on U × [0, T ], thenfor every t ∈ (0, T ] the set of points where the inward motion law fails
Γ− (u, t) := {x ∈ ∂Ω(t) : Vn (t, x) < 0, |p|2 > 1 − µ− for some p ∈ D− u(t, x)}
has Hd−1 measure zero.
We will only prove the result for Γ+ . The proof for Γ− is analogous.
Remark 7.2. The sub and superdifferential may both be empty at some freeboundary points. However, for energetic solutions, we can follow standard blow-up
arguments (see [6, Section 3.3]) to show that the sub and super-differential are both
non-trivial Hd−1 -a.e. on ∂Ω(u(t)).
Recall that ∂Ω(u(t)) is a finite perimeter set and ∂Ω(u(t)) has finite Hd−1 measure (Lemma 5.5) for each t > 0. Therefore the reduced boundary ∂∗ {u(t) > 0}
has full Hd−1 measure on ∂{u(t) > 0}.
The sub and superdifferentials are both non-trivial at every point of the reduced
boundary. More precisely, at x0 ∈ ∂∗ {u > 0} with measure theoretic normal n0
consider the blow-up sequenceu(x0 + rx)
rwhich are uniformly bounded, uniformly non-degenerate, and uniformly Lipschitz.
All subsequential blow up limits (not necessarily unique) must then be viscosity
solutions ofur (x) =∆v = 0 in {x · n0 > 0} and 1 − µ− ≤ |∇v|2 ≤ 1 + µ+ on ∂{x · n0 > 0}.
From this one can conclude that for any (subsequential, locally uniform) blow up
limit(1 − µ− )1/2 (x · n0 )+ ≤ v(x) ≤ (1 + µ+ )1/2 (n0 · x)+
and this implies that at least(1 + µ+ )1/2 n0 ∈ D+ u(x0 ) and (1 − µ− )1/2 n0 ∈ D− u(x0 ),i.e. the sub and superdifferential are nontrivial on the measure theoretic reduced
boundary.
Before we proceed to the details let us give a description of the proof, which also
explains why we can only prove the solution condition in an almost everywheresense.
We start by presenting the formal argument, valid in the case that everything is
C 1 in space and time. Computing the time derivative directly and integrating by
parts
ˆ
d
J (u(t)) =(1 − |∇u|2 )Vn dS + 2Ḟ (t)P (t)dt∂Ω(t)
but also differentiating the energy dissipation balance from Lemma 5.10 yieldsˆd
J (u(t)) = 2Ḟ (t)P (t) −µ+ (n)(Vn )+ + µ− (n)(Vn )− dS.
dt∂Ω(t)Combining the above two identities we findˆ
(7.1)
(1 + µ+ (n) − |∇u|2 )(Vn )+ + (|∇u|2 − 1 + µ− (n))(Vn )− dS = 0.
∂Ω(t)ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

51

Both terms in the above integral are non-negative (Corollary 5.4), and so they must
actually be zero pointwise.
Of course we cannot exactly use the formal argument. Even without jumps it istricky to justify taking a time derivative. Instead we need to make a similar kind ofenergy argument with discrete differences. Furthermore, even if we could justify theidentity (7.1), without continuity of ∇u we could only derive the slope condition in
the surface measure a.e. sense. Recall that we do not know C 1 regularity of generalenergy solutions. The regularity theory of Section 3 applies to viscosity solutions
and we are trying to prove a viscosity solution type property. This is why we canonly achieve the dynamic slope condition in the almost everywhere sense.
Although we do need to deal carefully with the jumps, the situation is actually
better when the free boundary jumps: in this case we can guarantee that slope
condition is satisfied pointwise everywhere.
Corollary 7.3. Let F > 0 and suppose that u ≥ uℓ (or u ≤ uℓ ) is a minimizer ofE(uℓ , u) = J (u) + Diss(uℓ , u) over F + H01 (U ).

(7.2)

Then for any x0 ∈ ∂Ω(u) \ Ω(uℓ ) (or x0 ∈ ∂Ω(u) ∩ Ω(uℓ )) then|p|2 ≥ (1 + µ+ ) for all p ∈ D+ u(x0 )(or |p|2 ≤ (1 − µ− ) for all p ∈ D− u(x0 )).

Proof. Apply Lemma 5.2 and Lemma 5.3.

□

In order to make the main ideas of the proof of Theorem 7.1 more clear we willstate in a separate lemma the construction of the energy competitor.
First we explain the idea of the energy competitor and then give a precise statement in Lemma 7.4 below. Suppose that the advancing part of the dynamic slope
condition fails at some time t0 on a set of positive Hd−1 measure. Then for times
t−1 = t0 − δ suﬀiciently close to t0 , we can perform an inward perturbation ofu(t0 ) in the normal direction in a region with measure at least O(δ), and withoutcrossing u(t−1 ). This perturbation will also have slope strictly smaller than thepinning interval endpoint value (1 + µ+ )1/2 in the perturbation region. This allows
us to estimate the change in the Dirichlet energy.
Lemma 7.4. Let u be upper semi-continuous on [0, T ], harmonic in Ω(u(t)) at eachtime, and satisfying the conclusions of Lemma 5.5 (e.g. u an upper semicontinuous
energy solution).
Suppose for some t0 ∈ (0, T ]


d−1
2
H
x ∈ ∂Ω(t0 ) : Vn (t0 , x) > 0, |p| < 1 + µ+ for some p ∈ D+ u(t0 , x)
> 0.
Then there exists c > 0 such that the following holds: for all suﬀiciently small
δ = t0 − t−1 > 0 there is a function uδ ∈ H 1 (U ) with uδ ≥ 0 and uδ = F (t0 ) on
∂U such thatΩ(u(t−1 )) ⊂ Ω(uδ ) ⊂ Ω(u(t0 )), |Ω(u(t0 )) \ Ω(uδ )| ≥ cδ

(7.3)
and

ˆ

ˆ
|∇uδ |2 dx −(7.4)Ω(uδ )|∇u(t0 )|2 dx ≤ (1 + µ+ − c)|Ω(u(t0 )) \ Ω(uδ )|.
Ω(t0 )

Remark 7.5. The constant c > 0 in above Lemma is not very easily quantified,
it depends delicately on the hypothesis that Hd−1 (Γ+ (u, t0 )) > 0 (not just on themeasure itself, but on the consequences of being positive measure).

52

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

We defer the proof of Lemma 7.4 to the end of the section. It is worth emphasizing that this part is the technical heart of the proof of Theorem 7.1.
Now we give the proof of the weak solution property Theorem 7.1 via an energy
argument using Lemma 7.4. Proposition 5.16 also plays a central role allowing us
to reduce to the case of left continuous times.
Proof of Theorem 7.1. Fix t0 ∈ (0, T ], we first assume that P (u(t)) is left continuous at t0 . Jump times will be considered at the end, building on the result at
left continuous times and using the upper semi-continuity of u as well as the otherregularity properties of energy solutions from Proposition 5.16.
Step 1. Assume for now that
|P (u(t0 )) − P (u(t))| ≤ σ(t0 − t) for t ≤ t0
with some modulus of continuity σ. Suppose that
Hd−1 Γ+ (u, t) > 0.
Then Lemma 7.4 yields uδ , t−1 , t0 for small δ > 0. We definev(x) :=

F (t−1 )F (t−1 )u(t0 , x) and vδ (x) :=uδ (x).
F (t0 )
F (t0 )

From the energy dissipation inequality (1.6) evaluated on [t−1 , t0 ], we deriveDiss(Ω(t−1 ), Ω(t0 )) ≤ J (u(t−1 )) − J (u(t0 )) +

ˆ t0
2Ḟ (t)P (t) dt
t−1= [J (u(t−1 )) − J (vδ (t0 ))] + [J (vδ (t0 )) − J (v(t0 ))]#"ˆ t0
2Ḟ (t)P (t) dt+ J (v(t0 )) − J (u(t0 )) +
t−1

(7.5)

= A + B + C.

We next explain, individually, how to bound A, B, C to yield a contradiction.
Since vδ = F (t−1 ) on ∂U , we can apply the stability property Definition 1.1(2)at time t−1 to obtain
A = J (u(t−1 )) − J (vδ ) ≤ Diss(Ω(t−1 ), Ωδ (t0 )).
Next the energy inequality (7.4) from Lemma 7.4 yields

B := J (vδ (t0 )) − J (v(t0 )) ≤F (t−1 )2(1 + µ+ − c) − 1 |Ω(t0 ) \ Ωδ (t0 )|.
F (t0 )2

Recall that c > 0 is independent of δ > 0, so from the Lipschitz continuity oflog F (t) we conclude thatB ≤ (µ+ − c + oδ (1))|Ω(t0 ) \ Ωδ (t0 )|.

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

53

Lastly we address C. We haveˆF (t−1 )J (u(t0 )) − J (v(t0 )) =(1 −)|Du|2 (t0 )dx
F (t0 )UF 2 (t0 ) − F 2 (t−1 )=P (t0 )
F (t0 )
ˆ t02Ḟ (t)P (t)dt − 2kḞ kL∞ σ(t0 − t−1 )|t0 − t−1 |,
≥
t−1

ˆ t0
(7.6)

=

2Ḟ (t)P (t)dt − oδ (1)δ.

t−1This is the key place where we use the left continuity hypothesis on P .
Now, combining the previous three inequalities into (7.5), we have obtainedDiss(Ω(t−1 ), Ω(t0 )) ≤ Diss(Ω(t−1 ), Ωδ (t0 )) + (µ+ − c + oδ (1))|Ω(t0 ) \ Ωδ (t0 )| + oδ (1)δOn the other hand, since Ω(u(t−1 )) ⊂ Ωδ (t0 ) ⊂ Ω(t0 ) from Lemma 7.4, we haveDiss(Ω(t−1 ), Ω(t0 )) − Diss(Ω(t−1 ), Ωδ (t0 )) = µ+ |Ω(t0 ) \ Ωδ (t−1 )| − µ+ |Ωδ (t0 ) \ Ω(t−1 )|= µ+ |Ω(t0 ) \ Ωδ (t0 )|.
Putting the above estimates together, it follows thatµ+ |Ω(t0 ) \ Ωδ (t0 )| ≤ (µ+ − c + oδ (1))|Ω(t0 ) \ Ωδ (t0 )| + oδ (1)δ.
Since Lemma 7.4 crucially guarantees that |Ω(t0 ) \ Ωδ (t0 )| ≥ cδ, we can divide
through by |Ω(t0 ) \ Ωδ (t0 )| and findµ+ ≤ µ+ − c + oδ (1)Sending δ = (t0 − t−1 ) to zero yields 0 ≤ −c, which is a contradiction.
Step 2. (Jump times) Suppose that P (u(t)) is not left continuous at t0 . By
Proposition 5.16 (i) u has left and right limits at time t0 , uℓ (t0 ) and ur (t0 ). Furthermore, by Proposition 5.16 part (iii) and since u is upper semicontinuous wemust have u(t0 ) = uℓ (t0 ) ∨ ur (t0 ).
By Proposition 5.16 part (iv)(u(t)
t < t0ũ(t) :=uℓ (t0 ) t = t0 .
is itself an energy solution on [0, t0 ] and ũ(t) and P (ũ(t)) are left continuous at t0 .
The previous arguments give
Hd−1 (Γ+ (ũ, t0 )) = 0.
We aim to show Γ+ (u, t0 ) ⊂ Γ+ (ũ, t0 ) which will complete the proof showingH(Γ+ (u, t0 )) = 0. By Corollary 7.3
d−1

Γ+ (u, t0 ) \ ∂Ω(uℓ (t0 )) = ∅.
On the other hand since u touches ũ from above at each x0 ∈ ∂Ωℓ (t0 ) ∩ ∂Ω(t0 )Vn+ (t0 , x0 ) ≥ Ṽn+ (t0 , x0 ) and D+ u(t0 , x0 ) ⊂ D+ ũ(t0 , x0 )Thus
Γ+ (u, t0 ) ⊂ Γ+ (ũ, t0 ),which concludes the proof.

□

54

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

Remark 7.6. The regularity of the pressure P (t) seems to play an important role.
In (7.6) the regularity of P (t) dictates the size of a positive error term which needs
to be balanced by the positive Hd−1 measure of the set where the viscosity solution
condition fails.
Before continuing to the proof of the barrier construction Lemma 7.4, it is useful to separate out one further measure-theoretic Lemma which says that if the
subsolution condition fails on a set of positive surface measure, then it fails soquantitatively.
Let us first define quantified versions of the velocity and sub/superdifferentials.
For given r0 , c, σ > 0 we define
o
n
σ,r0
D+
u(x0 ) := p ∈ Rd : u(x) ≤ p · (x − x0 ) + σ|x − x0 | in Br0 (x0 ) ∩ {u > 0} ,
and

o
n
σ,r0
D−
u(x0 ) := p ∈ Rd : u(x) ≥ p · (x − x0 ) − σ|x − x0 | in Br0 (x0 ) ∩ {u > 0} .

Lastly we define a quantified set where the dynamic slope condition strictly fails:

σ,r0
Vnr0 (t, x) ≥ η and ∃p ∈ D+u(t, x)(7.7) Γ+
(u,
t)
:=
x
∈
∂Ω(u(t)):.
η,σ,r0s.t. |p|2 − (1 + µ+ ) < −ηThe subdifferential version is defined similarly. In the above definition we are keeping more parameters than minimally necessary in order to clarify certain delicate
points in the proof.
Lemma 7.7. The family of sets Γ+η,r0 ,σ (u, t) is monotone decreasing with respectto η and r0 and monotone increasing with respect to σ. Furthermore[ \ [Γ+ (u, t) =Γ+η,σ,r0 (u, t)
η>0 σ>0 r0 >0

and thus, by monotone convergence theorem, if Hd−1 (Γ+ (u, t)) > 0 then there isη > 0 so that for all σ > 0 there is r0 (σ) > 0 such that Hd−1 (Γ+η,σ,r0 (u, t)) > 0.
Proof. The monotonicities follow from the set definition. Let us check the set
formula. Let x ∈ Γ+ (u, t), which means
Vn (t, x) > 0 and there is p ∈ D+ u(t, x) with |p|2 < 1 + µ+ .
By definition of D+ uu(t, x) ≤ p · x + o(|x − x0 |) for x ∈ Ω(u(t))
and so for every σ > 0 there is r1 (σ) > 0 suﬀiciently small so thatu(t, x) ≤ p · x + σ|x − x0 | for x ∈ Ω(u(t)) ∩ Br1 (x)σ,r1Namely p ∈ D+u(t, x). By the definition of Vn

Vnr2 (t, x) ≥ η2 > 0 for some r2 , η2 > 0.
Call η1 := (1 + µ+ ) − |p|2 > 0. Taking r0 = min{r1 , r2 } and η = min{η1 , η2 } wefind that x ∈ Γ+□
η,σ,r0 (u, t).

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

55

Proof of Lemma 7.4. By Lemma 7.7 and the hypothesis of the Lemma there isη > 0 such that for all σ > 0 there is positive r0 (σ) with
(7.8)

Hd−1 (Γ+η,σ,r0 (u, t0 )) ≥1 d−1 +H(Γ (u, t0 )) > 0.
2

We will take σ = c(d)η, where the dimensional constant c(d) will be determined in
the course of the proof (under (7.15)). This choice of σ also fixes r0 = r0 (d, η) > 0,
we will usually drop the dependence on the dimension when it is not important andwrite (for example) r0 (η). Let us consider δ ∈ (0, r0 (η)] and t−1 := t0 − δ for the
rest of the proof. By monotonicity of the sets Γ+η,σ,r0 (u, t0 ) in η, we may assume1
that η < 2 .
We outline the proof, which is divided into three steps. In the first step we use
the speed and slope condition at a single point to construct a strict supersolution
replacement in a local neighborhood which is below u(t0 ) by O(δ d ) in measurebut still above u(t−1 ). The construction is fairly standard, using the first order
asymptotic information from the quantified superdifferential, sliding the linearization inwards while also bending up in the tangential directions and bending downin the normal direction in order to create a local superharmonic perturbation. In
the second step we perform a typical Vitali-type covering argument. In the final
step we perform the local supersolution replacement in each disjoint ball from thecovering to create the perturbation uδ , which is now O(δ) below u(t0 ) because wehave shifted inwards by O(δ) on a set of positive surface measure.
Step 1. (Construction of a barrier based on single reference point) In orderto make certain parameter choices more clear in the proof below we will adopt
the notation ηt = ηx = η and use ηt when the lower bound of the velocity is
being used, and ηx when the lower bound of the slope condition is being used. Let
x 0 ∈ Γ+η,σ,r0 (u, t0 ). For simplicity we may assume that x0 = 0 and p0 = αed forsome α ≥ 0, so that
(7.9)

u(t0 , x) ≤ αxd + σ|x| in Br0 (0) ∩ Ω(t0 )and, since we can always increase α if necessary, we may assume thatα = (1 + µ+ )1/2 − ηx .
Note that, since u is nonnegative, (7.9) yields that α(xd )− ≤ σr on Br (0) ∩ Ω(t0 ),namely(7.10)σBr (0) ∩ Ω(t0 ) ⊂ {xd ≥ − r} for all r ≤ r0 .
αFurthermore, again from the definition of Γ+
η,σ,r0 (u, t0 ),(7.11)Bηt (t0 −t) (0) ⊂ Rd \ Ω(t) for t ≥ t0 − r0 .

Recall that we assumed that δ = t0 − t−1 is smaller than r0 (η) > 0.
Our perturbation will be performed in ed -aligned cylinders which are anisotropic,
slightly wider in the tangential directions than the normal direction. Denoting
x′ = (x1 , . . . , xd−1 ) and Br′ the ball of radius r in Rd−1 ,Cylr = Br′ × (−cd r, cd r).

56

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

The dimensional constant cd := √ 12(d−1)

is chosen so that on the side boundary of

the cylinder ∂side Cylr := ∂Br × (−cd r, cd r)(7.12)|x′ |2 − (d − 1)x2d ≥ r2 − (d − 1)c2d r2 ≥1 2
r on x ∈ ∂side Cylr .
2

We also call
∂top Cylr := Br × {xd = cd r} and ∂bot Cylr := Br × {xd = −cd r}.
√
Obviously Cylr ⊂ Br0 for r ≤ r0 / 2.
We aim to make a localized perturbation in Cyl √1 ηt δ . Note that this cylinder is2chosen since, by (7.11),
Cyl √1 ηt δ ⊂ Bηt δ ⊂ Rd \ Ω(t−1 ).

(7.13)

2

Since we assume δ = t0 − t−1 ≤ r0 and η < 1, so ηt δ ≤ r0 as well.
We shift the plane αxd inwards by O(δ) in the normal direction ed while also
bending upwards in the tangential directions so that the barrier is below u(t0 ) but
above u(t−1 ).
More precisely, define the barrier(7.14)

ψδ (x) := (α + η4x )(xd − a1 ηt δ) + a2 (ηt δ)−1 (|x′ |2 − (d − 1)x2d ).

We will see that if we choose ai := c(d, µ+ )ηx , i = 1, 2 then we can guarantee thefollowing important properties of ψδ :(i) ψδ is harmonic;(ii) ψδ (x) > u(t0 , x) on ∂Cyl √1 ηt δ ∩ Ω(t0 );2(iii) |∇ψδ (x)|2 ≤ 1 + µ+ − 12 ηx in Cyl √1 ηt δ ;2(iv) There is c > 0 depending on d and µ+ such thatBcηx ηt δ (0) ⊂ {ψδ ≤ 0}.
Property (i). This is straightforward, but we do emphasize that the superharmonicity is important and needed later to deal with the energy computation in Step
3 below.
Property (ii). We check the bottom, sides and top of the cylinder separately. On
the top boundary x ∈ ∂top Cyl √1 ηt δ we can use the extra room on the slope:2ψδ (x) ≥ αxd + c(d)ηx ηt δ − C(d)(a1 + a2 )ηt δ≥ αxd + c(d)[ηx − C(d)(a1 + a2 )]ηt δ.
Thus if we takea1 + a2 ≤ c(d)ηx(7.15)
then, again on x ∈ ∂top Cyl √1 ηt δ ,2ψδ (x) ≥ αxd + c(d)ηx ηt δ
≥ αxd + c(d)ηx |x| ≥ αxd + σ|x| ≥ u(t0 , x).
where this computation fixes the choice of the dimensional constant in σ = c(d)ηx .

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

57

On the sides ∂side Cyl √1 ηt δ ∩ Ω(t0 ) we can use the upwards bending in the tan2

gential directions. Using (7.12), we have.
1ψδ (x) ≥ αxd − 14 ηx (xd )− − (1 + µ+ )1/2 a1 ηt δ + a2 ηt δ
2
h
i
1
1
= αxd + σ|x| + 2 a2 ηt δ − σ|x| − 4 ηx (xd )− − (1 + µ+ )1/2 a1 ηt δ
|
{z
}
:= A

(7.10) yields thatA ≥ c [a2 − C(d)σ − C(µ+ , d)a1 ] ηt δ on x ∈ ∂side Cyl √1 ηt δ ∩ Ω(t0 ).
2

We want this to be non-negative, so along with condition (7.15) we need to choose
σ, a1 and a2 so that
a1 + a2 ≤ c(d)ηx and C(d)σ + C(µ+ , d)a1 ≤ a2 .
To satisfy both inequalities, we choose σ = c(d)ηx , a1 = c(d, µ+ )ηx and a2 = c(d)ηx .
Applying these choices of parameters, we findψδ (x) ≥ αxd + σ|x| ≥ u(t0 , x) on ∂side Cyl √1 ηt δ ∩ Ω(t0 ).
2

Finally, the bottom boundary ∂bot Cyl √1 ηt δ is compactly contained in the zero2

level set of u(t0 ). Namely the set (xd )− = cd ηt δ does not intersect Ω(t0 ) by (7.10)
and fixing the choice of dimensional constant in the original specification σ = c(d)η.
Property (iii). For this we just compute the gradient and bound with triangle
inequality|∇ψδ − (α + 14 ηx )ed | ≤ C(d)a2

in Cyl √1 ηt δ .
2Choosing the dimensional constant in a2 = c(d)ηx smaller if necessary,|∇ψδ | ≤ α + 12 ηx ≤ (1 + µ+ )1/2 − 21 ηx

in Cyl √1 ηt δ .
2Property (iv). Note that ψδ is monotone in Cyl √1 ηt δ on the cone of direc2

tions, namely for the directions e ∈ S d−1 such that e · ed ≥ 21 ≥ C(d)a2 . Sinceψδ (a1 ηt δed ) = 0 we find that
B2a1 ηt δed (0) ⊂ {ψδ ≤ 0}.
Step 2. (Covering set up) Since Hausdorff measure is inner regular, there is acompact subset K ⊂ Γ+
η,σ,r0 (u, t0 ) with1
1 d−1 +H(Γη,σ,r0 (u, t0 )) ≥ Hd−1 (Γ+ (u, t0 )).
2
4
Since K is compact, there is r1 > 0 such thatHd−1 (K) ≥(7.16)

inf{

∞
X
j=1

:K⊂
ρd−1
j

∞
[

Bρj (xj ), ρj ≤ r1 } ≥

j=1

1 d−1
H
(K).
2

Let 3ηt δ ≤ r1 . By the Vitali covering lemma there is a finite collection of disjointballs Bj of radius ηt δ centered at points xj ∈ K ⊂ ∂Ω(t0 ) with
[
3Bj .
K⊂
j

58

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

Call B̃j = Bcηx ηt δ (xj ) with constant 1 > c(d, µ+ ) > 0 from property (iv) above.
Since B̃j ⊂ Bj they are a disjoint collection as well. By (7.16)
X

3d−1 δ d−1 ≥

j

1 d−1
H
(K).
2

By measure non-degeneracy of the free boundary Lemma 5.5|Ω(t0 ) ∩ B̃j | ≥ cη −2d δ d ,where the constant c depends on d and µ+ .
Since B̃j′ s are disjoint, we obtain the following by absorbing universal constants
into c:X[X
cη −2d δ d ≥ cη −2d Hd−1 (K)δ,|Ω(t0 ) ∩ B̃j | ≥|Ω(t0 ) ∩B̃j | =
j

j

j

or,
|Ω(t0 ) ∩(7.17)[

B̃j | ≥ cη −2d Hd−1 (Γ+ (u, t0 ))δ.

j

Step 3. (Construction of the comparison functions) Let δ ≤ min{(3ηt )−1 r1 , r0 }so that both Step 1 and Step 2 apply, recall r1 was fixed by (7.16) and r0 (η) was
fixed at the beginning of the proof below (7.8). Now for each xj ∈ Γ+ (u, t0 ) fromthe covering constructed in Step 2, call ψxj ,δ to be the single-point barrier function
constructed in Step 1. The ψxj ,δ are defined in Bj , extend them to be equal to +∞outside of their respective Bj .
Consider the comparison function^
ψxj ,δ (x) defined for x ∈ Ω(t0 ).
uδ (x) = u(t0 , x) ∧j

By properties (i) and (ii), and by the disjointness of the Bj , we can conclude thatuδ is Lipschitz continuous and superharmonic in Ω(t0 ).
Now by property (iv) (first containment below), property (ii) (second containment below), and (7.13) (third containment below), applied for each j[[
Bj ⊂ Rd \ Ω(u(t−1 )).
B̃j ∩ Ω(u(t0 )) ⊂ Ω(u(t0 )) \ Ω(uδ ) ⊂(7.18)jj

Thus we conclude thatΩ(u(t−1 )) ⊂ Ω(uδ ) ⊂ Ω(u(t0 ))and also, by (7.17),|Ω(u(t0 )) \ Ω(uδ )| ≥X

|B̃j ∩ Ω(u(t0 ))| ≥ cη −2d Hd−1 (Γ+ (u, t0 ))δ.

j

The previous two displayed equations establish (7.3), the first claim of the Lemma.
To conclude we still need to establish the inequality on the difference of the
Dirichlet energy (7.4). Recall that uδ is superharmonic and Lipschitz continuous

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

59

in Ω(t0 ), so we can apply Lemma A.2 with v0 = uδ and v1 = u(t0 ). This results inˆˆˆ|∇uδ |2 dx −|∇u(t0 )|2 dx ≤|Duδ |2 dxΩ(uδ )Ω(t0 )Ω(t0 )\Ω(uδ )ˆ≤(1 + µ+ − 12 ηx ) dxΩ(t0 )\Ω(uδ )where we applied property (iii) for the second inequality. This proves (7.4) andfinishes the proof of the Lemma.
□
Appendix A. Technical results
A.1. Non-degeneracy. Non-degeneracy of Bernoulli problem solutions is much
more delicate than Lipschitz continuity, it typically needs a stronger condition thanjust the viscosity solution property. Here we give several cases where it is known.
Lemma A.1. Let u ∈ C(B2 ) satisfy one of the following:(i) Suppose that ∂{u > 0} is an L-Lipschitz graph in B2 and u solves∆u = 0 in {u > 0} ∩ B2 , and |∇u| ≥ 1 on ∂{u > 0} ∩ B2 .
(ii) Suppose that u is an inward minimizer of J in B2 .
(iii) Suppose that u is a minimal supersolution in B2∆u = 0 in {u > 0} ∩ B2 , and |∇u| ≤ 1 on ∂{u > 0} ∩ B2 .
(iv) Suppose d = 2 and u is a maximal subsolution in B2 of∆u = 0 in {u > 0} ∩ B2 , and |∇u| ≥ 1 on ∂{u > 0} ∩ B2 .
Then there is c0 depending on d (and on L if in the first case) such thatsup u(x) ≥ c0 r

for all x ∈ ∂{u > 0} ∩ B1 , Br (x) ⊂ B2 .

Br (x)

Proof. For part (i) see [15, Lemma 6.1], for part (ii) see [36, Lemma 4.4], for part(iii) see [6, Lemma 6.9], and for (iv) see [33].
□A.2. Energy difference quotient formulae. We give and prove several formulas
for the difference of the Dirichlet energy under varying positivity set. We have notseen these particular formulas before in the literature, but they provide a very
convenient way to go from energy minimization to viscosity solution properties.
Lemma A.2. Suppose that v0 , v1 ∈ H 1 (U ) with v0 ≤ v1 and v0 = v1 ≥ 0 on ∂U .
Call Ωj = {vj > 0} ∩ U .
• If v1 is subharmonic in Ω1 thenˆˆˆ|∇v1 |2 dx ≥|∇v0 |2 dx −
(A.1)
Ω0

Ω1

Ω1 \Ω0• If v0 is superharmonic in Ω1 thenˆˆˆ|∇v1 |2 dx ≤|∇v0 |2 dx −
(A.2)
Ω0

Ω1

Ω1 \Ω0|∇v1 |2 dx.

|∇v0 |2 dx

60

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

Proof. For (A.1) we start withˆˆˆ|∇v1 |2 dx =|∇v0 |2 dx −
Ω0

Ω1

Ω0

ˆ
|∇v0 |2 − |∇v1 |2 dx +

Ω1 \Ω0−|∇v1 |2 dx

Then we compute the first term on the rightˆˆ|∇v1 + ∇(v0 − v1 )|2 − |∇v1 |2 dx|∇v0 |2 − |∇v1 |2 dx =Ω0ˆΩ02∇v1 · ∇(v0 − v1 ) + |∇(v0 − v1 )|2 dx=Ω0ˆ
2∇v1 · ∇(v0 − v1 ) dx≥
Ω0ˆˆ2|∇v1 |2 dx
2∇v1 · ∇((v0 )+ − v1 ) dx +=
Ω1 \Ω0Ω1
ˆ2|∇v1 |2 dx.
≥Ω1 \Ω0The last inequality in the sequence above is is due to v1 being H 1 subharmonic inΩ1 . Note that v1 − (v0 )+ is a valid test function for the subharmonicity condition
since it is non-negative in Ω1 and in H01 (Ω1 ).
For (A.2) we start withˆˆˆ
ˆ−|∇v0 |2 dx|∇v0 |2 − |∇v1 |2 dx +|∇v1 |2 dx =|∇v0 |2 dx −
Ω1 \Ω0Ω1Ω1Ω0Then we compute the first term on the rightˆˆ|∇v0 |2 − |∇v0 + ∇(v1 − v0 )|2 dx|∇v0 |2 − |∇v1 |2 dx =Ω1
ˆΩ1
−2∇v0 · ∇(v1 − v0 ) − |∇(v1 − v0 )|2 dx=ˆˆΩ12|∇v0 |2 dx−2∇v0 · ∇(v1 − (v0 )+ ) dx +≤
Ω1 \Ω0Ω1
ˆ2|∇v0 |2 dx.
≤Ω1 \Ω0The last inequality in the sequence above is is due to v0 being H 1 superharmonic inΩ1 . Note that v1 − (v0 )+ is a valid test function for the superharmonicity condition
since it is non-negative in Ω1 and in H01 (Ω1 ).
In the smooth case one can also check the related identitiesˆˆ|∇v1 |2 dx|∇v0 |2 dx −
Ω1
Ω0
ˆ
ˆ
ˆ
|∇v1 |2 dx.
|∇(v1 − v0 )|2 dx +2∆v1 (v1 − (v0 )+ ) dx +=and

Ω1 \Ω0

Ω0

Ω1

ˆ

ˆ

|∇v1 |2 dx =Ω1ˆˆ|∇(v1 − v0 )|2 dx +2(v1 − (v0 )+ )∆v0 dx −|∇v0 |2 dx −
Ω0

ˆ
Ω1

Ω1

Ω1 \Ω0|∇v0 |2 dx.

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION61(A ∩ C) \ B

B \ (A ∪ C)

Figure 8. Diagram of the sharp triangle inequality for the dissipation distance. Here A ∩ C is the inner circle, A ∪ C is the outercircle, and B is the ellipse.
□
A.3. Sharp triangle inequality of the Dissipation distance. Here we givea sharp triangle inequality for the Dissipation distance. The triangle inequality
is important for most of the standard theory of rate independent energetic evolutions, although we remark that it does not seem to generalize well to the generalanisotropic case. The sharp triangle inequality is used in Proposition 5.16 to showthat if an energetic solution jumps multiple times then u(t) must always be in
between uℓ (t) and ur (t).
Lemma A.3 (Dissipation distance sharp triangle inequality). Let A, B, and C bearbitrary finite measure regions in Rn . ThenDiss(A, B) + Diss(B, C) − Diss(A, C) = (µ− + µ+ ) |B \ (A ∪ C)| + |(A ∩ C) \ B|
Proof. The result is purely set algebraic computations, see Figure 8 for the geometric idea. We compute(A.3)

Diss(A, B) + Diss(B, C) − Diss(A, C)ˆ= µ+ 1B\A + µ− 1A\B + µ+ 1C\B + µ− 1B\C − µ+ 1C\A − µ− 1C\A dxˆ= µ+ [1B\A + 1C\B − 1C\A ] + µ− [1A\B + 1B\C − 1A\C ] dx.

Now we just compute by force the first term in brackets on the previous line
1B\A + 1C\B − 1C\A = 1B − 1A∩B + 1C − 1B∩C − [1C − 1A∩C ]= 1B + 1A∩C − 1B∩C − 1B∩A
= 1B + 1A∩C 1B − 1B∩C − 1B∩A + 1A∩C (1 − 1B )= 1B (1 + 1A∩C − 1C − 1A ) + 1(A∩C)\B= 1B (1 − 1A∪C ) + 1(A∩C)\B= 1B\(A∪C) + 1(A∩C)\B
the second bracketed term in (A.3) works out quite similarly and is identical after
the second step above giving the same result.
□
Plugging these identities back into (A.3) gives the result.

62

W. M. FELDMAN, I. C. KIM, AND N. POŽÁR

A.4. Proof of Lemma 4.7. We just do the subsolution equivalence, the supersolution one is similar. Suppose u is a subsolution in the superdifferential sense
Definition 4.5. Suppose φ is a smooth test function with φ+ touching u from above
at x0 ∈ ∂Ω(u) ∩ U . Thenu(x) ≤ [∇φ(x0 ) · (x − x0 )]+ + o(|x − x0 |) so ∇φ(x0 ) ∈ D+ u(x0 )and so|∇φ(x0 )|2 ≥ Q.
Next suppose that u is a subsolution in the superdifferential sense Definition 4.6,
and p ∈ D+ u(x0 ) for some x0 ∈ ∂Ω(u) ∩ U . Without loss assume that p = αed .
Let δ, r > 0 and define
φ(x) := (1 + δ)α(x − x0 ) · ed + c(d)δr−1 (|(x − x0 )′ |2 − dx2d ).
This function is a superharmonic polynomial and using the superdifferential condition αed ∈ D+ u(x0 ) one can show that for all δ > 0 there is r > 0 so that u < φ on
∂C ∩ {u > 0} where C is a small open cylindrical neighborhood of x0 with radius≈ r0 , see the proof of Lemma 7.4 for a similar computation.
Then, since φ(x0 ) = u(x0 ) we can shift φ vertically φ(x) + s until (φ(x) + s)+
touches u from above at some x1 ∈ C. Since φ is strictly superharmonic thetouching must be on x1 ∈ ∂Ω(u) ∩ C. Thus finally the viscosity solution condition
implies thatQ ≤ |∇φ(x1 )|2 ≤ α + oδ (1).
Sending δ → 0 finishes the argument.

□

A.5. Uniqueness of the Bernoulli problem solution in the complement ofa star-shaped set. In the star-shaped case we have the following uniqueness result
for a Bernoulli problem with an obstacle. The result without obstacle for classicalsolutions in dimension two was proved in [35] based on an existence theorem by
Beurling [4] (more easily found reprinted here [5]). Here we give a proof for viscosity
solutions of the obstacle problem for convenience.
Theorem A.4. Suppose that U is a domain such that that K = Rd \ U is strongly
star-shaped with respect to a neighborhood of the origin, and that O is a strongly
star-shaped set with respect to the same neighborhood. If u, v ∈ C(U ) have compact
support and for some constants F, q > 0 are viscosity solutions of a Bernoulli
problem with obstacle O from below
∆u = 0 in {u > 0} ∩ U,





u > 0 in O,|∇u| = q on (∂{u > 0} \ O) ∩ U,|∇u| ≤ q on ∂{u > 0} ∩ U,u = F on ∂Uthen u = v.
The same result applies for a Bernoulli problem with obstacle O from above.
Proof. By a simple rescaling it is suﬀicient to consider F = q = 1. We show thatΩ(u) = Ω(v) which is suﬀicient by the uniqueness of the Laplace equation.

ON THE GEOMETRY OF RATE INDEPENDENT DROPLET EVOLUTION

63

We first check that Ω(u) ⊂ Ω(v). Suppose that Ω(u) \ Ω(v) 6= ∅. Consider thelargest λ < 1 such thatv λ (x) := v(λx)
satisfiesΩ(u) ⊂ Ω(v λ ) = λ−1 Ω(v).
Such λ exists since both Ω(u) and Ω(v) are open, Ω(u) is bounded and Ω(v) contains
a neighborhood of the origin. We have clearly λ < 1, and by the definition of λ there
exists x0 ∈ ∂Ω(u) ∩ ∂Ω(v λ ). Also note that x0 ∈/ O since O is strongly star-shaped
and O ∩ ∂Ω(v) = ∅.
If u and v are suﬀiciently smooth, we have|∇v λ |(x0 ) = λ|∇v|(λx0 ) ≤ λ < 1 = |∇u|(x0 ),a contradiction with the fact that u ≤ v λ by the maximum principle for the harmonic function v λ − u in Ω(v) ∩ λ−1 U since v λ − u > 0 on λ−1 ∂U ⊂ U by the strong
star-shapedness.
To make the argument work without assuming regularity, we consider the supand inf convolutions
vr (x) = inf v,

ur (x) = sup u

B r (x)B r (x)defined on U r where Ur := {x : B r (x) ⊂ U }. It is easy to see that vr is superharmonic on Ur ∩ Ω(vr ), ur is subharmonic on Ur ∩ Ω(ur ), and inf ∂Ur vr → 1 as r → 0
by continuity.
Let us call the λ < 1 above λ0 . For each r > 0 there exists a largest λr < 1
so that Ω(ur ) ⊂ λ−1r Ω(vr ). Furthermore λr → λ0 < 1 as r → 0. Therefore−1/2
inf ∂Ur vr > 1 andwe can choose r > 0 suﬀiciently small so that λr < 1, λr−1dist(O, λr ∂Ω(vr )) > r (by strong star-shapedness of O).
We fix such r > 0 and choose x0 ∈ ∂Ω(ur ) ∩ λ−1r ∂Ω(vr ). Let us set w(x) :=−1/2
vr (λr x). We have that both ur and w are defined on λ−1λr
r U r . We also notethat w − ur is superharmonic on λ−1
U
∩
{u
>
0}
with
w
−
ur > 0 on λ−1rrrr ∂Ur .
Therefore the slopes of ur and w in the normal direction at x0 are ordered.
We have the standard situation of boundaries of ur and w touching at a pointthat has both interior and exterior touching balls centered at the free boundaries
of u and v(λr x) respectively. By a standard barrier construction at these centers,
we arrive at a contradiction with the fact that u satisfies |∇u| ≥ 1 (since the center
1/2−1/2|∇v λr | ≤ λr 1 < 1.
is not in O by the choice of r) and v λr := v(λr x) satisfies λr
Therefore we conclude that Ω(u) ⊂ Ω(v).
The inclusion Ω(v) ⊂ Ω(u) can be shown by the same argument, swapping theroles of u and v. By the uniqueness of the Laplace equation we have u = v.
The proof for the problem with obstacle from above follows an analogous argument.
□
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Department of Mathematics, University of Utah, Salt Lake City, Utah, 84112, USA
Email address: feldman@math.utah.edu
Department of Mathematics, University of California, Los Angeles, California,
90095, USA
Email address: ikim@math.ucla.edu
Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa
University, Kakuma, Kanazawa 920-1192, Japan
Email address: npozar@se.kanazawa-u.ac.jp