LONG-TIME BEHAVIOR OF THE ONE-PHASE STEFANPROBLEM IN PERIODIC AND RANDOM MEDIANORBERT POŽÁR AND GIANG THI THU VU

Abstract. We study the long-time behavior of solutions of the one-phase
Stefan problem in inhomogeneous media in dimensions n ≥ 2. Using the technique of rescaling which is consistent with the evolution of the free boundary,
we are able to show the homogenization of the free boundary velocity as wellas the locally uniform convergence of the rescaled solution to a self-similar solution of the homogeneous Hele-Shaw problem with a point source. Moreover,by viscosity solution methods, we also deduce that the rescaled free boundary
uniformly approaches a sphere with respect to Hausdorff distance.

1. Introduction
We consider the one-phase Stefan problem in periodic and random media in adimension n ≥ 2. The aim of this paper is to understand the behavior of thesolutions and their free boundaries when time t → ∞.
Let K ⊂ Rn be a compact set with sufficiently regular boundary, for instance∂K ∈ C 1,1 , and assume that 0 ∈ int K. The one-phase Stefan problem (on an
exterior domain) with inhomogeneous latent heat of phase transition is to find afunction v(x, t) : Rn × [0, ∞) → [0, ∞) that satisfies the free boundary problem
vt − ∆v = 0
in {v > 0}\K,




v=1
on K,
(1.1)

Vν = g(x)|Dv| on ∂{v > 0},
v(x, 0) = v0
on Rn ,
where D and ∆ are respectively the spatial gradient and Laplacian, vt is the partial
derivative of v with respect to time variable t, Vν is the normal velocity of thefree boundary ∂{v > 0}. v0 and g are given functions, see below. Note that theresults in this paper can be trivially extended to general time-independent positive
continuous boundary data, 1 is taken only to simplify the exposition.
The one-phase Stefan problem is a mathematical model of phase transitions
between a solid and a liquid. A typical example is the melting of a body of ice
maintained at temperature 0, in contact with a region of water. The unknowns are
the temperature distribution v and its free boundary ∂{v(·, t) > 0}, which models
the ice-water interface. Given an initial temperature distribution of the water, thediffusion of heat in a medium by conduction and the exchange of latent heat will
govern the system. In this paper, we consider an inhomogeneous medium where
Date: May 11, 2017.
2000 Mathematics Subject Classification. 35B27 (35R35, 74A50, 80A22).
Key words and phrases. Stefan problem, homogenization, viscosity solutions, long-time
behavior.
1

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NORBERT POŽÁR AND GIANG THI THU VU

the latent heat of phase transition, L(x) = 1/g(x), and hence the velocity law
depend on position. The related Hele-Shaw problem is usually referred to in theliterature as the quasi-stationary limit of the one-phase Stefan problem when theheat operator is replaced by the Laplace operator. This problem typically describes
the flow of an injected viscous fluid between two parallel plates which form the socalled Hele-Shaw cell, or the flow in porous media.
In this paper, we assume that the function g satisfies the following two conditions,which guarantee respectively the well-posedness of (1.1) and averaging behavior ast → ∞:(1) g is a Lipschitz function in Rn , m ≤ g ≤ M for some positive constants mand M .
(2) g(x) has some averaging properties so that Lemma 3.1 applies, for instance,
one of the following holds:(a) g is a Zn -periodic function,(b) g(x, ω) : Rn × A → [m, M ] is a stationary ergodic random variable
over a probability space (A, F, P ).
For a detailed definition and overview of stationary ergodic media, we refer to [17,16]
and the references therein.
Throughout most of the paper we will assume that the initial data v0 satisfies(1.2)

v0 ∈ C 2 (Ω0 \K), v0 > 0 in Ω0 , v0 = 0, on Ωc0 := Rn \ Ω0 , and v0 = 1 on K,|Dv0 | =6 0 on ∂Ω0 , for some bounded domain Ω0 ⊃ K.

This will guarantee the existence of both the weak and viscosity solutions below andtheir coincidence, as well as the weak monotonicity (4.2). However, the asymptotic
limit, Theorem 1.1, is independent of the initial data, and therefore the result
applies to arbitrary initial data as long as the (weak) solution exists, satisfies thecomparison principle, and the initial data can be approximated from below andfrom above by data satisfying (1.2). For instance, v0 ∈ C(Rn ), v0 = 1 on K,
v0 ≥ 0, supp v0 compact is sufficient.
The Stefan problem (1.1) does not necessarily have a global classical solution
in n ≥ 2 as singularities of the free boundary might develop in finite time. The
classical approach to
R tdefine a generalized solution is to integrate v in time andintroduce u(x, t) := 0 v(x, s)ds [1, 6, 9, 7, 21, 23, 22]. If v is sufficiently regular, thenu solves the variation inequality(u(·, t) ∈ K(t),(1.3)(ut − ∆u)(ϕ − u) ≥ f (ϕ − u) a.e (x, t) for any ϕ ∈ K(t),where K(t) is a suitable functional space specified later in Section 2.2 and f is
v0 (x),v0 (x) > 0,1
(1.4)
f (x) =
−, v0 (x) = 0.
g(x)This parabolic inequality always has a global unique solution u(x, t) for initial datasatisfying (1.2) [9, 21, 23, 22]. The corresponding time derivative v = ut , if it exists,
is then called a weak solution of the Stefan problem (1.1). The main advantage ofthis definition is that the powerful theory of variational inequalities can be applied
for the study of the Stefan problem, and as was observed in [20, 15, 16] yields
homogenization of (1.3).

THE ONE-PHASE STEFAN PROBLEM

3

More recently, the notion of viscosity solutions of the Stefan problem was introduced and well-posedness was established by Kim [11]. Since this notion relieson the comparison principle instead of the variational structure, it allows for moregeneral, fully nonlinear parabolic operators and boundary velocity laws. Moreover,the pointwise viscosity methods seem more appropriate for studying the behavior of
the free boundaries. The natural question whether the weak and viscosity solutions
coincide was answered positively by Kim and Mellet [16] whenever the weak solution exists. In this paper we will use the strengths of both the weak and viscosity
solutions to study the behavior of the solution and its free boundary for large times.
The homogeneous version of this problem, i.e, when g ≡ const, was studied by
Quirós and Vázques in [19]. They obtained the result on the long-time convergenceof weak solution of the one-phase Stefan problem to the self-similar solution of theHele-Shaw problem. The homogenization of this type of problem was considered byRodrigues in [20] and by Kim-Mellet in [15, 16]. The long-time behavior of solutionof the Hele-Shaw problem was studied in detail by the first author in [17]. In
particular, the rescaled solution of the inhomogeneous Hele-Shaw problem converges
to the self-similar solution of the Hele-Shaw problem with a point-source, formally−∆v = Cδin {v > 0},




1
|Dv|2 on ∂{v > 0},
vt =
(1.5)
h1/gi



 v(·, 0) = 0,
where δ is the Dirac δ-function, C is a constant depending on K and n, and theconstant h1/gi will be properly defined later. Moreover, the rescaled free boundary
uniformly approaches a sphere.
Here we extend the convergence result to the Stefan problem in the inhomogeneous medium. Since the asymptotic behavior of radially symmetric solutions of
the Hele-Shaw and the Stefan problem are similar and the solutions are bounded,
we can take the limit t → ∞ and obtain the convergence for rescaled solutions andtheir free boundaries. However, solutions of the Hele-Shaw problem have a very
useful monotonicity in time, which is missing in the Stefan problem. This makessome steps more difficult. We instead take advantage of a weak monotonicity property (4.2), which holds for regular initial data satisfying (1.2) and then show theconvergence result for general initial data using the uniqueness of the limit andthe comparison principle. Moreover, the heat operator is not invariant under the
rescaling, unlike the Laplace operator. The rescaled parabolic equation becomes
elliptic when λ → ∞, which causes some issues when applying parabolic Harnack’s
inequality, for instance. Following [19, 17] we use the natural rescaling of solutionsof the formv λ (x, t) := λ(n−2)/n v(λ1/n x, λt)

if n ≥ 3,

and the corresponding rescaling for variational solutions
uλ (x, t) := λ−2/n u(λ1/n , λt)

if n ≥ 3(see Section 2.4 for n = 2). Then the rescaled viscosity solution satisfies the freeboundary velocity law
Vνλ = g(λ1/n x)|Dv λ |.

4

NORBERT POŽÁR AND GIANG THI THU VUHeuristically, if g has some averaging properties, such as in condition (2), thefree boundary velocity law should homogenize as λ → ∞. Since the latent heat ofphase transition 1/g should average out, the homogenized velocity law will be
1
Vν =
|Dv|,
h1/gi
where h1/gi represents the “average” of 1/g. More precisely, the quantity h1/gi isthe constant in the subadditive ergodic theorem such that
Z
Z  
1
1
u(x)dx for all u ∈ L2 (Rn ), for a.e. ω ∈ A.
u(x)dx →gg(λ1/n x, ω)Rn

Rn

In the periodic case, it is just the average of 1/g over one period. Since we always
work with ω ∈ A for which the convergence above holds, we omit it from thenotation in the rest of the paper.
This yields the first main result of this paper, Theorem 3.2, on the homogenization of the obstacle problem (1.3) for the rescaled solutions, with the correct
singularity of the limit function at the origin, and therefore the locally uniform
convergence of variational solutions. To prove the second main result in Theorem
4.2 on the locally uniform convergence of viscosity solutions and their free boundaries, we use pointwise viscosity solution arguments. In summary, we will show thefollowing theorem.
Theorem 1.1. For almost every ω ∈ A, the rescaled viscosity solution v λ of the
Stefan problem (1.1) converges locally uniformly to the unique self-similar solution
V of the Hele-Shaw problem (1.5) in (Rn \{0})×[0, ∞) as λ → ∞, where C depends
only on n, the set K and the boundary data 1. Moreover, the rescaled free boundary∂{(x, t) : v λ (x, t) > 0} converges to ∂{(x, t) : V (x, t) > 0} locally uniformly withrespect to the Hausdorff distance.
It P
is a natural question to consider more general linear divergence form operators i,j ∂xi (aij (x)∂xj ·) instead of the Laplacian in (1.1) so that the variational
structure is preserved. This was indeed the setting considered in [16], with g ≡ 1
and appropriate free boundary velocity law adjusted for the operator above. In thelimit λ → ∞, we expect that the rescaled solutions v λ to converge to the uniquesolution of the Hele-Shaw type problem with a point source with the homogenized
non-isotropic operator with coefficients āi,j . This question is a topic of ongoingwork.
Context and open problems. In recent years, there have been significant developments in the homogenization theory of partial differential equations like HamiltonJacobi and second order fully nonlinear elliptic and parabolic equations that havebeen made possible by the improvements of the viscosity solutions techniques, seefor instance the classical [8, 24, 5, 4] to name a few.
A common theme of these results is finding (approximate) correctors and use
the perturbed test function method to establish the homogenization result in the
periodic case, or use deeper properties in the random case, such as the variational
structure of the Hamilton-Jacobi equations or the strong regularity results for elliptic and parabolic equations, including the ABP inequality.
One of the goals of this paper is to illustrate the powerful combination of variational and viscosity solution techniques for some free boundary problems thatTHE ONE-PHASE STEFAN PROBLEM

5have a variational structure. By viscosity solution techniques we mean specificallypointwise arguments using the comparison principle.
Unfortunately, when the variational structure is lost, for instance, when thefree boundary velocity law is more general as in the problem with contact angle
dynamics Vν = |Dv| − g(x) so that the motion is non-monotone [13, 14], or evensimple time-dependence Vν = g(x, t)|Dv| [18], the comparison principle is all thatis left. Even in the periodic case, the classical correctors as solutions of a cell
problem are not available. This is in part the consequence of the presence of thefree boundary on which the operator is strongly discontinuous. [12, 13, 18] use avariant of the idea that appeared in [5] to replace the correctors by solutions ofcertain obstacle problems. However, the analysis of these solutions requires rather
technical pointwise arguments since there are almost no equivalents of the regularity
estimates for elliptic equations. An important tool in [18] to overcome this was
the large scale Lipschitz regularity of the free boundaries of the obstacle problem
solutions (called cone flatness there) that allows for the control of the oscillationsof the free boundary in the homogenization limit.
For the reasons above, the homogenization of free boundary problems is rather
challenging and there are still many open problems. Probably the most important
one is the homogenization of free boundary problems of the Stefan and Hele-Shaw
type that do not admit a variational structure, such as those mentioned above, in
random environments. Currently there is no known appropriate stationary subadditive quantity to which we could apply the subadditive ergodic theorem to recover
the homogenized free boundary velocity law, for instance. Other tools like concentration inequalities have so far not yielded an alternative.
Another important problem are the optimal convergence rates of the free boundaries in the Hausdorff distance. The techniques used in this paper do not provide
this information, however viscosity techniques were used to obtain non-optimal algebraic convergence rates in [14]. It is an interesting question what the optimalrate in the periodic case is, even for problems like (1.1). The large scale Lipschitz
estimate from [18] could possibly directly give only ε| log ε|1/2 -rate for velocity law
with g(x/ε), but there are some indications that a rate ε might be possible.
Outline. The paper is organized as follows: In Section 2, we recall the definitions
and well-known results for weak and viscosity solutions. We also introduce therescaling and state some results for radially symmetric solutions. In Section 3,
we recall the limit obstacle problem and prove the locally uniform convergence of
rescaled variational solutions. In Section 4, we focus on treating the locally uniform
convergence of viscosity solutions and their free boundaries.

2. Preliminaries
2.1. Notation. For a set A, Ac is its complement. Given a nonnegative functionv, we will use notations for its positive set and free boundary of v,Ω(v) := {(x, t) : v(x, t) > 0},
and for fixed time t,Ωt (v) := {x : v(x, t) > 0},(f )+ is the positive part of f : (f )+ = max(f, 0).

Γ(v) := ∂Ω(v),Γt (v) := ∂Ωt (v).

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NORBERT POŽÁR AND GIANG THI THU VU

2.2. Weak solutions. Let v(x, t) be a classical solution of the Stefan problem(1.1). Fix R, T > 0 and set B = BR (0), D = B\K. Following [9] it can be shown
Rt
that, if R is large enough (depending on T ), then the function u(x, t) := 0 v(x, s)ds
solves the following variational problem: Find u ∈ L2 (0, T ; H 2 (D)) such that ut ∈L2 (0, T ; L2 (D)) andu(·, t) ∈ K(t),0 < t < T,(ut − ∆u)(ϕ − u) ≥ f (ϕ − u), a.e (x, t) ∈ B × (0, T ) for any ϕ ∈ K(t),
(2.1)


u(x, 0) = 0 in D.
Here we set K(t) = {ϕ ∈ H 1 (D), ϕ ≥ 0, ϕ = 0 on ∂B, ϕ = t on K} and f was
defined in (1.4). We use the standard notation for Sobolev spaces H k , W k,p . Notethat u(x, t) is independent of the choice of B as long as R is large enough [16, Lemma
3.6]. If v is a classical solution of (1.1) then u is solution of (2.1), but the inverse
statement is not valid in general. However, we have the following result [9, 21].
Theorem 2.1 (Existence and uniqueness of variational problem). If v0 satisfies(1.2), then the problem (2.1) has a unique solution satisfyingu ∈ L∞ (0, T ; W 2,p (D)),

1 ≤ p ≤ ∞,∞ut ∈ L (D × (0, T )),and(ut − ∆u ≥ fu(ut − ∆u − f ) = 0

for a.e. (x, t) ∈ {u ≥ 0},a.e in D × (0, ∞).

We will thus say that if u is a solution of (2.1), then ut is a weak solution of thecorresponding Stefan problem (1.1). The theory of variational inequalities for an
obstacle problem is well developed, for more details, we refer to [9, 21, 15]. We now
collect some useful results on the weak solutions from [9, 21].
Proposition 2.2. The unique solution u of (2.1) satisfies
0 ≤ ut ≤ C a.e D × (0, T ),where C is a constant depending on f . In particular, u is Lipschitz with respect to t
and u is C α (D) with respect to x for all α ∈ (0, 1). Furthermore, if 0 ≤ t < s ≤ T ,then u(·, t) < u(·, s) in Ωs (u) and also Ω0 ⊂ Ωt (u) ⊂ Ωs (u).
Lemma 2.3 (Comparison principle for weak solutions). Suppose that f ≤ fˆ. Letu, û be solutions of (2.1) for respective f, fˆ. Then u ≤ û, moreover,
θ≡

∂ û
∂u
≤
≡ θ̂.
∂t
∂t

Remark 2.4. Regularity of θ and its free boundary has been studied quite extensively, including Caffarelli and Friedman (see [2, 3, 10]). It is known that a weaksolution is classical as long as Γt (u) has no singularity. The smoothness criterion(see [2, 10], [19, Proposition 2.4]) immediately leads to the following corollary.
Corollary 2.5. Radial weak solutions of the Stefan problem (1.1) are smooth classical solutions.

THE ONE-PHASE STEFAN PROBLEM

7

2.3. Viscosity solutions. The second notion of solutions we will use are the viscosity solutions introduced in [11]. First, for any nonnegative function w(x, t) we
define the semicontinuous envelopesw? (x, t) := lim inf w(y, s),w? (x, t) := lim sup w(y, s).

(y,s)→(x,t)(y,s)→(x,t)

We will consider solutions in the space-time cylinder Q := (Rn \K) × [0, ∞).
Definition 2.6. A nonnegative upper semicontinuous function v(x, t) defined in Q
is a viscosity subsolution of (1.1) if the following hold:a) For all T ∈ (0, ∞), the set Ω(v) ∩ {t ≤ T } ∩ Q is bounded.
2,1b) For every φ ∈ Cx,t(Q) such that v − φ has a local maximum in Ω(v) ∩ {t ≤t0 } ∩ Q at (x0 , t0 ), the following holds:i) If v(x0 , t0 ) > 0, then (φt − ∆φ)(x0 , t0 ) ≤ 0.
ii) If (x0 , t0 ) ∈ Γ(v), |Dφ(x0 , t0 )| =6 0 and (φt − ∆φ)(x0 , t0 ) > 0, then(2.2)

(φt − g(x0 )|Dφ|2 )(x0 , t0 ) ≤ 0.

Analogously, a nonnegative lower semicontinuous function v(x, t) defined in Q is aviscosity supersolution if (b) holds with maximum replaced by minimum, and withinequalities reversed in the tests for φ in (i–ii). We do not need to require (a).
Now let v0 be a given initial condition with positive set Ω0 and free boundary
Γ0 = ∂Ω0 , we can define viscosity subsolution and supersolution of (1.1) withcorresponding initial data and boundary data.
Definition 2.7. A viscosity subsolution of (1.1) in Q is a viscosity subsolution of(1.1) in Q with initial data v0 and boundary data 1 if:a) v is upper semicontinuous in Q̄, v = v0 at t = 0 and v ≤ 1 on K,b) Ω(v) ∩ {t = 0} = {x : v0 (x) > 0} × {0}.
A viscosity supersolution is defined analogously by requiring (a) with v lower semicontinuous and v ≥ 1 on Γ. We do not need to require (b).
And finally we can define viscosity solutions.
Definition 2.8. The function v(x, t) is a viscosity solution of (1.1) in Q (withinitial data v0 and boundary data 1) if v is a viscosity supersolution and v ? is aviscosity subsolution of (1.1) in Q (with initial data v0 and boundary data 1).
Remark 2.9. By standard argument, if v is the classical solution of (1.1) then it isa viscosity solution of that problem in Q with initial data v0 and boundary data 1.
The existence and uniqueness of a viscosity solution as well as its properties havebeen studied in great detail in [11]. One important feature of viscosity solutions isthat they satisfy a comparison principle for “strictly separated” initial data.
One of the main tools we will use in this paper is the following coincidence of
weak and viscosity solutions from [16].
Theorem 2.10 (cf. [16, Theorem 3.1]). Assume that v0 satisfies (1.2). Let u(x, t)
be the unique solution of (2.1) in B × [0, T ] and let v(x, t) be the solution ofvt − ∆v = 0
in Ω(u)\K,




v=0
on Γ(u),
(2.3)

v=1
in K,



v(x, 0) = v0 (x).

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NORBERT POŽÁR AND GIANG THI THU VU

Then v(x, t) is a viscosity solution of (1.1) in B × [0, T ] with initial data v(x, 0) =Rt
v0 (x), and u(x, t) = 0 v(x, s)ds.
Remark 2.11. The definition of the solution v of (2.3) must be clarified when Ω(u)is not smooth. Since u is continuous and Ω(u) is bounded at all times ( [16, Lemma3.6]) then the existence of solution of (2.3) is provided by Perron’s method as
v = sup{w|wt − ∆w ≤ 0 in Ω(u), w ≤ 0 on Γ(u), w ≤ 1 in K, w(x, 0) ≤ v0 (x)}.
Note that v might be discontinuous on Γ(u).
The coincidence of weak and viscosity solutions gives us a more general comparison principle.
Lemma 2.12 (cf. [16, Corollary 3.12]). Let v 1 and v 2 be, respectively, a viscosity
subsolution and supersolution of the Stefan problem (1.1) with continuous initial
data v01 ≤ v02 and boundary data 1. In addition, suppose that v01 (or v02 ) satisfies
condition (1.2). Then v?1 ≤ v 2 and v 1 ≤ (v 2 )? in Rn \K × [0, ∞).
2.4. Rescaling. We will use the following rescaling of solutions as in [17].
2.4.1. For n ≥ 3. For λ > 0 we use the rescaling
n−2

1

2

1

1

uλ (x, t) = λ− n u(λ n x, λt).

v λ (x, t) = λ n v(λ n x, λt),1

If we define K λ := K/λ n and Ωλ0 := Ω0 /λ n then v λ satisfies the problem
 2−n
λ n vtλ − ∆v λ = 0in Ω(v λ )\K λ ,




n−2

on K λ ,vλ = λ n(2.4)
vtλ = g λ (x)|Dv λ |2 on Γ(v λ ),v λ (·, 0) = v0λ ,1where g λ (x) = g(λ n x). And the rescaled uλ satisfies the obstacle problem(2.5)uλ (·, t) ∈ Kλ (t),



1

 (λ 2−nn uλ − ∆uλ )(ϕ − uλ ) ≥ f (λ n x)(ϕ − uλ )a.e (x, t) ∈ Rn × (0, ∞)
t








for any ϕ ∈ Kλ (t),uλ (x, 0) = 0,n−2where Kλ (t) = {ϕ ∈ H 1 (Rn ), ϕ ≥ 0, ϕ = λ n t on K λ }.
Remark 2.13. We can take the admissible set K λ (t) as above due to the continuity
with respect to the H 1 norm of all terms in the variational inequality and the factthat the variational solution u has a compact support in space at every time.
2.4.2. For n=2. For dimension n = 2, we use a different rescaling that preserves
the singularity of logarithm, namely
log R(λ)(2.6)
v λ (x, t) = log R(λ)v(R(λ)x, λt),uλ (x, t) =u(R(λ)x, λt),
λwhere R(λ) is the unique solution of R2 log R = λ, limλ→∞ R(λ) → ∞ (see [17]for more details). v λ and uλ satisfy rescaled problems analogous to (2.4) and (2.5).
In particular, the term λ(2−n)/n in front of the time derivatives is replaced by
1/ log(R(λ)) → 0 as λ → ∞.

THE ONE-PHASE STEFAN PROBLEM

92.5. Convergence of radially symmetric solutions. We will recall the resultson the convergence of radially symmetric solutions of (1.1) as derived in [19]. First,
we collect some useful facts of radial solution of the Hele-Shaw problem and then use
a comparison to have the information of radial solution of the Stefan problem. The
radially symmetric solution of the Hele-Shaw problem in the domain |x| ≥ a, t ≥ 0
is a pair of functions p(x, t) and R(t), where p is of the form

 n−2
Aa
|x|n−2 − Rn−2 (t) +, n ≥ 3,
  a2−n − R2−n (t)(2.7)
p(x, t) = A log R(t)

|x|

+


,
n = 2,
R(t)log aand R(t) satisfies a certain algebraic equation (see [19] for details).
This solution satisfies the boundary conditions and initial conditions(2.8)

p(x, t) = Aa2−n

for |x| = a > 0,p(x, t) = 0
1
R0 (t) = |Dp|
L
R(0) = b > a.

for |x| = R(t),for |x| = R(t),Furthermore,
R(t)
= 1,
t→∞ c∞ t1/n
R(t)
lim
= 1,
t→∞ c (t/ log t)1/2
∞
lim



An(n − 2)Lpc∞ = 2 A/L
c∞ =

1/n
if n ≥ 3,
if n = 2.

1
In dimension n = 2, we will also use lim loglogR(t)t = 2.
t→∞
The radial solution of the Stefan problem satisfies the corresponding conditions
similar to (2.8) together with the initial data

(2.9)

θ(x, 0) = θ0 (|x|) if |x| ≥ a.

The following results were shown in [19].
Lemma 2.14 (cf. [19, Proposition 6.1]). Let p and θ be radially symmetric solutions
to the Hele-Shaw problem and to the Stefan problem respectively, and let {|x| =Rp (t)}, {|x| = Rθ (t)} be the corresponding interfaces. If Rp (0) > Rθ (0), p(x, 0) ≥θ(x, 0) and, moreover, p(x, t) ≥ θ(x, t) on the fixed boundary, that is, for |x| =a, t > 0, then p(x, t) ≥ θ(x, t) for all |x| ≥ a and t ≥ 0.
This immediately leads to an upper bound for the free boundary of radial solutions of Stefan problem, see Corollary 6.2, Theorem 6.4, Theorem 7.1 in [19].
Lemma 2.15. Let {|x| = R(t)} be the free boundary of a radial solution to the
Stefan problem satisfying the corresponding conditions (2.8) and (2.9). There areconstants C, T > 0, such that, for all t ≥ T ,R(t) ≤ Ct1/n , n ≥ 3,orMoreover, we have
R(t)
1/n
lim
= (An(n − 2)/L), n ≥ 3,t→∞ t1/nR(t) ≤ C(t/log t)1/2 , n = 2.

or

p
R(t)
=
2
A/L, n = 2.
t→∞ (t/ log t)1/2
lim

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NORBERT POŽÁR AND GIANG THI THU VU

The solution of the Stefan problem is bounded for all time.
Lemma 2.16 (cf. [19, Lemma 6.3]). Let θ be a weak solution of the Stefan problem
for n ≥ 2. There is a constant C > 0 such that, for all t > 0, 0 ≤ θ(x, t) ≤ C|x|2−n .
Next, we define the solution of the Hele-Shaw problem with a point source, which
will appear as the limit function in our convergence results,A |x|2−n − ρ2−n (t) , n ≥ 3,+

(2.10)V (x, t) = VA,L (x, t) =A log ρ(t),n = 2,
|x|
+

where
(
1/n
(An(n − 2)t/L), n ≥ 3,ρ(t) = ρL (t) = R∞ =1/2
(2At/L) ,n = 2.
It is the unique solution of the Hele-Shaw problem with a point source,∆v = 0in Ω(v)\{0},





v(x, t)v(x, t)= A,n ≥ 3, orlim −= A, n = 2,
 lim
2−n
log(|x|)|x|→0 |x||x|→0
(2.11)
1
vt = |Dv|2 on ∂Ω(v),



L


v(x, 0) = 0
in Rn \{0}.
The asymptotic result for radial solutions of the Stefan problem follows from Theorem 6.5 and Theorem 7.2 in [19].
Theorem 2.17 (Far field limit). Let θ be the radial solution of the Stefan problem
satisfying the corresponding boundary and initial conditions (2.8), (2.9). Then
(2.12)

lim t(n−2)/n |θ(x, t) − V (x, t)| = 0t→∞

uniformly on sets of form {x ∈ Rn : |x| ≥ δt1/n }, δ > 0 if n ≥ 3, and(2.13)
!
r
r
2A
A
2A
q
lim log
R(t) θ(x, t) −log
R(t) − log |x|t→∞LLlog 2A R(t)
L

=0

+

uniformly on sets of form {x ∈ Rn : |x| ≥ δR(t)}, δ > 0 if n = 2.
Proof. Following the proof of Theorem 6.5 in [19] with recalling that we assume
θ = Aa2−n for |x| = a, we immediately get the result for n = 3.

R2
1 (t)
For n = 2, let R1 (t) be the solution of 21 log R1 − 21 = At
lim RR(t)=L with t→∞
q
q
2A
2A

L . Thus, we can replace R1 (t) in Theorem 7.2 in [19] by
L R(t).
Finally, we can improve Theorem 2.17 to have the following convergence result
for rescaled radial solutions of the Stefan problem which holds up to t = 0.
Lemma 2.18 (Convergence for radial case). Let θ(x, t) be a radial solution of the
Stefan problem satisfying the corresponding boundary and initial conditions (2.8)
and (2.9). Then θλ converges locally uniformly to VA,L in the set (Rn \{0})×[0, ∞).

THE ONE-PHASE STEFAN PROBLEM

11

Proof. We will prove the uniform convergence in the sets Q = {(x, t) : |x| ≥ ε, 0 ≤t ≤ T } for some ε, T > 0 and use notation V = VA,L . We consider the casen ≥ 3 first. Set ξ = λ1/n x, τ = λt then an easy computation leads to V (x, t) =λ(n−2)/n V (ξ, τ ). Let t0 = ρ−1 (ε/2). We split the proof into two cases:(a) When 0 ≤ t ≤ t0 : Clearly from the formula, we have V (x, t) = 0 in {(x, t) :|x| ≥ ε, 0 ≤ t ≤ t0 }. Besides, for λ large enough,Rλ (t) =R(λt0 )εR(λt)≤ 1/n < ρ(t0 ) + = ε (due to Proposition 2.15).
2
λ1/n
λ

Thus, θλ = 0 = V in {(x, t) : |x| ≥ ε, 0 ≤ t ≤ t0 } for λ large enough.
(b) When t0 ≤ t ≤ T , we have:(2.14)|θλ (x, t) − V (x, t)| = t(2−n)/n τ (n−2)/n |θ(ξ, τ ) − V (ξ, τ )|Since t0 ≤ t ≤ T , t(2−n)/n is bounded. From Theorem 2.17, the right handside of (2.14) converges to 0 uniformly in the sets {ξ ∈ Rn : |ξ| ≥ δτ 1/n } ={x ∈ Rn : |x| ≥ δt1/n } ⊃ {(x, t) : |x| ≥ ε, t0 ≤ t ≤ T } for fixed ε and δ > 0
small enough and thus we obtain the convergence for n ≥ 3.

R(τ )= t1/2For n = 2, we argue similar as the case n ≥ 3, but noting that lim R(λ)λ→∞

together with Theorem 2.17.



2.6. Some more results for viscosity solutions. Following [17, 19], we also canstate some results for viscosity solutions.
Lemma 2.19. For L = 1/m (resp. L = 1/M ), with m, M as in (1), let θ(x, t) be
the radial solution of Stefan problem (1.1) satisfying boundary conditions (2.8) andinitial condition (2.9) with g(x) = 1/L and a such that B(0, a) ⊂ K (resp. K ⊂B(0, a)). Then the function θ(x, t) is a viscosity subsolution (resp. supersolution)of the Stefan problem (1.1) in Q.
Proof. The statement follows directly from properties of radially solutions and thefact that a classical solution is also a viscosity solution.
Using viscosity comparison principle, we also can get the same estimates forfree boundary as in Proposition 2.15 and boundedness for a general viscosity
solution.
Lemma 2.20. Let v be a viscosity solution of (1.1). There exists t0 > 0 andconstant C, C1 , C2 > 0 such that for t ≥ t0 ,C1 t1/n < min |x| ≤ max |x| < C2 t1/n

if n ≥ 3,

C1 R(t) < min |x| ≤ max |x| < C2 R(t)if n = 2,

Γt (v)

Γt (v)

Γt (v)

Γt (v)

and for 0 ≤ t ≤ t0 , max |x| < C2 . Moreover, 0 ≤ v(x, t) ≤ C|x|2−n for all n ≥ 2.
Γt (v)

Proof. Argue as in [17] with using Lemma 2.15 and Lemma 2.16 above.



We also have the near field limit and the asymptotic behavior result as in [19].

12

NORBERT POŽÁR AND GIANG THI THU VU

Theorem 2.21 (Near-field limit). The viscosity solution v(x, t) of the Stefan problem (1.1) converges to the unique solution P (x) of the exterior Dirichlet problem
∆P = 0, x ∈ Rn \K,
P = 1, x ∈ Γ,
(2.15)

or P is bounded if n = 2,
 lim P (x) = 0 if n ≥ 3,|x|→∞

as t → ∞ uniformly on compact subsets of K c .
Proof. See proof of Theorem 8.1 in [19].

Lemma 2.22 (cf. [19, Lemma 4.5]). There exists a constant C∗ = C∗ (K, n) suchthat the solution P of problem (2.15) satisfies lim |x|n−2 P (x) = C∗ .
|x|→∞

3. Uniform convergence of variational solutions
3.1. Limit problem and the averaging properties of media. We first recall
the limit variational problem as introduced in [17] (see [17, section 5] for derivationRtand properties). Let UA,L (x, t) := 0 VA,L (x, s) ds. For given A, L > 0, [17,Theorem 5.1] yields that UA,L (x, t) is the unique solution of the limit obstacle
problemw ∈ Kt ,a(w, φ) ≥ h−L, φi, for all φ ∈ V,
(3.1)

 a(w, ψw) = hL, ψwi for all ψ ∈ W,
n
o
T
ϕ(x)
where Kt = ϕ ∈ ε>0 H 1 (Rn \Bε ) ∩ C(Rn \Bε ) : ϕ ≥ 0, lim UA,L(x,t) = 1 ,
|x|→0

(3.2)


V = φ ∈ H 1 (Rn ) : φ ≥ 0, φ = 0 on Bε for some ε > 0 ,(3.3)

W = V ∩ C 1 (Rn ),

and

Z

Z
Du · Dvdx,aΩ (u, v) :=Ωhu, viΩ :=

uvdx.
Ω

We omit the set Ω in the notation if Ω = Rn .
We also recall the following application of the subadditive ergodic theorem.
Lemma 3.1 (cf. [15, Section 4, Lemma 7], see also [17]). For given g satisfying(2), there exists a constant, denoted by h1/gi, such that if Ω ⊂ Rn is a boundedmeasurable set and if {uε }ε>0 ⊂ L2 (Ω) is a family of functions such that uε → ustrongly in L2 (Ω) as ε → 0, then
Z
Z  
1
1
ε
lim
u (x)dx =u(x)dx a.e. ω ∈ A.
ε→0g(x/ε, ω)gΩΩ

3.2. Uniform convergence of rescaled variational solutions. Now we areready to prove the first main result, similar to Theorem 6.2 in [17].
Theorem 3.2. Let u be the unique solution of variational problem (2.1) and uλ be
its rescaling. Let UA,L be the unique solution of limit problem (3.1) where A = C∗ asin Lemma 2.22, and L = h1/gi as in Lemma 3.1. Then the functions uλ converges
locally uniformly to UA,L as λ → ∞ on (Rn \{0}) × [0, ∞).

THE ONE-PHASE STEFAN PROBLEM

13

Proof. We argue as in [17]. Fix T > 0. By Lemma 2.20, we can bound Ωt (uλ ) by
Ω := Bδ (0) for some δ > 0, for all 0 ≤ t ≤ T and λ > 0. For some ε > 0, defineΩε := Ω\B(0, ε), Qε := Ωε × [0, T ] . We will prove the convergence in Qε .
Let v be the viscosity solution of the Stefan problem (1.1). We can find constants
0 < a < b such that K ⊂ Ba (0) and Ω0 ⊂ Bb (0). Set L = 1/M and A = max v0 .
Choose radially symmetric smooth θ0 ≥ 0 such that θ0 ≥ v0 on Ω0 \ Ba (0) andθ0 = 0 on Rn \ Bb (0). The radial solution θ of the Stefan problem on Rn \ Ba (0)with such parameters will be above v by the comparison principle. Thus, for λlarge enough, the rescaled solutions satisfy
0 ≤ v λ ≤ θλ in Qε/2 .
On the other hand, by Lemma 2.18, θλ converges to VA,L as λ → ∞ uniformlyon Qε/2 and VA,L is bounded in Qε/2 and therefore for λ large enough so that(0)
⊂ Bε/2 (0),(Ba (0))λ := Bλa1/n(3.4)

kuλt kL∞ (Qε/2 ) = kv λ kL∞ (Qε/2 ) ≤ C(ε).

Since uλ satisfies (2.5), we have∆uλ (ϕ − uλ ) ≤ λ(2−n)/n uλt − f (λ1/n x) (ϕ − uλ ) a.e for any ϕ ∈ Kλ (t).
As uλt is bounded, uλ satisfies the elliptic obstacle problem∆uλ (ϕ − uλ ) ≤ Cλ(2−n)/n − f (λ1/n x) (ϕ − uλ )a.e for any ϕ ∈ Kλ (t) such that ϕ − uλ ≥ 0.
Now we can use the standard regularity estimates for the obstacle problem (see[21, Proposition 2.2, chapter 5] for instance),k∆uλ (·, t)kLp (Ωε/2 ) ≤ Cλ(2−n)/n −1
gλ≤ C0 for all 1 ≤ p ≤ ∞,Lp (Ωε/2 )Rt
for all λ large so that also Ωλ0 ⊂ Bε/2 (0). Using (3.4) and uλ (x, t) = 0 v λ (x, s)ds,
we conclude kuλ (·, t)kLp (Ωε/2 ) is bounded uniformly in t ∈ [0, T ] and λ large.
Using elliptic interior estimate results for obstacle problem again (for example,[21, Theorem 2.5]), we can find constants 0 < α < 1 and C2 , independent oft ∈ [0, T ] and λ  1, such thatkuλ (·, t)kW 2,p (Ωε ) ≤ C2 ,kuλ (·, t)kC 0,α (Ωε ) ≤ C2 ,for all 0 ≤ t ≤ T, λ  1.

Moreover, using (3.4) again, we have |uλ (x, t) − uλ (x, s)| ≤ C3 |t − s|. Thus uλ is
Hölder continuous in x with 0 < α < 1 and Lipschitz continuous in t. In particular,
uλ satisfies
kuλ kC 0,α (Qε ) ≤ C4 (C2 , C3 ) for all λ ≥ λ0 .
The argument for case n = 2 is similar.
By the Arzelà-Ascoli theorem, we can find a function ū ∈ C((Rn \ {0}) × [0, ∞))and a subsequence {uλk } ⊂ {uλ } such thatuλk → ū locally uniformly on (Rn \ {0}) × [0, ∞) as k → ∞,Due to the compact embedding of H 2 in H 1 , we have, uλk (·, t) → ū(·, t) stronglyin H 1 (Ωε ) for all t ≥ 0, ε > 0.

14

NORBERT POŽÁR AND GIANG THI THU VU

To finish the proof, we need to show that the function ū is the solution of limit
problem (3.1) and then by the uniqueness of the limit problem, we deduce that theconvergence is not restricted to a subsequence.
Lemma 3.3 (cf. [17, Lemma 6.3]). For each t ≥ 0, w̄ := ū(·, t) satisfies(3.5)

a(w̄, φ) ≥ h−L, φi for all φ ∈ V,

(3.6)

a(w̄, ψ w̄) = h−L, ψ w̄i for all ψ ∈ W,

where L = h1/gi as in Lemma 3.1 and V, W as in (3.2) and (3.3).
Proof. Consider n ≥ 3. Following the techniques in [17], fix t ∈ [0, T ] and denote
wk := uλk (·, t). Take φ ∈ V first. Analogously to Remark 2.13, we only needto prove the inequality for functions φ with compact support, the conclusion forgeneral function φ in V will follow by the continuity of all terms in the inequality.
There exists k0 > 0 such that for all k ≥ k0 , Ωλ0 k ⊂ Bε (0) and φ = 0 on Bε (0). Set
ϕk = φ + wk ∈ Kλk (t). Substitute the function ϕk into the rescaled equation (2.5)
and integrate both sides and integrate by parts, which yields

D
E  1
(2−n)/nλk
λk
ut (·, t), φ + − λ , φ .
a(w , φ) ≥ −λk
g k
The linear functional w 7→ a(w, φ) is bounded in H 1 . Recalling Lemma 3.1 andthat uλt k is bounded, since wk → w̄ strongly in H 1 as k → ∞, we can send λk → ∞and obtain (3.5).
Now take ψ ∈ W . As above, we assume that ψ has compact support, and without
loss of generality we can also assume that 0 ≤ ψ ≤ 1, ψ = 0 on Bε (0) (otherwise
consider maxψRn ψ instead). Take k0 such that Ωλ0 k ⊂ Bε (0) for all k ≥ k0 . Sinceψ ∈ W then ψ w̄ ∈ V . As above we have a(w̄, ψ w̄) ≥ h−L, ψ w̄i. Moreover, consider
ϕk = (1 − ψ)wk ∈ Kλk (t), k ≥ k0 . Then,


D
E
1
(2−n)/nkkkkkkuλt k (·, t), ψwk .
a(w , ψw ) = −a(w , ϕ − w ) ≤ − λ , ψw − λkg kAgain using Lemma 3.1, boundedness in L∞ (Rn ) of wk and uλt k , the lower semicontinuity in H 1 of the map w 7→ a(w, ψw), and the fact that wk → w̄ strongly in
H 1 as k → ∞ we can conclude the equality (3.6).
Again, n = 2 is similar.
Finally, the next lemma establishes that the singularity of ū as |x| → 0 is correct.
Lemma 3.4 (cf. [17, Lemma 6.4]). We havelimū(x, t)|x|→0 UC∗ ,L(x, t) = 1

for every t ≥ 0, where L = h1/gi as in Lemma 3.1 and C∗ as in Lemma 2.22.
Proof. The proof follows from the proof of [17, Lemma 6.4] since the solutions of
the Stefan problem have the same near field limit (Lemma 2.15) as the Hele-Shawsolutions.
This finishes the proof of Theorem 3.2 .

THE ONE-PHASE STEFAN PROBLEM

15

4. Uniform convergence of rescaled viscosity solutions and freeboundaries
In this section, we will deal with the convergence of v λ and their free boundaries.
Let v be a viscosity solution of the Stefan problem (1.1) and v λ be its rescaling. Let
V = VC∗ ,L be the solution of Hele-Shaw problem with a point source as in (2.10),
where C∗ is the constant of Lemma 2.22 and L = h1/gi as in Lemma 3.1.
We define the half-relaxed limits in {|x| =6 0, t ≥ 0}:v ∗ (x, t) =lim supv λ (y, s),v∗ (x, t) =(y,s),λ→(x,t),∞lim infv λ (y, s),(y,s),λ→(x,t),∞Remark 4.1. V is continuous in {|x| =6 0, t ≥ 0}, therefore V∗ = V = V ∗ .
To complete Theorem 1.1, we prove a result similar to [17, Theorem 7.1.]
Theorem 4.2. The rescaled viscosity solution v λ of the Stefan problem (1.1) converges locally uniformly to V = VC∗ ,h1/gi in (Rn \{0}) × [0, ∞) as λ → ∞ andv∗ = v ∗ = V.
Moreover, the rescaled free boundary {Γ(v λ )}λ converges to Γ(V ) locally uniformly
with respect to the Hausdorff distance.
To prepare for the proof of Theorem 4.2, we need to collect some results which
are similar to the ones in [16] and [17] with some adaptations to our case. All the
results for n ≥ 3 we have in this section can be obtained for n = 2 by using limit
1
log R(λ) → 0 as λ → ∞. Thus, from here on we only consider case n ≥ 3, the results
for n = 2 are omitted.
4.1. Some necessary technical results.
Lemma 4.3 (cf. [16, Lemma 3.9]). The viscosity solution v of the Stefan problem(1.1) is strictly positive in Ω(u), satisfies Ω(v) = Ω(u) and Γ(v) = Γ(u).
Lemma 4.4. Let v λ be a viscosity solution of the rescaled problem (2.4). Thenv ∗ (·, t) is subharmonic in Rn \ {0} and v∗ (·, t) is superharmonic in Ωt (v∗ )\{0} in
viscosity sense.
Proof. This follows from a standard viscosity solution argument using test functions, see for instance [11].

The behavior of functions v ∗ , v∗ at the origin and their boundaries can be established by following the arguments in [17] and [16].
Lemma 4.5 (v ∗ and v∗ behave as V at the origin). The functions v ∗ , v∗ have asingularity at 0 with:v∗ (x, t)v ∗ (x, t)
(4.1)
lim
= 1,
lim
= 1, for each t > 0.
|x|→0+ V (x, t)|x|→0+ V (x, t)
Proof. See [17, Lemma 7.4].

Lemma 4.6 (cf. [16, Lemma 5.4]). Suppose that (xk , tk ) ∈ {uλk = 0} and (xk , tk , λk ) →(x0 , t0 , ∞). Then:a) U (x0 , t0 ) = 0,b) If xk ∈ Γtk (uλk ) then x0 ∈ Γt0 (U ),where U = UC∗ ,L is the limit function in Theorem 3.2.

16

NORBERT POŽÁR AND GIANG THI THU VU

Proof. See proof of [16, Lemma 5.4].

The rest of the convergence proof in [17] relies on the monotonicity of the solutions of the Hele-Shaw problem in time. Since the Stefan problem lacks thismonotonicity, we will show that sufficiently regular initial data satisfies a weak
monotonicity below. The convergence result for general initial data will then follow
by the uniqueness of the limit and the comparison principle.
Lemma 4.7. Suppose that v0 satisfies (1.2). Then there exist C ≥ 1 independentof x and t such that
(4.2)

v0 (x) ≤ Cv(x, t) in Rn \K × [0, ∞).

Proof. Let γ1 := min∂Ω0 |Dv0 |, γ2 := max∂Ω0 |Dv0 |. Note that 0 < γ1 ≤ γ2 < ∞.
For given ε > 0, let w be the solution of boundary value problem ∆w = 0 in Ω0 \K,w = ε on K, w = 0 on Ωc .
0

For x close to ∂Ω0 we have v0 (x) ≥ γ21 dist(x, ∂Ω0 ). Since γ1 > 0, v0 > 0 in Ω0and ∂Ω0 has a uniform ball condition, we can choose ε > 0 small enough such thatw ≤ v0 in Rn \K. By Hopf’s Lemma, γw := min∂Ω0 |Dw| > 0. It is clear that w
is a classical subsolution of the Stefan problem (1.1) and the comparison principle
yields(4.3)

w ≤ v in (Rn \K) × [0, ∞).

Now assume that (4.2) does not hold, that is, for every k ∈ N, there exists(xk , tk ) ∈ Rn \K × [0, ∞) such that1v0 (xk ) > v(xk , tk ).
(4.4)
kClearly xk ∈ Ω0 . {tk } is bounded by Theorem 2.21 since v0 is bounded. Therefore,
there exists a subsequence (xkl , tkl ) and a point (x0 , t0 ) such that (xkl , tkl ) →(x0 , t0 ). Since v0 is bounded, we get v(x0 , t0 ) ≤ 0 and thus x0 ∈ ∂Ω0 by (4.3).
Consequently, for kl large enough,


1
γw
γw
w(xkl ) ≥ γw dist(xkl , ∂Ω0 ) =2γ2 dist(xkl , ∂Ω0 ) ≥v0 (xkl ).
2
4γ2
4γ2
Combine this with (4.3) and (4.4) to obtainγw
1v0 (xkl ) >v0 (xkl )kl
4γ2
for every kl large enough, which yields a contradiction since v0 (xkl ) > 0.

Some of the following lemmas will hold under the condition (4.2).
Lemma 4.8. Let u be the solution of the variational problem (2.1), and v be theassociated viscosity solution of the Stefan problem, and suppose that (4.2) holds.
Then
(4.5)

u(x, t) ≤ Ctv(x, t).

Proof. The statement follows from checking that ũ := Ctv is a supersolution of theheat equation in Ω(u) and the classical comparison principle. Indeed, ũt − ∆ũ =Cv + Ct(vt − ∆v) ≥ v0 ≥ f = ut − ∆u in Ω(u) by (4.2).
THE ONE-PHASE STEFAN PROBLEM

17

Lemma 4.9 (cf. [16, Lemma 5.5]). The function v∗ satisfies Ω(V ) ⊂ Ω(v∗ ). In
particular v∗ ≥ V .
Proof. Assume that the inclusion does not hold, there exists (x0 , t0 ) ∈ Ω(V ) andv∗ (x0 , t0 ) = 0. By (4.2) and Lemma 4.8, there exists C > 1 such that u(x, t) ≤Ctv(x, t). This inequality is preserved under the rescaling, uλ (x, t) ≤ Ctv λ (x, t)
in (Rn \K λ ) × [0, ∞). Taking lim inf ∗ of both sides gives the contradiction 0 <U (x0 , t0 ) ≤ Ct0 v∗ (x0 , t0 ) = 0.
The inequality v∗ ≥ V follows from the elliptic comparison principle as v∗ issuperharmonic in Ω(v∗ ) \ {0} by Lemma 4.4 and behaves as V at the origin byLemma 4.5.
Lemma 4.10. There exists constant C > 0 independent of λ such that for every
x0 ∈ Ωt0 (uλ ) and Br (x0 ) ∩ Ωλ0 = ∅ for some r, for every λ large enough we havesupuλ (x, t0 ) > Cr2 .

x∈Br (x0 )Proof. Follow the arguments in [15, Lemma 3.1] while noting that since uλt isbounded then, for λ large enough, uλ is a strictly subharmonic function in Ωt0 (uλ )\Ωλ0 .

Corollary 4.11. There exists a constant C1 = C1 (n, M, λ0 ) such that if (x0 , t0 ) ∈Ω(v λ ) and Br (x0 ) ∩ Ωλ0 = ∅ and λ ≥ λ0 , we havesup v λ (x, t0 ) ≥Br (x0 )C1 r 2.
t0

Proof. The inequality follows directly from Lemma 4.8 and Lemma 4.10.

Lemma 4.12 (cf. [16, Lemma 5.6 ii]). We have the following inclusion:Γ(v ∗ ) ⊂ Γ(V ).
Proof. Argue as in [16, Lemma 5.6 ii] together with using Lemma 4.6 and Lemma
4.10 above.

Now we are ready to prove Theorem 4.2.
4.2. Proof of Theorem 4.2.
Proof. Step 1. We prove the convergence of viscosity solutions and the free boundaries under the conditions (1.2) and (4.2) first.
Lemma 2.20 yields that Ωt (v ∗ ) is bounded at all time t > 0. Since Ω(V ) is simply
connected set, Lemma 4.12 implies thatΩ(v ∗ ) ⊂ Ω(V ) ⊂ Ω(VC ∗ +ε,L ) for all ε > 0.
We see from Lemma 4.4, v ∗ (·, t) is a subharmonic function in Rn \ {0} for every∗(x,t)= 1 for all t ≥ 0 by Lemma 4.5, comparison principlet > 0 and lim|x|→0 vV (x,t)∗
yields v (x, t) ≤ VC∗ +ε,L (x, t) for every ε > 0.
By Lemma 4.9, V (x, t) ≤ v∗ and letting ε → 0+ we obtain by continuity
V (x, t) ≤ v∗ (x, t) ≤ v ∗ (x, t) ≤ V (x, t).
Therefore, v∗ = v ∗ = V and in particular, Γ(v∗ ) = Γ(v ∗ ) = Γ(V ).

18

NORBERT POŽÁR AND GIANG THI THU VU

Now we need to show the uniform convergence of the free boundaries with respect
to the Hausdorff distance. Fix 0 < t1 < t2 and denote:
Γ∞ := Γ(V ) ∩ {t1 ≤ t ≤ t2 },Γλ := Γ(v λ ) ∩ {t1 ≤ t ≤ t2 },a δ-neighborhood of a set A in Rn × R isUδ (A) := {(x, t) : dist((x, t), A) < δ}.
We need to prove that for all δ > 0, there exists λ0 > 0 such that:Γλ ⊂ Uδ (Γ∞ )

(4.6)

and

Γ∞ ⊂ Uδ (Γλ ),∀λ ≥ λ0 .

We prove the first inclusion in (4.6) by contradiction. Suppose therefore that we
can find a subsequence {λk } and a sequence of points (xk , tk ) ∈ Γλk such that
dist((xk , tk ), Γ∞ ) ≥ δ. Since Γλ is uniformly bounded in λ by Lemma 2.20, there
exists a subsequence {(xkj , tkj )} which converge to a point (x0 , t0 ). By Lemma
4.6, (x0 , t0 ) ∈ Γ(U ) = Γ(V ). Moreover, since t1 ≤ tkj ≤ t2 then t1 ≤ t0 ≤ t2 andtherefore, (x0 , t0 ) ∈ Γ∞ , a contradiction.
The proof of the second inclusion in (4.6) is more technical. We prove a pointwise
result first. Suppose that there exists δ > 0, (x0 , t0 ) ∈ Γ∞ and {λk }, λk → ∞,
δfor all k. Then there exists r > 0 such thatsuch that dist((x0 , t0 ), Γλk ) ≥2
Dr (x0 , t0 ) := B(x0 , r) × [t0 − r, t0 + r] satisfies either:Dr (x0 , t0 ) ⊂ {v λk = 0} for all k,(4.7)

or after passing to a subsequence,
Dr (x0 , t0 ) ⊂ {v λk > 0} for all k.

(4.8)

If (4.7) holds, clearly V = v∗ = 0 in Dr (x0 , t0 ) which is in a contradiction with theassumption that (x0 , t0 ) ∈ Γ∞ .
Thus we assume that (4.8) holds. In Dr (x0 , t0 ), v λk solves the heat equation(2−n)/n λk
λvt − ∆v λk = 0. Set(2−n)/nwk (x, t) := v λk (x, λkt)(n−2)/n(n−2)/n(t0 − r), λk(t0 + r)] andthen w > 0 in Drw (x0 , t0 ) := B(x0 , r) × [λk(n−2)/n rkwkkw satisfies wt − ∆w = 0 in Dr (x0 , t0 ). Since λk
2 → ∞ as k → ∞, bykHarnack’s inequality for the heat equation, for fixed τ > 0 there exists a constant(n−2)/n r
C1 > 0 such that for each t ∈ [t0 − 2r , t0 + 2r ] and λk such that τ < λk
4 we
have
(n−2)/n(n−2)/nsup wk (·, λkt − τ ) ≤ C1 inf wk (·, λkt).
B(x0 ,r/2)

B(x0 ,r/2)

This inequality together with Corollary 4.11 yields:
C2 r 2(2−n)/n≤ sup v λk (·, t − λkτ ) ≤ C1 inf v λk (·, t)(2−n)/nB(x0 ,r/2)
B(x
,r/2)
t − λkτ0
for all t ∈ [t0 − 2r , t0 + 2r ], λk ≥ λ0 large enough, where C2 only depends on n, M, λ0 .
Taking the limit when λk → ∞, the uniform convergence of {v λk } to V gives V > 0in B(x0 , 2r ) × [t0 − 2r , t0 + 2r ], which is a contradiction with (x0 , t0 ) ∈ Γ∞ ⊂ Γ(V ).
We have proved that every point of Γ∞ belongs to all Uδ/2 (Γλ ) for sufficiently

large λ. Therefore the second inclusion in (4.6) follows from the compactness ofΓ∞ .
This concludes the proof of Theorem 4.2 when (4.2) holds.

THE ONE-PHASE STEFAN PROBLEM

19

Step 2. For general initial data, we will find upper and lower bounds for the
initial data for which (4.2) holds, and use the comparison principle. For instance,
assume that v0 ∈ C(Rn ), v0 ≥ 0, such that supp v0 is bounded, v0 = 1 on K.
Choose smooth bounded domains Ω10 , Ω20 such that K ⊂ Ω10 ⊂ Ω10 ⊂ supp v0 ⊂2
Ω0 . Let v01 , v02 be two functions satisfying (1.2) with positive domains Ω10 , Ω20 , respectively, and v01 ≤ v0 ≤ v02 . If necessary, that is, when v0 is not sufficiently
regular at ∂K, we may perturb the boundary data for v01 , v02 on K as 1 − ε and
1 + ε, respectively, for some ε ∈ (0, 1).
Let v1 , v2 be respectively the viscosity solution of the Stefan problem (1.1) withinitial data v01 , v02 . By the comparison principle, we have v1 ≤ v ≤ v2 and after
rescaling v1λ ≤ v λ ≤ v2λ . By Step 1, we see that v1λ → VC∗,1−ε ,L and v2λ → VC∗,1+ε ,L .
Since C∗,1±ε → C∗ as ε → 0 by [19, Lemma 4.5], we deduce the local uniform
convergence of v λ → V = VC∗ ,L .
The convergence of free boundaries follows from the ordering Ω(v1 ) ⊂ Ω(v) ⊂Ω(v2 ) and the convergence of free boundaries of VC∗,1±ε ,L to the free boundary of
VC∗ ,L locally uniformly with respect to the Hausdorff distance.

Acknowledgments. The first author was partially supported by JSPS KAKENHI
Grant No. 26800068 (Wakate B). This work is a part of doctoral research of the second author. The second author would like to thank her Ph.D. supervisor Professor
Seiro Omata for his valuable support and advice.
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E-mail address: npozar@se.kanazawa-u.ac.jp(G. T. T. Vu) Graduate School of Natural Science and Technology, Kanazawa University, Kakuma, Kanazawa, 920-1192, Japan
E-mail address: vtgiang@vnua.edu.vn