# On Viscosity and Equivalent Notions of Solutions for Anisotropic Geometric Equations

Authors: Cecilia De Zan; Pierpaolo Soravia
Journal: Abstract and Applied Analysis
 (2020)

---

## Abstract

We prove that viscosity solutions of geometric equations in step two Carnot groups can be equivalently reformulated by restricting the set of test functions at the singular points. These are characteristic points for the level sets of the solutions and are usually difficult to deal with. A similar property is known in the Euclidian space, and in Carnot groups, it is based on appropriate properties of a suitable homogeneous norm. We also use this idea to extend to Carnot groups the definition of generalised flow, and it works similarly to the Euclidian setting. These results simplify the handling of the singularities of the equation, for instance, to study the asymptotic behaviour of singular limits of reaction diffusion equations. We provide examples of using the simplified definition, showing, for instance, that boundaries of strictly convex subsets in the Carnot group structure become extinct in finite time when subject to the horizontal mean curvature flow even if characteristic points are present.


---

## Body

## 1. Introduction

In this paper, we want to discuss the notion of viscosity solution for geometric equations, describing weak front propagation in step two Carnot groups, of the form(1)utx,t+Fx,t,Xux,t,X2ux,t=0,x,t∈ℝn×0,+∞.Here, the operatorF=Fx,t,q,A, F:ℝn×0,+∞×ℝm\0×Sm⟶ℝ is elliptic and geometric, meaning that it is positively homogeneous in the pair q,A∈ℝm\0×Sm and invariant in the last argument with respect to matrices of the form μq⊗q, μ∈ℝ, as we make it more precise later. The notation Sm indicates the set of symmetric m×m matrices, n,m≥2. Therefore, it is possible that F has a singularity at q=0, and we assume that it behaves nicely, namely,(2)F∗x,t,0,O=F∗x,t,0,O=0,where the stars above indicate the lower and upper semicontinuous envelopes, respectively. The notation Xu indicates the horizontal gradient with respect to a family of vector fields X1,…,Xm, seen as differential operators:(3)Xj=∑i=1nσi,jx∂i,j∈1,…,m,generators of a step two Carnot group. In particular, for a smooth function u, Xu=∇uσx, σ=σi,ji,j, and if m<n, equation (1) has singularities when Xu=0, i.e., at characteristic points of the level set of u and therefore on a subspace of positive dimension. Notation X2u indicates instead the horizontal hessian, namely, X2u=XiXjui,j=1,…,m∗, the symmetrised matrix of second derivatives. This compares to the usual Euclidian case when σ≡In is the identity matrix, where Xu≡∇u is the standard gradient and the singularity is just at the origin. In the special case when the operator F:ℝm\0×Sm⟶ℝ is defined as(4)Fq,A=−trI−qq⊗qqA,and moreover m=n and σ≡In, (1) reads as the well-known mean curvature flow equation:(5)utx,t−trI−∇u∇u⊗∇u∇uD2u=0.In a group setting instead, (5) becomes(6)utx,t−∑i,j=1mδij−Xiux,tXjux,t∑i=1mXiux,t2XiXjux,t=0,which is the horizontal mean curvature flow equation in the Carnot group.Due to the presence of singularities and the fact that we do not expect classical solutions in general in (1), we will use as usual the notion of viscosity solution, as given by Crandall et al. [1] and Chen et al. [2]. In our main result, we prove an equivalent notion of solution where we use a restricted class of test functions at singular points, with the property that if the horizontal gradient vanishes, then the horizontal hessian vanishes as well. This equivalent notion of solution simplifies the dealing with singularities and was first proved in the Euclidian setting for the mean curvature flow equation by Barles and Georgelin [3] to study the convergence of numerical schemes. We also use this approach to extend to our setting the notion of generalised flow, introduced as a general and flexible method to study singular limits in partial differential equations giving rise to propagating fronts by Barles and Souganidis [4] and applied in several situations in the Euclidian setting (see also Barles and Lio [5]). As a matter of fact, we will use this notion of solution in a forthcoming paper, when we discuss the singular limit of reaction diffusion equations for anisotropic and degenerate diffusions [6], while we develop here the preliminary needed tools on weak front propagation. This simplified approach, which is particularly helpful when studying approximations of (1) of different nature, therefore extends to the Carnot group setting with similar properties. Hopefully, it could also prove useful to tackle the comparison principle for viscosity solutions of (1), which is still missing in the literature in full generality. To achieve our goal, we need to modify the usual approach with the doubling of variables in viscosity solutions, by changing the test function, since the Euclidian norm does not work for singular anisotropic equations as (1), and replace it instead with a homogeneous norm, adapted to the Carnot group structure. As an application, we show how one can more easily check that functions are supersolutions or subsolutions of (1) especially at singular points, by providing explicit examples of supersolutions or subsolutions to be used as barriers. If in particular we consider the recent notion of v-convex functions with respect to the family of vector fields, we can prove, coupling our result with a comparison principle, that their level sets become extinct in finite time under the horizontal mean curvature flow equation, by constructing suitable supersolutions of (1).Equation (1) appears in the level set approach of the weak propagation of hypersurfaces, where we want to discuss the propagation of interfaces and boundaries of open sets, with prescribed normal velocity. In the Euclidian space, usually the velocity V=Vx,n,Dn, where n is the exterior normal. Indeed, if Ωt⊂ℝn is a family of open sets, Γt=∂Ωt is the propagating front, and there exists a smooth function u:ℝn×0,+∞ such that(7)Γt=x∈ℝn:ux,t=0,Ωt=x∈ℝn:ux,t>0,∇u≠0onΓt,one computes(8)V=ut∇u,n=−∇u∇u,Dn=−1∇uI−∇u⊗∇u∇u2D2u.Therefore, u formally satisfies(9)ut=Gx,t,∇u,D2u,where G is related to V by(10)Gx,t,p,A=pVx,t,−pp,−1pI−p⊗pp2A,x,p,A∈ℝn×ℝn\0×Sn.In our case, the anisotropy of the velocity will be for instance exploited by the fact that(11)Gx,t,p,A=Fpσx,tσxAσx,so that as an operator Gx,t,∇u,D2u=FXu,X2u. The novelty here with respect to the classical cases is that while in the Euclidian case σ=I and its square is a nondegenerate matrix, here the diffusion matrix σxtσx is not only anisotropic but also degenerate. When the family of vector fields does not span the whole ℝn at each point, this fact adds metric singularities to the usual one of geometric equations.The geometric property of the level set approach is based on the fact that ifu solves (1) and ψ:ℝ⟶ℝ is smooth and increasing, then ψu also solves the same equation. As a consequence, when a comparison principle holds true, it is easy to see that if uo1 and uo2 are two initial conditions such that(12)Γo=x:uo1x=0=x:uo2x=0,and u1andu2 are the corresponding solutions in (1), then one has(13)x:u1x,t=0=Γt=x:u2x,t=0,for allt>0.One can therefore define the family of closed setsΓtt to be the geometric flow of the front or interface Γo with the prescribed normal velocity.The notion of horizontal normal and horizontal mean curvature is due to Danielli et al. [7]. Recently, equation (1) has been studied by several authors. Existence results are available in the work of Capogna and Citti [8], who proved existence in Carnot groups by vanishing viscosity riemannian approximations. Dirr et al. [9] used stochastic approximations to show existence for more general Hörmander structures. Capogna et al. [10] prove uniform regularity estimates on the riemannian vanishing viscosity approximations for the flow of graphs, which also apply to prove existence for (1) in that case. On uniqueness results, the literature is far less complete. Capogna and Citti [8] proved a comparison principle if either one of the functions compared is uniformly continuous or their initial condition does not depend on the vertical coordinate, thus avoiding characteristic points in the initial front. A very recent paper by Baspinar and Citti [11] finds a comparison principle in Carnot groups of step two as a consequence of the fact that all solutions are limits of suitable families of riemannian regularisations. We remark the fact that in [8, 9], the authors use a notion of solution that differs from standard viscosity solutions at singular points. However, their notion of solution turns out to be equivalent to viscosity solutions as a consequence of our result. One of the referees pointed out to us the work of Ferrari et al. [12] where the authors use an approach similar to ours in the case of the horizontal mean curvature flow equation in the Heisenberg group, and they show a comparison principle for axisymmetric viscosity solutions.We recall that the level set method for geometric flows was proposed by Osher and Sethian [13] for numerical computations of geometric flows. The rigorous theory of weak front evolution started with the work by Evans and Spruck [14] for the mean curvature flow and by Chen et al. [2] for more general geometric flows. For the mathematical analysis of the level set method via viscosity solutions, the reader is referred to the book by Giga [15], where the approach is discussed in detail (see also Souganidis [16] and the references therein for the main applications of the theory, and [17] for equations with discontinuous coefficients).


## 2. Step Two Carnot Groups and Level Set Equations on the Group

In this paper, we consider inℝn a family of vector fields X=X1,…,Xm written as differential operators as in (3) and consider σ:ℝn⟶ℝn×m which is the matrix valued family of the coefficients. We will indicate σj the j−th column of σ so that(14)XjIx=σjx,j∈1,…,m,where Ix is the identity map in ℝn and in general Xj applied to a vector valued smooth function φ means the vector whose entries are given by Xj applied to the components of φ. The vector fields of the family are throughout the paper assumed to be generators of a step two Carnot group. To be more precise, we rely on the following definition (see the book by Bonfiglioli et al. [18], which we refer the reader to, for an introduction to the subject).Definition 1.
We say thatG=ℝn is a Lie group if ∘ is a group operation on ℝn and the map x,y⟼x−1∘y is smooth.
We then say thatG,∘,δλ is a step two Carnot group if we can split ℝn=ℝm×ℝn−m, x=xh,xv, m<n, and for all λ>0, the family of dilations δλx=λxh,λ2xv are automorphisms of the group (the group is homogeneous). Moreover, the family of vector fields X is left invariant on G with respect to the group operation, that is, for all φ∈C∞ℝn and all α∈ℝn, we have that(15)Xjφα∘x=Xjφταx,j∈1,…,m,where ταx=α∘x is the left translation, and the following Hörmander property is satisfied:(16)spanXix,Xj,Xkx:i,j,k∈1,…,m=ℝn,for allx∈ℝn,so that the family of vector fields X, together with their first order Lie brackets, generates ℝn at every point (the Carnot group is step two).
The vector fields of the familyX are said to be generators of the Carnot group.
Following [18], it is then well known that if X generates a step two Carnot group, then, by a suitable change of variables, we can suppose that(17)σx=ImBxht,where Im is the m×m identity matrix, Bxh=B1xh,…,Bn−mxh, and Bj, j∈1,…,n−m, are skew symmetric, linearly independent, m×m matrices. In addition, ℝn has the group structure with the following operation:(18)x∘y=xh+yh,xv+yv+Bxh,yh,with the notation Bxh,yh=B1xh⋅yh,…,Bn−mxh⋅yh. With this group operation, it is clear that x−1=−x and 0 is the identity element of the group.
Moreover, we notice that the Jacobian of the left translation has the following structure:(19)Dταx=ImOm×n−mBxhtIn−m,so the first m columns of the Jacobian give the matrix σx. It is also good to remember that for λ>0, the family X is homogeneous of degree one with respect to the dilations, namely,(20)Xjφδλx=λXφδλx,j∈1,…,n−m,for all φ∈C∞ℝn.Example 1.
The well-known example of the Heisenberg group comes fromℝ3=ℝ2×ℝ and the single matrix(21)B=01−10.
For our purposes, given a smooth functionu∈C2ℝn, we indicate the horizontal gradient (here, gradients are row vectors) as(22)Xux=∇uxσxand the horizontal hessian as(23)X2ux=XjXkuxj,k=1,…,n−m∗=tσxD2uxσx.
We just observe thatA∗=A+tA/2 indicates the symmetrisation and that the first-order terms in the second derivatives of X2 cancel out by direct computation since σ only depends on the first m variables.
Inℝn, taking advantage of the group structure of the family of vector fields, we want to study the problem of weak front propagation by extending the now classical-level set idea. Let F:ℝn×0,+∞×ℝm\0×Sm⟶ℝ be a continuous function, locally bounded at points of the form x,t,0,A, where Sm denotes the space of the m×m symmetric matrices. We assume on F the following structure conditions:(F1)F satisfies(24)F∗x,t,0,O=F∗x,t,0,O,for allx,t∈ℝn×0,+∞;(F2)F is elliptic, i.e., for any x,t∈ℝn×0,+∞,p∈ℝm\0 and A,B∈Sm:(25)Fx,t,p,A≤Fx,t,p,B,ifA≥B;(F3)F is geometric, i.e.,(26)Fx,t,λp,λA+μp⊗p=λFx,t,p,A,for allλ>0,μ∈ℝ,for everyx,t∈ℝn×0,+∞,p∈ℝm\0 and A∈Sm.
In the above equations, we are using the following notation for the lower semicontinuous extension ofF at the singular points:(27)F∗x,t,0,A=limr⟶0+infFy,t,q,B:q≠0,y,q,B−x,0,A≤r,and similarly for the upper semicontinuous extension F*. Note in particular that the geometric property of F implies F∗x,t,0,O=0 for all x,t∈ℝn×0,+∞.
We want to discuss the notion of solution for the equation(28)utx,t+Fx,t,Xu,X2u,x,t∈ℝn×0,+∞,where now only the horizontal first and second derivatives of the unknown function appear in the equation. Note that in our group setting, the operator F in (28), written in the usual coordinates of ℝn, becomes(29)Gx,t,p,A=Fx,t,pσx,tσxAσx,G:ℝn×0,+∞×ℝn\x,t,p:pσx=0×Sm⟶ℝ.Remark 1.
We easily show in a moment thatG preserves the assumptions (F1), (F2), and (F3); however, the singularities of G are not just at the origin but in the whole of the subset:(30)S=x,t,p,A∈ℝn×0,+∞×ℝm×Sm:pσx=0,where now for all x,t,A∈ℝn×0,+∞×Sm, the set p:x,t,p,A∈S is a varying subspace, not necessarily trivial if the family of vector fields X does not span ℝn at x. In this sense, the operator G is not covered by the standard theory of the anisotropic operators.
OperatorG is elliptic since if A≥B, then σtxAσx≥tσxBσx and thus Gx,t,p,A≤Gx,t,p,B.
OperatorG is also geometric since(31)Gx,t,λp,λA+μp⊗p=Fx,t,λpσx,λtσxAσx+μpσx⊗pσx=λFx,t,pσx,tσxAσx=λGx,t,p,A.
Thus, (F2) and (F3) hold true.
We now recall the usual definition of viscosity solution for level set equation (28).Definition 2.
An upper (respectively, lower) semicontinuous functionu:ℝn×0,+∞⟶ℝ is a viscosity subsolution (respectively, supersolution) of (28) if and only if for any ϕ∈C2ℝn×0,+∞, if x,t∈ℝn×0,+∞ is a local maximum (respectively, minimum) point for u−ϕ, we have(32)ϕtx,t+G∗x,t,∇ϕx,t,D2ϕx,t≤0,where G is given in (29). A viscosity solution of (28) is a continuous function u:ℝn×0,+∞⟶ℝ which is either a subsolution or a supersolution.Remark 2.
In the previous definition, the lower semicontinuous extension ofG at the singular points where pσx=0 is(33)G*x,t,p,A=limr⟶0+infGy,s,q,B:y,s,q,B∈domG,x−y,t−s,p−q,A−B≤r=limr⟶0+infFy,s,qσy,tσyBσy:qσy≠0,x−y,t−s,p−q,A−B≤r.
In particular, frompσx=0, we have(34)F∗x,t,0,tσxAσx≤G∗x,t,p,A≤G∗x,t,p,A≤F∗x,t,0,tσxAσx.
Thus, ifpσx=0 and σtxAσx=O, then G∗x,t,p,A=G∗x,t,p,A=0, so a counterpart of (F1) holds for G.
In Definition2, if Xϕx,t≠0, then (32) is equivalently written as(35)ϕtx,t+Fx,t,Xϕx,t,X2ϕx,t≤0,and the extended operator G* only appears when Xϕx,t=0. Therefore, at singular points, the notion of viscosity subsolution is stronger than one would require:(36)ϕtx,t+F∗x,t,Xϕx,t,X2ϕx,t≤0instead of (32). Note that in the special case (4), if pσx=0,(37)F*x,t,0,A=minp=1−trI−p⊗pA,and this is used in [8] or in [9] to define (weak-) subsolutions of the horizontal mean curvature flow equation, by requiring (36) instead of (32).


## 3. Viscosity Solutions

In this section, we consider equation (28) and prove an equivalent definition of viscosity solution. This result extends [3] to our setting and simplifies the treatment of singularities of equation (28) by restricting the family of test functions at characteristic points.When it will be necessary to emphasise the variablex in which we are computing the vector fields Xi (and with respect to computing the derivatives), we will denote the horizontal gradient and the horizontal Hessian matrix as Xx and Xx2 . For example if Hx,y is a C2 function defined in ℝn×ℝn and xo,yo is a generic point of ℝn×ℝn; we will denote with XxHxo,yo the horizontal gradient of H with respect to the variable x and with XyHxo,yo the horizontal gradient of H with respect to y, both computed at the point xo,yo. Analogous definitions hold for Xx2Hxo,yo and Xy2Hxo,yo. We consider an homogeneous (with respect to any dilatation δλ, λ>0) norm on ℝn:(38)xG=xh4+xv21/4,and we define a left invariant metric dG:ℝn×ℝn⟶0,+∞ as(39)dGx,y=x−1∘yG=yh−xh4+yv−xv−Bxh,yh21/4.Remark 3.
Here, we make some comments on definitions (38) and (39). Dealing with fully nonlinear partial differential equations with singularities poses a number of additional difficulties. Viscosity solutions theory can cope with these difficulties since the work of Evans and Spruck [14] and Chen et al. [2]. The horizontal mean curvature flow equation adds further difficulties since the singularity does not just appear when the gradient of the solution vanishes, but rather when the horizontal gradient vanishes, so when the gradient takes its values in a nontrivial subspace. In some key step of the proofs, the standard Euclidian distance does not work and one has to think something different. One natural choice would be to exchange the Euclidian distance with the Carnot–Caratheodory distance. This distance is not smooth, however being only locally Hölder continuous. Therefore, due to the nature of Carnot groups, one thinks of distance functions that are related to homogeneous norms, which are distance function equivalents to the Euclidian one but are smooth. One well-known example is the norm in (38). This one works well in step two groups, at least for the results we prove, but not for the comparison principle, one reason being that the group operation is not commutative and this makes the distance not symmetric. In groups of higher steps, one has a natural homogeneous distance with more terms, making the computations in this section more complex. Moreover, we are often using the structure (17), which is valid specifically in step two groups. There might be additional difficulties due to the fact that Carnot groups with steps higher than two differ in some important geometric properties. Nonetheless, step two groups already have important applications that make their study quite interesting as, for instance, in models of the visual cortex (see [11] and the references therein for details).
We start proving a nice property of the homogeneous metricdG defined in (39).Lemma 1.
PutNx=xG4 for any x∈ℝn. Then,(i)
x∈ℝn:XNx=0=x∈ℝn:X2Nx=O=x∈ℝn:xh=0(ii)
XxdG4x,y=XydG4x,y and Xx2dG4x,y=Xy2dG4x,y for any x,y∈ℝn; moreover, they all have as zero-set the set x,y∈ℝn×ℝn:xh=yhProof.
(i)
The proof of the first point follows by some simple computations. In fact, since
(40)
X
N
x
=
4
x
h
2
x
h
+
2
∑
k
=
1
n
−
m
x
v
k
B
k
x
h
,
and since the matricesBk are all skew symmetric, we have that,
(41)
X
N
x
2
=
16
x
h
6
+
4
∑
k
=
1
n
−
m
x
v
k
B
k
x
h
2
=
16
x
h
6
+
4
∑
k
,
l
=
1
n
−
m
x
v
k
x
v
l
B
k
x
h
,
B
l
x
h
.
Thus,XNx=0 if and only if xh=0. Moreover,
(42)
X
2
N
x
=
4
x
h
2
I
m
+
8
x
h
⊗
x
h
+
2
∑
k
=
1
n
−
m
B
k
x
h
⊗
B
k
x
h
,
which is null atxh=0.
(ii)
First of all we observe that, since the vector fieldsXi are invariant by left composition of the group operation, we have
(43)
X
y
d
G
4
x
,
y
=
X
y
N
x
−
1
∘
y
=
X
N
x
−
1
∘
y
,
(44)
X
y
2
d
G
4
x
,
y
=
X
2
N
x
−
1
∘
y
,
and so by point (i)XydG4x,y and Xy2dG4x,y are null if and only if x−1∘yh=0, i.e., yh=xh. To compute the horizontal gradient and the horizontal Hessian matrix with respect to the x variable, we observe that, since Nx−1=N−x=Nx, it holds dG4x,y=Nx−1∘y=Ny−1∘x and, by left invariance of the vector fields,
(45)
X
x
d
G
4
x
,
y
=
X
N
y
−
1
∘
x
,
X
x
2
d
G
4
x
,
y
=
X
2
N
y
−
1
∘
x
.
AgainXxdG4x,y and Xx2dG4x,y are null exactly when yh=xh.
Finally, we observe thatXydG4x,y2=XxdG4x,y2 and Xy2dG4x,y=Xx2dG4x,y.
We use the previous lemma to prove an equivalent definition of solution other than Definition2 which is the usual definition of viscosity solution for equation (28). The definition will only change at singular points of the differential operator.Theorem 1.
An upper (respectively, lower) semicontinuous functionu is a viscosity subsolution (respectively, supersolution) of (28) if and only if for any ϕ∈C2ℝn×0,+∞, if x,t∈ℝn×0,+∞ is a local maximum (respectively, minimum) point for u−ϕ, one has(46)∂ϕx,t∂t+Fx,t,Xϕx,t,X2ϕx,t≤0,ifXϕx,t≠0,(47)∂ϕx,t∂t≤0,ifXϕx,t=0,X2ϕx,t=0,respectively,(48)∂ϕx,t∂t+Fx,t,Xϕx,t,X2ϕx,t≥0,ifXϕx,t≠0,∂ϕx,t∂t≥0,ifXϕx,t=0,X2ϕx,t=0.Proof.
We only show the result for subsolutions, the other part being similar. It is clear that a viscosity subsolution will satisfy (47) since G*x,t,p,O=0 if pσx=0 by Remark 2 and (F1).
Letu be an upper semicontinuous function which satisfies (46) and (47). Consider ϕ∈C2ℝn×0,+∞ and x^,t^∈ℝn×0,+∞ a local maximum point for u−ϕ such that Xϕx^,t^=0 and X2ϕx^,t^≠0. Without loss of generality, we can assume that u is a strict local maximum point for u−ϕ. We need to prove that(49)∂ϕx^,t^∂t+G∗x,t,∇ϕx^,t^,D2ϕx^,t^≤0.
For anyε>0, we consider the function(50)ψεx,y,t=ux,t−dG4x,yε−ϕy,t.
By standard arguments, one proves that forε being sufficiently small, there is a family of local maxima xε,yε,tε of ψε such that xε,yε,tε converges to x^,x^,t^. Indeed, if xε,yε,tε are the maximum points of ψε in a small compact neighborhood of x^,x^,t^, xε,yε,tε will converge to some x¯,y¯,t¯ (passing to a subsequence if necessary). One first uses ψxε,yε,tε≥ψx^,x^,t^ to show that x¯=y¯ and next by taking the limit that x¯,t¯ is a maximum of u−ϕ in the neighborhood so that x¯,t¯=x^,t^.
Moreover, since the functiony⟼ψεxε,y,tε has a local maximum in yε, we have(51)∇ϕyε,tε=−DydG4xε,yεε,D2ϕyε,tε≥−Dy2dG4xε,yεε.
Thus,(52)Xϕyε,tε=−XydG4xε,yεε,X2ϕyε,tε≥−Xy2dG4xε,yεε.
Two cases may now occur:(1)
Xϕyε,tε=0 along a subsequence. This means that XydG4xε,yε=0 and by Lemma 1, xεh=yεh. Since the map x,t⟼ux,t−φx,t with φx,t=dG4x,yε/ε+ϕyε,t attains a maximum at xε,tε and(53)Xφx,t=0⟺xεh=yεh⟺X2φx,t=0,by (47), we get(54)∂φ∂txε,tε=∂tϕyε,tε≤0.For future reference, we remark that the test functionφ satisfies in a neighborhood of x^,t^: Xφ=0 implies X2φ=0. We proceed and by (52) and xεh=yεh, we get that X2ϕyε,tε≥O. Using the ellipticity of F and Remark 2, it holds(55)∂tϕyε,tε+G∗yε,tε,∇ϕyε,tε,D2ϕyε,tε≤∂tϕyε,tε+F∗yε,tε,Xϕyε,tε,X2ϕyε,tε≤∂tϕyε,tε+F∗yε,tε,0,Om×m=∂tϕyε,tε≤0,and we conclude by lettingε go to 0.(2)
Xϕyε,tε≠0 for all ε being sufficiently small. Using (52) and the previous lemma, this means yεh≠xεh. Moreover, the point xε,tε is a maximum for(56)x,t⟼ψεx,x∘xε−1∘yε,t=ux,t−dG4xε,yεε−ϕx∘xε−1∘yε,t≔ux,t−φx,t,sincedG4x,x∘xε−1∘yε=Nxε−1∘yε=dG4xε,yε. Let τ˜αx=x∘α be the right translation by α and Dτ˜αx≡Dτ˜α its Jacobian matrix. A simple computation shows that Dτ˜α has the form(57)Dτ˜α=ImOm×nttB1αh⋮InttBn−mαh=ImOm×nt−B1αh⋮Int−Bn−mαh=ImOm×n−tBαhIn.
By the chain rule, we get(58)Xφxε,tε=tσxεtDτ˜xε−1∘yε∇ϕτ˜xε−1∘yεxε,tε=tDτ˜xε−1∘yεσxε∇ϕyε,tε=tσ2xε−yε∇ϕyε,tε⟶Xϕx^,t^=0,asε⟶0,since xεh−xε−1∘yεh=xε∘yε−1∘xεh=2xε−yεh and(59)X2φxε,tε=tσxεtDτ˜xε−1∘yεD2ϕτ˜xε−1∘yεxε,tεDτ˜xε−1∘yεσxε=tσ2xε−yεD2ϕyε,tεσ2xε−yε⟶X2ϕx^,t^≠0,asε⟶0.
Moreover, we show thatXφxε,tε≠0. In fact, as uxε,t−1/εdG4xε,y−ϕy,t has a maximum at y,t=yε,tε,(60)Xφxε,tε=tσ2xε−yε∇ϕyε,tε=−ε−1tσ2xε−yε∇ydG4xε,yε=−ε−1tσ2xε−yεtDτxε−1∇Nxε−1∘yε=−ε−1tσxε−yε∇Nxε−1∘yε=ε−1tσxε−yε∇Nyε−1∘xε=ε−1XNyε−1∘xε.
By Lemma1, this is null if and only if yεh=xεh, and we already know that this cannot be true. Thus, by (32), it holds(61)∂φ∂txε,tε+Gxε,tε,∇φxε,tε,D2φxε,tε≤0,and we conclude by letting ε⟶0:(62)0≥liminfε⟶0∂φ∂txε,tε+Gxε,tε,∇φxε,tε,D2φxε,tε≥∂tϕx^,t^+G∗x^,t^,∇ϕx^,t^,D2ϕx^,t^.Remark 4.
By a remark during the previous proof, it is not restrictive to assume in Definition2 that if u (respectively, v) is an upper semicontinuous subsolution (respectively, a lower semicontinuous supersolution) of equation (28) and φ∈C2ℝn×0,+∞ is a test function for u (resp., for v) at the point x,t, then at any point y,s in a neighborhood of x,t such that(63)Xφy,s=Xφx,t=0,it holds(64)X2φy,s=0.
Complementing Theorem1 and Remark 2, we obtain the following consequence. It shows, in particular, that the notion of solution for the horizontal mean curvature flow equation used in [8] or in [9], which is different from viscosity solutions at characteristic points, is in fact equivalent to standard viscosity solutions and ours.Corollary 1.
Letu:ℝn×0,+∞⟶ℝ be an upper (respectively, lower) semicontinuous function. Function u is a viscosity subsolution (resp., supersolution) of (28) if and only if for any ϕ∈C2ℝn×0,+∞, if x,t∈ℝn×0,+∞ is a local maximum (respectively, minimum) point for u−ϕ, one has(65)∂tϕx,t+F∗x,t,Xϕx,t,X2ϕx,t≤0,respectively,(66)∂tϕx,t+F∗x,t,Xϕx,t,X2ϕx,t≥0.Proof.
Suppose thatx,t∈ℝn×0,+∞ is a local maximum point for u−ϕ. If ∇ux,t≠0, then Gx,t,∇ux,t,D2ux,t=Fx,t,Xux,t,X2x,t so there is nothing to prove. We therefore limit ourselves to discuss the case Xux,t=0.
Ifu is a viscosity subsolution, by Remark 2, we know that F*≤G*; therefore, (65) is satisfied.
If instead we suppose that (65) holds true, then by Theorem 1, we limit ourselves to test functions ϕ that satisfy the following: Xϕx,t=0 implies X2ϕx,t=O. In this case F*x,t,Xϕx,t,X2ϕx,t=0 and then ∂tϕx,t≤0. Thus, by Theorem 1, we know that u is a viscosity subsolution.Remark 5.
In [11], the authors require a subsolution u of the horizontal mean curvature flow equation to satisfy(67)∂tϕx,t−TrX2ϕx,t≤0,if u−ϕ has a maximum at x,t and Xϕx,t=0. If in particular ϕ is in the class of test functions such that Xϕx,t=0 implies X2ϕx,t=0, then ∂tϕx,t≤0. Therefore, u is a subsolution in the sense of Theorem 1, and then, it is a viscosity subsolution of (28).


## 4. Examples of Explicit Supersolutions or Subsolutions

In this section, we present examples of supersolutions and subsolutions of the geometric equation in the case of the horizontal mean curvature flow equation (mcfe) whenF is given in (4). From Theorem 1, we see that when we deal with functions with separated variables like ux,t=ϕt+Ux, it is easy to check the (mcfe) at singular points of the operator. If u−φ has a maximum/minimum at xo,to and Xφxo,to=0, then we only need to look at the sign of φtxo,to provided suitable test functions exist, i.e., X2φxo,to=O, otherwise we have nothing to check. We start with a general result in step two Carnot groups, based on the definition of convex functions in the group. The definition of v−convex function (as in viscosity-convex) is given by Bardi and Dragoni [19], where it is discussed and characterised and the reader can find explicit examples.Definition 3.
A continuous functionU:ℝn⟶ℝ is v−convex in the Carnot group if there is α≥0, and for all test functions, ϕ∈C2 such that U−ϕ has a maximum at xo, then X2ϕxo,to≥αI. If α>0, we say that U is strictly v-convex.
The idea is to build supersolutions of the (mcfe) from av−convex function.Proposition 1.
In a Carnot group of Step 2, letU∈CΩ be continuous and a strictly v−convex function. Then for c,r∈ℝ, the function ux,t=ct−Ux+r is a supersolution of (mcfe) for all c≥−m−1α, r∈ℝ. Suppose that U is nonnegative. Then, if c=−m−1α and r>0, the initial front x:ux,0=0=x:Ux=r becomes extinct before time t¯=r/m−1α.Proof.
In order to check the supersolution condition, we use the alternative definition as in Theorem1. Let φ∈C2ℝn×0,+∞ be such that u−φ has a minimum at xo,to. Since U−−φ⋅,to has a maximum at xo and U is strictly v−convex, then −X2φxo,to≥αI for some α>0. Therefore, it cannot be Xφxo,to=0 if φ is an appropriate test function, and then,(68)∂tφxo,t0−trI−Xφxo,toXφxo,to⊗Xφxo,toXφxo,toX2φxo,to≥c+m−1α≥0,provided c≥−m−1α.
The zero sublevel set of the supersolutionu becomes a barrier if a comparison principle holds. At time t, if c=−m−1α, it is given by x:ux,t≥0=x:Ux≤−m−1αt+r and becomes empty if t>r/m−1α.
In the previous proposition, the front may have characteristic points, as we see in some more explicit examples in the following.
To simplify, we now specialise Heisenberg-like groups. Building supersolutions seems to be easier than subsolutions in particular if characteristic points are present. Below, we consider as reference spaceℝn=ℝm×ℝ∋x=xh,xv, m≥2, and suppose that σxh,xv=tIm,Bxh, where Bt=−B=B−1 is an m×m matrix. Notice that Bxh⋅xh=0, B2=−Im, and Bxh=xh.Example 2.
In the first example, we avoid characteristic points. Forc,r∈ℝ, consider the family of functions wx,t=ct−xh2+r. We easily get that(69)Xwx,t=−2xh,X2wx,t=−2Im×m.
In particular,xh2 is strictly v−convex; we can compute exactly the operator(70)wtx,t−trX2wx,t−X2wXw⊗Xwx,tXwx,t2=c+2m−1,and thus by Theorem 1 and Proposition 1, w is a supersolution for c≥−2m−1 and a subsolution for c≤−2m−1 in ℝn×0,+∞, so w is a viscosity solution, for c=−2m−1. Note that for r>0, the zero level set of w is a cylinder with axis x:xh=0 and it goes extinct at time t=r/2m−1.
In general, it is not as easy to find explicit solutions.Example 3.
We consider a function built on the gauge function of the Heisenberg group, namely, a variation of the homogeneous norm:(71)ux,t=ct−Gxh,xv+r,whereGxh,xv=xh4+4xv2and candr are constants to be decided later. Note that the zero level set of u is (we will always regard r>0 for convenience)(72)x,t:ux,t=0=x,t:Gx,t=r+ct.Therefore, it is the boundary of a ball for the distance G1/4 centred at the origin. It has characteristic points, namely, points where XGx,t=0 precisely in its intersection with the axis xh=0, as we readily see in the following. We can easily compute (here, we will do complete calculations and not only the signature of X2G because we also want to check the subsolution condition)(73)XGx,t=σxt∇Gx,t=4xh2xh+8xvtBxh,XGx,t2=16xh2Gx,t,X2Gx,t=tσD2Gσx=Im×m,Bxh8xh⊗xh+4xh2Im×m008tIm×m,Bxh=8xh⊗xh+4xh2Im×m+8Bxh⊗Bxh≥0.
Therefore,G is v-convex but not strictly v-convex. Finally,(74)XG⋅X2Gx,tXGx,t=48xh4xh+96xvxh2Bxh⋅XGx,t=192xh4Gx,t.and since u is smooth, we conclude that, for xh≠0,(75)utx,t−trX2ux,t−X2uXu⊗Xux,tXux,t2=c+trX2Gx,t−X2GXG⊗XGx,tXGx,t2=c+8+4m+8xh2−12xh2=c+4nxh2.
Now, we can use our alternative definition to obtain that the viscosity supersolution/subsolution condition is satisfied also at points where the horizontal gradient vanishes. We conclude that(i)
Forc≥0, u is a global supersolution in ℝn×ℝ+, since ut≥0(ii)
Forc<0, u is a subsolution but only in the cylinder x:xh<−c/4n around the axis xh=0
Compared to the previous example, nowu may itself be a test function and then we cannot fulfill a supersolution or subsolution condition just by the lack of test functions.
Note that the level sets ofu in the supersolution case, which are propagating (super) fronts, have radius nondecreasing in time and it may even be stationary for c=0. Instead, the radius decreases in time in the subsolution case where however the diameter of the section of the domain of the subsolution vanishes with c. The zero level set at time t=0 is contained in the cylinder in (ii) provided −c>4nr, and it goes extinct at time t=−r/c.Example 4.
Similar calculations of the previous example can be made forwx,t=ct−x2+r. We get(76)Xwx,t=−2xh+2xvBxh,X2wx,t=−2Im×m,Xwx,t2=−4xh21+xv2,and therefore,(77)wtx,t−trX2wx,t−X2wXw⊗Xwx,tXwx,t2=c+2m−1+2xh21+xv2.
Again by Theorem1, Proposition 1, and since x2 is strictly v-convex, w is a supersolution in ℝn×0,+∞, for c≥−2m−1, and a subsolution for c<−2m−1 in the open sets x∈ℝn:xh2<ε1+xv2 if ε is sufficiently small.
We now construct a modification of the second example to build a global subsolution of the mean curvature flow equation whose level sets have characteristic points. We first prove a lemma on change of variables for the horizontal mean curvature operator.Lemma 2.
LetU∈C2ℝn and ψ:ℝ⟶ℝ be smooth with ψ′>0. Then, for W=ψU, if XU≠0, we have(78)−trX2W−X2WXW⊗XWxXWx2=−ψ′UtrX2U−X2UXU⊗XUxXUx2.Proof.
It is just a matter of computing terms. We obtain(79)∇W=ψ′U∇U,XWx=ψ′UXUx,XWx2=ψ′U2XUx2,D2Wx=ψ″U∇U⊗∇Ux+ψ′UD2Ux,X2Wx=ψ″UXU⊗XUx+ψ′UX2Ux,trX2Wx=ψ″UXU2+ψ′UtrX2Ux,X2WXW⋅XWx=ψ′U2ψ″UXUx4+ψ′UX2UXU⋅XUx.
Finally, putting things together the first two terms in the previous equations cancel out.Example 5.
In this example, we consider the function(80)vx,t=ct−Gxh,xv1/2+r,which is now not differentiable at points 0,t∈ℝn×0,+∞. However, v is locally Lipschitz continuous and is differentiable in the group of variables xh. Moreover, there is no smooth test function such that v−ϕ has a local minimum at 0,t, and if v−ϕ has a local maximum at 0,t, then ϕt0,t=c and ∇xhϕ0,t=0 so that Xϕ0,t=0 since σ0=Im×m,0. Therefore, to check the mean curvature flow equation at such points, we only need to look at the sign of c by Theorem 1.
We now proceed at points such thatxh≠0. We use the lemma with ψs=s1/2, so that ψ′s=1/2ψs, and the calculations of the previous example. Again, the zero level sets of v are(81)x,t:vx,t=0=x,t:Gx,t=r+ct2,and we check the equation at noncharacteristic points. We obtain, by the lemma,(82)vtx,t−trX2vx,t−X2vXv⊗Xvx,tXvx,t2=c+12Gx,t1/24nxh2.
We conclude thatv is a supersolution for c≥0 as before, but now, since xh2≤Gx,t1/2, v becomes a global subsolution for c≤−2n. If c=−2n, the extinction time of the zero level set of the subsolution is t=r/2n. Finally, note that all functions of the family share the same initial condition at time t=0 independently of c.


## 5. A Geometric Definition of Generalised Flow in Carnot Groups

In this section, we extend the definition of generalised superflows and subflows introduced by Barles and Souganidis [4], later revisited by Barles and Lio [5], to the setting of level set equations in Carnot groups, also in view of the ideas described in Section 3. This more geometric definition turns out to determine uniquely the geometric flow of a hypersurface if the usual level set equation determines a unique evolution with empty interior (no fattening). This definition has been proven to be much more efficient when dealing with singularly perturbed problems that give rise to geometric flows and we will use it in [6] to extend to the Carnot group setting the classical Allen–Cahn approach. In the following definition, we follow [5] with one modification (see Remark 6).Definition 4.
LetF:ℝn×0,+∞×ℝm\0×Sm be locally bounded and satisfy (F1), (F2), and (F3), and let G be defined as in (29). A family Ωtt∈0,T (resp., ℱtt∈0,T) of open (resp., close) subsets of ℝn is called a generalized superflow (resp., subflow) with normal velocity −F if, for any x0∈ℝn, t∈0,T, r>0 and h>0 so that t+h<T and for any smooth function, ϕ:Bx0,r×t,t+h⟶ℝ such that(i)
∂tϕx,s+G∗x,t,∇ϕx,s,D2ϕx,s<0 in Bx0,r×t,t+h (resp., ∂tϕx,s+G∗x,t,∇ϕx,s,D2ϕx,s>0 in Bx0,r×t,t+h)(ii)
For anys∈t,t+h, x∈Bx0,r:ϕx,s=0≠∅ and ∇ϕx,s≠0onx,s∈Bx0,r×t,t+h:ϕx,s=0(iii)
If there exists a pairx,s∈Bx0,r×t,t+h so that Xϕx,s=0, then it also holds X2ϕx,s=0(iv)
x∈Bx0,r:ϕx,t≥0⊂Ωt (resp., x∈Bx0,r:ϕx,t≤0⊂ℱtc)(v)
For alls∈t,t+h, x∈∂Bx0,r:ϕx,s≥0⊂Ωs (resp. x∈∂Bx0,r:ϕx,s≤0⊂ℱsc)Then, we have(83)x∈Bx0,r:ϕx,s>0⊂Ωs,resp.x∈Bx0,r:ϕx,s<0⊂ℱsc,for every s∈t,t+h.
A familyΩtt∈0,T of open subsets of ℝn is called a generalized flow with normal velocity −F if Ωtt∈0,T is a superflow and Ω¯tt∈0,T is a subflow.Remark 6.
The previous definition focuses on evolution of sets directly instead of looking at the level sets of the solutions of a differential equation. It does this by assuming local comparison with smooth evolutions. Indeed, checking if a collection of open sets provides a superflow (i) requires the smooth functionϕ to be a local strict subsolution, (ii) assumes that the zero level set of ϕ is smooth, (iv)-(v) requires compatible initial and boundary conditions in the local cylinder between the family of sets and the smooth evolution. Condition (iii) is new and we add it to restrict the family of test functions in view of what we did in Section 3. As we will see from the proof of the characterisation of Theorem 2 in the following, in view of our Theorem 1, the condition (iii) can be present or not and the corresponding definition would be equivalent.
It follows immediately by Definition4 that a family Ωtt∈0,T of open subsets of ℝn is a generalised superflow with normal velocity −F if and only if Ωtct∈0,T is a generalised subflow with normal velocity F.
We now state and prove the following result which describes the connection between generalised flows and solutions of (28).Theorem 2.
(i)
LetΩtt∈0,T be a family of open subsets of ℝn such that the set Ω≔∪t∈0,TΩt×t is open in ℝn×0,T. Then, Ωtt∈0,T is a generalised superflow with normal velocity −F if and only if the function χ=1Ω−1Ωc is a viscosity supersolution of (28).
(ii)
Letℱtt∈0,T be a family of closed subsets of ℝn such that the set ℱ≔∪t∈0,Tℱt×t is closed in ℝn×0,T. Then, ℱtt∈0,T is a generalised subflow with normal velocity −F if and only if the function χ¯=1ℱ−1ℱc is a viscosity subsolution of (28).Proof.
We adapt to our situation some of the ideas in [5] and only consider (i) as the other case is similar. We first assume that χ=1Ω−1Ωc is a supersolution of (28), and we show that Ωtt∈0,T is a generalised superflow. To do this, we consider a smooth function ϕ, a point x0,t∈ℝn×0,T, and r,h>0 satisfying conditions (i)–(v) in Definition 4. We assume that ϕ≤1 in Bx0,r×t,t+h (otherwise, we change ϕ with ηϕ for η>0 small enough and we use the homogeneity of F). We consider(84)m≔minχx,s−ϕx,s:x,s∈Bx0,r×t,t+h.
Sinceϕ satisfies condition (i), χ is a supersolution of equation (28) in Bx0,r×t,t+h and it is well known (see, e.g. [20]) that χ is therefore also a supersolution in Bx0,r×t,t+h; we deduce that the minimum m has to be attained either in ∂Bx0,r or at time t.
Letx,s∈∂Bx0,r×t,t+h∪Bx0,r×t. If x∈Ωs, then χx,s=1 and χ−ϕx,s≥0 because ϕ≤1 in Bx0,r×t,t+h. If instead x∉Ωs, then χx,s=−1 and, by (iv) and (v), χ−ϕx,s≥−1+δ for some δ>0. In any case, we can conclude that(85)χy,s−ϕy,s≥−1+δ,y,s∈Bx0,r×t,t+h.In particular, ϕy,s≤−δ, if y∉Ωs. This means that for every s∈t,t+h,(86)y∈Bx0,r:ϕy,s≥0∩Ωsc=∅,which implies that Ωtt∈0,T is a generalised superflow with normal velocity −F.
Conversely, we assume thatΩtt∈0,T is a generalised superflow and we show that χ is a supersolution of equation (28) in ℝn×0,T. We consider a point x,t∈ℝn×0,T and a function ϕ∈C∞ℝn×0,T so that x,t is a strict local minimum point of χ−ϕ and by adding a constant to ϕ if necessary, we may assume ϕx,t=0. We want to show that(87)∂tϕx,t+G∗x,t,∇ϕx,t,D2ϕx,t≥0.
By using the equivalent definition of viscosity solution with a restricted family of test functions, we will suppose thatX2ϕy,s=0 whenever Xϕy,s=0.
Whenx,t is in the interior of either χ=1 or χ=−1, then χ is constant in a neighborhood of x,t and therefore ∂tϕx,t=0, ∇ϕx,t=0, and D2ϕx,t≤0. Since F satisfies (F1) and (F2), then the inequality in (87) is true. Assume instead that x,t∈∂χ=1∩∂χ=−1. Thus, by the lower semicontinuity of χ, χx,t=−1. We suppose by contradiction that there exists an α>0 so that we have(88)∂tϕx,t+G∗x,t,∇ϕx,t,D2ϕx,t<−α.
We can findr,h>0 such that for all y,s∈Bx,r×t−h,t+h:(89)∂tϕy,s+G*y,s,∇ϕy,s,D2ϕy,s<−α2,(90)χx,t−ϕx,t=−1<χy,s−ϕy,s,y,s≠x,t.
We consider first the case∇ϕx,t≠0, and by choosing smaller r and h, we assume that ∇ϕ≠0 in Bx,r×t−h,t+h. We introduce the test function ϕδy,s≔ϕy,s+δs−t−h, for 0<δ≪1. Since ϕx,t=0 and ∇ϕx,t≠0, it is easy to see that if h and δ are small enough then, for any t−h≤s≤t+h, the set y∈Bx,r:ϕδy,s=0 is not empty. We observe that, for δ>0 small enough, by (89) and (90), we have(91)ϕδy,s−1<χy,s,for all y,s∈Bx,r×t−h∪∂Bx,r×t−h,t+h, and(92)∂tϕδy,s+G∗y,s,∇ϕδy,s,D2ϕδy,s<−α4,for all y,s∈Bx,r×t−h,t+h. Inequality (91) implies that(93)y∈Bx,r:ϕδy,t−h≥0⊂Ωt−h,y∈∂Bx,r:ϕδy,s≥0⊂Ωs,for all s∈t−h,t+h. Therefore, ϕδ satisfies (i), (ii), (iv), and (v) in Definition 4. Assumption (iii) holds as well by assumptions on function ϕ. The definition of superflow then yields(94)y∈Bx,r:ϕδy,s>0⊂Ωs,for every s∈t−h,t+h. Since ϕδx,t=δh>0, we deduce that x∈Ωt, and this is a contradiction with x,t∈∂χ=−1.
Now, we turn to the case when∇ϕx,t=0. In particular, Xϕx,t=0 and X2ϕx,t=O by our assumption, and therefore, to prove (87), it is then enough to show that(95)∂tϕx,t≥0.
We further observe that by the result in [3] corresponding to our Theorem 1, we could have restricted ϕ to the class of functions such that ∇ϕx,t=0 implies(96)∂2ϕ∂xi∂xjx,t=∂3ϕ∂xi∂xj∂xkx,t=∂4ϕ∂xi∂xj∂xk∂xlx,t=0,for any i,j,k,l∈1,…,n as we do now. Suppose by contradiction that a≔∂tϕx,t<0. Therefore, by Taylor formula(97)ϕy,s=∂tϕx,ts−t+os−t+y−x4,ass⟶t,y−x⟶0.
Thus, for allε>0, there exist r=rε,h=hε,andh′=hε′>0 such that(98)h′≤h,h<−εr4a,and, for any y,s∈Bx,r×t−h,t+h′,(99)ϕy,s≥as−t+a2s−t−εy−x4=a2s−t+as−t+−εy−x4≥a2s−t−εy−x4+ah′.
LetdGx,y=x−1∘yG be the distance function defined in (39). For any compact set K⊂ℝn, by known results (see, e.g., Proposition 5.15.1 in [18]), there exists a positive constant CK>0 so that(100)x−yCK≤dGx,y≤CKx−y1/2,for any x,y∈K. Thus, if we put Cr=CBx,r4, we get(101)x−y4Cr≤Nx−1∘y≤Crx−y2,and by definition of N,(102)ϕy,s≥a2s−t−εCrNx−1∘y+ah′,for any y,s∈Bx,r×t−h,t+h′. By (90), we can take β>0 such that(103)2β+ϕy,s−1<χy,s,for all y,s∈Bx,r×t−h∪∂Bx,r×t−h,t+h′. By taking β smaller, we may also suppose β<εr4/2. We now proceed similarly as before and consider the function ψβy,s=a/2s−t−εCrNx−1∘y+β. Since we can take h′ smaller, we assume from now on that h′≤−β/a. Combining the last two displayed inequalities and the assumptions on β,h,h′, and r, we get(104)ψβy,s−1<χy,s,for all y,s∈Bx,r×t−h∪∂Bx,r×t−h,t+h′. Thus, with a reasoning similar to the one that we used in the previous case, it is possible to prove that ψβ satisfies conditions (iv) and (v) in Definition 4. Furthermore, we consider a fixed s∈t−h,t+h′. We have ψβx,s=as−t/2+β≥ah′/2+β>0 while for y−x=r,(105)ψβy,s=a2s−t−εCrdGx,y4+β≤a2s−t−εy−x4+β≤−ah2−εr4+β≤−ah+εr42≤0.
Thus, the sety∈Bx,r:ψβy,s=0 is not empty. Let y∈Bx,r; we compute(106)∇ψβy,s=−εCr4yh−xh2yh−xh−2∑i=1n−mym+i−xm+i−Bixh,yhBixh2yv−xv−Bxh,yh..
Thus, since the matricesBi are skew-symmetric, ∇ψβy,s=0 if and only if y=x and therefore ∇ψβy,s≠0 for every y,s∈Bx,r×t−h,t+h′:ψβy,s=0. This proves that ψβ satisfies (ii) in Definition 4. Moreover, it also satisfies (iii) since, by Lemma 1,(107)Xψβy,s=0⟺yh=xh⟺X2ψβy,s=0.
It remains to prove that (i) holds. SinceG* is upper semicontinuous, G*y,s,0,O=0 and G is geometric, and we have that(108)∂tψβy,s+G*y,s,∇ψβy,s,D2ψβy,s=a2+G*y,s,−εCr∇yNx−1∘y,−εCrDyy2Nx−1∘y<0,for y,s∈Bx,r×t−h,t+h′ and ε is small enough.
Thus, sinceΩtt∈0,T is a generalised superflow, we have(109)y∈Bx,r:ψβy,s>0⊂Ωs,for any s∈t−h,t+h′. But again, ψβx,t=β>0, and this means x∈Ωt, which is a contradiction.Remark 7.
When we define generalised flows as in Definition4, then Theorem 2 provides a discontinuous solution of (28). The discontinuous solution χ bears a natural initial condition at t=0 in the following way. Since χ is lower semicontinuous, we can extend it at t=0 by lower semicontinuity and then define a lower semicontinuous initial condition as(110)χox=χ∗x,0.
In order to better understand the nature of Definition4, we comment briefly on the previous result by recalling the connection between the discontinuous solution of (28) that appears in Theorem 2 and usual viscosity solutions of (28) (see, e.g., Souganidis [16] and the references therein). Suppose that u∈Cℝn×0,+∞ is a viscosity solution of (28). Then, we can define the following family of sets, for t≥0,(111)Γt=x,t:ux,t=0,Dt+=x,t:ux,t>0,Dt−=x,t:ux,t<0.
The following result is well known in the theory and contains as a consequence of Theorem2 an existence result for geometric flows. The second part of the statement is based on the validity of a comparison principle, which at the moment for equation (28) is valid under some restrictions as we discussed in the introduction.Theorem 3.
Suppose thatu∈Cℝn×0,+∞ is a viscosity solution of (28). With the notation in (111), the two functions χ¯x,t=1Dt+∪Γtx−1Dt−x and χ¯x,t=1Dt+x−1Dt−∪Γtx are viscosity solutions of (28) associated, respectively, with the discontinuous initial data:(112)w¯o=1Do+∪Γo−1Do−,w¯o=1Do+−1Do−∪Γo,respectively. In particular, the family of sets Dt+t>0 and Dt+∪Γtt>0o are generalised flows and they coincide if and only if the no-interior condition holds: Γt=∂Dt+=∂Dt−, for all t≥0.
If moreoverΓo has an empty interior and a comparison principle holds for equation (28), then χ¯andχ¯ are, respectively, the maximal subsolution and the minimal supersolution of the Cauchy problem coupling (28) with the initial condition wo=1Do+−1Do− and it has a unique discontinuous solution if and only if the no-interior condition holds. The unique solution is given by the following function:(113)χx,t=1Dt+x−1Dt−x.



---

*Source: 7545983-2020-01-08.xml*