Research Article | Open Access
On Viscosity and Equivalent Notions of Solutions for Anisotropic Geometric Equations
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.
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
Here, the operator , is elliptic and geometric, meaning that it is positively homogeneous in the pair and invariant in the last argument with respect to matrices of the form , , as we make it more precise later. The notation indicates the set of symmetric matrices, . Therefore, it is possible that F has a singularity at , and we assume that it behaves nicely, namely,where the stars above indicate the lower and upper semicontinuous envelopes, respectively. The notation indicates the horizontal gradient with respect to a family of vector fields , seen as differential operators:generators of a step two Carnot group. In particular, for a smooth function u, , , and if , equation (1) has singularities when , i.e., at characteristic points of the level set of u and therefore on a subspace of positive dimension. Notation indicates instead the horizontal hessian, namely, , the symmetrised matrix of second derivatives. This compares to the usual Euclidian case when is the identity matrix, where is the standard gradient and the singularity is just at the origin. In the special case when the operator is defined asand moreover and , (1) reads as the well-known mean curvature flow equation:
In a group setting instead, (5) becomeswhich 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.  and Chen et al. . 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  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  and applied in several situations in the Euclidian setting (see also Barles and Lio ). 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 , 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 , where n is the exterior normal. Indeed, if is a family of open sets, is the propagating front, and there exists a smooth function such thatone computesTherefore, u formally satisfieswhere G is related to V by
In our case, the anisotropy of the velocity will be for instance exploited by the fact thatso that as an operator . The novelty here with respect to the classical cases is that while in the Euclidian case and its square is a nondegenerate matrix, here the diffusion matrix is not only anisotropic but also degenerate. When the family of vector fields does not span the whole 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 if u solves (1) and is smooth and increasing, then also solves the same equation. As a consequence, when a comparison principle holds true, it is easy to see that if and are two initial conditions such thatand are the corresponding solutions in (1), then one has
One can therefore define the family of closed sets to be the geometric flow of the front or interface with the prescribed normal velocity.
The notion of horizontal normal and horizontal mean curvature is due to Danielli et al. . Recently, equation (1) has been studied by several authors. Existence results are available in the work of Capogna and Citti , who proved existence in Carnot groups by vanishing viscosity riemannian approximations. Dirr et al.  used stochastic approximations to show existence for more general Hörmander structures. Capogna et al.  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  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  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.  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  for numerical computations of geometric flows. The rigorous theory of weak front evolution started with the work by Evans and Spruck  for the mean curvature flow and by Chen et al.  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 , where the approach is discussed in detail (see also Souganidis  and the references therein for the main applications of the theory, and  for equations with discontinuous coefficients).
2. Step Two Carnot Groups and Level Set Equations on the Group
In this paper, we consider in a family of vector fields written as differential operators as in (3) and consider which is the matrix valued family of the coefficients. We will indicate the th column of σ so thatwhere is the identity map in and in general applied to a vector valued smooth function φ means the vector whose entries are given by 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. , which we refer the reader to, for an introduction to the subject).
Definition 1. We say that is a Lie group if is a group operation on and the map is smooth.
We then say that is a step two Carnot group if we can split , , , and for all , the family of dilations are automorphisms of the group (the group is homogeneous). Moreover, the family of vector fields is left invariant on G with respect to the group operation, that is, for all and all , we have thatwhere is the left translation, and the following Hörmander property is satisfied:so that the family of vector fields , together with their first order Lie brackets, generates at every point (the Carnot group is step two).
The vector fields of the family are said to be generators of the Carnot group.
Following , it is then well known that if generates a step two Carnot group, then, by a suitable change of variables, we can suppose thatwhere is the identity matrix, , and , , are skew symmetric, linearly independent, matrices. In addition, has the group structure with the following operation:with the notation . With this group operation, it is clear that and 0 is the identity element of the group.
Moreover, we notice that the Jacobian of the left translation has the following structure:so the first m columns of the Jacobian give the matrix . It is also good to remember that for , the family is homogeneous of degree one with respect to the dilations, namely,for all .
Example 1. The well-known example of the Heisenberg group comes from and the single matrixFor our purposes, given a smooth function , we indicate the horizontal gradient (here, gradients are row vectors) asand the horizontal hessian asWe just observe that indicates the symmetrisation and that the first-order terms in the second derivatives of cancel out by direct computation since σ only depends on the first m variables.
In , 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 be a continuous function, locally bounded at points of the form , where denotes the space of the symmetric matrices. We assume on F the following structure conditions: (F1) F satisfies (F2) F is elliptic, i.e., for any and : (F3) F is geometric, i.e., for every and .In the above equations, we are using the following notation for the lower semicontinuous extension of F at the singular points:and similarly for the upper semicontinuous extension . Note in particular that the geometric property of F implies for all .
We want to discuss the notion of solution for the equationwhere 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 , becomes
Remark 1. We easily show in a moment that G 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:where now for all , the set is a varying subspace, not necessarily trivial if the family of vector fields does not span at x. In this sense, the operator G is not covered by the standard theory of the anisotropic operators.
Operator G is elliptic since if , then and thus .
Operator G is also geometric sinceThus, (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 function is a viscosity subsolution (respectively, supersolution) of (28) if and only if for any , if is a local maximum (respectively, minimum) point for , we havewhere G is given in (29). A viscosity solution of (28) is a continuous function which is either a subsolution or a supersolution.
Remark 2. In the previous definition, the lower semicontinuous extension of G at the singular points where isIn particular, from , we haveThus, if and , then , so a counterpart of (F1) holds for G.
In Definition 2, if , then (32) is equivalently written asand the extended operator only appears when . Therefore, at singular points, the notion of viscosity subsolution is stronger than one would require:instead of (32). Note that in the special case (4), if ,and this is used in  or in  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  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 variable x in which we are computing the vector fields (and with respect to computing the derivatives), we will denote the horizontal gradient and the horizontal Hessian matrix as and . For example if is a function defined in and is a generic point of ; we will denote with the horizontal gradient of H with respect to the variable x and with the horizontal gradient of H with respect to y, both computed at the point . Analogous definitions hold for and . We consider an homogeneous (with respect to any dilatation , ) norm on :and we define a left invariant metric as
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  and Chen et al. . 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  and the references therein for details).
We start proving a nice property of the homogeneous metric defined in (39).
Lemma 1. Put for any . Then,(i)(ii) and for any ; moreover, they all have as zero-set the set
Proof. (i)The proof of the first point follows by some simple computations. In fact, since and since the matrices are all skew symmetric, we have that, Thus, if and only if . Moreover, which is null at .(ii)First of all we observe that, since the vector fields are invariant by left composition of the group operation, we have and so by point (i) and are null if and only if , i.e., . To compute the horizontal gradient and the horizontal Hessian matrix with respect to the x variable, we observe that, since , it holds and, by left invariance of the vector fields,Again and are null exactly when .
Finally, we observe that and .
We use the previous lemma to prove an equivalent definition of solution other than Definition 2 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 function u is a viscosity subsolution (respectively, supersolution) of (28) if and only if for any , if is a local maximum (respectively, minimum) point for , one hasrespectively,
Proof. We only show the result for subsolutions, the other part being similar. It is clear that a viscosity subsolution will satisfy (47) since if by Remark 2 and (F1).
Let u be an upper semicontinuous function which satisfies (46) and (47). Consider and a local maximum point for such that and . Without loss of generality, we can assume that u is a strict local maximum point for . We need to prove thatFor any , we consider the functionBy standard arguments, one proves that for ε being sufficiently small, there is a family of local maxima of such that converges to . Indeed, if are the maximum points of in a small compact neighborhood of , will converge to some (passing to a subsequence if necessary). One first uses to show that and next by taking the limit that is a maximum of in the neighborhood so that .
Moreover, since the function has a local maximum in , we haveThus,Two cases may now occur:(1) along a subsequence. This means that and by Lemma 1, . Since the map with attains a maximum at and by (47), we get For future reference, we remark that the test function φ satisfies in a neighborhood of : implies . We proceed and by (52) and , we get that . Using the ellipticity of F and Remark 2, it holds and we conclude by letting ε go to 0.(2) for all ε being sufficiently small. Using (52) and the previous lemma, this means . Moreover, the point is a maximum for since . Let be the right translation by α and its Jacobian matrix. A simple computation shows that has the formBy the chain rule, we getsince andMoreover, we show that . In fact, as has a maximum at ,By Lemma 1, this is null if and only if , and we already know that this cannot be true. Thus, by (32), it holdsand we conclude by letting :
Remark 4. By a remark during the previous proof, it is not restrictive to assume in Definition 2 that if u (respectively, ) is an upper semicontinuous subsolution (respectively, a lower semicontinuous supersolution) of equation (28) and is a test function for (resp., for ) at the point , then at any point in a neighborhood of such thatit holdsComplementing Theorem 1 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  or in , which is different from viscosity solutions at characteristic points, is in fact equivalent to standard viscosity solutions and ours.
Corollary 1. Let be an upper (respectively, lower) semicontinuous function. Function u is a viscosity subsolution (resp., supersolution) of (28) if and only if for any , if is a local maximum (respectively, minimum) point for , one hasrespectively,
Proof. Suppose that is a local maximum point for . If , then so there is nothing to prove. We therefore limit ourselves to discuss the case .
If u is a viscosity subsolution, by Remark 2, we know that ; 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: implies . In this case and then . Thus, by Theorem 1, we know that u is a viscosity subsolution.
Remark 5. In , the authors require a subsolution u of the horizontal mean curvature flow equation to satisfyif has a maximum at and . If in particular ϕ is in the class of test functions such that implies , then . 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) when F is given in (4). From Theorem 1, we see that when we deal with functions with separated variables like , it is easy to check the (mcfe) at singular points of the operator. If has a maximum/minimum at and , then we only need to look at the sign of provided suitable test functions exist, i.e., , 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 convex function (as in viscosity-convex) is given by Bardi and Dragoni , where it is discussed and characterised and the reader can find explicit examples.
Definition 3. A continuous function is convex in the Carnot group if there is , and for all test functions, such that has a maximum at , then . If , we say that U is strictly -convex.
The idea is to build supersolutions of the (mcfe) from a convex function.
Proposition 1. In a Carnot group of Step 2, let be continuous and a strictly convex function. Then for , the function is a supersolution of (mcfe) for all , . Suppose that U is nonnegative. Then, if and , the initial front becomes extinct before time .
Proof. In order to check the supersolution condition, we use the alternative definition as in Theorem 1. Let be such that has a minimum at . Since has a maximum at and U is strictly convex, then for some . Therefore, it cannot be if φ is an appropriate test function, and then,provided .
The zero sublevel set of the supersolution u becomes a barrier if a comparison principle holds. At time t, if , it is given by and becomes empty if .
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 space , , and suppose that , where is an matrix. Notice that , , and .
Example 2. In the first example, we avoid characteristic points. For , consider the family of functions . We easily get thatIn particular, is strictly convex; we can compute exactly the operatorand thus by Theorem 1 and Proposition 1, is a supersolution for and a subsolution for in , so is a viscosity solution, for . Note that for , the zero level set of is a cylinder with axis and it goes extinct at time .
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:and are constants to be decided later. Note that the zero level set of u is (we will always regard for convenience)Therefore, it is the boundary of a ball for the distance centred at the origin. It has characteristic points, namely, points where precisely in its intersection with the axis , as we readily see in the following. We can easily compute (here, we will do complete calculations and not only the signature of because we also want to check the subsolution condition)Therefore, G is -convex but not strictly -convex. Finally,and since u is smooth, we conclude that, for ,