Eikonal equations on ramified spaces
Abstract.
We generalize the results in [16] to higher dimensional ramified spaces. For this purpose we introduce ramified manifolds and, as special cases, locally elementary polygonal ramified spaces (LEP spaces). On LEP spaces we develop a theory of viscosity solutions for HamiltonJacobi equations, providing existence and uniqueness results.
Key words and phrases:
HamiltonJacobi equation; ramified space; viscosity solution; comparison principle.1991 Mathematics Subject Classification:
Primary 49L25; Secondary 58G20, 35F201. Introduction
In [15], [16] a theory of viscosity solution for HamiltonJacobi equations of eikonal type on topological networks was developed providing existence, uniqueness and stability results. In this paper we generalize these results to higher dimensional ramified spaces.
In literature, many different ways of introducing ramified spaces (cf. [11], [13], [14]) or branched manifolds (cf. [17]) are available. The definitions vary in different aspects, depending on the kind of theory to be developed. In a general approach, subsets of classic differentiable manifolds are glued together along parts of their boundaries by means of the topological gluing operation. Another, more specific, definition requires the uniqueness of the tangent space at ramification points (cf. [17]) by describing how the branches should be situated relatively to each other in the ambient space.
Here we choose an approach which is very similar to the concept of a manifold with boundary. The basic idea is that, in contrast to classic topological manifolds, besides points at which it is locally homeomorphic to an Euclidean space (simple points), a ramified topological manifold may also contain ramification points at which it is locally homeomorphic to some kind of “Euclidean ramified space”. The latter, called elementary ramified space, can be visualized as a collection of closed Euclidean half spaces glued together at their boundary hyperplanes. Consequently, small neighborhoods of a given ramification point split up into different branches corresponding to the branches of the homeomorphic elementary ramified space. If we endow these ramified topological manifolds with suitable differentiable structures, then we end up with an extension of the concept of tangent space at ramification points. This generalization should have the property that a real function defined in a neighborhood of a ramification point can be differentiated in the direction of each branch (of course, incident at the ramification point). In other words, each branch contributes a different tangent space.
Once we have introduced the differentiable structure on ramified spaces, we will see that for each of the branches emanating from a fixed ramification point , a normal direction at on this branch is welldefined. The possibility to differentiate in the normal directions at ramification points is crucial for our theory, as it will turn out that a general definition of viscosity solutions on ramified manifolds depends on this very possibility. In fact, the notion of viscosity solutions introduced in [15], [16] differs from its classical origin by the transition conditions we have additionally imposed at ramification points. The concept of test functions (see definition 3.7) allows us to ignore the ramification set by treating two branches as a single connected one; indeed, the differentiability links the two derivatives of a function with respect to a given pair of branches which are incident at the same ramification point. It suggests itself to apply this pattern in case of manifolds of dimension which have a certain manifold of dimension in common, as long as this manifold is smooth enough to ensure that we have welldefined normal derivatives with respect to each incident branch manifold.
In order not to get lost in too general approaches, we restrict ourselves to a rather simple, but still sufficiently general, kind of ramified manifolds, the socalled locally elementary polygonal ramified spaces (briefly, LEP spaces), which are characterized by two main criteria: on the one hand, LEP spaces are ramified spaces in the sense of Lumer [11] (see definition 2.1) meeting the additional requirement that each branch is a flat dimensional submanifold of . On the other hand, they are ramified manifolds in the sense described above. Hence they can be visualized as polygonal subsets of hyperplanes in which are glued together along certain edges, with the restriction that corner points cannot occur. The term “locally elementary” refers to the fact that they are locally homeomorphic to an open subset either of a dimensional Euclidean space or of an elementary ramified space. Once the notion of viscosity solutions has been correctly extended to LEP spaces, the development of the theory follows the line of the one devised for topological networks in [16]. Consequently we prove a comparison principle giving the uniqueness of the continuous viscosity solution. Moreover we show existence of the viscosity solution via an adaptation of the Perron’s method and we also provide a representation formula for the solution of the Dirichlet problem.
We mention that HamiltonJacobi equation and viscosity solutions on differentiable manifolds have been studied in [2], [12]. The theory of linear and semilinear differential equations on nonsmooth manifolds such as ramified spaces has been developed, since the seminal paper [11], in a large extent [9] and it is currently an active field of research ([10], [6]). For fully nonlinear equations such as HamiltonJacobi equations, the theory is at the beginning and, besides [16] and the companion paper [5], different approaches have been pursued for the case of networks in the recent papers [1] and [8] and for stratified domains in [4] . The present paper can been seen as a first attempt to extend the theory of viscosity solutions to general ramified spaces.
The paper is organized as follows. In section 2 we introduce the definition and give various examples of ramified spaces. In section 3 we study the differential structure of a ramified space. Section 4 is devoted to the notion of viscosity solution, while in section 5 and 6 we prove uniqueness and, respectively, existence of a viscosity solution. In section 7 we consider the Dirichlet problem and we obtain a representation formula for its solution.
2. Ramified spaces
In this section we introduce the geometric objects we will study in this paper. The general definition of ramified space is due to Lumer [11].
Definition 2.1.
Let be a nonempty, locally compact space with a countable basis. Let be a countable family of non empty open subsets of and let be a closed, possible empty, subset of with the property that it contains each point of which is contained in the boundary of exactly one . Then is a ramified space (induced by ) if

for all , ,

,

is locally finite in ,

is connected.
The set is called the boundary of while the set the ramification space of . We set and .
We also consider polygonal ramified space.
Definition 2.2.
A ramified space is said a dimensional polygonal ramified space if

with the endowed topology,

For each , there is a hyperplane such that is a bounded subset of ,

All , , are pairwise distinct.
We give some examples of ramified spaces and polygonal ramified space.
Example 2.1.
A topological network is a collection of pairwise different points in connected by continuous, non selfintersecting curves. More precisely (see [16]), let be a finite collection of pairwise different points in and let be a finite collection of continuous, non selfintersecting curves in given by . Defined , and , assume that

for all ,

for all ,

, and for all , .

For all there is a path with endpoints and (i.e. a sequence of edges such that and , ).
Then is called a (finite) topological network in .
A topological network is a ramified space with , , , any subset of containing
all the vertices with only one incident edge and .
Example 2.2.
If the edges of a topological network are segments, then and the corresponding dimensional topological networks defined as in example 2.2 are polygonal ramified spaces in the sense of definition 2.2. See figure 1.
Example 2.3.
Let be the surface of the dimensional cube and let , , be its open faces. Furthermore let be any closed (possible empty) subset of the union of the edges of the cube with the property that is connected. Then is a polygonal ramified space.
An important example of ramified space is the elementary ramified space, since it is the space of the parameters for LEP spaces and ramified manifolds we will define in the following.
Definition 2.3.
Given and , a dimensional elementary ramified space of order , denoted by , is the union of
half spaces , , of dimension which are included in and have in common.
If we set , then we can identify and with
where is the equivalence relation which for each choice of identifies the points for . The set is said the (closed) branch of while the set
is called the ramification space of .
Endowed with the topology induced by the path distance, is a connected, separable, locally compact topological space. Observe that can be identified with if .
In order to give the definition of ramified manifolds, we need to introduce the notion of diffeomorphism on .
Definition 2.4.
1) Let be an open set and . Then, for , is said differentiable at if the following holds:

If , for some , then is times continuously differentiable at in the standard sense.

If , then for each , there is a domain and such that and on (having identified with ).
2) Let be open sets and an homeomorphism. Then is said a diffeomorphism if for all the respective restrictions of and to and to are differentiable.
We are now ready to give the definition of topological ramified manifold and differentiable ramified manifold.
Definition 2.5.
A set is called a dimensional topological ramified manifold if it is endowed with a Hausdorff topology and if for any , there is a neighborhood of such that there is an integer , an open set with and a homeomorphism with .
The number is called ramification order of . A point is said a simple point if , a ramification point if . The set of all ramification points is denoted by and it is called ramification space of . If , we set .
Remark 2.1.
Observe that, since can be identified with , topological ramified manifolds are locally homeomorphic to a dimensional Euclidean space at simple points.
Definition 2.6.
A set is called a dimensional differentiable ramified manifold if is a dimensional topological ramified manifold and there is a family of local charts , i.e. open set and injective mappings , with the following properties

For any with , the sets and are open in and , respectively. Moreover the map given by is a diffeomorphism in the sense of definition 2.4.

.

The family is maximal with respect to the conditions i) and ii).
We introduce a class of flat ramified manifolds.
Definition 2.7.
A dimensional polygonal ramified space (see definition 2.2) is called locally elementary if it is also a differentiable manifold. Locally elementary ramified space will be called LEP spaces in the following.
Example 2.4.
Topological networks and dimensional topological networks are topological ramified manifolds. If the maps in the definition of are diffeomorphisms, they are also differentiable ramified manifolds. If the edges of are segments, a dimensional topological network is a LEP space.
The set of ramification points is given by for a topological network and by where for a dimensional topological network. Note that for a LEP space.
Example 2.5.
The cube in the example 2.3 is not locally homeomorphic to an elementary ramified space at the corner points. It is a LEP space if all the corner points are contained in .
3. The differential structure of a ramified manifold
In this section we extend the notion of tangent space to a differentiable ramified manifold. In fact, the interpretation of tangent vectors as equivalence classes of curves in can be easily transferred to ramification points.
Throughout this section, and stand respectively for a dimensional differentiable ramified manifold and for its ramification set. Let us now introduce some definitions regarding the differential structure of .
Definition 3.1.
A continuous function is said to be differentiable at if for any local chart around , the function is differentiable in sense of definition 2.4.
Definition 3.2.
Let and . Let with be a continuous curve and . We say that reaches from the branch whenever there exists a chart with and such that
(3.1) 
We denote by the set of all the curves reaching from the branch and we set
Definition 3.3.
Let , and , . We say that and are equivalent if for all functions which are differentiable at we have
where the derivatives are leftsided. We denote the set of equivalence classes by , the tangent space of at , and we say that is a tangent vector at , , if contains a curve reaching from the branch . We set
The set of all tangent vectors at (tangent space at ) is denoted by . The set is called the tangent space at and any is said a tangent vector at .
Remark 3.1.
If , the tangent space is not a vector space. Instead it can be identified with an elementary ramified space , where
Hence can be identified with and with .
Definition 3.4.
Let , continuously differentiable at (see def. 3.1) and a basis of . We define the gradient of at by
(3.2) 
We consider the case of an elementary ramified space and we introduce some notations for the derivatives at the ramification set.
Definition 3.5.
Let and let . Let be continuously differentiable at . We denote by the directional derivatives of at with respect to the canonical basis of . For we denote by the directional derivative of at with respect to the inward unit normal of at . See figure 2.
We now restrict our attention to LEP spaces, which have the important property that around any given point we can always choose a chart induced by the canonical identification with the Euclidean space or a suitable elementary ramified space. This is stated in the following proposition.
Proposition 3.1.
Let be a LEP space and its ramification set. For any , there is a neighborhood of and a canonical identification where and

if then ,

if then with .
In the latter case, induces a bijective map between the index set and the set .
We now consider derivatives of a function on a LEP space at ramification points. For any , we always fix a canonical identification chart as defined in proposition 3.1 and all the concepts will be expressed in terms of the chart for sake of simplicity. However it is easy to verify that they in fact do not depend on the choice of specific chart.
Definition 3.6.
For any function and each we denote by the restriction of to , i.e.
We denote by the space of continuous function on . This in particular implies that and
In a similar way we define the space of upper semicontinuous functions and the space of lower semicontinuous functions , respectively.
In [16], we introduced the concept of test function on a topological network , treating two edges and incident at a vertex as one connected edge and imposing that the derivatives in the direction of the incident edges, taking into account their orientations, coincide at . In other terms, a test function, considered as a function defined on , is differentiable at the interior point .
Here we follow a similar idea for LEP spaces, linking the two normal derivatives of a test function for a given couple of branch manifolds incident at a point .
Definition 3.7.
Let , , , . Then is said to be differentiable at if is differentiable at and if we have
(3.4) 
Definition 3.8.
Let and let be .

Let and let . We say that is an upper (lower) test function of at if is differentiable at and attains a local maximum (minimum) at .

Let and , . We say that is a upper (lower) test function of at if is differentiable at and attains a local maximum (minimum) at with respect to .
4. Viscosity solutions
Since now on, and stand respectively for a LEP space and for its ramification set. We introduce the class of HamiltonJacobi equations of eikonal type we consider in this paper. An Hamiltonian is a family of mappings with (recall: , ). By means of the canonical identification map around a fixed point we can think of as a mapping defined by the identification
(4.1) 
(see (3.2) and (3.3)) where is a neighborhood of or provided that or , respectively. In the sequel we will speak of under the canonical identification (around ) whenever we refer to in the sense of (4.1). We assume the Hamiltonian fulfills the following properties
(4.2)  
(4.3)  
(4.4)  
(4.5)  
(4.6) 
Remark 4.1.
Assumptions (4.2)–(4.3) are standard conditions in viscosity solution theory (see f.e. [7]) to ensure existence and uniqueness of the solution. Assumptions (4.5) and (4.6) represent compatibility conditions across the ramification set; the former guarantees a continuity condition at for the Hamiltonians defined on two different branches while the latter states the invariance with respect to orientation of the inward normal . Under hypotheses (4.3) and (4.6), assumption (4.4) is fulfilled provided that, for , is convex. Observe that thanks to the identification (4.1), (4.4)–(4.6) induce corresponding properties for the Hamiltonian .
Example 4.1.
A typical example of Hamiltonian satisfying the previous assumptions is given by the family where the functions are continuous, non negative and satisfies the compatibility condition if .
We introduce the definition of viscosity solution for the HamiltonJacobi equation of eikonal type
(4.7) 
For , we define by the projection on the tangent space of , i.e.
Definition 4.1.
A function is called a (viscosity) subsolution of (4.7) in if the following holds:

For any , , and for any upper test function of at we have

For any , for any and for any upper test function of at we have
A function is called a (viscosity) supersolution of (4.7) in if the following holds:

For any , , and for any lower test function of at we have

For any , (see definition 3.3) and , there exists , such that
for any lower test function of at satisfying .
A continuous function is called a (viscosity) solution of (4.7) if it is both a viscosity subsolution and a viscosity supersolution.
Remark 4.2.
i) Near the set is locally diffeomorphic to with a tangent space given by . The tangent space is composed by vectors with and . Taking into account the condition of differentiability (3.4), we see that if is a test function of at , then by (4.5)(4.6),
In other terms, at a ramification point, the equation (4.7) can be equivalently replaced by one of the two couples and .
ii)
The definitions of subsolution and supersolution are not symmetric at a ramification point. It turns out that solutions of eikonal equations are
distancelike functions from the boundary (see section 7); this definition of supersolution reflects the idea these functions
have to be solutions of (4.7). Since there is always a shortest path from a ramification point to the boundary, for any branch
and for any we can connect the to the boundary contained in the branch by a shortest path. Then lower test functions satisfies the definition of supersolution.
It is also worth to observe that, for , our definition of solution (resp., sub or supersolution) coincides with the standard notion of viscosity solution (resp., sub or supersolution).
Remark 4.3.
The definition of viscosity solution is not affected by the fact that belongs or not to the ramification set . At ramification points, it requires a suitable class of test functions (different from the classical one) with appropriate differentiability properties. In this respect, our approach is different from the one in [1, 4, 8] which require to suitably redefine the Hamiltonian at ramification points.
5. A comparison result
This section is devoted to the proof of a comparison theorem for (4.7). The coerciveness of the Hamiltonian (stated in (4.3)) implies the Lipschitzcontinuity of subsolutions; such a regularity which will be exploited in the comparison theorem.
Lemma 5.1.
Let be a compact subset of and let be a subsolution of (4.7). Then there exists a constant depending only on such that
The proof is standard in viscosity theory and we refer the reader to [3, Prop.II.4.1].
Theorem 5.1.
Let and be respectively a supersolution to (4.7) and a subsolution of
(5.1) 
with and for all . If on , then in .
Proof.
Assume by contradiction that there exists such that
(5.2) 
For define by
where is the geodesic distance between two points and on the space . Since is an upper semicontinuous function, there exists a maximum point for in . By we get
(5.3) 
for some . Hence
(5.4) 
It follows that there exists such that . Owing to (5.3), there holds: ; passing to the limit, we infer and, in particular, . Whence, for sufficiently small, we get: . By (5.3) and the Lipschitz continuity of (see lemma 5.1) we get
and therefore
(5.5) 
Since , there exists such that
(5.6) 
and for sufficiently small, . Since is compact and composed by a finite number of branch , we have
Hence, for sufficiently small, we can assume that if and , then .
¿From now on we set , and we define and and we work with the canonical identification (4.1).
Case 1. for some : Consider and set
If , then is differentiable at , and
If , then we have
We conclude that for any , , is differentiable at . A corresponding property holds for .
Since has a maximum point at and has a minimum point at , we get
Moreover
Fix and denote by , the modulus of continuity of with respect to . Hence, for sufficiently small we have
for some . By (5.5) we get a contradiction for .