Consider the Euclidean space with the orthogonal action of a compact Lie group G. We prove that a locally Lipschitz G-invariant mapping f from to can be uniformly approximated by G-invariant smooth mappings g in such a way that the gradient of g is a graph approximation of Clarkés generalized gradient of f. This result enables a proper development of equivariant gradient degree theory for a class of set-valued gradient mappings.
It is well known that various versions of degree theory are very useful in nonlinear analysis; see the books [1,2,3] and their extensive references. A powerful special degree for maps commuting with an action of a Lie group gives many multiplicity results; see the book . In order to extend the degrees to set-valued maps, a graph approximation method in the spirit of  appeared to be very successful. In , a series of selection and graph approximation results for convex-valued mappings was obtained in the presence of symmetries given by a compact group action as extensions of many classical results. Some of them can be basic tools in the construction of equivariant degree theory. An equivariant version of the Cellina approximation theorem for convex-valued upper semicontinuous mappings was used in  to define the equivariant degree, which extended the one from [4,8]. The degree theory was applied in  to obtain nontrivial solutions to multivalued boundary value problems. See also [9,10] for further applications of that degree to obtain multiple solutions to some implicit functional differential equations.
On the other hand, in many applications, the considered maps are additionally gradients of smooth functionals, and special invariants for this class are involved like the gradient equivariant degree and Conley index; see [11,12]. There are plenty of papers on various applications, and we mention only a few of them concerning second order ODEs , symmetry-breaking , bifurcations of the Neumann problem  and symmetric Newtonian systems . In the non-smooth case, one can use Clarkés generalized gradient notion  for locally Lipschitz functions, which is a convex-valued u.s.c. map in this case (see Definition 3 or  for definition of u.s.c. for multifunction). It appears that an approximation result is valid for such functions defined in (see ), which may be treated as an analogue of the Whitney approximation theorem.
The main purpose of this paper is to prove the equivariant version of the approximation theorem of Ćwiszewski and Kryszewski. Then, we use this to provide an equivariant gradient degree theory with all the usual properties including the Hopf classification theorem.
It is worth pointing out that the proof is strictly finite-dimensional, and it is a challenging problem to extend the approximation theorem to Hilbert spaces. Nevertheless, the degree can be defined for compact vector fields, i.e., maps of the form , by standard Leray–Schauder-type techniques. One can think also of the perturbations of unbounded self-adjoint operators in Hilbert spaces; see . We postpone the details to another paper, as well as applications to set-valued variational problems.
Let us recall some basic information on group actions. For a more detailed description, see [21,22]. Let G be a group. Recall that a G-set is a pair , where X is a set and is the action of G on X, i.e., a map such that:
for and ;
for , where is the group unit.
In the sequel, we write instead of , , , unless it leads to ambiguity.
Given G-sets X and Y, a map is G-equivariant if for any and . If the G-action on Y is trivial, then we say that f is G-invariant.
A subset of a G-set X is G-invariant if for all . The set is called the orbit through , and denotes the set of all orbits. Observe that if , then the set is G-invariant.
If G is a topological group, X is a topological space and a G-set, then X is a G-space, provided the action is (jointly) continuous. We say that a real (resp. complex) Banach space is a real (resp. complex) Banach representation of G if is a G-space, and for each , the map is linear and bounded. Throughout the whole paper, we assume that G is a compact Lie group.
Set-valued maps are the main object of our studies. Recall that given sets X and Y, a set-valued map from X into Y (written ) is a map that assigns to each the value , being a non-empty subset of Y. If X and Y are topological spaces and, for any closed (resp. open) set , the preimage is closed (resp. open), then we say that is upper (resp. lower) semicontinuous; is continuous if it is upper and lower semicontinuous simultaneously.
If Y is a metric space, then is lower semicontinuous if and only if for any , the function is upper semicontinuous (as a real function) or, equivalently, given and , .
A similar characterization of upper semicontinuity is not true, i.e., the lower semicontinuity of does not imply in general that is upper semicontinuous. However, if has closed values, is locally compact, i.e., each point has a neighbourhood U such that is compact, and d is lower semicontinuous, then is upper semicontinuous (with compact values). The graph of an upper semicontinuous map with closed values is closed; is upper semicontinuous with compact values if and only if the projection is perfect (recall that a continuous map is perfect if it is closed and is compact for any ). We say that a map is compact if it is upper semicontinuous and the closure of the image is compact. For other details on set-valued maps, see  or .
Let be metric spaces and . Recall the notion of graph approximations.
A continuous map is an ε-approximation of if for every x, there exists such that and .
One can formulate the above condition as follows: for all . It is also equivalent to the condition that the graph of f is contained in the -neighbourhood of the graph of . It is well known that such approximations are a good tool for extending topological invariants, and they have been proven to exist for example for convex-valued u.s.c. maps; see . We are interested in equivariant versions of the results.
Let X and Y be G-sets. A set-valued map is G-equivariant (resp. G-invariant) if (resp. ) for all and .
Note that is G-equivariant if and only if for all and (or for all and ). Moreover, it is easy to see that is G-equivariant if and only if its graph is a G-invariant subset of with a natural action , , . Observe that if , where Y is a topological space, is G-equivariant, then so is its closure, i.e., the map given by , .
Let us collect simple examples of G-equivariant set-valued maps.
(1) Let , where and is an open G-invariant subset of a Banach G-representation , be G-equivariant, i.e., for and . Consider a differential inclusion (under suitable assumptions assuring the existence of solutions):
Consider the space of continuous maps from to with the G-action defined by , where for , and . If is a solution to the above problem, i.e., there is an integrable function such that , , then is also a solution to this problem with the initial condition . Therefore, the solution map that assigns to each initial value the set of all solutions is G-equivariant, whenever well defined.
(2) Let , where is a real Banach representation of G, be a convex function. For each , the subdifferential:
is defined. Let f be G-invariant. Then, the map is G-equivariant. Clearly, is invariant, and if , then for all , . Hence:
which gives the assertion. In a similar manner, one shows that the Clarke generalized gradient , where U is a G-invariant open in and is a G-invariant locally Lipschitz function, is G-equivariant.
2.2. Gradient approximations
In , we shall use various analytic characterizations of graph-approximations. For , we recall the notion of a support function given by:
One easily checks that is finite and continuous if A is bounded. Moreover, .
We also define:
(, Lemma 2.2) Let , with closed convex values, , and . The following conditions are equivalent:
g is an ε-approximation of φ;
for every , there is such that:
for every , there is such that:
Let be an open subset, and consider a locally Lipschitz function . A generalized directional Clarke derivative:
is defined for all . Clarkés generalized gradient at x is the set:
It is well known that the map is an u.s.c. mapping with compact convex values (see  for this and more of the properties of the generalized gradient). One observes also that for . Recall only the local u.s.c. property:
The map is upper semicontinuous at if:
The following approximation result was proved in :
Let be locally Lipschitz and a continuous function. Then, there exists a -map such that:
for all ,
for all .
In other words, g is a uniform approximation of f, and is an -approximation of . We shall call such pairs Whitney approximations.
Suppose , where compact, is a locally Lipschitz function. If satisfies:
for all (uniform ε-approximation),
for all (ε-approximation (on the graph)),
for a continuous function , then we say that is the ε-WT-approximation (ε-Whitney-type-approximation) of .
Let us assume that V is an n-dimensional orthogonal representation of a compact Lie group G and is open and G-invariant. Assume that a locally Lipschitz function is G-invariant, i.e., for all . It is easy to verify that is then equivariant, i.e., as sets.
We now modify the proof from  in order to obtain an equivariant version of Theorem 1.
Let us start from the remark that since G acts orthogonally, there exists an invariant -mollifier, i.e., a nonnegative invariant function with the support and such that:
We can take, e.g., a function given by:
and then, we normalize it:
Let be an invariant L-Lipschitz map with a compact support, and let . We define the -regularization of f by the formula:
The following is a straightforward consequence of the definition.
The regularization satisfies the following properties:
is a uniform -approximation of f;
The following technical lemma was proven in detail in , Lemma 3.4.
Let be Lipschitz functions of common rank L and with compact supports. For any , there exists sufficiently small with the property that for every , there exists such that:
for each .
A local approximation result is a consequence of the above.
Let be G-invariant functions as in Lemma 2. Then, for any , there exists such that:
for any , the regularization of is a uniform ε-approximation of , for each ;
for every , there exists such that for all and :
We put , and let . Then, Property (i) follows from Proposition 1. We prove (ii) by means of Property (iii) from Lemma 1. For this, we fix a point x and . There exists a such that, for all i, we have:
On the other hand, we have:
and by the use of Lemma 2, we find such that:
Therefore, we have the inequality . This ends the proof because of the equality . □
Let be a locally G-invariant map as in Theorem 1. Then, for every , there exists a -smooth G-invariant map such that is an ε-WT-approximation of .
Since U is G-invariant, there exists an increasing family of G-invariant compact sets such that:
For each k = 1, the set is also G-invariant. Therefore, the family is an open G-invariant covering of U. Let be a -smooth and G-invariant partition of unity subordinated to .
Thus, the functions are globally Lipschitz, G-invariant, and they can be considered as defined on .
We define a sequence of positive numbers:
Now, we can apply Proposition 2 for each . We find such that for any invariant -regularizations: of and of are uniform -approximations. Moreover, for every , there is such that:
for all . We can assume that , and the sequence is nonincreasing.
The desired map is defined by the formula:
It is obviously well defined, since at most two terms are different from zero at given x. By definition, it is -smooth and G-invariant. We have to check the approximation conditions. For a fixed , there is a unique such that . If , then in some neighbourhood of x, and the claim is clear from the choice of .
If , then in some neighbourhood of x contained in , we have and . Therefore:
By the properties of the generalized gradient, we obtain:
Because of Lemma 1, the last condition means that is a graph -approximation of . This ends the proof. □
Let X be a compact G-invariant subset of V. Suppose that is locally Lipschitz and G-invariant. Then, for all , there exists a with and such that, for every two -approximations of , the pair is an -approximation of , where:
is a linear homotopy.
For all , is upper semicontinuous (1) at x, so we can apply Definition 3 with a given and obtain an appropriate, small .
We can assume that for , and . Then, we have . For all , we observe that implies . The family is an open covering of X. We can choose a finite subcovering:
Let be a Lebesgue number for the covering , i.e.,:
and let .
Fix , and suppose that are -approximations of . Then, we have:
Then, from (2) and (3), we get:
From (2) and (3), we have:
We can use the upper semicontinuity property of in :
for points , which are close enough to :
Now, we have:
However, is a convex set, and therefore:
Finally, from (2) and (4), we have:
This ends the proof. □
2.3. Equivariant Gradient Degree
In this section, we outline an axiomatic approach to the equivariant gradient degree. To this end, some preliminaries are in order.
Recall that the Euler ring of a compact Lie group G as an abelian group is equal to , where is the set of all conjugacy classes of normal subgroups of G. The multiplicative structure of is quite involved (see  or  for details on the ring structure).
Let V be an orthogonal G-representation. Denote by the space of G-invariant real -functions on V. Clearly, if , then the gradient map is G-equivariant. Let be an open, bounded and G-invariant subset of V and . A pair is said to be a G-gradient -admissible pair if for all , and there exists such that . Denote by the set of all G-gradient -admissible pairs, and put . In an obvious way, one can define a G-gradient -admissible homotopy between two G-gradient -admissible maps.
The notion of equivariant gradient degree we consider here was defined originally by K. Gȩba . We use an axiomatic approach here adapted from .
There exists a unique map , which assigns to every an element :
satisfying the following properties ( are generators of the Euler ring, and are integer coefficients that may be nonzero only if the orbit type is present in the representation V):
(Existence) If , i.e., there is a nonzero coefficient , then such that and .
(Additivity) Let be two disjoint, open, G-invariant subsets of Ω and . Then:
(Homotopy) If is an admissible gradient G-equivariant homotopy, then:
(Normalization) Let be a special Ω-Morse function such that and . Then,
where “” stands for the total dimension of eigenspaces for negative eigenvalues of a (symmetric Hessian) matrix.
(Multiplicativity) For all ,
where the multiplication “⋆” is taken in the Euler ring .
(Suspension) If W is an orthogonal G-representation and an open bounded invariant neighbourhood of , then:
(Hopf property) Assume that is the unit ball of an orthogonal G-representation V, and for , one has:
Then, and are G-gradient -admissible homotopic.
Let us remark that in the case of the trivial action of G, the above degree reduces to the classical Brouwer degree, as well as the usual non-gradient G-equivariant degree (comp. ). This follows from the celebrated A. Parusiński theorem (see ), which says that every two gradient maps that are -admissible homotopic are also gradient -admissible homotopic. However, this is no longer true for equivariant maps.
Take with the natural -action. Clearly, and are -equivariant. Moreover, is a -admissible -equivariant homotopy between Φ and Ψ. Clearly, this homotopy is not a gradient. Moreover, it follows from  that such a gradient -equivariant homotopy does not exist.
Since the equivariant gradient degree satisfies the Hopf property with respect to gradient G-homotopies, as well as the ordinary equivariant degree with respect to equivariant homotopies, the gradient degree is more delicate homotopy invariant.
The ordinary equivariant degree for multivalued convex-valued u.s.c. maps was defined in  by means of the equivariant version of the Cellina approximation theorem (see ). The approximation results from the previous section enable us to develop the equivariant degree theory in quite an analogous way.
Let V be a finite dimensional orthogonal representation of a compact Lie group G, and let be an open, bounded G-invariant subset of V. Let be a locally Lipschitz G-invariant map such that its Clarkés generalized gradient is -admissible, i.e., for . Since the map is u.s.c., the image of the boundary is compact, and its distance from zero is positive. The same is true for a sufficiently fine neighbourhood of . Thus, there exists an such that for every , each -WT-approximation of is -admissible. Moreover, Theorem 3 assures that there exists a smaller such that all -approximations are -admissible homotopic by a linear homotopy. Take such a small .
We define the equivariant degree of Ω-admissible pair as follows:
where is a δ-WT-approximation (with g being G-invariant) of .
The properties of the degree are consequences of Theorem 4. The first three of them are the most important for applications. Properties 1 and 2 are quite straightforward. The homotopy property needs some comment. Recall that the generalized gradient of a locally Lipschitz map is a multivalued u.s.c. map with convex compact values. Considering homotopies as continuous families of maps, we need here a stronger regularity than the u.s.c. property, which may admit quite a rapid change of values.
A gradient G-homotopy is a pair , where are locally Lipschitz G-invariant maps and the family of maps is continuous with respect to , as well as the family .
Observe that in the most popular case of linear homotopy , the Clarke gradient with respect to the x variable satisfies the condition:
In some cases, an equality holds, e.g., if at least one of the is of class , or if both of them are convex functions.
Let the linear homotopy of locally Lipschitz maps be Ω-admissible, and let the condition be satisfied for all x and . Then:
The proof is similar to the proof of Theorem 3. For all , are upper semicontinuous (1) at x, so we can apply Definition 3 with a given and obtain an appropriate .
Let and . Then, .
For all , we have that implies and is an open covering of X. We can choose a finite subcovering:
Let be a Lebesgue number for the covering , i.e.,:
and let .
Fix , and suppose that are -approximations of , respectively. Then, we have:
Then, from (5) and (6), we get:
From (5) and (6), we have:
We can use the upper semicontinuity of in :
for points :
Now, we have:
This means that there exist such that:
and by our assumption. Thus:
Finally, from (5) and (7), we have:
This ends the proof. □
The above theorem is satisfactory in many applications. In particular, the Hopf property follows from this weak version of the homotopy property.
Assume that is the unit ball of an orthogonal representation V and are Ω-admissible pairs such that:
Then, the pairs are Ω-admissible homotopic.
It is enough to observe that if we choose a -WT-approximation of defining the equivariant degree, then the linear formula:
defines an -admissible homotopy satisfying the assumptions of Theorem 5. Therefore, we apply this to both pairs. Now, we can apply the Hopf property for smooth maps . The transitivity property of the homotopy relation for single-valued maps ends the proof. □
We are ready to prove the full homotopy property now.
Let be an Ω-admissible gradient G-homotopy. Then:
First, we observe that because of the continuity of the homotopy with respect to , for each and every fixed t, there exist a and a Whitney approximation of , which is also an -Whitney approximation of for . Therefore, is linearly homotopic to , as well as to . Thus, by Theorem 5, the gradient degree is constant in the interval . Now, we consider the open covering of by the intervals (for , we put half-open intervals). We find a finite subcovering and apply the transitivity of the homotopy property to finish the proof. □
3. Discussion and Conclusions
The main technical result of this paper is contained in Theorem 2. It says that for a symmetric (G-invariant, where G acts orthogonally on ) and locally Lipschitz function , there exists an arbitrarily fine smooth and G-invariant uniform approximation g of f such that the gradient is a graph approximation of Clarkés generalized gradient . This is the equivariant version of the result from . It is known that a version of the Whitney smooth approximation theorem is valid not only in finite-dimensional spaces, but also in Hilbert spaces. However, the presented proof does not work in infinite dimensions because it uses the compactness of closed balls. Moreover, the generalized Clarke gradient is only a weakly upper semicontinuous map with bounded closed values in general. Thus, it as an open question if a similar approximation theorem is true in general (it is also an open problem in the non-equivariant case).
As an application, we developed a gradient equivariant degree theory for the class of set-valued equivariant maps, which are generated by locally Lipschitz potentials together with the Hopf-type classification theorem. It should be mentioned that thanks to our approximation theorem, the classes of homotopy equivalence of such maps can be represented by smooth equivariant gradient maps, and these are different from the usual equivariant homotopy classes.
Further applications into nonlinear analysis are expected, for example some multiplicity results for Hamiltonian systems as those considered for -maps in [8,16] and some references therein. By means of an infinite-dimensional extension of our degree theory by Leray–Schauder-type finite-dimensional approximations, we can expect similar results under a bit weaker smoothness assumptions. In search of periodic solutions for differential equations (inclusions), a natural action of the group is considered besides some additional spatial symmetries as one can see in the book  and the references therein. Compact multivalued perturbations of the identity operator in a Hilbert space will be considered, as well as perturbations of some unbounded self-adjoint operators (comp. ). We treat our paper as the preliminary step towards this direction.
The authors are grateful to the anonymous reviewers for careful reading and the improvements of the manuscript.
This is a joint research project of both authors. Conceptualization, Z.D.; methodology, Z.D.; formal analysis, Z.D. and T.G.; investigation, Z.D. and T.G.; writing–original draft preparation, Z.D.; writing–review & editing, T.G.; supervision, Z.D.; project administration, T.G. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Conflicts of Interest
The authors declare no conflict of interest.
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