On topological characteristics of Lipschitz functionals (Q1281518)

Let \(X\) be a real separable reflexive Banach space. A bounded domain \(\Omega\subset X\) is said to be strongly Lipschitzian if for each \(x\in \partial\Omega\) there exist a neighborhood \(V\) and a locally Lipschitzian function \(h:G\to\mathbb{R}\) such that \(G\cap\Omega= \{y\in G;\;h(y)<0\}\), \(h(x)=0\) and \(0\notin\partial h(x)\) (the generalized gradient of \(h\) at \(x)\). We denote by \(S(M)\) the set of all set-valued operators \(A:M\to X^*\), \(M\subset X\), where \(X^*\) is the dual space of \(X\), which satisfy the following properties: (i) \(A(x)\) is a nonempty closed subset of \(X^*\); (ii) if \((x_n)\subset M\), \(x_n^*\in A(x_n)\); \(x_n @> w>> x\); \(x_n^*@>w^*>>x^*\) such that \(\lim_{n \to\infty}(x_n,x^*_n) \leq(x,x^*)\), then \(x_n @>\|\cdot\|>> x\) and \(x^*\in A(x)\). An operator \(A\) is said to be nondegenerate on \(M_1\subset M\) if \(0\notin A(M_1)\). Given a metric space \(Y\) we denote by \(\chi(Y)\) the Euler-Poincaré characteristic of \(Y\) and by \(H_q(Y)\), \(q=0\), \(1,2,\dots\) the integral homology group with compact supports of \(Y\). The author gives without proof the following results: Theorem 1. Let \(\Omega\) be a strongly Lipschitzian domain. If \(A\in S(\overline\Omega)\) is a nondegenerate operator and \(x\in\partial\Omega\), \(w\in X\) such that \((w,x^*)\prec 0\) for any \(x^*\in A(x)\), then \(w\) is a hypertangent to \(\overline\Omega\) at the point \(x\). Moreover, if \(E\) is a finite-dimensional subspace of \(X\) and \(v\in C(\partial \Omega,X)\) is a continuous vector field such that \(v(x)\in E\) and \((x-v(x), \;x^*)\succ 0\) for all \(x\in\partial\Omega\) and \(x^*\in A(x)\) then (i) the groups \(H_q(\overline\Omega)\) and \(H_q(\overline\Omega\cap E)\) are isomorphic \((q=0,1,2,\dots)\) and (ii) \(\gamma(A,\Omega) =\chi (\overline\Omega)\), where \(\gamma(A,\Omega)\) is the rotation of the field \(A\) on \(\partial\Omega\). Theorem 2. Let \(f\) be a functional on \(X\) such that \(\partial f\in S(X)\). If \(N_a =\{x\in X;\;f(x)\leq a\}\) is a bounded set and \(A\) is the restriction to \(N_a\) of the gradient \(\partial f\), where \(a\) is a regular value of \(f\), the properties (i), (ii) of Theorem 1 hold. Moreover, if \(A\) is nondegenerate on \(N_a\setminus\text{argmin} f\), then \(H_q(N_a)\) and \(H_q(\text{argmin} f)\) are isomorphic \((q=0,1,2,\dots)\) and \(\gamma(A,\Omega)= \chi(\text{argmin} f) \). Theorem 3. If \(\partial f\in S(X)\), \(N_a\) is bounded for any \(a<\infty\) and the critical values of \(f\) do not exceed some number \(a_0\prec\infty\), then \(H_q (N_a)\) and \(H_q(X)\) are isomorphic \((a>a_0,\;q=0,\;1,2,\dots)\) and \(\gamma (\partial f,{\overset\circ N}_a)=1\). The last equality in Theorem 1 is an infinite-dimensional version of the Poincaré-Hopf theorem. As the author mentions ``the results on the stabilization of homology of Lebesgue sets are new even for smooth functionals that arise in the classical calculus of variations''.