Topological index of extremals of multidimensional variational problems (Q1822391)

Let \(\Omega\) be a bounded domain with smooth boundary in \(R^ n\) and \({\mathfrak M}\) be isolated and bounded in \(\overset\circ W^ m_ 2(\Omega)\) set of extremals of the functional \[ f(u)=\int_{\Omega}F(x,u,...,D^ mu)dx\quad (D^ ku=\{D^{\alpha}u: \alpha_ 1+...+\alpha_ n=k\}). \] Under certain natural assumptions f(u) is differentiable on \(\overset\circ W^ m_ 2(\Omega)\) and the topological index ind(\({\mathfrak M};f)\) relative to the gradient field \(\nabla f(u)\) is defined. The main result: Let \({\mathfrak M}\) be a compact connected smooth finite-dimensional manifold without edge. If \({\mathfrak M}\) realizes a local minimum of f(u), then ind(\({\mathfrak M};f)=\chi ({\mathfrak M})\), where \(\chi\) (\({\mathfrak M})\) is the Euler-Poincaré characteristic of \({\mathfrak M}\).