A sharp attainment result for nonconvex variational problems
The authors study the existence of solutions to problems of calculus of variations of the form: Minimize \(I(u)\) over the set \(u_0+ W_0^{1,p}(\Omega)\), where \(I(u) = \int_\Omega f(u(x),\nabla u(x))\,dx\), \(\Omega\) is an open bounded subset of \({\mathbb R}^N\), \(N\geq 2\), \(f: {\mathbb R}\times {\mathbb R}^N \mapsto [0,\infty)\) is a continuous function which can be nonconvex with respect to \(\nabla u\), and \(u_0\in W^{1,p}(\Omega)\). Existence results in the case when \(f\) is of the form \(f(u,\nabla u)=g(u)+h(\nabla u)\), with \(h\) nonconvex, have already been proved in the literature. The authors obtain new existence results in classes of functions \(f\) more general than the ones already considered in previous papers.
scientific article; zbMATH DE number 2123232