Diagonal non-semicontinuous variational problems

Abstract

We study the minimum problem for non sequentially weakly lower semicontinuos functionals of the form F(u)=∫If(x,u(x),u′(x))dx, defined on Sobolev spaces, where the integrand f:I×ℝm×ℝm→ℝ is assumed to be non convex in the last variable. Denoting by f̅ the lower convex envelope of f with respect to the last variable, we prove the existence of minimum points of F assuming that the application p↦f̅(⋅,p,⋅) is separately monotone with respect to each component pi of the vector p and that the Hessian matrix of the application ξ↦f̅(⋅,⋅,ξ) is diagonal. In the special case of functionals of sum type represented by integrands of the form f(x, p, ξ) = g(x, ξ) + h(x, p), we assume that the separate monotonicity of the map p↦h(⋅, p) holds true in a neighbourhood of the (unique) minimizer of the relaxed functional and not necessarily on its whole domain.