Necessary conditions for an extremum for general functionals of calculus of variations with a nonsmooth integrand
The variational problem in a Sobolev space is considered, i.e. \[ J_ F(u)=\int_{G}F[x,u(x),u'(x)]dx\to \inf, \] where the function F(x,s,\(\cdot)\) is convex for all x,s. The authors show that if u is a minimizer then there is a function h: \(G\times {\mathbb{R}}\to {\mathbb{R}}^ n\) such that \[ \int_{G}\{\frac{\partial F}{\partial S}[x,u(x),u'(x)]\phi (x)+<h(x,u(x)),\phi '(x)>\}dx=0 \] for all \(\phi \in C_ 0^{\infty}(G)\), where h(x,u(x)) belongs to the subdifferential of the function F(x,u(x),\(\cdot)\) at \(u'(x)\) for almost all \(x\in G\).
scientific article; zbMATH DE number 4173766