Integral functionals determined by their minima
The authors prove that the integrand \(f=f(x,p)\) of an integral functional of the type \[ F(u,A)=\int_{A}f(x,Du(x))dx \] may be found by the knowledge of the minima of the Dirichlet problems for F with linear boundary values. More precisely, they show that f(x,p) may be obtained by a differentiation process of the set function \(A\to m(p,A)\) where A is a bounded open subset of \(R^ n\) which belongs to a particular family and \(m(p,A)=\min \{F(u,A):\) \(u(x)=p\cdot x\), \(x\in \partial A\}\).
scientific article; zbMATH DE number 3992292