Solvability of boundary-value problems and criteria for a minimum of integral functionals
The author considers the functional (1) \(F(u)=\int_{\Omega}f(x,u,u_ x)dx\), \(u|_{\partial \Omega}=0\) and the associated Dirichlet problem for the Euler equation (2) \(\sum^{n}_{i=1}df_{p_ i}/dx_ i-f_ u=0\), \(u|_{\partial \Omega}=0\). It is shown that under some circumstances the solution u of (2) is a weak (or) strong local minimum for (1). Applications to variational problems are given.
scientific article; zbMATH DE number 3946996