Conditions nécessaires et suffisantes d'existence de solutions en calcul des variations. (Necessary and sufficient conditions for the existence of solutions in the calculus of variations) (Q1100721)

A variational problem of the form \[ P: Inf\{\int^{b}_{a}f(t,x(t),x'(t))dt: x\in W^{1,q}(a,b),\quad x(a)=\alpha,\quad x(b)=\beta \}, \] where \(1<q<+\infty\), is considered in the work. Necessary and sufficient conditions are formulated for the regular and non convex function f under which the problem P has at least one solution for any boundary conditions.