Existence of scalar minimizers for nonconvex simple integrals of sum type
The paper deals with an existence result for a Lagrange problem of the calculus of variations with a nonconvex Lagrangian. Specifically, it is proved that the problem \[ \min\biggl\{\int_0^1h(x'(t))+\varphi(x(t)) dt:\;x\in AC(0,1), x(0)=A, x(1)=B\biggr\} \] has at least one solution provided \(\varphi\) is a non negative lower semicontinuous function, \(h\) has a superlinear growth at infinity and \(h(0)=h^{**}(0)\), where \(h^{**}\) is the convex envelope of \(h\). The existence proof is obtained passing through the analysis of the relaxed variational problem, involving \(h^{**}\) in place of \(h\).
scientific article; zbMATH DE number 1217028