Regularity of minimizers for nonconvex vectorial integrals with $p$-$q$ growth via relaxation methods

Abstract

Local Lipschitz continuity of local minimizers of vectorial integrals ${\int }_{\Omega }f\left(x,Du\right)dx$ is proved when $f$ satisfies $p$-$q$ growth condition and $\xi \mapsto f\left(x,\xi \right)$ is not convex. The uniform convexity and the radial structure condition with respect to the last variable are assumed only at infinity. In the proof, we use semicontinuity and relaxation results for functionals with nonstandard growth.