AN OPTIMAL GRADIENT ESTIMATE FOR ASYMPTOTICALLY REGULAR VARIATIONAL INTEGRALS WITH MULTI-PHASE

Abstract

We devote this article to the Calderón-Zygmund estimate for the minimizers of asymptotically regular variational integrals with multi-phase integrands. A local gradient $L^q$-estimate for such problem under a sharp regular assumption is obtained by approximating the solutions of asymptotically regular problems to the solutions of regular problems based on a new perturbation method, while the gradients of solutions close to infinity.