Asymptotically quasiconvex functionals with general growth conditions

TL;DR

This paper proves local regularity of minimizers for vectorial integrals with general growth conditions, assuming asymptotic quasiconvexity, and establishes $C^{1,eta}$ and Lipschitz regularity results under $ riangle_2$ conditions.

Contribution

It introduces asymptotic quasiconvexity assumptions for regularity results in calculus of variations with general growth conditions.

Findings

$C^{1,eta}$ regularity at points with large gradient

Lipschitz regularity on a dense set

Results valid for all Young functions satisfying $ riangle_2$

Abstract

We establish local regularity results for minimizers of autonomous vectorial integrals of Calculus of Variations, assuming $\psi$-growth conditions and imposing $\varphi$-quasiconvexity only in an asymptotic sense, both in the sub-quadratic and super-quadratic case. In particular, we obtain $C^{1,\alpha}$ regularity at points $x_0$ where $Du$ is sufficiently large in a neighborhood of $x_0$, as well as Lipschitz regularity on a dense set. \ The results hold for all pairs of Young functions $(\varphi, \psi)$ satisfying the $\Delta_2$ condition.