Regularity for minimizers of a class of non-autonomous functionals with sub-quadratic growth

TL;DR

This paper establishes higher differentiability and regularity results for minimizers of non-autonomous integral functionals with sub-quadratic growth, under Sobolev regularity assumptions on the integrand's oscillation.

Contribution

It extends regularity theory to functionals with sub-quadratic growth, showing Lipschitz continuity and higher integrability under new conditions.

Findings

Higher differentiability of minimizers

Lipschitz regularity for q > n

Higher integrability of gradients for q = n

Abstract

We consider a class of integral functionals with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space W^{1;q}. We prove a result of higer differentiability for the minimizers. We also infer a result of Lipschitz regularity of minimizers if q > n, and a result of higher integrability for the gradient if q = n. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.