Regularity for minimizers of non-autonomous non-quadratic functionals in the case 1 < p < 2: an a priori estimate

TL;DR

This paper establishes an a priori estimate for second derivatives of local minimizers of certain integral functionals with subquadratic growth, advancing regularity theory in calculus of variations.

Contribution

It provides the first a priori second derivative estimate for minimizers of non-autonomous, non-quadratic functionals with 1 < p < 2, under Sobolev regularity assumptions.

Findings

Proves second derivative bounds for minimizers with subquadratic growth

Extends regularity results to non-autonomous functionals with oscillating integrands

Demonstrates applicability of Sobolev space conditions to regularity estimates

Abstract

We prove an a priori estimate for the second derivatives of local minimizers of integral functionals of calculus of variation 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. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.