Partial regularity for local minimizers of variational integrals with lower order terms

TL;DR

This paper proves partial regularity of local minimizers for a class of variational integrals with lower order terms, extending previous results to cases where the integrand depends on the function itself, under certain regularity and positivity conditions.

Contribution

It extends partial regularity results to variational integrals with lower order terms and u-dependence, using a direct strategy instead of blow-up arguments.

Findings

Strong local minimizers are of class C^{1,β} on a full measure subset of the domain.

Partial regularity also applies to certain weak local minimizers with positive second variation.

The approach avoids blow-up arguments, providing a new direct proof method.

Abstract

We consider functionals of the form $$\mathcal{F}(u):=\int_\Omega\!F(x,u,\nabla u)\,\mathrm{d} x,$$ where $\Omega\subseteq\mathbb{R}^n$ is open and bounded. The integrand $F\colon\Omega\times\mathbb{R}^N\times\mathbb{R}^{N\times n}\to\mathbb{R}$ is assumed to satisfy the classical assumptions of a power $p$-growth and the corresponding strong quasiconvexity. In addition, $F$ is H\"older continuous with exponent $2\beta\in(0,1)$ in its first two variables uniformly with respect to the third variable, and bounded below by a quasiconvex function depending only on $z\in\mathbb{R}^{N\times n}$. We establish that strong local minimizers of $\mathcal{F}$ are of class $\mathrm{C}^{1,\beta}$ in an open subset $\Omega_0\subseteq\Omega$ with $\mathcal{L}^n(\Omega\setminus\Omega_0)=0$. This partial regularity also holds for a certain class of weak local minimizers at which the second variation is strongly positive and satisfying a $\mathrm{BMO}$-smallness condition. This extends the partial regularity result for local minimizers by Kristensen and Taheri (2003) to the case where the integrand depends also on $u$. Furthermore, we provide a direct strategy for this result, in contrast to the blow-up argument used for the case of homogeneous integrands.