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

Judith Campos Cordero

Published 2021-08-19Version 1

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 polynomial $p$-growth and strong quasiconvexity. In addition, $F$ is H\"older continuous 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, for every $\alpha\in(0,1)$, strong local minimizers of $\mathcal{F}$ are of class $\mathrm{C}^{1,\alpha}$ in a subset $\Omega_0\subseteq\Omega$ with $\Omega_0$ of full $n$-dimensional measure. This extends the partial regularity result for strong 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.