Partial Regularity for Local Minimizers of Variational Integrals With Lower-Order Terms

Abstract

Abstract We consider functionals of the form $$\begin{equation*} \mathcal{F}(u):=\int_\Omega\!F(x,u,\nabla u)\,\mathrm{d} x, \end{equation*}$$ 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 Hölder 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 $\operatorname{C}^{1,\beta}$ 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 bounded mean oscillation (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.