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

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 $F:\Omega \times {\mathbb{R}}^N\times {\mathbb{R}}^{N\times n}\to \mathbb{R}$ 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 ${\mathrm{C}}^{1,\beta }$ 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.

35 pp

Cited by in corpus (2)

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