Partial regularity of minimizers for double phase functionals with variable exponents

Abstract

The aim of this article is to study partial regularity of a minimizer \(\varvec{u:\Omega \subset \mathbb {R}^n \rightarrow \mathbb {R}^N}\) for a double phase functional with variable exponents:

$$\begin{aligned} \varvec{\int \left( \vert Du\vert _A^{p(x)} + a(x) { \vert } Du{ \vert } _A^{q(x)}\right) dx,} \end{aligned}$$

where \(\varvec{{ \vert } \cdot { \vert }_A}\) stands for the norm deduced from a positive definite sufficiently continuous tensor field \(\varvec{A:=\big (A_{\alpha \beta }^{ij} (x,u)\big )~~((x,u) \in \Omega \times \mathbb {R}^N)}\). We show that a minimizer \(\varvec{u}\) is in the class \(\varvec{C^{1,\gamma }(\Omega _1;\mathbb {R}^N)}\) for some constant \(\varvec{\gamma \in (0,1)}\) and open subset \(\varvec{\Omega _1 \subset \Omega }\). We obtain also an estimate for the Hausdorff dimension of \(\varvec{\Omega \setminus \Omega _1}\).