Partial Regularity of Solutions to $\bm{p(x)}$-Laplacian PDEs with Discontinuous Coefficients

Abstract

For $\Omega\subseteq\mathbb{R}^{n}$ an open and bounded region we consider solutions $u\in W_{\text{loc}}^{1,p(x)}\big(\Omega;\mathbb{R}^{N}\big)$, with $N>1$, of the $p(x)$-Laplacian system \begin{equation} \nabla\cdot\left(a(x)|Du|^{p(x)-2}Du\right)=0\text{, a.e. }x\in\Omega,\notag \end{equation} where concerning the coefficient function $x\mapsto a(x)$ we assume only that \begin{equation} a\in W^{1,q}(\Omega)\cap L^{\infty}(\Omega),\notag \end{equation} where $q>1$ is essentially arbitrary. This implies that the coefficient in the PDE can be highly irregular, and yet in spite of this we still recover that \begin{equation} u\in\mathscr{C}_{\text{loc}}^{0,\alpha}\big(\Omega_0\big),\notag \end{equation} for each $0<\alpha<1$, where $\Omega_0\subseteq\Omega$ is a set of full measure. Due to the variational methodology that we employ, our results apply to the more general question of the regularity of the integral functional \begin{equation} \int_{\Omega}a(x)|Du|^{p(x)}\ dx.\notag \end{equation}