Partial Regularity of Solutions to $\bm{p(x)}$-Laplacian PDEs with Discontinuous Coefficients
For $Ω\subseteq\mathbb{R}^{n}$ an open and bounded region we consider solutions $u\in W_{\text{loc}}^{1,p(x)}\big(Ω;\mathbb{R}^{N}\big)$, 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\inΩ,\notag \end{equation} where concerning the coefficient function $x\mapsto a(x)$ we assume only that \begin{equation} a\in W^{1,q}(Ω)\cap L^{\infty}(Ω),\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,α}\big(Ω_0\big),\notag \end{equation} for each $0<α<1$, where $Ω_0\subseteqΩ$ 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_Ωa(x)|Du|^{p(x)}\ dx.\notag \end{equation}