# Regularity of Poisson problem with rough coefficients and mixed boundary conditions

Let $d \in \mathbb N$ and $\Omega \subseteq \mathbb{R}^d$ be open, bounded and connected. Let the boundary $\partial \Omega$ be piecewise Lipschitz and partitioned into a Neumann part $\partial \Omega_N$ and a Dirichlet part $\partial \Omega_D$ and let $\nu$ be an outer unit normal vector. Let $A\colon \Omega \rightarrow \mathbb{R}^{d\times d}$ be uniformly elliptic, bounded, measurable and rough (i.e. discontinuous). Let $f \in L^2(\Omega$) and $u_D \in H^{1/2}(\partial \Omega_D)$.

Under what conditions (should the conditions above not suffice) on $f$, $u_D$ and $\Omega$ is the solution $u$ of the Poisson problem

\begin{alignat}{2} -\nabla \cdot (A(x)\nabla u(x)) & = f(x) &\qquad& \forall x \in \Omega,\\ u(x) & = u_D(x) && \forall x \in \partial \Omega_D,\\ \nu \cdot A(x)\nabla u(x)& = 0 && \forall x \in \partial \Omega_N, \end{alignat} regular enough such that $u \in L^\infty (\Omega)$?

So far, I haven't been able to find literature on this. The closest I've gotten is Theorem 4.1 in this paper: Denote by $B_1$ the unit ball, then if $u \in W^{1,p}(B_1)$ is a solution to $$-\nabla \cdot (A(x)\nabla u(x)) = f(x) \qquad \forall x \in B_1,$$ with $f \in L_{\text{weak}}^{\theta \cdot d/p}(B_1), 1 < \theta < p,$ then $u \in C^{\alpha}(B_{1/2})$ for some $\alpha > 0$. The case where $f=0$ seems to be a somewhat classical result due to De Giorgi.

The question seems quite natural to me, but unfortunately I have not had a lot of success with my literature review.