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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.