Pathological solutions to elliptic problems in divergence form with continuous coefficients

TL;DR

This paper constructs explicit solutions to divergence form elliptic equations with continuous coefficients that exhibit irregular regularity properties, including solutions that are not in certain Sobolev spaces.

Contribution

It provides explicit examples of solutions with pathological regularity behavior for divergence form elliptic equations with continuous coefficients.

Findings

Existence of solutions in W^{1,1}_{loc} that are not in W^{1,p}_{loc} for p > 1.

Existence of solutions in W^{1,1}_{loc} that are in W^{1,p}_{loc} for all p < ∞ but not in W^{1,∞}_{loc}.

Continuous coefficients do not guarantee higher regularity of solutions.

Abstract

We construct a function $u \in W^{1,1}_{\mathrm{loc}} (B(0,1))$ which is a solution to $\Div (A \nabla u)=0$ in the sense of distributions, where $A$ is continuous and $u \not \in W^{1,p}_{\mathrm{loc}} (B(0,1))$ for $p > 1$. We also give a function $u \in W^{1,1}_{\mathrm{loc}} (B(0,1))$ such that $u \in W^{1,p}_{\mathrm{loc}}(B(0,1))$ for every $p < \infty$, $u$ satisfies $\Div (A \nabla u)=0$ with $A$ continuous but $u \not \in W^{1, \infty}_{\mathrm{loc}}(B(0,1))$.