Pathological solutions to elliptic problems in divergence form with continuous coefficients

Abstract

We construct a function $u\in W_{\mathrm{loc}}^{1,1}(B(0,1))$ which is a solution to $\Div (A \nabla u)=0$ in the sense of distributions, where $A$ is continuous and $u\notin W_{\mathrm{loc}}^{1,p}(B(0,1))$ for $p>1$. We also give a function $u\in W_{\mathrm{loc}}^{1,1}(B(0,1))$ such that $u\in W_{\mathrm{loc}}^{1,p}(B(0,1))$ for every $p<\infty$, $u$ satisfies $\Div (A \nabla u)=0$ with $A$ continuous but $u\notin W_{\mathrm{loc}}^{1,\infty }(B(0,1))$.

6 pages

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