Pathological solutions to elliptic problems in divergence form with continuous coefficients

Details Pathological solutions to elliptic problems in divergence form with continuous coefficients Pathological solutions to elliptic problems in divergence form with continuous coefficients

2009-04-10

arxiv.org/abs/0904.1674v1

C. R. Math. Acad. Sci. Paris 347 (2009), no. 13-14, 773-778

Mathematics

Analysis of PDEs

6 pages

Scientific paper

10.1016/j.crma.2009.05.008

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 $u \not \in W^{1, \infty}_{\mathrm{loc}}(B(0,1))$.

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