Uniqueness for an overdetermined boundary value problem for the p-Laplacian

Bahrami, Farid; Shahgholian, Henrik

doi:10.1090/S0002-9939-98-04087-8

Uniqueness for an overdetermined boundary value problem for the p-Laplacian
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by Farid Bahrami and Henrik Shahgholian PDF

Proc. Amer. Math. Soc. 126 (1998), 745-750 Request permission

Abstract:

For $p>1$ set $\Delta _p u = {\mathrm {div}}(|\nabla u|^{p-2}\nabla u)$, and let $\mu$ be a measure with compact support. Suppose, for $j=1,2$, there are functions $u_j \in W^{1,p}$ and (bounded) domains $\Omega _j$, both containing the support of $\mu$ with the property that $\Delta _p u_j =\chi _{\Omega _j} - \mu$ in $\mathbf {R}^N$ (weakly) and $u_j=0$ in the complement of $\Omega _j$. If in addition $\Omega _1 \cap \Omega _2$ is convex, then $\Omega _1 \equiv \Omega _2$ and $u_1\equiv u_2$.

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