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

Authors: Farid Bahrami and Henrik Shahgholian
Journal: Proc. Amer. Math. Soc. 126 (1998), 745-750
MSC (1991): Primary 31B20, 35J05, 35R35
MathSciNet review: 1422844
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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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Additional Information

Farid Bahrami
Affiliation: Department of Mathematics, University of Tehran, P.O. Box 13145-1873, Tehran, Iran

Henrik Shahgholian
Affiliation: Department of Mathematics, The Royal Institute of Technology, 100 44 Stockholm, Sweden

Keywords: Inverse domain problem, p-Laplacian, uniqueness
Received by editor(s): April 3, 1996
Received by editor(s) in revised form: August 28, 1996
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1998 American Mathematical Society

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