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Local Overdetermined Linear Elliptic Problems in Lipschitz Domains with Solutions Changing Sign
Canuto, Bruno
Rial, Diego
2009
Abstract
We prove that the only domain $\Omega$ such that there exists a solution to the following overdetermined problem $\Deltau+\omega2u=−1$ in in $\Omega$, u = 0 on $\partial\Omega$, and $\partialnu = c$ on $\partial\Omega$, is the ball B1, independently on the sign of u, if we assume that the boundary $\partial\Omega$ is a perturbation (no necessarily regular) of the unit sphere $\partialB1$ of Rn. Here $\omega2 \neq (\lambdan)n\geq1$ (the eigenvalues of $−\Delta$ in B1 with Dirichlet boundary conditions), and $\omega \Lambda$, where $\Lambda$ is a enumerable set of R+, whose limit points are the values $\lambda1m$, for some integer $m\geq1$, $\lambda1m$ being the mth-zero of the first-order Bessel function I1.
Series
Rendiconti dell'Istituto di Matematica dell'Università di Trieste. An International Journal of Mathematics
40 (2008)
Publisher
EUT - Edizioni Università di Trieste
Source
Bruno Canuto, Diego Rial, "Local Overdetermined Linear Elliptic Problems in Lipschitz Domains with Solutions Changing Sign", in Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 40 (2008), pp. 1-27.
Languages
en
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