Overdetermined problems with fractional Laplacian

By M. Fall, Sven Jarohs. 2013

109 citations recorded by the source corpus.

Let $N\geq 1$ and $s\in (0,1)$. In the present work we characterize bounded open sets $\Omega$ with $ C^2$ boundary (\textit{not necessarily connected}) for which the following overdetermined problem \begin{equation*} ( -\Delta)^s u = f(u) \text{ in $\Omega$,} \qquad u=0 \text{ in $\mathbb{R}^N\setminus \Omega$,} \qquad(\partial_{\eta})_s u=Const. \text{ on $\partial \Omega$} \end{equation*} has a nonnegative and nontrivial solution, where $\eta $ is the outer unit normal vectorfield along $\partial\Omega$ and for $x_0\in\partial\Omega$ \[ \left(\partial_{\eta}\right)_{s}u(x_{0})=-\lim_{t\to 0}\frac{u(x_{0}-t\eta(x_0))}{t^s}. \] Under mild assumptions on $f$, we prove that $\Omega$ must be a ball. In the special case $f\equiv 1$, we obtain an extension of Serrin's result in 1971. The fact that $\Omega$ is not assumed to be connected is related to the nonlocal property of the fractional Laplacian. The main ingredients in our proof are maximum principles and the method of moving planes.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/cec499876f15e77425b8abfe4983abc949bc6e50

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Citations in the corpus

Uniqueness (2021, unknown): “The strict positivity u > 0 in Ω follows from the maximum principle; see, for example, [39,48].”
Hopf’s lemma and constrained radial symmetry for the fractional Laplacian (2016, unknown): “is considered in [12]: assuming that Ω is a bounded open set with a C2 boundary, in this paper the authors prove that such a problem is solvable only if the domain is a ball, thus extending to the fractional case the celebrated result by Serrin [30] and Weinberger [32] (see also [10] for the case N ”
On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results (2016, background): “2], and constituted the base point in the study of overdetermined problems for the fractional Laplacian, see [17,24,30,34,44].”
A Hopf's lemma and a strong minimum principle for the fractional p-Laplacian (2016, background): “alized Hopf lemma for fractional harmonic functions. Whereas, in [30], under the assumption Ω is a smooth bounded domain, it is proven a Hopf lemma for weak solutions of a Dirichlet problem. See also [15, 29]. For a Hopf lemma with mixed boundary condition, see [2]. Finally, Theorems 1.4 and 1.5 are”