Overdetermined problems with fractional laplacian

Fall, Mouhamed Moustapha; Jarohs, Sven

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 924-938.

Let $N\ge 1$ and $s\in (0,1)$. In the present work we characterize bounded open sets $\Omega$ with $C^2$ boundary (not necessarily connected) for which the following overdetermined problem

$$\begin{array}{c}\hfill \left\{\begin{array}{cccc}\hfill {(-\Delta )}^{s}u& =f\left(u\right)\hfill & & \text{in}\Omega ;\hfill \\ \hfill u& =0\hfill & & \text{in}{ℝ}^N\setminus \Omega ;\hfill \\ \hfill {\left({\partial }_{\eta }\right)}_{s}u& =\mathrm{Const}.\hfill & & \text{on}\partial \Omega \hfill \end{array}\right.\end{array}$$

has a nonnegative and nontrivial solution, where $\eta$ is the outer unit normal vectorfield along $\partial \Omega$ and for $x_0\in \partial \Omega$

$$\begin{array}{c}\hfill {\left({\partial }_{\eta }\right)}_{s}u\left(x_0\right)=-\underset{t\to 0}{lim}\frac{u(x_0-t\eta \left(x_0\right))}{t^s}.\end{array}$$

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.

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