## An overdetermined problem in Riesz-potential and fractional Laplacian.(English)Zbl 1236.31004

This paper addresses two open questions raised by W. Reichel [Ann. Mat. Pura Appl. (4) 188, No. 2, 235–245 (2009; Zbl 1180.31008)] concerning the characterization of balls in terms of Riesz potentials and fractional Laplacian. More precisely, let $$u(x)=\int_{\Omega} |x-y|^{\alpha-N}dy$$ if $$N\neq \alpha$$ and $$u(x)=\int_{\Omega} \log |x-y|^{-1}dy$$ otherwise. The first result of the paper establishes that if $$\alpha>1$$, $$\Omega$$ is a $$C^1$$ bounded domain and $$u$$ is constant on $$\partial\Omega$$, then $$\Omega$$ is a ball. Next, a similar result is obtained for $$v(x)=\int_{\Omega} |x-y|^{\alpha-N}\log |x-y|^{-1}dy$$. More precisely, if $$\alpha>N$$, $$\Omega$$ is a $$C^1$$ bounded domain with diam$$(\Omega)<e^{1/(N-\alpha)}$$ and $$v$$ is constant on $$\partial\Omega$$, then $$\Omega$$ is a ball. An even more general result is obtained by the authors which is as follows:
Let $$w(x)=\int_{\Omega} g(|x-y|)dy$$, where $$\Omega$$ is a $$C^1$$ bounded domain in $$\mathbb R^N$$ and $$g$$ is a $$C^1(0,\infty)$$ function such that $$g'$$ has constant sign on (0,diam$$(\Omega)$$) and
(i) $$\varepsilon \int_0^{\varepsilon} |g(r)|r^{N-1}dr\rightarrow 0$$ as $$\varepsilon \rightarrow 0$$;
(ii) $$\int_0^{\varepsilon} |g'(r)|r^{N-1}dr\rightarrow 0$$ as $$\varepsilon \rightarrow 0$$.
If $$w$$ is constant on $$\partial\Omega$$ then $$\Omega$$ has to be a ball.
The second part of the paper is concerned with integral equations of the form $u(x)=\int_G \frac{f(u)}{|x-y|^{N-\alpha}}dy\,,$ where $$1<\alpha<N$$ and $$G$$ is an exterior domain whose complement is a bounded and connected $$C^1$$ domain. Under some natural assumptions on $$f$$ it is proved that the above integral equation has no solutions $$u$$ which are constant on $$\partial G$$.

### MSC:

 31B10 Integral representations, integral operators, integral equations methods in higher dimensions 35N25 Overdetermined boundary value problems for PDEs and systems of PDEs

Zbl 1180.31008
Full Text:

### References:

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