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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

Citations:

Zbl 1180.31008
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References:

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