An overdetermined problem in Riesz-potential and fractional Laplacian
69 citations recorded by the source corpus.
The main purpose of this paper is to address two open questions raised by W. Reichel on characterizations of balls in terms of the Riesz potential and fractional Laplacian. For a bounded $C^1$ domain $\Omega\subset \mathbb R^N$, we consider the Riesz-potential $$u(x)=\int_{\Omega}\frac{1}{|x-y|^{N-\alpha}} \,dy$$ for $2\leq \alpha \not =N$. We show that $u=$ constant on the boundary of $\Omega$ if and only if $\Omega$ is a ball. In the case of $ \alpha=N$, the similar characterization is established for the logarithmic potential. We also prove that such a characterization holds for the logarithmic Riesz potential. This provides a characterization for the overdetermined problem of the fractional Laplacian. These results answer two open questions of W. Reichel to some extent.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/afa1357bef30d7fc04c26519c2c6283eb6ade35c
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Computational Theory and Mathematics
In Mathematical Physics
Citations in the corpus
- Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian (2015, background): “For more Liouville type theorems, please see [7, 16, 17, 18, 19, 21, 22, 23, 28] and the references therein.”