Loading...
Please wait, while we are loading the content...
Similar Documents
An overdetermined problem in Riesz-potential and fractional Laplacian
| Content Provider | Semantic Scholar |
|---|---|
| Author | Lu, Guozhen Zhu, Jiuyi |
| Copyright Year | 2011 |
| Abstract | 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. |
| Starting Page | 3036 |
| Ending Page | 3048 |
| Page Count | 13 |
| File Format | PDF HTM / HTML |
| DOI | 10.1016/j.na.2011.11.036 |
| Alternate Webpage(s) | http://www.math.wayne.edu/~gzlu/papers/LuZhu_NA.pdf |
| Alternate Webpage(s) | http://www.math.wayne.edu/~gzlu/papers/LuZhu_NA_Online.pdf |
| Alternate Webpage(s) | https://arxiv.org/pdf/1101.1649v2.pdf |
| Alternate Webpage(s) | https://doi.org/10.1016/j.na.2011.11.036 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
