@Article{AAM-37-3,
author = {Tianling, Jin and Kriventsov, Dennis and Xiong, Jingang},
title = {On a Rayleigh-Faber-Krahn Inequality for the Regional Fractional Laplacian},
journal = {Annals of Applied Mathematics},
year = {2021},
volume = {37},
number = {3},
pages = {363--393},
abstract = {<p style="text-align: justify;">We study a Rayleigh-Faber-Krahn inequality for regional fractional Laplacian operators. In particular, we show that there exists a compactly supported nonnegative Sobolev function $u_0$ that attains the infimum (which will be a positive real number) of the set$$\left\{\iint_{\{u&gt;0\}\times\{u&gt;0\}}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2\sigma}}dxdy:u\in H^{\sigma}(\mathbb{R}^n), \int_{\mathbb{R}^n}u^2=1, |\{u&gt;0\}|\le 1\right\}$$Unlike the corresponding problem for the usual fractional Laplacian, where the domain of the integration is $\mathbb{R}^n \times \mathbb{R}^n$, symmetrization techniques may not apply here. Our approach is instead based on the direct method and new a priori diameter estimates. We also present several remaining open questions concerning the regularity and shape of the minimizers, and the form of the Euler-Lagrange equations.</p>},
issn = {},
doi = {https://doi.org/10.4208/aam.OA-2021-0005},
url = {https://global-sci.com/article/72648/on-a-rayleigh-faber-krahn-inequality-for-the-regional-fractional-laplacian}
}