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| On a Rayleigh-Faber-Krahn Inequality forthe Regional Fractional Laplacian |
| Tianling Jin,Dennis Kriventsov,Jingang Xiong |
| (Department of Mathematics, The Hong Kong University of Science and
Technology, Clear Water Bay, Kowloon, Hong Kong;Department of Mathematics, Rutgers University, 110 Frelinghuysen Road,
Piscataway, NJ 08854, USA;School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University, Beijing 100875, China) |
| DOI： |
| Abstract: |
| 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
\[
\left\{ \iint_{\{u > 0\}\times\{u>0\}} \frac{|u(x) - u(y)|^2}{|x - y|^{n + 2 \sigma}}\ud x\ud y : u \in \mathring H^\sigma(\R^n), \int_{\R^n} u^2 = 1, |\{u > 0 \}| \leq 1\right\}.
\]
Unlike the corresponding problem for the usual fractional Laplacian, where the domain of the integration is $\R^n \times \R^n$, symmetrization techniques may not apply here. Our approach is instead based on the direct method and new \emph{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. |
| Key words: Rayleigh-Faber-Krahn inequality, regional fractional Laplacian, first eigen value |