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- W3194927135 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{ intint_{{u > 0}times{u>0}} frac{|u(x) - u(y)|^2}{|x - y|^{n + 2 sigma}}d x d y : u in mathring H^sigma(mathbb{R}^n), int_{mathbb{R}^n} u^2 = 1, |{u > 0 }| leq 1right}. ] 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." @default.
- W3194927135 created "2021-08-30" @default.
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- W3194927135 date "2021-08-18" @default.
- W3194927135 modified "2023-09-24" @default.
- W3194927135 title "On a Rayleigh-Faber-Krahn inequality for the regional fractional Laplacian" @default.
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- W3194927135 doi "https://doi.org/10.48550/arxiv.2108.07967" @default.
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