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On a conjectured reverse Faber-Krahn inequality for a Steklov--type Laplacian eigenvalue

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  • For a given bounded Lipschitz set $\Omega$, we consider a Steklov--type eigenvalue problem for the Laplacian operator whose solutions provide extremal functions for the compact embedding $H^1(\Omega)\hookrightarrow L^2(\partial \Omega)$. We prove that a conjectured reverse Faber--Krahn inequality holds true at least in the class of Lipschitz sets which are ``close'' to a ball in a Hausdorff metric sense. The result implies that among sets of prescribed measure, balls are local minimizers of the embedding constant.
    Mathematics Subject Classification: Primary: 46E35, 35P15; Secondary: 28A75.


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