Spectral estimates of the $p$-Laplace Neumann operator in conformal regular domains

Vladimir Gol'dshtein; Alexander Ukhlov

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arXiv · 1601.01533

Spectral estimates of the $p$-Laplace Neumann operator in conformal regular domains

Vladimir Gol'dshtein·Alexander Ukhlov

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Abstract

In this paper we study spectral estimates of the $p$-Laplace Neumann operator in conformal regular domains $Ω\subset\mathbb R^2$. This study is based on (weighted) Poincaré-Sobolev inequalities. The main technical tool is the composition operators theory in relation with the Brennan's conjecture. We prove that if the Brennan's conjecture holds then for any $p\in (4/3,2)$ and $r\in (1,p/(2-p))$ the weighted $(r,p)$-Poincare-Sobolev inequality holds with the constant depending on the conformal geometry of $Ω$. As a consequence we obtain classical Poincare-Sobolev inequalities and spectral estimates for the first nontrivial eigenvalue of the $p$-Laplace Neumann operator for conformal regular domains.

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Vladimir Gol'dshtein, Alexander Ukhlov. 2016-01-17. Spectral estimates of the $p$-Laplace Neumann operator in conformal regular domains. https://arxiv.org/abs/1601.01533

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