On the First Eigenvalue of the Degenerate $p$-Laplace Operator in Non-Convex Domains

V. Gol'dshtein; V. Pchelintsev; A. Ukhlov

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arXiv · 1707.08867

On the First Eigenvalue of the Degenerate $p$-Laplace Operator in Non-Convex Domains

V. Gol'dshtein·V. Pchelintsev·A. Ukhlov

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Abstract

In this paper we obtain lower estimates of the first non-trivial eigenvalues of the degenerate $p$-Laplace operator, $p>2$, in a large class of non-convex domains. This study is based on applications of the geometric theory of composition operators on Sobolev spaces that permits us to estimates constants of Poincar\'e-Sobolev inequalities and as an application to derive lower estimates of the first non-trivial eigenvalues for the Alhfors domains (i.e. to quasidiscs). This class of domains includes some snowflakes type domains with fractal boundaries.

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V. Gol'dshtein, V. Pchelintsev, A. Ukhlov. 2017-07-27. On the First Eigenvalue of the Degenerate $p$-Laplace Operator in Non-Convex Domains. https://arxiv.org/abs/1707.08867

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