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Spectral estimates of the Dirichlet-Laplace operator in conformal regular domains (2309.13616v1)

Published 24 Sep 2023 in math.AP

Abstract: In this paper we consider conformal spectral estimates of the Dirichlet-Laplace operator in conformal regular domains $\Omega \subset \mathbb R2$. This study is based on the geometric theory of composition operators on Sobolev spaces that permits us to estimate constants of the Poincar\'e-Sobolev inequalities. On this base we obtain lower estimates of the first eigenvalue of the Dirichlet-Laplace operator in a class of conformal regular domains. As a consequence we obtain conformal estimates of the ground state energy of quantum billiards.

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