Spectral estimates of the Dirichlet-Laplace operator in conformal regular domains

2023/09/24 by И. М. Колесников, Valerii Pchelintsev, Kolesnikov, Ivan +1
Mathematics · #30C35 #35P15 #46E35 #81Q50 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2309.13616

openalex publication_date 2023/09/24 · openalex created_date 2023/09/27 · openalex updated_date 2026/07/28

Abstract

In this paper we consider conformal spectral estimates of the Dirichlet-Laplace operator in conformal regular domains Ω⊂ \mathbb R². This study is based on the geometric theory of composition operators on Sobolev spaces that permits us to estimate constants of the Poincaré-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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