On the Weyl Asymptotic Formula for Euclidean Domains of Finite Volume

TL;DR

This paper provides a straightforward proof of the Weyl asymptotic formula for eigenvalues of various differential operators in Euclidean domains of finite volume, without boundary assumptions.

Contribution

It introduces a simple proof technique for the Weyl asymptotic formula applicable to multiple operators in finite-volume Euclidean domains.

Findings

Proof of Weyl asymptotic formula for Dirichlet Laplacian

Extension to buckling problem and Dirichlet bi-Laplacian

No boundary assumptions required

Abstract

We give a simple proof of the Weyl asymptotic formula for eigenvalues of the Dirichlet Laplacian, the buckling problem, and the Dirichlet bi-Laplacian in Euclidean domains of finite volume, with no assumptions about the boundary.