L_1-Estimates for Eigenfunctions of the Dirichlet Laplacian

TL;DR

This paper derives $L_1$-norm estimates for eigenfunctions of the Dirichlet Laplacian below the essential spectrum and applies these estimates to compare heat content and heat trace in open sets.

Contribution

It provides new $L_1$-estimates for eigenfunctions and uses them to relate heat content and heat trace, extending understanding of spectral properties of the Dirichlet Laplacian.

Findings

Established $L_1$-estimates for eigenfunctions below the essential spectrum.

Derived two-sided bounds relating heat content and heat trace.

Proved all eigenfunctions associated with discrete eigenvalues are in $L_1( ext{Omega})$.

Abstract

For $d \in \N$ and $\Omega \ne \emptyset$ an open set in $\R^d$, we consider the eigenfunctions $\Phi$ of the Dirichlet Laplacian $-\Delta_\Omega$ of $\Omega$. If $\Phi$ is associated with an eigenvalue below the essential spectrum of $-\Delta_\Omega$ we provide estimates for the $L_1$-norm of $\Phi$ in terms of its $L_2$-norm and spectral data. These $L_1$-estimates are then used in the comparison of the heat content of $\Omega$ at time $t>0$ and the heat trace at times $t' > 0$, where a two-sided estimate is established. We furthermore show that all eigenfunctions of $-\Delta_\Omega$ which are associated with a discrete eigenvalue of $H_\Omega$, belong to $L_1(\Omega)$.