Short note on energy maximization property of the first eigenfunction of the Laplacian

TL;DR

This paper investigates the energy maximization problem for solutions of the Laplacian with Dirichlet boundary conditions, revealing that the maximizers are the first eigenfunctions corresponding to the eigenvalues.

Contribution

It establishes that the first eigenfunctions uniquely maximize the Dirichlet energy among all unit-norm functions in L^2(D).

Findings

First eigenfunctions are the unique maximizers of the Dirichlet energy.

Maximizers are explicitly characterized as the first eigenfunctions with boundary conditions.

The result links eigenfunction properties to energy optimization in PDEs.

Abstract

We consider the Dirichlet-energy maximization problem of the solution $u_f$ of (\ref{main}), among all functions $f\in L^2(D)$, such that $\|f\|_2= 1$. We show that the two maximizers are the first eigenfunctions of the Laplacian with Dirichlet boundary condition $f=\pm u_1$.