Some Key Properties of Eigenfunctions Linked to Degenerate Elliptic Differential Operators

TL;DR

This paper extends classical eigenfunction properties, including Courant's nodal domain theorem, to degenerate elliptic operators where the weight vanishes on part of the boundary, showing key spectral properties still hold.

Contribution

It proves that the residual set of perturbations making eigenvalues simple persists for degenerate elliptic operators, despite their loss of uniform ellipticity.

Findings

Established Courant's nodal domain theorem for degenerate elliptic operators.

Proved the residual set of perturbations with simple eigenvalues exists for these operators.

Demonstrated key spectral properties are retained despite degeneracy.

Abstract

In this study, we address the eigenvalue problem given by: \begin{equation*} \begin{cases} -\Div (w\nabla u_i)=\la_iu_i &\text{in } \Om\subset \mathbb{R}^n,\\ u_i=0 &\text{on } \pt \Om, \end{cases} \end{equation*} where $w > 0$ within $\Om$ and $w = 0$ on part of $\partial \Omega$. We establish Courant's nodal domain theorem for the corresponding degenerate elliptic differential operator $\mathcal{A}$. Unlike uniformly elliptic operators, degenerate cases often result in the loss of many advantageous properties. Despite this, we show that the essential property that the set $\{\rho \in L^\infty(\Omega) \colon \mathcal{A} + \rho \text{ has simple eigenvalues}\}$ forms a residual subset within $(L^\infty(\Omega), |\cdot|_\infty)$ still holds for the degenerate elliptic differential operator $\mathcal{A}$.