Courant's Nodal Domain Theorem for Positivity Preserving Forms

TL;DR

This paper generalizes the concept of nodal domains to positivity preserving forms and proves a Courant nodal domain theorem within this broader framework using analytical techniques.

Contribution

It introduces a new notion of nodal domains for positivity preserving forms and establishes a Courant-type theorem in this generalized setting.

Findings

Generalization of nodal domains to positivity preserving forms

Proof of Courant's nodal domain theorem in the new setting

Analytical methods used for the proof

Abstract

We introduce a notion of nodal domains for positivity preserving forms. This notion generalizes the classical ones for Laplacians on domains and on graphs. We prove the Courant nodal domain theorem in this generalized setting using purely analytical methods.