An indefinite Laplacian on a rectangle

TL;DR

This paper studies a nonelliptic differential operator called the indefinite Laplacian on a rectangle, addressing the challenge of defining a selfadjoint operator in L^2, relevant for modeling metamaterials with negative permittivity or permeability.

Contribution

It provides a method to associate a selfadjoint operator with the indefinite Laplacian on a rectangle, a problem previously unresolved.

Findings

Successfully constructs a selfadjoint operator for the indefinite Laplacian

Clarifies the mathematical foundation for modeling metamaterials with negative parameters

Enhances understanding of nonelliptic differential operators in bounded domains

Abstract

In this note we investigate the nonelliptic differential expression A=-div sgn grad on a rectangular domain in the plane. The seemingly simple problem to associate a selfadjoint operator with the differential expression A in an L^2 setting is solved here. Such indefinite Laplacians arise in mathematical models of metamaterials characterized by negative electric permittivity and/or negative magnetic permeability.