Semilinear elliptic equations on rough domains

TL;DR

This paper develops a framework for analyzing semilinear elliptic equations on irregular domains using Banach lattice theory, avoiding traditional boundary regularity assumptions and employing Kato's inequality for key results.

Contribution

It introduces a novel approach leveraging Banach lattices and positive operators to handle minimal boundary regularity in semilinear elliptic equations.

Findings

Proves a comparison theorem for eigenfunctions without boundary regularity.

Establishes existence, uniqueness, and stability of positive solutions for a degenerate logistic equation.

Applies the theory to various operators including Dirichlet and Robin Laplacians on irregular domains.

Abstract

The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the usual arguments based on maximum principles do not apply. A key result is an application of Kato's inequality to prove a comparison theorem for eigenfunctions that only requires interior regularity and avoids the use of the Hopf boundary maximum principle. We demonstrate the theory on an abstract degenerate logistic equation by proving the existence, uniqueness and stability of non-trivial positive solutions. Examples of operators include the Dirichlet Laplacian on arbitrary bounded domains, a simplified construction of the Robin Laplacian on arbitrary domains with boundary of finite measure and general elliptic operators in divergence form.