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An operator theoretic approach to uniform (anti-)maximum principles

Sahiba Arora, Jochen Glück

Abstract

Maximum principles and uniform anti-maximum principles are a ubiquitous topic in PDE theory that is closely tied to the Krein--Rutman theorem and kernel estimates for resolvents. We take up a classical idea of Takáč - to prove (anti-)maximum principles in an abstract operator theoretic framework - and combine it with recent ideas from the theory of eventually positive operator semigroups. This enables us to derive necessary and sufficient conditions for (anti-)maximum principles in a very general setting. Consequently, we are able to either prove or disprove (anti-)maximum principles for a large variety of concrete differential operators. As a bonus, for several operators that are already known to satisfy or to not satisfy anti-maximum principles, our theory gives a very clear and concise explanation of this behaviour.

An operator theoretic approach to uniform (anti-)maximum principles

Abstract

Maximum principles and uniform anti-maximum principles are a ubiquitous topic in PDE theory that is closely tied to the Krein--Rutman theorem and kernel estimates for resolvents. We take up a classical idea of Takáč - to prove (anti-)maximum principles in an abstract operator theoretic framework - and combine it with recent ideas from the theory of eventually positive operator semigroups. This enables us to derive necessary and sufficient conditions for (anti-)maximum principles in a very general setting. Consequently, we are able to either prove or disprove (anti-)maximum principles for a large variety of concrete differential operators. As a bonus, for several operators that are already known to satisfy or to not satisfy anti-maximum principles, our theory gives a very clear and concise explanation of this behaviour.

Paper Structure

This paper contains 11 sections, 30 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Upper and lower estimates by rank-$1$ operators
  3. Compactness of the resolvent
  4. Estimating the resolvent and its powers
  5. Eventual positivity of resolvents and their powers
  6. Examples and applications
  7. Laplace operators on intervals
  8. An anti-maximum principle for a Laplace operator on a metric graph
  9. Polyharmonic operators with Dirichlet boundary conditions revisited
  10. Differential operators of odd order
  11. A delay differential operator

Key Result

Theorem 1.2

Assume that both the domination and the spectral assumption are satisfied in Setting sett:general. Then every spectral value of $A$ is isolated and the following assertions hold for the eigenvalue $\lambda_0$.

Theorems & Definitions (60)

  • Theorem 1.2
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • Corollary 2.7
  • ...and 50 more