Invariant cones for linear elliptic systems with gradient coupling

TL;DR

This paper investigates conditions under which linear elliptic systems with gradient and zero order couplings preserve certain invariant cones, providing counterexamples and establishing algebraic criteria for maximum principle validity.

Contribution

It introduces a general algebraic condition ensuring invariant cones for coupled elliptic systems, extending the understanding of maximum principle applicability.

Findings

Counterexamples to weak Maximum Principle in coupled systems

Algebraic conditions guarantee invariant cones

Reduction to nonlinear scalar equations supports results

Abstract

We discuss counterexamples to the validity of the weak Maximum Principle for linear elliptic systems with zero and first order couplings and prove, through a suitable reduction to a nonlinear scalar equation, a quite general result showing that some algebraic condition on the structure of gradient couplings and a cooperativity condition on the matrix of zero order couplings guarantee the existence of invariant cones in the sense of Weinberger [21].