A Strong Maximum Principle for Parabolic Systems in a Convex Set with Arbitrary Boundary

TL;DR

This paper establishes a strong maximum principle for parabolic systems in convex sets with arbitrary boundary regularity, utilizing viscosity solutions to extend applicability to all convex sets.

Contribution

It introduces a novel approach that removes boundary regularity restrictions, broadening the scope of maximum principles for parabolic systems.

Findings

Maximum principle holds for all convex sets regardless of boundary regularity

Viscosity solutions are effective in establishing maximum principles in this context

Results apply to a wide class of parabolic systems in convex domains

Abstract

In this paper we prove a strong maximum principle for certain parabolic systems of equations. In particular, our methods place no restriction on the regularity of the boundary of the convex set in which the system takes its values, and therefore our results hold for any convex set. We achieve this through the use of viscosity solutions and their corresponding strong maximum principle.