A general form of the weak maximum principle and some applications

TL;DR

This paper introduces new generalized forms of the weak and Omori-Yau maximum principles for linear and non-linear operators, demonstrating their applications in PDEs and hypersurface theory.

Contribution

It presents novel generalized maximum principles for linear and non-linear operators, expanding their applicability in geometric analysis and PDEs.

Findings

New weak and Omori-Yau maximum principles for trace type operators

Applications demonstrated in PDEs and hypersurface theory

Generalization to a broad class of non-linear operators

Abstract

The aim of this paper is to introduce new forms of the weak and Omori-Yau maximum principles for linear operators, notably for trace type operators, and show their usefulness, for instance, in the context of PDE's and in the theory of hypersurfaces. In the final part of the paper we consider a large class of non-linear operators and we show that our previous results can be appropriately generalized to this case.