A Strong Maximum Principle for Weak Solutions of Quasi-Linear Elliptic Equations with Applications to Lorentzian and Riemannian Geometry

TL;DR

This paper proves a strong maximum principle for weak solutions of certain quasi-linear elliptic equations, including applications to Lorentzian and Riemannian geometry, notably a splitting theorem in Lorentzian manifolds.

Contribution

It establishes a maximum principle for weak solutions of quasi-linear elliptic equations, extending classical results to less regular solutions in geometric contexts.

Findings

Maximum principle holds for weak solutions in Lorentzian and Riemannian settings.

Application to Lorentzian warped product splitting theorem.

Extension of maximum principles to $C^0$ spacelike hypersurfaces.

Abstract

The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for $C^0$ spacelike hypersurfaces in a Lorentzian manifold. As one application a Lorentzian warped product splitting theorem is given.