Viscosity subsoltions of Hamilton-Jacobi equations and Invariant sets of contact Hamilton systems

TL;DR

This paper investigates viscosity subsolutions of contact Hamilton-Jacobi equations on closed manifolds, focusing on solution semigroup monotonicity, invariance properties, and implications for strict subsolutions.

Contribution

It introduces new results on the monotonicity and invariance of viscosity subsolutions and their semigroups in the context of contact Hamilton-Jacobi equations.

Findings

Monotonicity of solution semigroups for viscosity subsolutions

Positive invariance of the epigraph of viscosity subsolutions

Results for strict viscosity subsolutions on manifolds

Abstract

The objective of this paper is to present some results about viscosity subsolutions of the contact Hamiltonian-Jacobi equations on connected, closed manifold $M$ $$ H(x,\partial_x u,u)= 0, \quad x\in M. $$ Based on implicit variational principles introduced in [12,14], we focus on the monotonicity of the solution semigroups on viscosity subsolutions and the positive invariance of the epigraph for viscosity subsolutions. Besides, we show a similar consequence for strict viscosity subsolutions on $M$.