Generalized comparison principle for contact Hamilton-Jacobi equations

TL;DR

This paper addresses the generalized comparison problem for the contact Hamilton-Jacobi equation $H(x, d_x u, u)=c$ on a closed manifold $M$, with a Hamiltonian convex in $p$, coercive in $p$, and non-decreasing in $u$. It develops a Mather-measures-based framework within viscosity and weak KAM theory to establish a comparison principle and to characterize the set of admissible ergodic constants $\mathfrak C$ via an ergodic-constant function $c(\theta)$ and intervals $I(c)$. The authors introduce two kinds of Mather measures, prove invariance properties under the contact dynamics, and show how multiplicity of solutions can arise when certain measures are nonempty, supported by explicit examples. They also derive regularity and structural results for $c(\theta)$, including left/right derivatives and convexity under stronger hypotheses, linking these to the geometry of the Mather sets. Overall, the work connects generalized Hamilton-Jacobi theory with Mather/weak KAM theory in the nonlinear $u$-dependent setting, providing a principled way to order and classify viscosity solutions and to understand the underlying dynamical structure. The results have implications for non-equilibrium dynamics and variational approaches to contact Hamiltonian systems where $u$-dependence plays a crucial role.

Abstract

In this paper, we discuss all the possible pairs $(u,c)\in C(M,\mathbb R)\times\mathbb R$ solving (in the sense of viscosity) the contact Hamilton-Jacobi equation \[ H (x, d_xu, u) = c,\quad x\in M \] of which $M$ is a closed manifold and the continuous Hamiltonian $H: (x,p,u)\in T^*M\times\mathbb R\rightarrow\mathbb R$ is convex, coercive in $p$ but merely non-decreasing in $u$. Firstly, we propose a comparison principle for solutions by using the dynamical information of Mather measures. We then describe the structure of $\mathfrak C$ containing all the $c\in\mathbb R$ makes previous equation solvable. We also propose examples to verify the optimality of our approach.

Figures (2)

• Figure 1: When $\lambda=0$, the solution $u_\lambda$ is actually $C^1-$smooth, which therefore is a classical solution. As $\lambda\geq 0$ increases, $u_\lambda$ descends in the way comforming to item 3) of Theorem \ref{['thm:g-cp']}.
• Figure 2: A couple of classical solutions exists once $0\neq\mathfrak M_-^\theta\subsetneq\mathfrak M^\theta$.

Theorems & Definitions (22)

• Proposition 1.1: ergodic constant
• Theorem 1.2: Local
• Theorem 1.3: Global
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Corollary 1.8
• Definition 2.1
• Lemma 2.2