Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Static class-guided selection of elementary solutions in non-monotone vanishing discount problems

Panrui Ni, Jun Yan, Maxime Zavidovique

TL;DR

The paper analyzes a non-monotone vanishing discount problem for Hamilton–Jacobi equations on a closed manifold by introducing a generalized discount term with a signed coefficient $a(x)$. By choosing $a(x)>0$ on a selected static class and $a(x)<0$ elsewhere, it proves that the maximal discounted solution $u_\lambda$ converges uniformly to an elementary solution of the critical equation, represented as $h^\infty(x_0,x)+C$ with $x_0$ in the chosen static class and $C$ determined by a Mather-measure-based infimum. Importantly, by varying the sign pattern of $a$ across the finite static classes, all elementary solutions can be recovered as limits, revealing a purely dynamical mechanism for selection beyond monotonicity. The analysis connects vanishing discount limits with large-time behavior, leveraging the Peierls barrier $h^\infty$, static-class partitions, and Mather measures to characterize the limit. This provides a principled method to select multiple critical solutions in non-monotone settings, extending weak KAM/Mather theory to a broader class of discounted Hamilton–Jacobi problems.

Abstract

We study a generalized vanishing discount problem for Hamilton--Jacobi equations, removing the standard monotonicity assumption, either in a global sense or when integrated against all Mather measures. Specifically, we consider \[ λa(x)u(x)+H(x,Du(x))-Aλ=c_0, \] with a suitably chosen constant $A>0$. By appropriately changing the signs of the function $a(x)$ on different static classes associated with $H$, we show that the maximal viscosity solution converges uniformly as $λ\to 0^+$ and that all elementary solutions of the stationary equation \[ H(x,Du(x))=c_0 \] can be selected as limits. This provides the first result for selecting multiple viscosity solutions in vanishing discount problems beyond the usual monotonicity and integral assumptions, as long as $a(x)$ is positive on one static class. Our results highlight the crucial role of static classes in controlling the asymptotic behavior of viscosity solutions. Previously, under usual monotonicity assumptions, only a single solution could be selected (as discussed in \cite{GL}), whereas our approach allows controlled selection of multiple solutions via static class-guided discount coefficients.

Static class-guided selection of elementary solutions in non-monotone vanishing discount problems

TL;DR

The paper analyzes a non-monotone vanishing discount problem for Hamilton–Jacobi equations on a closed manifold by introducing a generalized discount term with a signed coefficient . By choosing on a selected static class and elsewhere, it proves that the maximal discounted solution converges uniformly to an elementary solution of the critical equation, represented as with in the chosen static class and determined by a Mather-measure-based infimum. Importantly, by varying the sign pattern of across the finite static classes, all elementary solutions can be recovered as limits, revealing a purely dynamical mechanism for selection beyond monotonicity. The analysis connects vanishing discount limits with large-time behavior, leveraging the Peierls barrier , static-class partitions, and Mather measures to characterize the limit. This provides a principled method to select multiple critical solutions in non-monotone settings, extending weak KAM/Mather theory to a broader class of discounted Hamilton–Jacobi problems.

Abstract

We study a generalized vanishing discount problem for Hamilton--Jacobi equations, removing the standard monotonicity assumption, either in a global sense or when integrated against all Mather measures. Specifically, we consider with a suitably chosen constant . By appropriately changing the signs of the function on different static classes associated with , we show that the maximal viscosity solution converges uniformly as and that all elementary solutions of the stationary equation can be selected as limits. This provides the first result for selecting multiple viscosity solutions in vanishing discount problems beyond the usual monotonicity and integral assumptions, as long as is positive on one static class. Our results highlight the crucial role of static classes in controlling the asymptotic behavior of viscosity solutions. Previously, under usual monotonicity assumptions, only a single solution could be selected (as discussed in \cite{GL}), whereas our approach allows controlled selection of multiple solutions via static class-guided discount coefficients.
Paper Structure (3 sections, 38 theorems, 146 equations, 1 figure)

This paper contains 3 sections, 38 theorems, 146 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1.1

Assume (H1), (H2), and ($\diamond$). Let $v_0$ be a viscosity solution of E0 and take $A > \|a\|_\infty \|v_0\|_\infty$ in E. Then E admits a maximal solution $u_\lambda$. Fix $i_0 \in \{1,\dots,k\}$ and assume Then the maximal solution $u_\lambda$ converges uniformly as $\lambda \to 0^+$ to where $x_0 \in M_{i_0}$ is arbitrary and the constant $C$ is given by with the infimum over all projecte

Figures (1)

  • Figure 1: Two cases causing difficulties in the proof of Lemma \ref{['e1e2']}.

Theorems & Definitions (67)

  • Theorem 1.1
  • Example 1.2
  • Example 1.3
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Lemma 2.7
  • ...and 57 more