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

TL;DR

This paper introduces a novel method for selecting multiple solutions in non-monotone vanishing discount problems for Hamilton-Jacobi equations by leveraging static classes and class-guided coefficients, extending beyond traditional monotonicity assumptions.

Contribution

It presents the first approach to select multiple viscosity solutions in non-monotone vanishing discount problems using static class-guided coefficients, removing the need for monotonicity.

Findings

Uniform convergence of maximal viscosity solutions as discount vanishes.

All elementary solutions of the stationary Hamilton-Jacobi equation can be obtained as limits.

Static classes play a key role in controlling the asymptotic behavior of solutions.

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 \[ \lambda a(x)u(x)+H(x,Du(x))-A\lambda=c_0, \] 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 $\lambda\to 0^+$ 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.