Limit of solutions for semilinear Hamilton-Jacobi equations with degenerate viscosity

TL;DR

This paper investigates the behavior of viscosity solutions to a class of degenerate viscous Hamilton-Jacobi equations as a parameter approaches zero, establishing convergence and characterizing the limit via stochastic Mather measures.

Contribution

It proves the convergence of solutions for degenerate viscous Hamilton-Jacobi equations and characterizes the limit using stochastic Mather measures under specific conditions.

Findings

Viscosity solutions converge as the parameter tends to zero.

The limit solution is characterized by stochastic Mather measures.

The results apply to equations with degenerate viscosity and convex Hamiltonians.

Abstract

In the paper we prove the convergence of viscosity solutions $u_{\lambda}$ as $\lambda\rightarrow0_+$ for the parametrized degenerate viscous Hamilton-Jacobi equation \[ H(x,d_x u, \lambda u)=\alpha(x)\Delta u,\quad \alpha(x)\geq 0,\quad x\in \mathbb T^n \] under suitable convex and monotonic conditions on $H: T^*M\times\mathbb R\rightarrow\mathbb R$. Such a limit can be characterized in terms of stochastic Mather measures associated with the critical equation \[ H(x,d_x u,0)=\alpha(x)\Delta u. \]