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

Abstract

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

References in corpus (5)

• Convergence of the solutions of the discounted equation: the discrete case
• Convergence of viscosity solutions of generalized contact Hamilton-Jacobi equations
• Convergence of solutions for some degenerate discounted Hamilton--Jacobi equations
• Vanishing discount problem and the additive eigenvalues on changing domains
• Convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function

Cited by in corpus (1)

• Vanishing discount limits for first-order fully nonlinear Hamilton-Jacobi equations on noncompact domains