Qualitative properties of generalized principal eigenvalues for superquadratic viscous Hamilton-Jacobi equations

TL;DR

This paper studies the behavior of generalized principal eigenvalues in superquadratic viscous Hamilton-Jacobi equations, showing convergence to a limit and exploring qualitative properties under potential perturbations.

Contribution

It establishes the convergence of eigenvalues as the exponent m approaches infinity and characterizes the limit as an eigenvalue of a constrained ergodic problem, also analyzing perturbation effects.

Findings

Eigenvalues converge to a constant as m → ∞

Limit eigenvalue matches that of a gradient-constrained problem

Different qualitative behaviors are identified for m=2, 2<m<∞, and m=∞

Abstract

This paper is concerned with the ergodic problem for superquadratic viscous Hamilton-Jacobi equations with exponent m \textgreater{} 2. We prove that the generalized principal eigenvalue of the equation converges to a constant as m $\rightarrow$ $\infty$, and that the limit coincides with the generalized principal eigenvalue of an ergodic problem with gradient constraint. We also investigate some qualitative properties of the generalized principal eigenvalue with respect to a perturbation of the potential function. It turns out that different situations take place according to m = 2, 2 \textless{} m \textless{} $\infty$, and the limiting case m = $\infty$.