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

Abstract

This paper is concerned with the ergodic problem for superquadratic viscous Hamilton-Jacobi equations with exponent m > 2. We prove that the generalized principal eigenvalue of the equation converges to a constant as m → ∞, 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 < m < ∞, and the limiting case m = ∞.

Sharp estimates of the generalized principal eigenvalue for superlinear viscous Hamilton-Jacobi equations with inward drift

“…As mentioned, ergodic problem (EP) is a nonlinear extension of linear eigenvalue problem (1.3). From this point of view, a similar criticality theory is discussed in [2,3,7,8] for viscous Hamilton-Jacobi equations of the form (EP) (see also [9] for a discrete analogue of it). The present paper is closely related to [3] where the asymptotic behavior of λ max (β) as β → ∞ is investigated under some conditions including (1.1); it is proved that the spectral function is non-decreasing and concave in [0, ∞) with λ max (0) = 0, and that the limit of λ max (β) as β → ∞ is determined according to the strength of the inward-pointing drift b.…”

Ergodic Problems for Viscous Hamilton--Jacobi Equations with Inward Drift

“…We only give a sketch of the proof since the proof of this proposition is standard (cf. [9,19,20]). We first claim that, if (λ 0 , ϕ 0 ) ∈ R × C 3 (R d ) is a subsolution of (EP) and ε > 0 is arbitrary, then there exists an eigenfunction ϕ ∈ C 3 (R d ) of (EP) associated with λ 0 − ε.…”

“…Then we easily see that the function β → λ max (β) is nondecreasing and concave. Such kind of qualitative properties of λ max (β) with respect to β have been studied by [9,19,20] in connection with the so-called criticality theory for ergodic problem (EP) (see also [3] for related results). In this paper, we investigate the asymptotic behavior of λ max = λ max (β) as β → ∞.…”