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

Emmanuel Chasseigne (1, 2) , Naoyuki Ichihara (3)

1 FDP - Fédération de recherche Denis Poisson
2 LMPT - Laboratoire de Mathématiques et Physique Théorique
3 Université Aoyama Gakuin, Tokyo

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 = ∞.

Mots clés: viscous Hamilton-Jacobi equation; viscosity solution; ergodic problem; gradient constraint

Domaines: Equations aux dérivées partielles [math.AP]

Emmanuel Chasseigne, Naoyuki Ichihara. Qualitative properties of generalized principal eigenvalues for superquadratic viscous Hamilton-Jacobi equations. Nonlinear Differential Equations and Applications, 2016, 23, pp.1-17. ⟨hal-01291015⟩