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 \(m\rightarrow \infty \), 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 \(m=2\), \(2<m<\infty \), and the limiting case \(m=\infty \).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barles, G., Jakobsen, E.R.: On the convergence rate of approximation schemes for Hamilton–Jacobi–Bellman equations. M2AN Math. Model. Numer. Anal. 36, 33–54 (2002)
Berestycki, H., Capuzzo Dolcetta, I., Porretta, A., Rossi, L.: Maximum principle and generalized principal eigenvalue for degenerate elliptic operators. J. Math. Pures Appl. (9) 103, 1276–1293 (2015)
Berestycki, H., Nirenberg, L., Varadhan, S.R.S.: The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Commun. Pure Appl. Math. 47, 47–92 (1994)
Capuzzo Dolcetta, I., Leoni, F., Porretta, A.: Hölder estimates for degenerate elliptic equations with coercive Hamiltonians. Trans. Am. Math. Soc. 362, 4511–4536 (2010)
Crandall, M., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27, 1–67 (1992)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Grundlehren Math. Wiss. 224, 2nd edn. Springer, Berlin (1988)
Hynd, R.: The eigenvalue problem of singular ergodic control. Commun. Pure Appl. Math. 65, 649–682 (2012)
R. Hynd, An eigenvalue problem for fully nonlinear elliptic equations with gradient constraints (preprint). arXiv:1412.8011v3 [math.AP]
Ichihara, N.: Large time asymptotic problems for optimal stochastic control with superlinear cost. Stoch. Process. Appl. 122, 1248–1275 (2012)
Ichihara, N.: Criticality of viscous Hamilton-Jacobi equations and stochastic ergodic control. J. Math. Pures Appl. 100, 368–390 (2013)
Ichihara, N.: The generalized principal eigenvalue for Hamilton–Jacobi–Bellman equations of ergodic type. Ann. I. H. Poincaré-AN 32, 623–650 (2015)
Juutinen, P., Parviainen, M., Rossi, J.: Discontinuous gradient constraints and the infinity Laplacian. Int. Math. Res. Not. IMRN 8, 2451–2492 (2016)
Koike, S.: A Beginner’s Guide to the Theory of Viscosity Solutions, MSJ Memoirs 13. Mathematical Society of Japan, Tokyo (2004)
Krylov, N.V.: On the rate of convergence of finite-difference approximations for Bellman’s equations with variable coefficients. Probab. Theory Rel. Fields 117, 1–16 (2000)
Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)
Lions, P.-L.: Résolution de problèmes elliptiques quasilinéaires. Arch. Ration. Mech. Anal. 74, 335–353 (1980)
Menaldi, J.-L., Robin, M., Taksar, M.I.: Singular ergodic control for multidimensional Gaussian processes. Math. Control Signals Syst. 5, 93–114 (1992)
Pinchover, Y.: Maximum and anti-maximum principles and eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations. Math. Ann. 314, 555–590 (1999)
Pinsky, R.G.: Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics, vol. 45. Cambridge University Press, Cambridge (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
E. Chasseigne research was partially supported by Spanish Grant MTM2011-25287. N. Ichihara research was partially supported by JSPS KAKENHI Grant Number 15K04935.
Rights and permissions
About this article
Cite this article
Chasseigne, E., Ichihara, N. Qualitative properties of generalized principal eigenvalues for superquadratic viscous Hamilton–Jacobi equations. Nonlinear Differ. Equ. Appl. 23, 66 (2016). https://doi.org/10.1007/s00030-016-0422-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-016-0422-x