The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus

Al-Aidarous, 2016, A convergence result for the ergodic problem for Hamilton–Jacobi equations with Neumann-type boundary conditions, Proc. R. Soc. Edinb., 146A, 225, 10.1017/S0308210515000517 Armstrong, 2009, The Dirichlet problem for the Bellman equation at resonance, J. Differ. Equ., 247, 931, 10.1016/j.jde.2009.03.007 Armstrong, 2015, Viscosity solutions of general viscous Hamilton–Jacobi equations, Math. Ann., 361, 647, 10.1007/s00208-014-1088-5 Bardi, 1997, Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations, 10.1007/978-0-8176-4755-1 Caffarelli, 1989, Interior a priori estimates for solutions of fully nonlinear equations, Ann. Math. (2), 130, 189, 10.2307/1971480 Capuzzo-Dolcetta, 2010, Hölder estimates for degenerate elliptic equations with coercive Hamiltonians, Trans. Am. Math. Soc., 362, 4511, 10.1090/S0002-9947-10-04807-5 Crandall, 1992, User's guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc., 27, 1, 10.1090/S0273-0979-1992-00266-5 Davini, 2016, Convergence of the solutions of the discounted Hamilton–Jacobi equation, Invent. Math., 206, 29, 10.1007/s00222-016-0648-6 Donsker, 1975, On a variational formula for the principal eigenvalue for operators with maximum principle, Proc. Natl. Acad. Sci. USA, 72, 780, 10.1073/pnas.72.3.780 Donsker, 1976, On the principal eigenvalue of second-order elliptic differential operators, Commun. Pure Appl. Math., 29, 595, 10.1002/cpa.3160290606 Evans, 1982, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Commun. Pure Appl. Math., 35, 333, 10.1002/cpa.3160350303 Evans, 2002, Linear programming interpretations of Mather's variational principle, ESAIM Control Optim. Calc. Var., 8, 693, 10.1051/cocv:2002030 Gomes, 2005, Duality principles for fully nonlinear elliptic equations, vol. 61, 125 Gomes, 2008, Generalized Mather problem and selection principles for viscosity solutions and Mather measures, Adv. Calc. Var., 1, 291, 10.1515/ACV.2008.012 Ishii, 1990, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differ. Equ., 83, 26, 10.1016/0022-0396(90)90068-Z Ishii Krylov, 1982, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR, Ser. Mat., 46, 487 Krylov, 1979, An estimate for the probability of a diffusion process hitting a set of positive measure, Dokl. Akad. Nauk SSSR, 245, 18 P.-L. Lions, G. Papanicolaou, S.R.S. Varadhan, Homogenization of Hamilton–Jacobi equations, unpublished work (1987). Mañé, 1996, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, 9, 273, 10.1088/0951-7715/9/2/002 Mather, 1991, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207, 169, 10.1007/BF02571383 Mitake Sion, 1958, On general minimax theorems, Pac. J. Math., 8, 171, 10.2140/pjm.1958.8.171 Terkelsen, 1972, Some minimax theorems, Math. Scand., 31, 405, 10.7146/math.scand.a-11441 Trudinger, 1989, On regularity and existence of viscosity solutions of nonlinear second order, elliptic equations, vol. 2, 939