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Selection problems for a discount degenerate viscous Hamilton-Jacobi equation. (English) Zbl 1357.35090
Summary: We prove that the solution of the discounted approximation of a degenerate viscous Hamilton-Jacobi equation with convex Hamiltonians converges to that of the associated ergodic problem. We characterize the limit in terms of stochastic Mather measures by using the nonlinear adjoint method and deriving a commutation lemma. This convergence result was first proven by Davini, Fathi, Iturriaga, and Zavidovique for first order Hamilton-Jacobi equations.

##### MSC:
 35F21 Hamilton-Jacobi equations 35B40 Asymptotic behavior of solutions to PDEs 37J50 Action-minimizing orbits and measures (MSC2010) 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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##### References:
 [1] Al-Aidarous, E. S.; Alzahrani, E. O.; Ishii, H.; Younas, A. M.M., A convergence result for the ergodic problem for Hamilton-Jacobi equations with Neumann type boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 146, 225-242, (2016) · Zbl 1350.35063 [2] Ambrosio, L., Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158, 227-260, (2004) · Zbl 1075.35087 [3] Anantharaman, N.; Iturriaga, R.; Padilla, P.; Sanchez-Morgado, H., Physical solutions of the Hamilton-Jacobi equation, Discrete Contin. Dyn. Syst. Ser. B, 5, 3, 513-528, (2005) · Zbl 1081.37036 [4] Armstrong, S. N.; Tran, H. V., Viscosity solutions of general viscous Hamilton-Jacobi equations, Math. Ann., 361, 3, 647-687, (2015) · Zbl 1327.35058 [5] Bessi, U., Aubry-Mather theory and Hamilton-Jacobi equations, Comm. Math. Phys., 235, 3, 495-511, (2003) · Zbl 1023.37034 [6] Cagnetti, F.; Gomes, D.; Mitake, H.; Tran, H. V., A new method for large time behavior of convex Hamilton-Jacobi equations: degenerate equations and weakly coupled systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32, 183-200, (2015) · Zbl 1312.35020 [7] Cagnetti, F.; Gomes, D.; Tran, H. V., Aubry-Mather measures in the non convex setting, SIAM J. Math. Anal., 43, 6, 2601-2629, (2011) · Zbl 1258.35059 [8] Cagnetti, F.; Gomes, D.; Tran, H. V., Adjoint methods for obstacle problems and weakly coupled systems of PDE, ESAIM Control Optim. Calc. Var., 19, 3, 754-779, (2013) · Zbl 1273.35090 [9] Crandall, M. G.; Ishii, H.; Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27, 1, 1-67, (1992) · Zbl 0755.35015 [10] Davini, A.; Fathi, A.; Iturriaga, R.; Zavidovique, M., Convergence of the solutions of the discounted equation, Invent. Math., 1-27, (2016) [11] Di Perna, R. J.; Lions, P.-L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98, 511-547, (1989) · Zbl 0696.34049 [12] Evans, L. C., Adjoint and compensated compactness methods for Hamilton-Jacobi PDE, Arch. Ration. Mech. Anal., 197, 1053-1088, (2010) · Zbl 1273.70030 [13] Evans, L. C., Envelopes and nonconvex Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 50, 257-282, (2014) · Zbl 1302.49038 [14] Fathi, A., Théorème KAM faible et théorie de Mather sur LES systèmes lagrangiens, C. R. Acad. Sci. Paris Ser. I Math., 324, 9, 1043-1046, (1997) · Zbl 0885.58022 [15] A. Fathi, Weak KAM Theorem in Lagrangian Dynamics. · Zbl 0885.58022 [16] Fathi, A.; Siconolfi, A., PDE aspects of aubry-Mather theory for quasiconvex Hamiltonians, Calc. Var. Partial Differential Equations, 22, 2, 185-228, (2005) · Zbl 1065.35092 [17] Gomes, D. A., A stochastic analogue of aubry-Mather theory, Nonlinearity, 15, 3, 581-603, (2002) · Zbl 1073.37078 [18] Gomes, D. A., Duality principles for fully nonlinear elliptic equations, (Trends in Partial Differential Equations of Mathematical Physics, Progr. Nonlinear Differential Equations Appl., vol. 61, (2005), Birkhäuser Basel), 125-136 · Zbl 1237.35053 [19] Gomes, D. A., Generalized Mather problem and selection principles for viscosity solutions and Mather measures, Adv. Calc. Var., 1, 291-307, (2008) · Zbl 1181.37090 [20] Gomes, D. A.; Mitake, H.; Tran, H. V., The selection problem for discounted Hamilton-Jacobi equations: some non-convex cases, submitted for publication · Zbl 1403.35082 [21] Ishii, H., On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial. Ekvac., 38, 1, 101-120, (1995) · Zbl 0833.35053 [22] Iturriaga, R.; Sanchez-Morgado, H., On the stochastic aubry-Mather theory, Bol. Soc. Mat. Mexicana (3), 11, 1, 91-99, (2005) · Zbl 1100.37040 [23] Iturriaga, R.; Sanchez-Morgado, H., Limit of the infinite horizon discounted Hamilton-Jacobi equation, Discrete Contin. Dyn. Syst. Ser. B, 15, 623-635, (2011) · Zbl 1217.37053 [24] Jensen, R., The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Ration. Mech. Anal., 101, 1-27, (1988) · Zbl 0708.35019 [25] Jensen, R.; Lions, P.-L.; Souganidis, P. E., A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations, Proc. Amer. Math. Soc., 102, 4, 975-978, (1988) · Zbl 0662.35048 [26] Le Bris, C.; Lions, P.-L., Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients, Comm. Partial Differential Equations, 33, 7-9, 1272-1317, (2008) · Zbl 1157.35301 [27] Lions, P.-L., Control of diffusion processes in $$\mathbb{R}^N$$, Comm. Pure Appl. Math., 121-147, (1981) · Zbl 0441.49020 [28] P.-L. Lions, G. Papanicolaou, S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations, 1987, unpublished work. [29] Mañé, R., Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, 9, 2, 273-310, (1996) · Zbl 0886.58037 [30] Mather, J. N., Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207, 2, 169-207, (1991) · Zbl 0696.58027 [31] Mitake, H.; Tran, H. V., Large-time behavior for obstacle problems for degenerate viscous Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 54, 2039-2058, (2015) · Zbl 1327.35065 [32] Tran, H. V., Adjoint methods for static Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 41, 301-319, (2011) · Zbl 1231.35043
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