Abstract
We present comparison principles, Lipschitz estimates and study state constraints problems for degenerate, second-order Hamilton–Jacobi equations.
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Armstrong, S.N., Cardaliaguet, P.: Quantitative stochastic homogenization of viscous Hamilton–Jacobi equations. Comm. Partial Differ. Equ. (to appear)
Armstrong, S.N., Souganidis, P.E.: Stochastic homogenization of Hamilton–Jacobi and degenerate Bellman equations in unbounded environments. J. Math. Pures Appl. (9) 97(5), 460–504 (2012)
Armstrong, S.N., Tran, H.V.: Stochastic homogenization of viscous Hamilton–Jacobi equations and applications. arXiv:1310.1749 [math.AP]
Barles, G.: A weak Bernstein method for fully nonlinear elliptic equations. Differ. Integral Equ. 4(2), 241–262 (1991)
Barles, G., Biton, S., Bourgoing, M., Ley, O.: Uniqueness results for quasilinear parabolic equations through viscosity solutions’ methods. Calc. Var. Partial Differ. Equ. 18(2), 159–179 (2003)
Barles, G., Da Lio, F.: On the generalized Dirichlet problem for viscous Hamilton–Jacobi equations. J. Math. Pures Appl. (9) 83(1), 53–75 (2004)
Barles, G., Perthame, B.: Discontinuous solutions of deterministic optimal stopping time problems. RAIRO Modél. Math. Anal. Numér. 21(4), 557–579 (1987)
Bernstein, S.: Sur la généralisation du problème de Dirichlet. Math. Ann. 69(1), 82–136 (1910)
Crandall, M.G., 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), 1–67 (1992)
Da Lio, F., Ley, O.: Uniqueness results for second-order Bellman–Isaacs equations under quadratic growth assumptions and applications. SIAM J. Control Optim. 45(1), 74–106 (2006, electronic)
Da Lio, F., Ley, O.: Convex Hamilton–Jacobi equations under superlinear growth conditions on data. Appl. Math. Optim. 63(3), 309–339 (2011)
Dolcetta, I.C., Leoni, F., Porretta, A.: Hölder estimates for degenerate elliptic equations with coercive hamiltonians. Trans. Am. Math. Soc. 362(9), 4511–4536 (2010)
Fathi, A., Siconolfi, A.: Existence of \(C^1\) critical subsolutions of the Hamilton–Jacobi equation. Invent. Math. 155(2), 363–388 (2004)
Kobylanski, M.: Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28(2), 558–602 (2000)
Koike, S., Ley, O.: Comparison principle for unbounded viscosity solutions of degenerate elliptic PDEs with gradient superlinear terms. J. Math. Anal. Appl. 381(1), 110–120 (2011)
Lasry, J.-M., Lions, P.-L.: Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem. Math. Ann. 283(4), 583–630 (1989)
Lions, P.-L.: Generalized Solutions of Hamilton–Jacobi Equations. Research Notes in Mathematics, vol. 69. Pitman (Advanced Publishing Program). Boston (1982)
Lions, P.-L.: Quelques remarques sur les problèmes elliptiques quasilinéaires du second ordre. J. Anal. Math. 45, 234–254 (1985)
Mitake, H.: Asymptotic solutions of Hamilton–Jacobi equations with state constraints. Appl. Math. Optim. 58(3), 393–410 (2008)
Acknowledgments
S. Armstrong thanks the Forschungsinstitut für Mathematik (FIM) of ETH Zürich for support. H. Tran is supported in part by NSF Grant DMS-1361236.
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Armstrong, S.N., Tran, H.V. Viscosity solutions of general viscous Hamilton–Jacobi equations. Math. Ann. 361, 647–687 (2015). https://doi.org/10.1007/s00208-014-1088-5
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DOI: https://doi.org/10.1007/s00208-014-1088-5