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Fully nonlinear Neumann type boundary conditions for first-order Hamilton-Jacobi equations. (English) Zbl 0736.35023
The authors investigate fully nonlinear Neumann type boundary conditions for first-order Hamilton-Jacobi equations, i.e. they handle the problem \[ H(x,u,\nabla u)=0 \quad\hbox { in } \Omega,\qquad F(x,u,\nabla u)=0 \quad\hbox { on } \partial\Omega, (*) \] where \(H\) and \(F\) have to satisfy certain structure conditions. Their main result is a uniqueness and existence result for \((*)\).
Reviewer: N.Jacob (Erlangen)

MSC:
35F30 Boundary value problems for nonlinear first-order PDEs
70H20 Hamilton-Jacobi equations in mechanics
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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[1] Barles, G.; Perthame, B., Discontinuous solutions of deterministic optimal stopping time problems, Math. mod. analysis num., 21, 557-579, (1987) · Zbl 0629.49017
[2] Barles, G.; Perthame, B., Exit time problems in optimal control and vanishing viscosity method, SIAM J. control optim., 26, 1133-1148, (1988) · Zbl 0674.49027
[3] Barles G. & Perthame B., Comparison principle for Dirichlet type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations, Applied Math. Optim. (to appear). · Zbl 0691.49028
[4] Capuzzo-Dolcetta I. & Lions P.L., Viscosity solutions of Hamilton-Jacobi equations and state constraints, Trans. Am. math. Soc. · Zbl 0702.49019
[5] Crandall M.G., Semidifferentials, quadratic forms and fully nonlinear elliptic equations of second-order, Ann. Inst. H. Poincaré Analyse non Linéaire (to appear). · Zbl 0734.35033
[6] Crandall, M.G.; Evans, L.C.; Lions, P.L., Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. am. math. soc., 282, 487-502, (1984) · Zbl 0543.35011
[7] Crandall, M.G.; Lions, P.L., Viscosity solutions of Hamilton-Jacobi equations, Trans. am. math. soc., C.r. hebd. Séanc. acad. sci. Paris, 292, 183-186, (1981), see also · Zbl 0469.49023
[8] Dupuis P. & Ishii H., in preparation.
[9] Dupuis P., Ishii H. & Soner M., A viscosity solution approach to the asymptotic analysis of quencing systems, Annl. Prob. · Zbl 0715.60035
[10] Fleming W.H., Ishii H. & Menaldi J.L., in preparation.
[11] Ishii, H., Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. fac. sci. engng. chuo univ., 28, 33-77, (1985) · Zbl 0937.35505
[12] Ishii H., A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations. · Zbl 0701.35052
[13] Ishii H., On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic P.D.E.’s, Communs. pure appl. Math. (to appear).
[14] Ishii, H., Perron’s method for Hamilton-Jacobi equations, Duke math. J., 55, 369-384, (1987) · Zbl 0697.35030
[15] Ishii H. & Lions P.L., Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. diff. Eqns (to appear). · Zbl 0708.35031
[16] Jensen R., The maximum principle for viscosity solutions of fully nonlinear second-order partial differential equations, Arch. ration. Mech. Analysis. · Zbl 0708.35019
[17] Jensen, R.; Lions, P.L.; Souganidis, P.E., A uniqueness result for viscosity solutions of second-order fully nonlinear partial differential equations, Proc. am. math. soc., 102, 975-978, (1988) · Zbl 0662.35048
[18] Lions, P.L., Generalized solutions of Hamilton-Jacobi equations, (1982), Pitman London · Zbl 1194.35459
[19] Lions, P.L., Neumann type boundary conditions for Hamilton-Jacobi equations, Duke math. J., 52, 793-829, (1985) · Zbl 0599.35025
[20] Perthame B. & Sanders R., The Neumann problem for nonlinear second-order singular perturbation problems. · Zbl 0664.35003
[21] Soner, H.M., Optimal control with state-space constraints I, SIAM J. control optim., 24, (1986)
[22] Soner, H.M., Optimal control with state-space constraints II, SIAM J. control optim., 24, (1986)
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