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Neumann type boundary conditions for Hamilton-Jacobi equations. (English) Zbl 0599.35025
The concept of viscosity solutions to first order fully nonlinear partial differential equations, first introduced by M. G. Crandall and the author [Trans. Am. Math. Soc. 277, 1-42 (1983; Zbl 0599.35024)] is extended here to problems with Neumann boundary conditions. This problem naturally arises in deterministic optimal control with state constraints and elsewhere. Since viscosity solutions are not required to have derivatives of any sort the problem is to make sense out of a condition like \(\partial u/\partial v=0.\)
The following definition is given. Let \(u\in C({\bar \Omega})\). Then u is a viscosity subsolution of \(H(x,u,Du)=0\) in \(\Omega\), \(\partial u/\partial v=0\) on \(\partial \Omega\) if for every \(\phi \in C^ 1({\bar \Omega})\), if u-\(\phi\) has a local max at \(x_ 0\) in \({\bar \Omega}\), then \(H(x_ 0,u(x_ 0),D\phi (x_ 0))\leq 0\) if \(x_ 0\in \Omega\), \(H(x_ 0,u(x_ 0)\), \(D\phi (x_ 0))\leq 0\) if \(x_ 0\in \partial \Omega\) and \(\partial \phi /\partial v(x_ 0)\geq 0\). For u to be a viscosity supersolution replace max by min and reverse the inequalities. A solution is both a sub and supersolution. The theory of viscosity solutions is then extended to this case.
Applications to control theory and differential games and an ergodic result are also presented.
Reviewer: E.Barron

MSC:
35F20 Nonlinear first-order PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35F30 Boundary value problems for nonlinear first-order PDEs
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[1] R. F. Anderson and S. Orey, Small random perturbation of dynamical systems with reflecting boundary , Nagoya Math. J. 60 (1976), 189-216. · Zbl 0324.60063
[2] C. Bardos, Y. LeRoux, and J. C. Nédelec, Rapport de l’Ecole Polytechnique , Palaiseau.
[3] G. Barles, Controle impulsionnel déterministe, inéquation quasi-variationnelles et equations de Hamilton-Jacobi du premier ordre , Thése de 3e Cycle, Université de Paris-Dauphiné, 1983.
[4] E. N. Barron, L. C. Evans, and R. Jensen, Viscosity solutions of Isaacs’ equations and differential games with Lipschitz controls , J. Differential Equations 53 (1984), no. 2, 213-233. · Zbl 0548.90104 · doi:10.1016/0022-0396(84)90040-8
[5] A. Bensoussan and J.-L. Lions, Controle impulsionnel et inequations quasi variationnelles , Méthodes Mathématiques de l’Informatique [Mathematical Methods of Information Science], vol. 11, Gauthier-Villars, Paris,Dunod, 1982. · Zbl 0491.93002
[6] C. Burch and J. Goldstein,
[7] M. G. Crandall, L. C. Evans, and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations , Trans. Amer. Math. Soc. 282 (1984), no. 2, 487-502. JSTOR: · Zbl 0543.35011 · doi:10.2307/1999247 · links.jstor.org
[8] M. G. Crandall and P.-L. Lions, Condition d’unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre , C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 3, 183-186. · Zbl 0469.49023
[9] M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations , Trans. Amer. Math. Soc. 277 (1983), no. 1, 1-42. · Zbl 0599.35024 · doi:10.2307/1999343
[10] M. G. Crandall and P.-L. Lions, Solutions de viscosité non bornées des équations de Hamilton-Jacobi du premier ordre , C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 10, 217-220. · Zbl 0565.49022
[11] M. G. Crandall and P.-L. Lions, On existence and uniqueness of solutions of Hamilton-Jacobi equations , Nonlinear Anal. 10 (1986), no. 4, 353-370. · Zbl 0603.35016 · doi:10.1016/0362-546X(86)90133-1
[12] M. G. Crandall and P.-L. Lions, Remarks on existence and uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations . · Zbl 0678.35009
[13] M. G. Crandall and P.-L. Lions, Hamilton-Jacobi equations in infinite dimensions , to appear in J. Fund. Anal. · Zbl 0739.49015
[14] M. G. Crandall and R. Newcomb, Viscosity solutions of Hamilton-Jacobi equations at the boundary , to appear. JSTOR: · Zbl 0575.35008 · doi:10.2307/2045392 · links.jstor.org
[15] Robert J. Elliott and Nigel J. Kalton, The existence of value in differential games of pursuit and evasion , J. Differential Equations 12 (1972), 504-523. · Zbl 0244.90046 · doi:10.1016/0022-0396(72)90022-8
[16] R. J. Elliott and N. J. Kalton, Cauchy problems for certain Isaacs-Bellman equations and games of survival , Trans. Amer. Math. Soc. 198 (1974), 45-72. · Zbl 0302.90074 · doi:10.2307/1996746
[17] L. C. Evans and H. Ishii, Differential games and nonlinear first-order PDE on bounded domains , · Zbl 0559.35013 · doi:10.1007/BF01168747 · eudml:155038
[18] L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations , Indiana Univ. Math. J. 33 (1984), no. 5, 773-797. · Zbl 1169.91317 · doi:10.1512/iumj.1984.33.33040 · www.iumj.indiana.edu
[19] W. H. Fleming and P. E. Souganidis,
[20] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order , Springer-Verlag, Berlin, 1977. · Zbl 0361.35003
[21] F. Gimbert, <i/> Thése de 3\(^\mathrme\) cycle, Univ. Paris-Dauphiné and to appear in J. Funct. Anal., 1984.
[22] H. Ishii, Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations , Indiana Univ. Math. J. 33 (1984), no. 5, 721-748. · Zbl 0551.49016 · doi:10.1512/iumj.1984.33.33038
[23] H. Ishii, Remarks on existence of viscosity solutions of Hamilton-Jacobi equations , Bull. Fac. Sci. Engrg. Chuo Univ. 26 (1983), 5-24. · Zbl 0546.35042
[24] J. M. Lasry, Contróle stochastique ergodique , Thése d’Etat, Université Paris-Dauphiné, 1974.
[25] P.-L. Lions, Generalized solutions of Hamilton-Jacobi equations , Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass., 1982. · Zbl 0497.35001
[26] P.-L. Lions, Existence results for first-order Hamilton-Jacobi equations , Ricerche Mat. 32 (1983), no. 1, 3-23. · Zbl 0552.70012
[27] P. L. Lions, Optimal control and viscosity solutions , In Proc. Dynamic Programming Conf. in Roma, March 1984, · doi:10.1007/BFb0074782
[28] P.-L. Lions, Équations de Hamilton-Jacobi et solutions de viscosité , Ennio De Giorgi colloquium (Paris, 1983), Res. Notes in Math., vol. 125, Pitman, Boston, MA, 1985, pp. 83-97. · Zbl 0579.35009
[29] 1 P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications , Comm. Partial Differential Equations 8 (1983), no. 10, 1101-1174. · Zbl 0716.49022 · doi:10.1080/03605308308820297
[30] 2 P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. II. Viscosity solutions and uniqueness , Comm. Partial Differential Equations 8 (1983), no. 11, 1229-1276. · Zbl 0716.49023 · doi:10.1080/03605308308820301
[31] P.-L. Lions, Résolution de problèmes elliptiques quasilinéaires , Arch. Rational Mech. Anal. 74 (1980), no. 4, 335-353. · Zbl 0449.35036 · doi:10.1007/BF00249679
[32] P. L. Lions, Quelques remarques sur les problémes elliptiques quasilineaires du second ordre , · Zbl 0614.35034 · doi:10.1007/BF02792551
[33] P. L. Lions and B. Perthame, Quasi-variational inequalities and ergodic impulse control ,
[34] P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions , Comm. Pure Appl. Math. 37 (1984), no. 4, 511-537. · Zbl 0598.60060 · doi:10.1002/cpa.3160370408
[35] P.-L. Lions, J.-L. Menaldi, and A.-S. Sznitman, Construction de processus de diffusion réfléchis par pénalisation du domaine , C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 11, 559-562. · Zbl 0468.60073
[36] A. Sayah, Thése de 3\(^\mathrme\) cycle , 1984, Univ. Paris VI.
[37] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables , Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 413-496. · Zbl 0181.38003 · doi:10.1098/rsta.1969.0033
[38] P. E. Souganidis, Existence of viscosity solutions of Hamilton-Jacobi equations , J. Differential Equations 56 (1985), no. 3, 345-390. · Zbl 0506.35020 · doi:10.1016/0022-0396(85)90084-1
[39] P. E. Souganidis, Thesis , 1983, Univ. of Wisconsin-Madison.
[40] H. Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions , Hiroshima Math. J. 9 (1979), no. 1, 163-177. · Zbl 0423.60055
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