Hostname: page-component-6bf8c574d5-t27h7 Total loading time: 0 Render date: 2025-02-28T15:10:39.881Z Has data issue: false hasContentIssue false
  • English
  • Français

On Hamilton–Jacobi equations in bounded domains

Published online by Cambridge University Press:  14 November 2011

Hans Engler
Hans Engler
Affiliation:
Department of Mathematics, Georgetown University, Washington, D.C. 20057, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Initial-boundary value problems for nonlinear first order partial differential equations ∂tu + H(x, t, u, Dxu) = 0 and corresponding boundary value problems H(x, u, Dxu) = 0 are studied in bounded sets, using Crandal's and Lions' notion of viscosity solutions. We give pointwise conditions on the boundary data that guarantee the existence of such solutions and estimate their moduli of continuity in terms of continuity properties of the data. The results are applied to properties of the value function for certain differential games.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 102 , Issue 3-4 , 1986 , pp. 221 - 242
Copyright
Copyright © Royal Society of Edinburgh 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Amann, H.. Invariant sets and existence theorems for semilinear parabolic and elliptic systems. J. Math. Anal. Appl. 65 (1978), 432–467.CrossRefGoogle Scholar
2Barles, G.. Existence results for first order Hamilton-Jacobi equations. Ann. Inst. H. Poincaré, Analyse Non Liniéire, 5 (1984), 325–340.CrossRefGoogle Scholar
3Barron, E. N., Evans, L. C. and Jensen, R.. Viscosity solutions of Isaacs equations and differential games with Lipschitz controls. J. Differential Equations 53 (1984), 213–233.CrossRefGoogle Scholar
4Bensoussan, A., Lions, J. L. and Papanicolaou, G.. Asymptotic analysis for Periodic Structures (Amsterdam-New York-Oxford: North Holland, 1978).Google Scholar
5Benton, S. H.. The Hamilton-Jacobi Equation: A Global Approach (New York: Academic Press, 1977).Google Scholar
6Bergh, J. and Löfström, J.. Interpolation Spaces (Berlin, Heidelberg, New York: Springer, 1976).CrossRefGoogle Scholar
7Behrens, H. and Butzer, P. L.. Semigroups of Operators and Approximation (Berlin, Heidelberg, New York: Springer, 1967).Google Scholar
8Capuzzo-Dolcetta, I. and Evans, L. C.. Optimal switching for ordinary differential equations. SIAM J. Control Optim. 22 (1984), 143–161.CrossRefGoogle Scholar
9Chow, P. L. and Williams, S. A.. Nonlinear reaction-diffusion models for interacting populations. J. Math. Anal. Appl. 62 (1978), 157–169.Google Scholar
10Chueh, K. N., Conley, C. and Smoller, J.. Positively invariant regions for systems of nonlinear diffusion equations. Indiana Univ. Math. J. 26 (1977), 373–392.CrossRefGoogle Scholar
11Courant, R. and Hilbert, D.. Methoden der Mathematischen Physik (Berlin: Springer, 1931).CrossRefGoogle Scholar
12Crandall, M. G., Evans, L. C. and Lions, P. L.. Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984), 487–502.CrossRefGoogle Scholar
13Crandall, M. G. and Lions, P. L.. Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983), 1–42.CrossRefGoogle Scholar
14Crandall, M. G. and Lions, P. L.. On Existence and Uniqueness of Solutions of Hamilton-Jacobi Equations. (Madison: MRC-TRS 2719, 1984). J. Nonlin. Analysis Theory, Methods Applications, to appear.CrossRefGoogle Scholar
15Crandall, M. G. and Newcomb, R.. Viscosity Solutions of Hamilton-Jacobi Equations at the Boundary. Proc. Amer. Math. Soc. 94 (1985), 283–290.CrossRefGoogle Scholar
16Elliott, R. J. and Kalton, N. J.. The existence of value in differential games. Mem. Amer. Math. Soc. 126 (1972), 000–000.CrossRefGoogle Scholar
17Evans, L. C. and Souganidis, P.. Differential games and representation formulas for solutions of Hamilton-Jacobi equations. Indiana Univ. Math. J. 33 (1984), 771–797.Google Scholar
18Evans, L. C. and Ishii, H.. Differential games and nonlinear first order PDE on bounded domains. Manuscripta Math. 49 (1984), 109–139.CrossRefGoogle Scholar
19Fleming, W. H.. Exit probabilities and optimal stochastic control. Appl. Math. Optim. 4 (1978), 329–346.CrossRefGoogle Scholar
20Friedman, A.. Partial Differential Equations of Parabolic Type. (Englewood Cliffs: Prentice-Hall, 1967).Google Scholar
21Friedman, A.. Differential Games. CBMS Reg. Conf. Series in Math. #18, (Providence: American Mathematical Society, 1974).Google Scholar
22Ladyzenskaya, O. A., Solonnikov, V. A. and Uraltseva, N. N.. Linear and Quasilinear Equations of Parabolic Type (Providence: American Mathematical Society, 1968).CrossRefGoogle Scholar
23Ladyzenskaya, O. A. and Uraltseva, N. N.. Équations aux Dérivées Partielles de Type Elliptique (Paris: Dunod, 1968).Google Scholar
24Lions, P. L.. Generalized Solutions of Hamilton-Jacobi Equations (Boston: Pitman, 1982).Google Scholar
25Lions, P. L.. Existence for first order Hamilton-Jacobi equations. Ricerche Mat. 32 (1983), 3–23.Google Scholar
26Protter, M. H. and Weinberger, H. F.. Maximum Principles in Differential Equations (Englewood Cliffs: Prentice-Hall, 1967).Google Scholar
27Schwartz, J. T.. Nonlinear Functional Analysis (New York: Gordon & Breach, 1969).Google Scholar
28Sperb, R.. Maximum Principles and their Applications (New York: Academic Press, 1981).Google Scholar