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Barles, G.

Existence results for first order Hamilton Jacobi equations. (English) Zbl 0574.70019

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 325-340 (1984).
The author gives in this paper existence results for global solutions in viscosity sense of first-order Hamilton-Jacobi equations with Dirichlet type boundary conditions. The striking feature of these results is that existence is obtained under exactly the same generality as the optimal uniqueness results due to M. G. Crandall and the reviewer [C. R. Acad. Sci., Paris, Sér. I 292, 183-186 (1981; Zbl 0469.49023)]. The method of proofs uses various clear substitutions and approximations combined with the uniqueness and comparison results.
Reviewer: P.L.Lions

Cited in 24 Documents

MSC:

70H20 Hamilton-Jacobi equations in mechanics
35F30 Boundary value problems for nonlinear first-order PDEs
49L99 Hamilton-Jacobi theories

Keywords:

viscosity solutions; existence results for global solutions; Dirichlet type boundary conditions; uniqueness and comparison results

Citations:

Zbl 0469.49023
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References:

[1] Benton, S. H., The Hamilton Jacobi Equation. A global Approach (1977), Academic Press: Academic Press New York · Zbl 0418.49001
[2] Conway, E. D.; Hopf, E., Hamilton’s theory and generalized solutions of Hamilton Jacobi equations, J. Math. Mech., t. 12, 939-986 (1964) · Zbl 0178.11002
[5] Douglis, A., The continuous dependance of generalized solutions of non linear partial differential equations upon initial data, Comm. Pure Applied Math., t. 14, 267-284 (1961) · Zbl 0117.31102
[6] Fleming, W. H., The Cauchy problem for a non linear partial differential, J. Diff. equations, t. 5, 515-530 (1969) · Zbl 0172.13901
[7] Fleming, W. H., Nonlinear partial differential equations-probabilistie and game theoretic method, (CIME, Problems in Nonlinear Analysis (1971), Cremonese: Cremonese Roma) · Zbl 0225.35020
[8] Fleming, W. H., The Cauchy problem for degenerate parabolic equations, J. Math. Mech., t. 13, 987-1008 (1964) · Zbl 0192.19602
[9] Friedman, A., The Cauchy problem for first order partial differential equations, Indiana Univ. Mat. J., t. 23, 27-40 (1973) · Zbl 0243.35014
[10] Kruzkov, S. N., Generalized solutions of Hamilton Jacobi equations of Eikonal type, I. USSR Sbornik, t. 27, 406-446 (1975) · Zbl 0369.35012
[11] Kruzkov, S. N., Generalized solutions of nonlinear first order equations and certain quasilinear parabolic equations Vestnik Moscow unio Ser, I. Math. Mech., t. 6, 67-74 (1964), In Russian
[12] Kruskov, S. N., Generalized solutions of first order nonlinear equations in several independant variables, I. Mat. Sb.. I. Mat. Sb., II. Mat. Sb. (N. S.), t. 72, 114, 93-116 (1967), in Russian · Zbl 0164.40802
[13] Lions, P. L., Generalized solutions of Hamilton Jacobi equations (1982), Pitman: Pitman London · Zbl 1194.35459
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