On viscosity solutions of Hamilton-Jacobi equations
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Abstract:
We consider the Dirichlet problem for Hamilton-Jacobi equations and prove existence, uniqueness and continuous dependence on boundary data of Lipschitz continuous maximal viscosity solutions.References
- Guy Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 17, Springer-Verlag, Paris, 1994 (French, with French summary). MR 1613876
- Martino Bardi and Italo Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. With appendices by Maurizio Falcone and Pierpaolo Soravia. MR 1484411, DOI 10.1007/978-0-8176-4755-1
- Alberto Bressan and Fabián Flores, On total differential inclusions, Rend. Sem. Mat. Univ. Padova 92 (1994), 9–16. MR 1320474
- Fabio Camilli and Antonio Siconolfi, Maximal subsolutions for a class of degenerate Hamilton-Jacobi problems, Indiana Univ. Math. J. 48 (1999), no. 3, 1111–1131. MR 1736967, DOI 10.1512/iumj.1999.48.1678
- I. Capuzzo-Dolcetta and P.-L. Lions, Hamilton-Jacobi equations with state constraints, Trans. Amer. Math. Soc. 318 (1990), no. 2, 643–683. MR 951880, DOI 10.1090/S0002-9947-1990-0951880-0
- P. Cardaliaguet, B. Dacorogna, W. Gangbo, and N. Georgy, Geometric restrictions for the existence of viscosity solutions, Ann. Inst. H. Poincaré C Anal. Non Linéaire 16 (1999), no. 2, 189–220 (English, with English and French summaries). MR 1674769, DOI 10.1016/S0294-1449(99)80012-2
- 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. MR 732102, DOI 10.1090/S0002-9947-1984-0732102-X
- Bernard Dacorogna and Paolo Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Math. 178 (1997), no. 1, 1–37. MR 1448710, DOI 10.1007/BF02392708
- Bernard Dacorogna and Paolo Marcellini, Viscosity solutions, almost everywhere solutions and explicit formulas, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4643–4653. MR 2067137, DOI 10.1090/S0002-9947-04-03506-8
- Bernard Dacorogna and Paolo Marcellini, Implicit partial differential equations, Progress in Nonlinear Differential Equations and their Applications, vol. 37, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1702252, DOI 10.1007/978-1-4612-1562-2
- Francesco Saverio De Blasi and Giulio Pianigiani, On the Dirichlet problem for first order partial differential equations. A Baire category approach, NoDEA Nonlinear Differential Equations Appl. 6 (1999), no. 1, 13–34. MR 1674778, DOI 10.1007/s000300050062
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
- Pierre-Louis Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 667669
- A. Visintin, Strong convergence results related to strict convexity, Comm. Partial Differential Equations 9 (1984), no. 5, 439–466. MR 741216, DOI 10.1080/03605308408820337
- Sandro Zagatti, On the minimum problem for nonconvex scalar functionals, SIAM J. Math. Anal. 37 (2005), no. 3, 982–995. MR 2191784, DOI 10.1137/040612506
- Sandro Zagatti, Solutions of vectorial Hamilton-Jacobi equations and minimizers of nonquasiconvex functionals, J. Math. Anal. Appl. 335 (2007), no. 2, 1143–1160. MR 2346897, DOI 10.1016/j.jmaa.2007.02.034
- Sandro Zagatti, Uniqueness and continuous dependence on boundary data for integro-extremal minimizers of the functional of the gradient, J. Convex Anal. 14 (2007), no. 4, 705–727. MR 2350812
- Sandro Zagatti, Minimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations 31 (2008), no. 4, 511–519. MR 2372904, DOI 10.1007/s00526-007-0124-7
Additional Information
- Sandro Zagatti
- Affiliation: Department of Mathematics, Scuola Internazionale Superiore di Studi Avanzati, Via Beirut, 2/4, I-34014 Trieste, Italy
- Received by editor(s): August 21, 2006
- Published electronically: August 19, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 41-59
- MSC (2000): Primary 49L25
- DOI: https://doi.org/10.1090/S0002-9947-08-04557-1
- MathSciNet review: 2439397