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Transactions of the American Mathematical Society

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Viscosity solutions, almost everywhere solutions and explicit formulas

Authors: Bernard Dacorogna and Paolo Marcellini
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 4643-4653
MSC (2000): Primary 34A60, 35F30, 49L25
Published electronically: January 23, 2004
MathSciNet review: 2067137
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Abstract: Consider the differential inclusion $Du\in E$ in $\mathbb{R} ^{n}$. We exhibit an explicit solution that we call fundamental. It also turns out to be a viscosity solution when properly defining this notion. Finally, we consider a Dirichlet problem associated to the differential inclusion and we give an iterative procedure for finding a solution.

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  • 1. M. Bardi and I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Birkhäuser, 1997. MR 99e:49001
  • 2. F. Camilli and A. Siconolfi, Maximal subsolutions for a class of degenerate Hamilton-Jacobi problems, Indiana Univ. Math. J., 48 (1999), 1111-1131. MR 2001a:49028
  • 3. P. Cardaliaguet, B. Dacorogna, W. Gangbo and N. Georgy, Geometric restrictions for the existence of viscosity solutions, Annales Institut Henri Poincaré, Analyse Non Linéaire, 16 (1999), 189-220. MR 99k:35023
  • 4. B. Dacorogna, R. Glowinski and T.W. Pan, Numerical methods for the solution of a system of eikonal equations with Dirichlet boundary conditions, Comptes Rendus Acad. Sci. Paris, 336 (2003), 511-518.
  • 5. B. Dacorogna and P. Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial case, Acta Mathematica, 178 (1997), 1-37. MR 98d:35029
  • 6. B. Dacorogna and P. Marcellini, Implicit partial differential equations, Progress in Nonlinear Differential Equations and their Applications, 37, Birkhäuser, Boston, 1999. MR 2000f:35005
  • 7. L. Hörmander, Notions of convexity, Birkhaüser, 1994. MR 95k:00002
  • 8. H. Ischii and P. Loreti, Relaxation in an $L^{\infty} $-optimization problem, Proc. Royal Soc. Edinburgh, 133 (2003), 599-615.
  • 9. H. Ischii and M. Ramaswamy, Uniqueness results for a class of Hamilton-Jacobi equations with singular coefficients, Comm. Par. Diff. Eq., 20 (1995), 2187-2213. MR 96k:35026
  • 10. P. L. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Math. 69, Pitman, London, 1982. MR 84a:49038
  • 11. P. Marcellini, Non convex integrals of the calculus of variations, in: Methods of nonconvex analysis, ed. Cellina A., Lecture Notes in Math. 1446, Springer-Verlag, Berlin, 1990, 16-57. MR 91j:49002
  • 12. R.T. Rockafellar, Convex analysis, Princeton University Press, Princeton, 1970. MR 43:445
  • 13. R. Webster, Convexity, Oxford University Press, Oxford, 1994. MR 98h:52001

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Additional Information

Bernard Dacorogna
Affiliation: Départment de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland

Paolo Marcellini
Affiliation: Dipartimento di Matematica U. Dini, Università di Firenze, Firenze, Italy

Keywords: Almost everywhere solutions, viscosity solutions of nonlinear first order partial differential equations
Received by editor(s): December 2, 2002
Received by editor(s) in revised form: August 21, 2003
Published electronically: January 23, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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