Geodesic
Duality methods for the study of Hamilton–Jacobi equations
Jean-Paul PENOT1 ; Michel VOLLE2
1 Université de Pau, Faculté des Sciences, Laboratoire de Mathématiques Appliquées, CNRS UMR  5142 Av. de l'Université 64000 PAU, France.
2 Université d'Avignon, Faculté des Sciences, Département de Mathématiques, 33 rue Louis Pasteur 84000 AVIGNON, France.
ESAIM. Proceedings, Tome 17 (2007), pp. 96-142.

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We present a survey of recent results about explicit solutions of the first-order Hamilton–Jacobi equation. We take advantage of the methods of asymptotic analysis, convex analysis and of nonsmooth analysis to shed a new light on classical results. We use formulas of the Hopf and Lax–Oleinik types. In the quasiconvex case the usual Fenchel conjugacy is replaced by quasiconvex conjugacies known for some years and the usual inf-convolution is replaced by a sublevel convolution. Inasmuch we use weak generalized convexity and continuity assumptions, some of our results are new; in particular, we do not assume that the data are finite-valued, so that equations derived from attainability or obstacle problems could be considered.
DOI : 10.1051/proc:071708

Jean-Paul PENOT 1 ; Michel VOLLE 2

1 Université de Pau, Faculté des Sciences, Laboratoire de Mathématiques Appliquées, CNRS UMR  5142 Av. de l'Université 64000 PAU, France.
2 Université d'Avignon, Faculté des Sciences, Département de Mathématiques, 33 rue Louis Pasteur 84000 AVIGNON, France. 
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     title = {Duality methods for the study of {Hamilton{\textendash}Jacobi} equations},
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Jean-Paul PENOT; Michel VOLLE. Duality methods for the study of Hamilton–Jacobi equations. ESAIM. Proceedings, Tome 17 (2007), pp. 96-142. doi : 10.1051/proc:071708. http://geodesic.mathdoc.fr/articles/10.1051/proc:071708/

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