Duality methods for the study of Hamilton-Jacobi equations

Jean-Paul Penot (1) , Michel Volle (2)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 LANLG - Laboratoire d'Analyse non linéaire et Géométrie

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.

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Jean-Paul Penot, Michel Volle. Duality methods for the study of Hamilton-Jacobi equations. 2007. ⟨hal-00175205⟩