Convex Analysis techniques for Hopf-Lax formulae in Hamilton-Jacobi equations

Cyril Imbert (1)

1 MIP - Mathématiques pour l'Industrie et la Physique

The purpose of the present paper is to prove, solely using Convex (and Nonsmooth) analysis techniques, that Hopf-Lax formulae provide explicit solutions for Hamilton-Jacobi equations with merely lower semicontinuous initial data. The substance of these results appears in a paper by Alvarez, Barron and Ishii (1999) but the proofs are fundamentally different (we do not use the comparison principle) and a distinct notion of discontinuous solutions is used. Moreover we give a maximum principle for the Lax function. This approach permits us to fully understand the role of the convexity of the data.

Mots clés: lsc initial data; epi-sum; Legendre-Fenchel conjugate; Clarke-Ledyaev mean value inequality; Hopf-Lax functions; Convex analysis; lsc solutions

Domaines: Equations aux dérivées partielles [math.AP]

Cyril Imbert. Convex Analysis techniques for Hopf-Lax formulae in Hamilton-Jacobi equations. Journal of Nonlinear and Convex Analysis, 2001, 2 (3), pp.333-343. ⟨hal-00176512⟩