Hopf-Lax Formulas for Semicontinuous Data

O. Alvarez*, E. N. Barron, Hitoshi Ishii

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

43 Citations (Scopus)


The equations ut + H(Du) = 0 and ut + H(u, Du) = 0, with initial condition u(0, x) = g(x) have an explicit solution when the hamiltonian is convex in the gradient variable (Lax formula) or the initial data is convex, or quasiconvex (Hopf formula). This paper extends these formulas to initial functions g which are only lower semicontinuous (lsc), and possibly infinite. It is proved that the Lax formulas give a lsc viscosity solution, and the Hopf formulas result in the minimal supersolution. A level set approach is used to give the most general results.

Original languageEnglish
Pages (from-to)993-1035
Number of pages43
JournalIndiana University Mathematics Journal
Issue number3
Publication statusPublished - 1999 Sept
Externally publishedYes


  • Hopf and Lax formulas
  • Level sets
  • lsc viscosity solutions

ASJC Scopus subject areas

  • Mathematics(all)


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