Hopf-Type Estimates and Formulas For Nonconvex Nonconcave Hamilton--Jacobi Equations
- 1 September 1998
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 29 (5) , 1067-1086
- https://doi.org/10.1137/s0036141096309629
Abstract
Simple explicit estimates are presented for the viscosity solution of the Cauchy problem for the Hamilton--Jacobi equation where either the Hamiltonian or the initial data are the sum of a convex and a concave function. The estimates become equalities whenever a "minmax" equals a "maxmin" and thus a representation formula for the solution is obtained, generalizing the classical Hopf formulas as well as some formulas of Kruzkov [Functional Anal. Appl., 2 (1969), pp. 128--136].Keywords
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