Viscosity solutions of Hamilton-Jacobi equations
Thomas Strömberg
ABSTRACT.
Hamilton-Jacobi equations play a pivotal role in various fields of
mathematics including calculus of variations, optimal control theory,
and classical mechanics. It was however as late as in the early 80's, in
a series of papers by M. G. Crandall and P.-L. Lions, that the correct
concept of a solution was introduced and examined, namely that of a
"viscosity solution." The development of this theory has resulted in
existence and uniqueness results of substantial generality for uniformly
continuous solutions.
The aim of this talk is to present definitions and survey some
of the most important known theorems, but also to discuss uniqueness
for locally Lipschitz continuous solutions of the Cauchy problem
in which the initial condition is formulated in terms of a prescribed
function that is merely assumed to be lsc and proper.