Regularity results for first order Hamilton-Jacobi equations
Guy Barles · Differential and Integral Equations · 1990
We are interested here in regularity results for the first-order Hamilton-Jacobi equations (1)where x E IRn, y E IRm, H is a given continuous function and u is a real-valued function that we will always assume to be bounded uniformly continuous (BUC in short).Our work relies on the notion of viscosity solutions, introduced by M.G.Crandall and P.L. Lions [10] (see also M.G.Crandall, L.C.Evans and P.L. Lions [8] and P.L. Lions [24]) and in all the following, all the equations or inequalities have to be understood in viscosity sense.We refer the reader to the bibliography for references concerning the study of the properties of viscosity solutions (existence, uniqueness, stability, . . . ) but our list is by no means complete.Our aim is to give general sufficient conditions on H, ensuring a Holder modulus of continuity for u in x only.We will also consider the Cauchy problem