General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases

Abstract

F(Du(x)) =0, a.e. xe~, (1.1) u(x) = r ~ e 0a, where D is a (bounded) open set of R n, F: R m and ~E W 1'~ (D; Rm). We emphasize that u: DcRn--*R m, with m, n~>l, is a vector valued function if m> 1 (otherwise, if m--l, we say that u is a scalar function). As usual Du denotes the gradient of u. This problem (1.1) has been intensively studied, essentially in the scalar case in many relevant articles such as Lax [28], Douglis [23], Kru2kov [27], Crandall-Lions [16], Crandall-Evans-Lions [14], Capuzzo Dolcetta-Evans [8], Capuzzo Dolcetta-Lions [9],