A Class Of Implicit Partial Differential Equations The Dirichlet Problem

In this article, we give sufficient conditions for the existence of solutions to the vectorial Hamilton-Jacobi equations with Dirichlet boundary condition:obtaining, in addition, an application to the theory of existence of minimizers for a class of non-convex variational problems. Under suitable compatibility conditions, the above problem can be reduced to the following differential inclusions problem:By this way no convexity, continuity or growth condition assumption on function g are needed. We can show the existence of solutions to the differential inclusionsproblem by Baire category method, and so the formal problem. The main steps of using Baire category method are as follows. First we construct a complete metric space V .Then with the help of the likelihood functional, we obtain a series of openand dense subset Vs in V .Finally, by Baire category theorem, we know that the subset Vs is dense in V .