Mathematics

In this paper we address the question of solvability of the differential inclusions Du(.) \in K(.,u(.)), u is Lischitz, u = f in the boundary, where Df(.) \in F(.,f(.)) a.e. in the domain and F is a relevant multi-valued function. Our approach to these problems is based on the idea to construct a sequence of approximate solutions which converges strongly and makes use of Gromov's idea (following earlier work of Nash and Kuiper) to control convergence of gradients by appropriate selection of the elements of the sequence. In this paper we identify an optimal setting of this method. We discuss some applications of the result: Hamilton-Jacobi systems of equations, convex multi-valued mappings F (here we identify the minimal sets K to solve all inclusions), and a nonhomogeneous case of phase transitions K(.) = SO(2)A(.) U SO(2)B(.).