Optimal Existence Theorems for Nonhomogeneous Differential Inclusions

In this paper we address the question of solvability of the differential inclusions (1.1). Our approach to these problems is based on the idea of constructing a sequence of approximate solutions which converges strongly and makes use of Gromovʹs idea (following earlier work of Nash and Kuiper) to control convergence of the gradients by appropriate selection of the elements of the sequence. In this paper we identify an optimal setting of this method. In particular we show that the existence result holds for general upper semicontinuous functions H without extra requirements like quasiconvexity of H with respect to Du, which was assumed in previous works, where the idea to apply the Baire category lemma to the sets of approximate solutions was developed. We also apply our result to identify the minimal sets, where the function H should vanish to guarantee solvability of the inclusions.