Comparing two methods of resolving homogeneous differential inclusions

Abstract.

Given a compact set $K \subset{\bf R}^{m \times n}$ we consider the differential inclusion

$Du \in K, u \in W^{1,\infty}(\Omega;{\bf R}^m), u\Big|_{\partial\Omega} = f\Big|_{\partial \Omega}.$

We show how to use the main idea of the method of convex integration [ N], [G], [K] (to control convergence of the gradients of a sequence of approximate solutions by appropriate selection of the sequence) to obtain an optimal existence result. We compare this result with the ones available by the Baire category approach applied to the set of admissible functions with $L^{\infty}$ topology.

A byproduct of our result is attainment in the minimization problems

$J(u) = \int_{\Omega} L(Du(x)) dx \to \min, u\Big|_{\partial \Omega} = f, u \in W^{1,1}(\Omega;{\bf R}^m)$

with integrands L having quasiaffine quasiconvexification that was, in fact, the reason of our interest to differential inclusions. This result can be considered as a first step towards characterization of those minimization problems which are solvable for all boundary data. This problem was solved in [S1] in the scalar case m=1.