Solving partial differential inclusions

We are interested in mappings whose gradient takes values in a given (finite) set of matrices.
Existence results rely on constructions via laminations and hence rank-one convexity, where as non-existence (or regularity) uses properties of general gradients and is more related to quasiconvexity and, in particular, quasiconformality.
It turns out that these two approaches complement each other surprisingly well when we consider either sets without rank-one connections or the interface problem.