On the Vectorial Hamilton-jacobi System

We discuss some recent developments in the study of regularity and stability for a first order Hamilton-Jacobi system: ∇u(x) ∈ K, where K is a closed set of n × m-matrices and u is a map from a domain Ω ⊂ R to R. For regularity of solutions, we obtain a higher integrability from a very weak integral coercivity condition known as the L-mean coercivity. For the stability, we study W sequences {uj} for which {∇uj} converges weakly and approaches the set K in some point-wise sense, and describe a new approach to study the weak limits by the so-called W -quasiconvex hull of K. Computation of quasiconvex hulls is usually extremely hard, but some important new developments in the nonlinear partial differential equations turn out to be greatly useful for our study.