A partial differential equation for the rank one convex envelope

In this article we introduce a Partial Differential Equation (PDE) for the rank one convex envelope. Rank one convex envelopes arise in non-convex vector valued variational problems \cite{BallElasticity, kohn1986optimal1, BallJames87, chipot1988equilibrium}. More generally, we study a PDE for directional convex envelopes, which includes the usual convex envelope \cite{ObermanConvexEnvelope} and the rank one convex envelope as special cases. Existence and uniqueness of viscosity solutions to the PDE is established. Wide stencil elliptic finite difference schemes are built. Convergence of finite difference solutions to the viscosity solution of the PDE is proven. Numerical examples of rank one and other directional convex envelopes are presented. Additionally, laminates are computed from the rank one convex envelope.