A partial differential equation for the rank one convex envelope

TL;DR

This paper introduces a PDE framework for computing the rank one convex envelope, providing theoretical guarantees and numerical methods, with applications in variational problems and laminate computations.

Contribution

It develops a PDE for the rank one convex envelope, proves existence and uniqueness of solutions, and presents convergent finite difference schemes with numerical examples.

Findings

Established existence and uniqueness of viscosity solutions.

Developed convergent finite difference schemes.

Numerical examples demonstrate practical computation of envelopes.

Abstract

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.