Archive for Rational Mechanics and Analysis

, Volume 224, Issue 3, pp 955–984 | Cite as

A Partial Differential Equation for the Rank One Convex Envelope

  • Adam M. Oberman
  • Yuanlong Ruan


A partial differential equation (PDE) for the rank one convex envelope is introduced. The existence and uniqueness of viscosity solutions to the PDE is established. Elliptic finite difference schemes are constructed and convergence of finite difference solutions to the viscosity solution of the PDE is proven. Computational results are presented and laminates are computed from the envelopes. Results include the Kohn–Strang example, the classical four gradient example, and an example with eight gradients which produces nontrivial laminates.


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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.McGill UniversityMontrealCanada
  2. 2.Department of MathematicsBeihang UniversityBeijingChina

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