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A Partial Differential Equation for the Rank One Convex Envelope

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Abstract

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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Authors and Affiliations

  1. McGill University, Montreal, Canada

    Adam M. Oberman

  2. Department of Mathematics, Beihang University, Beijing, China

    Yuanlong Ruan

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  1. Adam M. Oberman
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  2. Yuanlong Ruan
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Correspondence to Adam M. Oberman.

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Communicated by I. Fonseca

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Oberman, A.M., Ruan, Y. A Partial Differential Equation for the Rank One Convex Envelope. Arch Rational Mech Anal 224, 955–984 (2017). https://doi.org/10.1007/s00205-017-1092-5

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