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BMO and uniform estimates for multi-well problems

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Abstract

We establish optimal local regularity results for vector-valued extremals and minimizers of variational integrals whose integrand is the squared distance function to a compact set K in matrix space \({{{\mathbb M}^{N \times n}}}\). The optimality is illustrated by explicit examples showing that, in the nonconvex case, minimizers need not be locally Lipschitz. This is in contrast to the case when the set K is suitably convex, where we show that extremals are locally Lipschitz continuous. The results rely on the special structure of the integrand and elementary Cordes–Nirenberg type estimates for elliptic systems.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Georg Dolzmann

  2. Mathematical Institute, University of Oxford, 24–29 St. Giles’, Oxford, OX1 3LB, UK

    Jan Kristensen

  3. Department of Mathematics, Swansea University, Singleton Park, Swansea, Wales, SA2 8PP, UK

    Kewei Zhang

Authors
  1. Georg Dolzmann
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  2. Jan Kristensen
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  3. Kewei Zhang
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Correspondence to Jan Kristensen.

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Dolzmann, G., Kristensen, J. & Zhang, K. BMO and uniform estimates for multi-well problems. manuscripta math. 140, 83–114 (2013). https://doi.org/10.1007/s00229-012-0531-8

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  • DOI: https://doi.org/10.1007/s00229-012-0531-8

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