Manuscripta Mathematica

, Volume 140, Issue 1–2, pp 83–114 | Cite as

BMO and uniform estimates for multi-well problems

  • Georg Dolzmann
  • Jan KristensenEmail author
  • Kewei Zhang


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.

Mathematics Subject Classification (1991)

49Q20 35B05 


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  2. 2.Mathematical InstituteUniversity of OxfordOxfordUK
  3. 3.Department of MathematicsSwansea UniversitySwanseaUK

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