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Quantitative stratification and higher regularity for biharmonic maps

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Abstract

In this paper we prove quantitative regularity results for stationary and minimizing extrinsic biharmonic maps. As an application, we determine sharp, dimension independent L p bounds for \({\nabla^k f}\) that do not require a small energy hypothesis. In particular, every minimizing biharmonic map is in W 4,p for all \({1 \le p < 5/4}\). Further, for minimizing biharmonic maps from \({\Omega \subset \mathbb{R}^5}\), we determine a uniform bound on the number of singular points in a compact set. Finally, using dimension reduction arguments, we extend these results to minimizing and stationary biharmonic maps into special targets.

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Authors and Affiliations

  1. Department of Mathematics, Fordham University, Bronx, NY, 10458, USA

    Christine Breiner

  2. Institute for Analysis, Karlsruhe Institute of Technology (KIT), Kaiserstr. 89-93, 76133, Karlsruhe, Germany

    Tobias Lamm

Authors
  1. Christine Breiner
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  2. Tobias Lamm
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Correspondence to Christine Breiner.

Additional information

The first author was supported in part by NSF grant DMS-1308420.

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Breiner, C., Lamm, T. Quantitative stratification and higher regularity for biharmonic maps. manuscripta math. 148, 379–398 (2015). https://doi.org/10.1007/s00229-015-0750-x

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  • DOI: https://doi.org/10.1007/s00229-015-0750-x

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