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A boundary monotonicity inequality for variationally biharmonic maps and applications to regularity theory

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Abstract

We derive a boundary monotonicity formula for a class of biharmonic maps with Dirichlet boundary conditions. A monotonicity formula is crucial in the theory of partial regularity in supercritical dimensions. As a consequence of such a boundary monotonicity formula, one is able to show partial regularity for variationally biharmonic maps and full boundary regularity for minimizing biharmonic maps.

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Notes

  1. One distinguishes between extrinsically and intrinsically biharmonic maps. We say that a map is intrinsically biharmonic iff it is a critical point of \(\mathcal {E}(u)=\int _{\mathcal {M}}\vert \nabla Du\vert ^2\text {d}\mu _{\mathcal {M}}\). The energy \(\mathcal {E}\) does not depend on the embedding \(\mathcal {N}\hookrightarrow \mathbb {R}^n\) while \(E_2\) does. Therefore, the distinction between extrinsically and intrinsically biharmonic maps.

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Acknowledgements

I would like to thank Prof. Dr. Christoph Scheven for his much helpful advice.

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  1. Fakultät für Mathematik, Universität Duisburg-Essen, 45117, Essen, Germany

    Serdar Altuntas

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  1. Serdar Altuntas
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Correspondence to Serdar Altuntas.

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Altuntas, S. A boundary monotonicity inequality for variationally biharmonic maps and applications to regularity theory. Ann Glob Anal Geom 54, 489–508 (2018). https://doi.org/10.1007/s10455-018-9610-8

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  • DOI: https://doi.org/10.1007/s10455-018-9610-8

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