The Oseen–Frank Energy Functional on Manifolds

Abstract

We observe that for a unit tangent vector field uTM on a 3-dimensional Riemannian manifold M, there is a unique unit cotangent vector field ATM associated to u such that we can define the curl of u by dA. Through a unit cotangent vector field ATM, we define the Oseen–Frank energy functional on 3-dimensional Riemannian manifolds. Moreover, we prove partial regularity of minimizers of the Oseen–Frank energy on 3-dimensional Riemannian manifolds.

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Acknowledgements

I would take this opportunity to thank Professors Mariano Giaquinta and Enrico Giusti for their strong influence and support; in particular, the main idea of this paper was taught by them. Part of the research was supported by the Australian Research Council grant DP150101275.

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Correspondence to Min-Chun Hong.

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Dedicated to Professor Jürgen Jost on the occasion of his 65th birthday.

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Hong, MC. The Oseen–Frank Energy Functional on Manifolds. Vietnam J. Math. (2021). https://doi.org/10.1007/s10013-020-00468-2

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Keywords

  • The Oseen–Frank energy
  • Partial regularity

Mathematics Subject Classification (2010)

  • 35J50
  • 58J05