Regularity of Minimizers of a Tensor-valued Variational Obstacle Problem in Three Dimensions
Abstract
Motivated by Ball and Majumdar's modification of Landau-de Gennes model for nematic liquid crystals, we study energy-minimizer of a tensor-valued variational obstacle problem in a bounded 3-D domain with prescribed boundary data. The energy functional is designed to blow up as approaches the obstacle. Under certain assumptions, especially on blow-up profile of the singular bulk potential, we prove higher interior regularity of , and show that the contact set of is either empty, or small with characterization of its Hausdorff dimension. We also prove boundary partial regularity of the energy-minimizer.
Keywords
Cite
@article{arxiv.1908.10889,
title = {Regularity of Minimizers of a Tensor-valued Variational Obstacle Problem in Three Dimensions},
author = {Zhiyuan Geng and Jiajun Tong},
journal= {arXiv preprint arXiv:1908.10889},
year = {2019}
}
Comments
We added the detailed proof of the two inequalities (2.25) in the Appendix. Also, for the log potential, we proposed a weaker assumption (1.12) which is satisfied by the Ball-Majumdar bulk potential. We also modify the result (1.19) and the proof (Page 18) of the second part of Theorem 1.4 accordingly