三维张量值变分障碍问题极小元的正则性
偏微分方程分析
2019-12-19 v2
摘要
受 Ball 和 Majumdar 对向列相液晶的 Landau-de Gennes 模型修正的启发，我们研究在有界三维区域中具有给定边界数据的张量值变分障碍问题的能量极小元 。该能量泛函被设计为当 接近障碍时发散。在若干假设下，特别是对奇异体势的发散剖面的假设，我们证明了 的更高内部正则性，并表明 的接触集要么为空，要么较小且其 Hausdorff 维数可被刻画。我们还证明了能量极小元的边界部分正则性。
引用
@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}
}
备注
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