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arXiv · 2607.26354

Contact set of solutions to gradient flow of Landau-de Gennes energy with a singular entropy potential

Yuning Liu·Xiang Xu
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Abstract

This paper investigates the convexity structure and contact set of the $Q$-tensor flow with anisotropic Landau--de Gennes elastic energy and a singular entropy potential $f(Q)$. The flow naturally exhibits a coercivity property, yielding the $H^1$ regularity of $f(Q)$ for almost every time. Combining this $H^1$ regularity with spherical averaging and capacity theory, the contact set $$\mathcal{C}(Q)\triangleq \{x\mid f(Q(x))=\infty\}$$ is characterized. In particular, it is shown to be empty in two spatial dimensions and to have Hausdorff dimension at most one in three spatial dimensions.

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Yuning Liu, Xiang Xu. 2026-07-29. Contact set of solutions to gradient flow of Landau-de Gennes energy with a singular entropy potential. https://arxiv.org/abs/2607.26354

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