具有奇异熵势的Landau-de Gennes能量梯度流解的接触集
Contact set of solutions to gradient flow of Landau-de Gennes energy with a singular entropy potential
浏览论文内容
中文总结 AI 辅助
本文研究带奇异熵势的Landau-de Gennes能量梯度流解的接触集，结合正则性等理论刻画其性质，证明二维时空接触集为空、三维时空其Hausdorff维数至多为1。
中文摘要 AI 辅助
本文研究具有各向异性Landau-de Gennes弹性能量和奇异熵势$f(Q)$的$Q$张量流的凸性结构与接触集。该流自然呈现强制性性质，使得几乎每个时刻$f(Q)$都具有$H^1$正则性。结合该$H^1$正则性、球平均与容量理论，刻画了接触集$\boldsymbol{\textit{C}}(Q)\triangleq \boldsymbol{\textit{\textbf{x}}}\boldsymbol{|}f(Q(x))=\boldsymbol{\textit{\textbf{\textbackslash infty}}}\boldsymbol{\textit{\textbf{\textbackslash}}}$。特别地，证明了在二维空间中该接触集为空，在三维空间中其Hausdorff维数至多为1。
英文摘要
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.
