论文标题
关于列液晶液滴的变分问题
On a variational problem of nematic liquid crystal droplets
论文作者
论文摘要
令$μ> 0 $为固定常数，我们证明将最小化器最小化到以下能量功能\ begin {align*} e_f（u，ω）：= \int_Ω| \ nabla u |^2+μp（ω）\ end \ end {align*}之间存在于对$（ω，u）$中，使得$ω$是$ m $ m $ - 统一的域，具有有限的和固定体积，$ u \ in h^1（$ u \ in^$ω，\ umω，\ umω，\ ymath^uc} { =ν_Ω$，测量理论外部单元正常，几乎在$ω$的降低边界上的任何地方。还获得了各种设置中最佳配置的唯一性。此外，我们考虑\ begin {align*}给出的一般能量功能 e_f（u，ω）：= \int_Ω| \ nabla u（x）|^2 \，dx + \ int _ {\ partial^*ω} f \ big big（x）\cdotν_Ω（x） $ω$和$ f $的减少边界是$ \ mathbb r $的凸正功能。我们证明，$ e_f $的最小化器也存在于$ m $均匀的外部最小化域中$ω$，带有固定音量，$ u \ in h^1（ω，\ mathbb {s}^2）$。
Let $μ>0$ be a fixed constant, and we prove that minimizers to the following energy functional \begin{align*} E_f(u,Ω):=\int_Ω|\nabla u|^2+μP(Ω) \end{align*}exist among pairs $(Ω,u)$ such that $Ω$ is an $M$-uniform domain with finite perimeter and fixed volume, and $u \in H^1(Ω,\mathbb{S}^2)$ with $u =ν_Ω$, the measure-theoretical outer unit normal, almost everywhere on the reduced boundary of $Ω$. The uniqueness of optimal configurations in various settings is also obtained. In addition, we consider a general energy functional given by \begin{align*} E_f(u,Ω):=\int_Ω |\nabla u(x)|^2 \,dx + \int_{\partial^* Ω} f\big(u(x)\cdot ν_Ω(x)\big) \,d\mathcal{H}^2(x), \end{align*}where $\partial^* Ω$ is the reduced boundary of $Ω$ and $f$ is a convex positive function on $\mathbb R$. We prove that minimizers of $E_f$ also exist among $M$-uniform outer-minimizing domains $Ω$ with fixed volume and $u \in H^1(Ω,\mathbb{S}^2)$.
