Cascade of Minimizers for a Nonlocal Isoperimetric Problem in Thin Domains
Cascade of Minimizers for a Nonlocal Isoperimetric Problem in Thin Domains
复制标题
薄域中非局部等周问题的极小化级联
DOI：
10.1137/130932594
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
P. Sternberg
中科院分区：
文献类型：
--
作者：
M. Morini;P. Sternberg
对于$\Omega_\varepsilon=(0,\varepsilon)\乘以（0,1）$矩形，我们考虑由$\inf_u E^{\gamma} {\Omega_\varepsilon}(u)$给出的二维非局部等距问题的最小化，其中$E^{\gamma} {\Omega_\varepsilon}(u)$:= P_{\Omega_\varepsilon} (u (x)=1)+\gamma\int_{\Omega_\varepsilon}\左\vert{\nabla{v}}\右\vert^2\，dx$，最小化被竞争对手$u\in BV(\Omega_\varepsilon)；\{\pm 1\})$满足质量约束$\fint_{\Omega_\varepsilon}u=m$对于一些$m\in(-1,1)$。这里$P_{\Omega_\varepsilon}(\{u(x)=1\})$表示集合$\{u(x)=1\}$ in $\Omega_\varepsilon$, $\fint$表示积分平均值，$v$表示泊松问题的解$-\Delta v=u-m\；\Omega_\varepsilon,\quad\nabla v\cdot n_{\partial\Omega_\varepsilon}=0\;\mbox{on}\;\partial\Omega_\varepsilon,\quad\int_{\Omega_\varepsilon}v=0。我们展示了条纹图案是$\varepsilon\ll 1$的最小化器，条纹的数量像$\gamma一样增长…
For $\Omega_\varepsilon=(0,\varepsilon)\times (0,1)$ a thin rectangle, we consider minimization of the two-dimensional nonlocal isoperimetric problem given by $\inf_u E^{\gamma}_{\Omega_\varepsilon}(u)$, where $E^{\gamma}_{\Omega_\varepsilon}(u):= P_{\Omega_\varepsilon} (\{u(x)=1\})+\gamma\int_{\Omega_\varepsilon}\left\vert{\nabla{v}}\right\vert^2\,dx$ and the minimization is taken over competitors $u\in BV(\Omega_\varepsilon;\{\pm 1\})$ satisfying a mass constraint $\fint_{\Omega_\varepsilon}u=m$ for some $m\in (-1,1)$. Here $P_{\Omega_\varepsilon}(\{u(x)=1\})$ denotes the perimeter of the set $\{u(x)=1\}$ in $\Omega_\varepsilon$, $\fint$ denotes the integral average, and $v$ denotes the solution to the Poisson problem $-\Delta v=u-m\;\mbox{in}\;\Omega_\varepsilon,\quad\nabla v\cdot n_{\partial\Omega_\varepsilon}=0\;\mbox{on}\;\partial\Omega_\varepsilon,\quad\int_{\Omega_\varepsilon}v=0.$ We showthat a striped pattern is the minimizer for $\varepsilon\ll 1$ with the number of stripes growing like $\gamma...
具有长程相互作用的等周问题的均匀能量分布
影响因子：
3.9
作者：
G. Alberti;R. Choksi;F. Otto
通讯作者：
F. Otto