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While the hexagonal lattice is ubiquitous in two dimensions, the body centered cubic lattice and the face centered cubic lattice are both commonly observed in three dimensions. A geometric variational problem motivated by the diblock copolymer theory consists of a short range interaction energy and a long range interaction energy. In three dimensions, and when the long range interaction is given by the nonlocal operator $$(-\Delta)^{-3/2}$$, it is proved that the body centered cubic lattice is the preferred structure.
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abstract: In the past years, there has been a new light shed on the harmonic map problem with free boundary in view of its connection with nonlocal equations. Here we fully exploit this link, considering the harmonic map flow with free boundary $$ (0.1)\hskip77pt\cases{u_t=\Delta u& in $$\Bbb{R}^2_+\times (0,T)$$,\cr u(x,0,t)\in\Bbb{S}^1& for all $$(x,0,t)\in\partial\Bbb{R}^2_+\times (0,T)$$,\cr {du\over dy}(x,0,t)\perp T_{u(x,0,t)}\Bbb{S}^1& for all $$(x,0,t)\in\partial\Bbb{R}^2_+\times (0,T)$$,\cr u(\cdot, 0)=u_0& in $$\Bbb{R}^2_+$} $$ for a function $$u:\Bbb{R}^2_+\times [0,T)\to\Bbb{R}^2$$. Here $$u_0 :\Bbb{R}^2_+\to\Bbb{R}^2$$ is a given smooth map and $$\perp$$ stands for orthogonality. We prove the existence of initial data $$u_0$$ such that (0.1) blows up at finite time with a profile being the half-harmonic map. This answers a question raised by Chen and Lin.more » « less
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We consider the initial boundary value problem of a simplified nematic liquid crystal flow in a bounded, smooth domain $$\Omega\subset\mathbb R^2$$. Given any k distinct points in the domain, we develop a new inner-outer gluing method to construct solutions that blow up exactly at those k points as t goes to a finite time T. Moreover, we obtain a precise description of the blowupmore » « less
