Speaker:Xiaofeng Ren, Department of Mathematics, George Washington University
Title: The spectrum of the torus profile to a geometric variational problem with long range interaction

Abstract: The profile problem for the Ohta-Kawasaki diblock copolymer theory is a geometric variational problem. The energy functional is defined on sets in $\mathbb{R}^3$ of prescribed volume and the energy of an admissible set is its perimeter plus a long range interaction term related to the Newtonian potential of the set. This problem admits a solution, called a torus profile, that is a set enclosed by an approximate torus of the major radius 1 and the minor radius q. The torus profile is both axially symmetric about the z-axis and reflexively symmetric about the xy-plane. There is a way to set up the profile problem in a function space as a partial differential-integro equation. The linearized operator ${\cal L}$ of the problem at the torus profile is decomposed into a family of linear ordinary differential-integro operators ${\cal L}^m$ where the index $m=0,1,2,...$ is called a mode. The spectrum of ${\cal L}$ is the union of the spectra of the ${\cal L}^m$'s. It is proved that for each m, when q is sufficiently small, ${\cal L}^m$ is positive definite. (0 is an eigenvalue for both ${\cal L}^0$ and ${\cal L}^1$, due to the translation and rotation invariance.) As q tends to 0, more and more ${\cal L}^m$'s become positive definite. However no matter how small q is, there is always a mode m of which ${\cal L}^m$ has a negative eigenvalue. This mode grows to infinity like $q^{-3/4}$ as $q \rightarrow 0$. This is joint work with Juncheng Wei.

Time: Friday, September 23, 2016, 1:30-2:20 p.m.

Place: Exploratory Hall, Room 4106

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