Peter Sternberg

On the global minimizers of a nonlocal isoperimetric problem in two dimensions

In this article we analyze the minimization of a nonlocal isoperimetric problem (NLIP) posed on the flat 2-torus. After establishing regularity of the free boundary of minimizers, we show that when the parameter controlling the influence of the nonlocality is small, there is an interval of values for the mass constraint such that the global minimizer is exactly lamellar; that is, the free boundary consists of two parallel lines. In other words, in this parameter regime, the global minimizer of the 2d (NLIP) coincides with the global minimizer of the local periodic isoperimetric problem.

Periodic phase separation: the periodic Cahn-Hilliard and isoperimetric problems

We consider two well known variational problems associated with the phenomenon of phase separation: the isoperimetric problem and minimization of the Cahn‐Hilliard energy. The two problems are related through a classical result in -convergence and we explore the behavior of global and local minimizers for these problems in the periodic setting. More precisely, we investigate these variational problems for competitors defined on the flat 2- or 3-torus. We view these two problems as prototypes for periodic phase separation. We give a complete analysis of stable critical points of the 2-d periodic isoperimetric problem and also obtain stable solutions to the 2-d and 3-d periodic Cahn‐Hilliard problem. We also discuss some intriguing open questions regarding triply periodic constant mean curvature surfaces in 3-d and possible counterparts in the Cahn‐Hilliard setting.

Critical points via Gamma-convergence: general theory and applications

It is well-known that Γ-convergence of functionals provides a tool for studying global and local minimizers. Here we present a general result establishing the existence of critical points of a Γ-converging sequence of functionals provided the associated Γ-limit possesses a nondegenerate critical point, subject to certain mild additional hypotheses. We then go on to prove a theorem that describes suitable non-degenerate critical points for functionals, involving the arclength of a limiting singular set, that arise as Γ-limits in a number of problems. Finally, we apply the general theory to prove some new results, and give new proofs of some known results, establishing the existence of critical points of the 2d Modica-Mortola (Allen-Cahn) energy and 3d Ginzburg-Landau energy with and without magnetic field, and various generalizations, all in a unified framework.

The Dirichlet Problem for Functions of Least Gradient

For a given domain Ω⊂ R n, we consider the variational problem of minimizing the L1-norm of the gradient on Ωof a function with prescribed continuous boundary values. Under certain weak conditions on the boundary of the domain Ω, it is shown that the BV solution is continuous and unique.

Cascade of minimizers for a nonlocal isoperimetric problem in thin domains

For

Local minimisers of a three-phase partition problem with triple junctions

\Omega_\varepsilon=(0,\varepsilon)\times (0,1)

The Resistive State in a Superconducting Wire: Bifurcation from the Normal State

a thin rectangle, we consider minimization of the two-dimensional nonlocal isoperimetric problem given by

GINZBURG--LANDAU MODEL IN THIN LOOPS WITH NARROW CONSTRICTIONS

\inf_u E^{\gamma}_{\Omega_\varepsilon}(u)

Dimension Reduction for the Landau-de Gennes Model in Planar Nematic Thin Films

, where

Convergence of a Particle Method for Diffusive Gradient Flows in One Dimension

E^{\gamma}_{\Omega_\varepsilon}(u):= P_{\Omega_\varepsilon} (\{u(x)=1\})+\gamma\int_{\Omega_\varepsilon}\left\vert{\nabla{v}}\right\vert^2\,dx