On minimizers of an anisotropic liquid drop model

TL;DR

This paper investigates the conditions under which Wulff shapes minimize an anisotropic liquid drop model, demonstrating convergence to Wulff shapes in the small nonlocality regime and providing stability and energy expansion results.

Contribution

It establishes that Wulff shapes are minimizers only in isotropic cases, and quantifies the convergence and stability of minimizers for smooth anisotropies and crystalline tensions.

Findings

Wulff shapes minimize the model only when surface energy is isotropic.

Minimizers converge to Wulff shapes in the small nonlocality regime with quantifiable rates.

Exact energy expansions are obtained for crystalline surface tensions.

Abstract

We consider a variant of Gamow's liquid drop model with an anisotropic surface energy. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. We show that for smooth anisotropies, in the small nonlocality regime, minimizers converge to the Wulff shape in $C^1$-norm and quantify the rate of convergence. We also obtain a quantitative extension of the energy of any minimizer around the energy of a Wulff shape yielding a geometric stability result. For certain crystalline surface tensions we can determine the global minimizer and obtain its exact energy expansion in terms of the nonlocality parameter.