On minimizers of an anisotropic liquid drop model

Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, USA.

^* Corresponding author: iatopaloglu@vcu.edu

Received: 23 December 2019
Accepted: 7 October 2020

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 C¹-norm and quantify the rate of convergence. We also obtain a quantitative expansion 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.