Calculus of Variations and Geometric Measure Theory

O. Misiats - I. Topaloglu

On minimizers of an anisotropic liquid drop model

created by topaloglu1 on 04 Jan 2020
modified on 01 Apr 2021


Published Paper

Inserted: 4 jan 2020
Last Updated: 1 apr 2021

Journal: ESAIM: COCV
Volume: 27
Pages: S20
Year: 2021
Doi: 10.1051/cocv/2020068

This is a post-peer-review, pre-copyedit version of an article published in ESAIM: COCV. The final authenticated version is available online.


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 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.