Anisotropic liquid drop models

TL;DR

This paper explores anisotropic variants of Gamow's liquid drop model, analyzing the existence, shape of minimizers, and conditions under which Wulff shapes are optimal, revealing differences between isotropic and crystalline surface tensions.

Contribution

It introduces anisotropic surface energies into the liquid drop model and characterizes the conditions for Wulff shapes to be minimizers, including new results for crystalline tensions.

Findings

Wulff shapes are minimizers if and only if surface tension is isotropic.

Wulff shapes are the unique minimizers for certain crystalline tensions.

Wulff shapes minimize models with anisotropic repulsion in the small mass regime.

Abstract

We introduce and study certain variants of Gamow's liquid drop model in which an anisotropic surface energy replaces the perimeter. After existence and nonexistence results are established, the shape of minimizers is analyzed. 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. In sharp contrast, Wulff shapes are the unique minimizers for certain crystalline surface tensions. We also introduce and study several related liquid drop models with anisotropic repulsion for which the Wulff shape is the minimizer in the small mass regime.