Minimizers of Anisotropic Surface Tensions Under Gravity: Higher Dimensions via Symmetrization

TL;DR

This paper studies the shape of liquid drops and crystals under gravity with anisotropic surface tensions, proving existence, symmetry, convexity, and uniqueness of minimizers using symmetrization techniques and ODE analysis.

Contribution

It introduces a variational model incorporating anisotropic surface tensions and applies symmetrization methods to establish key properties of minimizers, including their uniqueness in smooth cases.

Findings

Existence of minimizers for the model.

Convexity and symmetry of minimizers.

Uniqueness of minimizers for smooth surface tensions.

Abstract

We consider a variational model describing the shape of liquid drops and crystals under the influence of gravity, resting on a horizontal surface. Making use of anisotropic symmetrization techniques, we establish existence, convexity and symmetry of minimizers for a class of surface tensions admissible to the symmetrization procedure. In the case of smooth surface tensions, we obtain uniqueness of minimizers via an ODE characterization.