The anisotropic Cahn--Hilliard equation: regularity theory and strict separation properties

TL;DR

This paper establishes existence, uniqueness, regularity, and separation properties for solutions to the anisotropic Cahn--Hilliard equation with logarithmic free energy, addressing a gap in the mathematical understanding of these physically relevant models.

Contribution

It provides the first comprehensive analytical results for the anisotropic Cahn--Hilliard equation with logarithmic free energy, including regularity and separation properties.

Findings

Proved existence and uniqueness of weak solutions.

Established regularity results for the solutions.

Demonstrated separation properties of solutions.

Abstract

The Cahn--Hilliard equation with anisotropic energy contributions frequently appears in many physical systems. Systematic analytical results for the case with the relevant logarithmic free energy have been missing so far. We close this gap and show existence, uniqueness, regularity, and separation properties of weak solutions to the anisotropic Cahn--Hilliard equation with logarithmic free energy. Since firstly, the equation becomes highly non-linear, and secondly, the relevant anisotropies are non-smooth, the analysis becomes quite involved. In particular, new regularity results for quasilinear elliptic equations of second order need to be shown.