THE ANISOTROPIC CAHN-HILLIARD EQUATION: REGULARITY THEORY AND STRICT SEPARATION PROPERTIES

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.

Automated summary

This article investigates the Cahn-Hilliard equation with anisotropic energy contributions, specifically the case with logarithmic free energy. Analytical results are presented for the existence, uniqueness, regularity, and separation properties of weak solutions. The analysis becomes intricate due to the highly non-linear nature of the equation and the non-smoothness of the relevant anisotropies, requiring new regularity results for quasilinear elliptic equations of second order.