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

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.

41 pages

References in corpus (2)

• A diffuse-interface approach for solid-state dewetting with anisotropic surface energies
• Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow

Cited by in corpus (1)

• Strong well-posedness and separation properties for a bulk-surface convective Cahn--Hilliard system with singular potentials