The existence and stability of solutions for liquid crystal two-phase interface problems

• Published in: Calculus of variations and partial differential equations Vol. 64; no. 7; p. 233
• Main Author: Wu, Qin
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.09.2025
• Subjects: Eigenvectors; Energy; Equilibrium; Geometry; Liquid crystals; Maximum principle; Metric space; Numerical analysis; Point defects; Stability; Three dimensional analysis
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-025-03087-2

In this paper, we study the existence and energetic stability of equilibrium solutions to the Landau-de Gennes energy functional subject to various anchoring conditions. Building on the work of Park et al. (Calc Variat Partial Differ Equ 56(41):1–15, 2017), which established the energetic stability of a uniaxial solution with homeotropic anchoring for 1D variation perturbation when the anisotropic coefficient we extend the analysis to three-dimensional cases. Specifically, we demonstrate the 3D energetic stability in the optimal range of under the nonnegative physical hypothesis. The key techniques involve the decomposition of squared terms and the application of integration by parts for cross-terms. We also prove, for the first time, the existence of a smooth biaxial solution under the planar anchoring condition using the metric geometry method. By applying the maximum principle and exploiting its minimality in a certain sense, we derive properties of the biaxial solution. We also establish a sufficient condition for its energetic stability when which relies on the negativity of a specific integral, consistent with our numerical results.