2015/09/16 by Timothy J. Healey, Healey, Timothy J., Sanjay Dharmavaram +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Caveolin-1 and cellular processes #FOS: Mathematics #Lipid Membrane Structure and Behavior #math.AP
paper · pdf · doi:10.48550/arxiv.1509.05074
arXiv admin note: substantial text overlap with arXiv:1402.2314
arxiv created 2015/09/16 · openalex publication_date 2015/09/16 · arxiv updated 2015/09/18 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We study a model for lipid-bilayer membrane vesicles exhibiting phase separation, incorporating a phase field together with membrane fluidity and bending elasticity. We prove the existence of a plethora of equilibria in the large, corresponding to symmetry-breaking solutions of the Euler-Lagrange equations, via global bifurcation from the spherical state. To the best of our knowledge, this constitutes the first rigorous existence results for this class of problems. We overcome several difficulties in carrying this out. Due to inherent surface fluidity combined with finite curvature elasticity, neither the Eulerian (spatial) nor the Lagrangian (material) description of the model lends itself well to analysis. This is resolved via a singularity-free radial-map description, which effectively eliminates the grossly underdetermined in-plane deformation. The resulting governing equations comprise a quasi-linear elliptic system with nonlinear constraints. We show the equivalence of the problem to that of finding the zeros of compact vector field. The latter is not routine. Using spectral and a-priori estimates together with Fredholm properties, we demonstrate that the principal part of the quasi-linear mapping defines an operator with compact resolvent. We then combine well known group-theoretic ideas for symmetry breaking with global bifurcation theory to obtain our results.