2014/02/06 by Timothy J. Healey, Healey, Timothy J., Sanjay Dharmavaram +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Medicine · #Analysis of PDEs (math.AP) #Caveolin-1 and cellular processes #Erythrocyte Function and Pathophysiology #FOS: Mathematics #Lipid Membrane Structure and Behavior
paper · pdf · doi:10.48550/arxiv.1402.2314
openalex publication_date 2014/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a well known model for lipid-bilayer membrane vesicles exhibiting\nphase separation, incorporating a phase field with finite curvature elasticity.\nWe prove the existence of a plethora of equilibria, corresponding to\nsymmetry-breaking solutions of the Euler-Lagrange equations, via global\nbifurcation from the spherical state. To the best of our knowledge, this\nconstitutes the first rigorous existence results for this class of problems. We\novercome several difficulties in carrying this out. Due to inherent in-plane\nfluidity combined with finite curvature elasticity, neither the Eulerian\n(spatial) nor the Lagrangian (material) description of the model lends itself\nwell to analysis. Instead we adopt a singularity-free radial-map description\nthat effectively eliminates the grossly underdetermined in-plane fluid\ndeformation. The resulting governing equations comprise a quasi-linear elliptic\nsystem with lower-order nonlinear constraints. We then show the equivalence of\nour problem to that of finding the zeros of compact vector field. The latter is\nnot routine; we obtain certain spectral estimates and then shift the principle\npart of the operator. With this in hand, we combine well known group-theoretic\nideas for symmetry-breaking with global bifurcation theory to obtain our\nresults.\n