Calculus of Variations and Geometric Measure Theory

K. Brazda - L. Lussardi - U. Stefanelli

Existence of varifold minimizers for the multiphase Canham-Helfrich functional

created by lussardi on 02 Dec 2019
modified on 25 May 2020


Published Paper

Inserted: 2 dec 2019
Last Updated: 25 may 2020

Journal: Calc. Var. Partial Differential Equations
Volume: 59
Number: 3
Pages: 1-26
Year: 2020


We address the minimization of the Canham-Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers is here assumed. Correspondingly, the problem is reformulated and solved in the weaker frame of oriented curvature varifolds. We present a lower semicontinuity result and prove existence of single- and multiphase minimizers under area and enclosed-volume constrains. Additionally, we discuss regularity of minimizers and establish lower and upper diameter bounds.

Keywords: Canham-Helfrich functional, curvature varifolds, biological membranes