Inserted: 30 apr 2019
Last Updated: 30 apr 2019
Most cell membranes of living organisms are made of lipid bilayer, which is a thin polar membrane consisting of two opposite oriented layers of lipid molecules. In the early 70s, Canham and Helfrich introduced their energy which is a linear combination of the integrated squared mean curvature, called Willmore energy, the first moment of the mean curvature, and the area. We prove existence and regularity of minimisers for the Canham-Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the $2$-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973. On the way to prove the main results we answer some open question raised by R\"oger in 2008.