Stability of membranes (Q6611140)

Helfrich energy models the morphology of biological membranes. This energy is given by the surface integral of \((a[H+c]^2+bK)\), being the \(K\) Gaussian and \(H\) the mean curvatures of the surface modeling the membrane, and \(a,b,c\) parameters depending on the membrane.\N\NThe Euler--Lagrange equation of the functional is a fourth-order nonlinear PDE. In their previous work [\textit{B. Palmer} and \textit{Á. Pámpano}, Calc. Var. Partial Differ. Equ. 61, No. 3, Paper No. 79, 28 p. (2022; Zbl 1490.49030)] the authors showed that solutions to a certain second-order equation (called \textit{reduced membrane equation}) automatically satisfy the Euler-Lagrange equation of the Helfrich energy. This provides a sufficient, but in general not necessary condition, unless some extra assumptions are added, namely, asking the membrane to be an axially symmetric topological disc or topological sphere which is sufficiently regular.\N\NSolutions of the reduced membrane equation can be thought of as capillary surfaces in the hyperbolic space with gravity. In the paper, the authors write down the second variation formula for the Helfrich functional in the case that the surface satisfies the reduced membrane equation. In such a case, the fourth order operator for the Helfrich functional factors into a product of two second order operators, in analogy to what happens for the Willmore energy when assuming the surface to be conformal to a minimal one in the three sphere.\N\NThe second variation formula is then used to study the Euler-Helfrich functional, that is, the sum of the Helfrich energy and for the elastic energy of the boundary of the surface. A stability analysis is performed in some special cases.