Minimizing Configurations for Elastic Surface Energies with Elastic Boundaries

On p-Willmore Disks with Boundary Energies

“…A similar expression for the Helfrich energy, i.e. P (H) = σ (H − c o ) 2 with σ > 0 and c o ∈ R, with elastic boundary can be found in [27].…”

“…We point out here that a special class of axially symmetric Willmore surfaces was studied in [37], although it is not related to our approach. More closely related to the present problem is Conjecture 4.1 of [27], which states that axially symmetric critical disks attain the infimum (whenever it exists) of the Helfrich energy with elastic boundary. In Section 3, a first integral of the Euler-Lagrange equation ( 5) on the interior of the surface is computed, and it is shown that there are three essentially different families of axially symmetric disks critical for E[X], (4), depending on the boundary circle.…”

“…Then, for each term in the energy, we have the following variation formulas (see e.g. Appendix of [13] or [27]):…”

“…Note that this theorem assumes that the differential equation has real analytic coefficients which follows from P being real analytic. Moreover, from Proposition 5.1 of [27] (a constant mean curvature immersion whose boundary is a circle on which τ g ≡ 0 holds is axially symmetric) it follows that Σ is a part of a Delaunay surface, i.e. Σ is either a planar disk or a spherical cap.…”

“…Since these two cases will be analyzed in detail later, now we will focus on the case P H = 0. In this case, the differential equation ( 27) is a second order differential equation, and the profile curve γ(ς) satisfies the system of differential equations ( 25)- (27).…”