On the area-preserving Willmore flow of small bubbles sliding on a domain’s boundary

We consider the area-preserving Willmore evolution of surfaces ϕ that are close to a half-sphere with a small radius, sliding on the boundary S of a domain Ω while meeting it orthogonally. We prove that the flow exists for all times and keeps a “half-spherical” shape. Additionally, we investigate the asymptotic behavior of the flow and prove that for large times the barycenter of the surfaces approximately follows an explicit ordinary differential equation. Imposing additional conditions on the mean curvature of S, we then establish convergence of the flow.