Confined Willmore energy and the Area functional

TL;DR

This paper studies the minimization of a functional combining Willmore energy and area for surfaces confined in a bounded domain, providing explicit solutions, convergence analysis, and existence results for minimizers.

Contribution

It explicitly solves the minimization problem for confined surfaces, analyzes the behavior of minimizers depending on parameters, and proves existence of smooth minimizers for small weights.

Findings

Explicit solutions for the minimization problem in a unit ball.

Description of convergence of minimizers to varifolds based on parameter b3.

Existence of smooth minimizers for small b3.

Abstract

We consider minimization problems of functionals given by the difference between the Willmore functional of a closed surface and its area, when the latter is multiplied by a positive constant weight $\Lambda$ and when the surfaces are confined in the closure of a bounded open set $\Omega\subset\mathbb{R}^3$. We explicitly solve the minimization problem in the case $\Omega=B_1$. We give a description of the value of the infima and of the convergence of minimizing sequences to integer rectifiable varifolds, depending on the parameter $\Lambda$. We also analyze some properties of these functionals and we provide some examples. Finally we prove the existence of a $C^{1,\alpha}\cap W^{2,2}$ embedded surface that is also $C^\infty$ inside $\Omega$ and such that it achieves the infimum of the problem when the weight $\Lambda$ is sufficiently small.