A phase-field approximation of the Willmore flow with volume and area constraints

TL;DR

This paper establishes the well-posedness of a phase-field model for the constrained Willmore flow, using a variational approach that handles nonlinear area constraints, contributing to the mathematical understanding of geometric flows.

Contribution

It introduces a new phase-field approximation for the Willmore flow with volume and area constraints and proves its well-posedness under specific conditions, advancing the mathematical theory of constrained geometric flows.

Findings

Well-posedness of the phase-field approximation is proven.

The approach handles nonlinear area constraints effectively.

The gradient flow structure is key to the existence proof.

Abstract

The well-posedness of a phase-field approximation to the Willmore flow with area and volume constraints is established when the functional approximating the area has no critical point satisfying the two constraints. The existence proof relies on the underlying gradient flow structure of the problem: the time discrete approximation is solved by a variational minimization principle. The main difficulty stems from the nonlinearity of the area constraint.