Local Well-Posedness for Volume-Preserving Mean Curvature and Willmore Flows with Line Tension

TL;DR

This paper proves short-time existence and uniqueness of solutions for volume-preserving mean curvature and Willmore flows with line tension effects, addressing complex boundary conditions and nonlinear PDE challenges.

Contribution

It establishes well-posedness results for these geometric flows with line tension, including a novel application of the Hanzawa transformation for analysis.

Findings

Proves short-time existence and uniqueness for volume-preserving MCF with line tension.

Extends results to Willmore flow with line tension, a fourth-order nonlinear PDE.

Uses Hanzawa transformation to analyze the flows as graphs over a fixed hypersurface.

Abstract

We show the short-time existence and uniqueness of solutions for the motion of an evolving hypersurface in contact with a solid container driven by volume-preserving mean curvature flow (MCF) taking line tension effects on the boundary into account. Difficulties arise due to dynamic boundary conditions and due to the contact angle and the non-local nature of the resulting second order, nonlinear PDE. In addition, we prove the same result for the Willmore flow with line tension, which results in a nonlinear PDE of fourth order. For both flows we will use a Hanzawa transformation to write the flows as graphs over a fixed reference hypersurface.