Existence of the classical solution to the fractional mean curvature flow with capillary-type boundary conditions

TL;DR

This paper establishes the short-term existence of solutions for a fractional mean curvature flow with capillary boundary conditions, extending classical results to a fractional setting using fixed point methods.

Contribution

It introduces a fractional mean curvature flow with capillary boundary conditions and proves short-time existence for $C^{1,1}$-regular hypersurfaces using a fixed point approach.

Findings

Proved short-time existence of solutions for fractional mean curvature flow.

Extended classical mean curvature flow results to fractional setting.

Utilized fixed point methods for existence proof.

Abstract

Wang, Weng and Xia[Math. Ann. 388 (2024), no. 2] studied a mean curvature type flow for the smooth, embedded capillary hypersurfaces with a constant contact angle $\theta\in(0,\pi)$ and confirmed the existence of solutions by the standard PDE theory. In the present paper, we study a fractional mean curvature flow for $C^{1,1}$-regular hypersurfaces with a capillary-type boundary condition and obtain the short time existence by the fixed point argument.