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

TL;DR

This work establishes short-time existence for the fractional mean curvature flow of capillary-type hypersurfaces with constant contact angle $\theta$, starting from a $C^{1,1}$ initial hypersurface. By parametrizing the evolving hypersurface via a radial function $\rho$ on the unit upper sphere $\mathbb{S}^{n}_{+}$, the authors derive a nonlocal parabolic PDE for $\rho$ with a boundary condition reflecting capillarity: $\partial_{\eta}\rho=\cos\theta\sqrt{\rho^{2}+|\nabla_{\tau}\rho|^{2}}$. The core method combines a detailed kernel analysis and Schauder estimates for the nonlocal operator $\Delta^{\frac{1+s}{2}}$ with a fixed-point argument to obtain a strong solution, then leverages bootstrapping to achieve higher regularity. This advances the understanding of nonlocal geometric flows under capillary-type boundary conditions and provides a robust framework for local well-posedness in low-regularity settings.

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 $θ\in(0,π)$ 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.

Theorems & Definitions (19)

• Theorem 1.1
• Remark 1.1
• Lemma 2.1
• Lemma 3.1
• Proposition 3.2
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3