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

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.

💡 Research Summary

The paper addresses the short‑time existence of classical solutions to the fractional mean curvature flow (FMCF) for hypersurfaces that meet a flat boundary with a prescribed constant contact angle (a capillary‑type boundary condition). The authors consider an initial hypersurface that is $C^{1,1}$‑regular, star‑shaped, and embedded in the half‑space $\mathbb{R}^{n+1}+$, intersecting the flat boundary $\partial\mathbb{R}^{n+1}+$ at a fixed angle $\theta\in(0,\pi)$.

The fractional mean curvature $H_s$ (with $s\in(0,1)$) is defined by the principal value integral
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