Calculus of Variations and Geometric Measure Theory

F. Onoue

On the shape of hypersurfaces with boundary which have zero fractional mean curvature

created by onoue on 17 Oct 2023
modified on 05 Jul 2024


Accepted Paper

Inserted: 17 oct 2023
Last Updated: 5 jul 2024

Journal: The Jounral of Geometric Analysis
Year: 2023


We consider compact hypersurfaces with boundary in $\mathbb{R}^N$ that are the critical points of the fractional area introduced by Paroni, Podio-Guidugli, and Seguin in 2018. In particular, we study the shape of such hypersurfaces in several simple settings. First we consider the critical points whose boundary is a smooth, orientable, closed manifold $\Gamma$ of dimension $N-2$ and lies in a hyperplane $H \subset \mathbb{R}^N$. Then we show that the critical points coincide with a smooth manifold $\mathcal{N} \subset H$ of dimension $N-1$ with $\partial \mathcal{N} = \Gamma$. Second we consider the critical points whose boundary consists of two smooth, orientable, closed manifolds $\Gamma_1$ and $\Gamma_2$ of dimension $N-2$ and suppose that $\Gamma_1$ lies in a hyperplane $H$ perpendicular to the $x_N$-axis and that $\Gamma_2 = \Gamma_1 + d \, e_N$ ($d >0$ and $e_N = (0,\cdots,0,1) \in \mathbb{R}^N$). Then, assuming that $\Gamma_1$ has a non-negative mean curvature, we show that the critical points do not coincide with the union of two smooth manifolds $\mathcal{N}_1 \subset H$ and $\mathcal{N}_2 \subset H + d \, e_N$ of dimension $N-1$ with $\partial \mathcal{N}_i = \Gamma_i$ for $i \in \{1,2\}$. Moreover, the interior of the critical points does not intersect the boundary of the convex hull in $\mathbb{R}^N$ of $\Gamma_1$ and $\Gamma_2$, while this can occur in the codimension-one situation considered by Dipierro, Onoue, and Valdinoci in 2022. We also obtain a quantitative bound which may tell us how different the critical points are from $\mathcal{N}_1 \cup \mathcal{N}_2$. Finally, in the same setting as in the second case, we show that, if $d$ is sufficiently large, then the critical points are disconnected and, if $d$ is sufficiently small, then $\Gamma_1$ and $\Gamma_2$ are in the same connected component of the critical points when $N \geq 3$. Moreover, by computing the fractional mean curvature of a cone whose boundary is $\Gamma_1 \cup \Gamma_2$, we also obtain that the interior of the critical points does not touch the cone if the critical points are contained in either the inside or the outside of the cone.