The author studies the fattening phenomena for nonlocal curvature flow of sets in Euclidean space. Given a rotationally invariant kernel \(K:\mathbb{R}^n\setminus\{0\}\to [0,\infty)\) satisfying some mild integrability assumption and a set \(E\subset \mathbb{R}^n\), the \(K\)-curvature of the set \(E\) at a boundary point \(x\in\partial E\) is defined by \[ H_E^K(x):=\lim_{\varepsilon\to 0}\int_{\mathbb{R}^n\setminus B_{\varepsilon}(x)}\left(\chi_{\mathbb{R}^n\setminus E}(y)-\chi_E(y)\right)K(x-y)dy \] which comes from the first variation formula of the nonlocal perimeter functional \[ \mathrm{Per}_K(E):=\int_E\int_{\mathbb{R}^n\setminus E}K(x-y)dxdy. \] When the kernel is homogeneous \(K(x)=|x|^{-n-s}\) for some \(s\in (0,1)\), the \(K\)-curvature is just the fractional mean curvature.
The evolution of sets driven by the \(K\)-curvature is interpreted as the level set flow \[ \partial_tu_E(x,t)+|Du_E(x,t)|H^K_{\{y|u_E(y,t)\geq u_E(x,t)}(x)=0\tag{1} \] with initial condition \(u_E(x,0)=u_E(x)\), where \(u_E:\mathbb{R}^n\to \mathbb{R}\) characterizes the initial set \(E\subset \mathbb{R}^n\). The (viscosity) solution of the level set flow starting from \(E\) is denoted by \[ \Sigma_E(t):=\{x\in \mathbb{R}^n:~u_E(x,t)=0\}. \] If \(\Sigma_E(t)\) has nonempty interior, we say that fattening occurs at \(t>0\). The nonfattening phenomenon is in fact corresponding to the uniqueness of the geometric evolution.
The fattening phenomenon has been studied for the mean curvature flow by many authors. But this problem has not yet been studied for nonlocal curvature flows, apart from the result in [A. Chambolle et al., Interfaces Free Bound. 19, No. 3, 393–415 (2017; Zbl 1380.53070)] for nonfattening for a convex initial set along the fractional mean curvature flow. In this paper, the authors first show that fattening cannot occur for regular initial set with positive \(K\)-curvature along the flow (1), similar to the evolution by mean curvature flow.
In Sections 3–5, the authors study the evolution of cross along the flow (1). The results show that the fattening phenomenon is very sensitive to the properties of the kernel \(K\) in the sense that if \(K\) has sufficiently large mass near the origin, then the fattening occurs; while the fattening cannot occur for kernels that are sufficiently weak near the origin.
In the second part (Sections 6–8) the authors focus on the fractional mean curvature flow. Firstly, it is proved that for strictly star-shaped sets the fattening cannot occur along the fractional mean curvature flow, similar to the case of the mean curvature flow. Then the authors provide two different examples to show that for general star-shaped sets we can expect either fattening or nonfattening. In particular, along the evolution of two tangent balls by fractional mean curvature flow, the fattening phenomenon cannot occur, in complete contrast to the evolution by the mean curvature flow.