Area-preserving anisotropic mean curvature flow in two dimensions (Q6657164)

The authors study the long-term behavior of the flat flow solution to the area-preserving anisotropic mean curvature flow in the plane. The anisotropic mean curvature flow of sets \(E_{t}\subset \mathbb{R}^{N}\), which preserves the volume \(\left\vert E_{t}\right\vert \), is given by \[V_{t}=\psi ^{o}(\nu _{E_{t}})(-\kappa _{E_{t}}^{\phi }+\lambda _{t})\] on the boundary \(\partial E_{t}\), where \(V_{t}\) is the outward normal velocity along \(\partial E_{t}\), \(\nu _{E_{t}}\) is the outward normal vector, \(\phi \) and \(\psi ^{o}:\mathbb{R}^{N}\rightarrow \lbrack 0,\infty )\) are norms which represent surface energy density and mobility, respectively, \[\lambda _{t}=\int_{\partial E_{t}}\kappa _{E_{t}}^{\phi }\psi ^{o}(\nu _{E_{t}})d \mathcal{H}^{N-1}/\int_{\partial E_{t}}\psi ^{o}(\nu _{E_{t}})d\mathcal{H} ^{N-1},\] and \(\kappa _{E_{t}}^{\phi }\) is the anisotropic mean curvature that represents the first variation of the anisotropic perimeter functional defined as \[P_{\phi }(E)=\int_{\partial E}\phi (\nu _{E})d\mathcal{H}^{N-1}\] for a set \(E\subset \mathbb{R}^{N}\).\N\NThe minimizer of \(P_{\phi }\) among all sets of finite perimeter with a prescribed volume is a scaled and translated version of the Wulff shape \[W_{\phi }=\{x\in \mathbb{R}^{N}:x\cdot v\leq \phi (v)\ \ \ \forall \phi \in \mathbb{R}^{N}\}.\]\N\NThe authors introduce the space \(\mathcal{M}(\mathbb{R}^{N})\) of norms on \(\mathbb{R}^{N}\), and the space \[\mathcal{M}^{2,\alpha }(\mathbb{R}^{N}),\ \alpha \in (0,1]\] of all regular elliptic integrands on \(\mathbb{R}^{N}\) which belong to \(C^{2,\alpha }(\mathbb{R}^{N-1})\). The first main result considers \(\phi \in \mathcal{M }^{2,1}(\mathbb{R}^{2})\), \(\psi \in \mathcal{M}(\mathbb{R}^{2})\), and \( \{E(t)\}t\geq 0\) an area-preserving flat \((\phi ,\psi )\)-flow starting from a bounded set of finite perimeter \(E_{0}\subset \mathbb{R}^{2}\). The authors refer to the paper by \textit{L. Mugnai} et al. [Calc. Var. Partial Differ. Equ. 55, No. 1, Paper No. 18, 23 p. (2016; Zbl 1336.53082)] for the definition of a flat \((\phi ,\psi )\)-flow solution to the above problem. This main result proves the existence of a disjoint union of scaled and translated version of Wulff shapes \(E_{\infty }=\cup _{j=1}^{d}W_{\phi }(x_{j},r)\) such that \[\left\vert E_{0}\right\vert =\left\vert E_{\infty }\right\vert , P_{\sigma }(E_{\infty })\leq P_{\phi }(E_{0})\] and \[\sup_{E(t)\bigtriangleup E_{\infty }}d_{E_{\infty }}^{\psi }\leq Ce^{-t/C_{0}}\] for some positive constants \(C,C_{0}\) and all \(t\geq 0\) . Here \(W_{\phi }(x_{j},r)=x_{j}+rW_{\phi }\).\N\NFor the proof, the authors recall the notions of regular elliptic integrand for a norm on \(\mathbb{R} ^{N}\). They prove properties of the mean \(\phi \)-curvature, starting with an anisotropic version of the Gauss-Bonnet theorem for curves using the Cahn-Hoffman map to parametrize boundary components as small perturbations of the Wulff shape and some compactness result. \N\NThe second main result proves that certain reflection comparison symmetries are preserved by the flat flow.. It considers \(N=2,3\), \(\phi \in \mathcal{M}^{2,1}(\mathbb{R}^{N}) \), \(\psi \in \mathcal{M}(\mathbb{R}^{N})\) strictly convex, a half-space \(H \) with normal vector \(\nu \), such that \(\phi ,\psi \) are compatible with \( \nu \). If \(E_{0}\subset \mathbb{R}^{N}\) is a bounded set of finite perimeter which satisfies \(\Psi (E)\cap H\subseteq E\cap H\) and \(\partial \Psi (E)\cap \partial E\subset \partial H\), where \(\Psi \) denotes the reflection across \( \partial H\), and is \(C^{1}\) near \(\partial H\), then any flat \((\phi ,\psi )\) -flow \(E(t)\) with initial set \(E_{0}\) satisfies \(\Psi (E(t))\cap H\subseteq E(t)\cap H\) for all \(t\geq 0\). The authors can also weaken the hypotheses on the initial set \(E_{0}\). The authors introduce approximate flat \((\phi ,\psi )\)-flows and they prove some uniform estimates.