Motion of level sets by inverse anisotropic mean curvature (Q6053819)

Let $F\in C^\infty(\mathbb{R}^n\backslash\{0\})$ be a Minkowski norm on $\mathbb{R}^n$, i.e., $F$ is a norm on $\mathbb{R}^n$ with $D^2(\frac{1}{2}F^2)$ positive definite on $\mathbb{R}^n\backslash\{0\}$. The authors consider a family $X(\cdot,t):M \times [0,T) \rightarrow \mathbb{R}^n$ of smooth embeddings of a closed manifold $M$ in $\mathbb{R}^n$ evolving according to the anisotropic inverse mean curvature flow (AIMCF) \[ \frac{\partial}{\partial t} X(x,t) = \frac{1}{H_F(x,t)} \nu_F(x,t) \tag{1} \] where $H_F(\cdot,t)>0$ is the anisotropic mean curvature of the hypersurface $N_t=X(M,t)$ and $\nu_F(x,t)$ is the unit anisotropic outer normal. For the classical inverse mean curvature flow (IMCF) where $F$ is the Euclidean norm on $\mathbb{R}^n$, it has been shown by \textit{C. Gerhardt} [J. Differ. Geom. 32, No. 1, 299--314 (1990; Zbl 0708.53045)] and the reviewer [Math. Z. 205, No. 3, 355--372 (1990; Zbl 0691.35048)] that any smooth, starshaped, strictly mean convex initial hypersurface evolves under the IMCF for all time $t>0$ and after appropriate rescaling converges in the $C^\infty$ sense to a round sphere. An analogous result for the AIMCF has been proved by the third author [Adv. Math. 315, 102--129 (2017; Zbl 1368.53046)]. For more general initial hypersurfaces the situation is more complicated: singularities may develop in finite time so one needs a formulation of the flow that allows for this. A theory of weak solutions was developed by \textit{G. Huisken} and \textit{T. Ilmanen} [J. Differ. Geom. 59, No. 3, 353--437 (2001; Zbl 1055.53052)] for the IMCF using the level set approach previously introduced by \textit{L. C. Evans} and \textit{J. Spruck} [J. Differ. Geom. 33, No. 3, 635--681 (1991; Zbl 0726.53029)] and by \textit{Y.-G. Chen} et al. [J. Differ. Geom. 33, No. 3, 749--786 (1991; Zbl 0696.35087)] for the mean curvature flow. If the hypersurfaces $N_t$ are given by the level sets of a smooth function $u:\mathbb{R}^n\rightarrow\mathbb{R}$, then (1) is equivalent to the degenerate elliptic equation \[ \mathrm{div}\left(F_\xi(\nabla u)\right)=F(\nabla u). \tag{2} \] In the spirit of Huisken and Ilmanen, the authors define a weak solution of (2) as follows. Let $\Omega\subset\mathbb{R}^n$ be an open set. A function $u\in C^{0,1}_{\mathrm{loc}}(\Omega)$ is called a weak solution of (2) if $J_{F,u}(u) \le J_{F,u}(\varphi)$ for every precompact set $K\subset\Omega$ and every test function $\varphi\in C^{0,1}_{\mathrm{loc}}(\Omega)$ with $\varphi=u$ in $\Omega\backslash K$, where \[ J_{F,u}(\varphi) = \int_K [F(\nabla \varphi)+\varphi F(\nabla u)]\, dx. \] Moreover, $u$ is a proper solution if in addition $\lim_{|x|\rightarrow \infty}u(x)=\infty$. The authors' main result is the following. Let $\Omega\subset\mathbb{R}^n$ be an open set with smooth boundary such that $\Omega^c$ is bounded. Then there exists a unique proper weak solution $u\in C^{0,1}_{\mathrm{loc}}(\overline\Omega)$ of (2) such that $u=0$ on $\partial\Omega$. Moreover, $u$ satisfies $F(\nabla u(x)) \le \sup_{\partial\Omega} H^+_F$, $x\in \overline\Omega$, and $F(\nabla u(x)) \le H^+_F(x)$, $x\in\partial\Omega$, where $H^+_F$ denotes the positive part of $H_F$. The authors also prove an analogous result for the Finsler-$p$-Laplacian approximations \[ \mathrm{div}\left((F^{p-1}(\nabla u) F_\xi(\nabla u)\right) = F(\nabla u)^p \text{ in }\Omega,\quad u=0\text{ in }\Omega^c, \quad u(x)\rightarrow\infty\text{ as }x\rightarrow \infty. \] These approximations are inspired by \textit{R. Moser}'s $p$-Laplacian approximations in the isotropic case [J. Eur. Math. Soc. (JEMS) 9, No. 1, 77--83 (2007; Zbl 1116.53040)].