The streamlines of ∞-harmonic functions obey the inverse mean curvature flow

Abstract

Given an ∞-harmonic function $u_{\infty }$ on a domain $\Omega \subseteq {\mathbb{R}}^2,$ consider the function $w=-\hspace{0.17em}\text{log}\hspace{0.17em}|\nabla u_{\infty }|.$ If $u_{\infty }\in C^2(\Omega )$ with $\nabla u_{\infty }\ne 0$ and $\nabla |\nabla u_{\infty }|\ne 0,$ then it is easy to check that

• the streamlines of $u_{\infty }$ are the level sets of w and

• w solves the level set formulation of the inverse mean curvature flow.

For less regular solutions, neither statement is true in general, but even so, w is still a weak solution of the inverse mean curvature flow under far weaker assumptions. This is proved through an approximation of $u_{\infty }$ by p-harmonic functions, the use of conjugate $p\prime$-harmonic functions, and the known connection of the latter with the inverse mean curvature flow. A statement about the regularity of $|\nabla u_{\infty }|$ arises as a by-product.

Keywords:

• infinity-harmonic functions
• inverse mean curvature flow
• viscosity solution
• weak solution

1. Introduction

Let $\Omega \subseteq {\mathbb{R}}^2$ be an open set. A function $u_{\infty }\in C^0(\Omega )$ is called ∞-harmonic if it is a viscosity solution of the Aronsson equation (1) $${\left(\frac{\partial u_{\infty }}{\partial x_1}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_1^2}+2\frac{\partial u_{\infty }}{\partial x_1}\frac{\partial u_{\infty }}{\partial x_2}\frac{{\partial }^2{u}_{\infty }}{\partial x_1\partial x_2}+{\left(\frac{\partial u_{\infty }}{\partial x_2}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_2^2}=0.$$(1)

This equation was introduced by Aronsson [1, 2], motivated by optimal Lipschitz extensions of the boundary data, and has been studied extensively since then. Highlights of the theory include existence [3] and uniqueness [4] of solutions for boundary value problems associated to (1)(1) $${\left(\frac{\partial u_{\infty }}{\partial x_1}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_1^2}+2\frac{\partial u_{\infty }}{\partial x_1}\frac{\partial u_{\infty }}{\partial x_2}\frac{{\partial }^2{u}_{\infty }}{\partial x_1\partial x_2}+{\left(\frac{\partial u_{\infty }}{\partial x_2}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_2^2}=0.$$(1) , regularity results [5, 6], and connections to stochastic tug-of-war games [7].

For $u_{\infty }\in C^2(\Omega ),$ Eq. (1)(1) $${\left(\frac{\partial u_{\infty }}{\partial x_1}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_1^2}+2\frac{\partial u_{\infty }}{\partial x_1}\frac{\partial u_{\infty }}{\partial x_2}\frac{{\partial }^2{u}_{\infty }}{\partial x_1\partial x_2}+{\left(\frac{\partial u_{\infty }}{\partial x_2}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_2^2}=0.$$(1) (1) $${\left(\frac{\partial u_{\infty }}{\partial x_1}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_1^2}+2\frac{\partial u_{\infty }}{\partial x_1}\frac{\partial u_{\infty }}{\partial x_2}\frac{{\partial }^2{u}_{\infty }}{\partial x_1\partial x_2}+{\left(\frac{\partial u_{\infty }}{\partial x_2}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_2^2}=0.$$(1) may alternatively be represented as $$\nabla u_{\infty }·\nabla |\nabla u_{\infty }{|}^2=0.$$

It is then obvious that the function $|\nabla u_{\infty }|$ is constant along the streamlines of $u_{\infty },$ i.e., along the curves in Ω arising through the solutions of the ordinary differential equation $\dot{\gamma }(t)=\nabla u_{\infty }(\gamma (t)).$ This is one of the reasons why the streamlines of an ∞-harmonic function are of particular interest and have received some attention in the literature [2, 8, 9]. In general, however, viscosity solutions of (1)(1) $${\left(\frac{\partial u_{\infty }}{\partial x_1}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_1^2}+2\frac{\partial u_{\infty }}{\partial x_1}\frac{\partial u_{\infty }}{\partial x_2}\frac{{\partial }^2{u}_{\infty }}{\partial x_1\partial x_2}+{\left(\frac{\partial u_{\infty }}{\partial x_2}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_2^2}=0.$$(1) are not C²-regular. It was shown by Evans and Savin [5] that they are of class $C^{1,\alpha }$ for some $\alpha >0.$ As the example $u_{\infty }(x)=|x_1{|}^{4/3}-|x_2{|}^{4/3}$ of Aronsson [10] shows, no exponent better than $\alpha =1/3$ can be expected. Nevertheless, at least in an annular domain with boundary values 0 and 1, respectively, on the two boundary components, it was shown by Lindgren and Lindqvist [8, 9] that $|\nabla u_{\infty }|$ is constant along streamlines that are generic in some sense. There is also a weaker statement that is true in general (see, e.g., the description by Crandall [11, Section 6]).

If $u_{\infty }\in C^2(\Omega )$ is a solution of (1)(1) $${\left(\frac{\partial u_{\infty }}{\partial x_1}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_1^2}+2\frac{\partial u_{\infty }}{\partial x_1}\frac{\partial u_{\infty }}{\partial x_2}\frac{{\partial }^2{u}_{\infty }}{\partial x_1\partial x_2}+{\left(\frac{\partial u_{\infty }}{\partial x_2}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_2^2}=0.$$(1) with $\nabla u_{\infty }\ne 0$ and $\nabla |\nabla u_{\infty }|\ne 0,$ then we also conclude that $$\frac{\nabla |\nabla u_{\infty }|}{|\nabla |\nabla u_{\infty }\left|\right|}=\pm \frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|},$$ where we write ${\nabla }^{\perp }=(-\frac{\partial }{\partial x_2},\frac{\partial }{\partial x_1}).$ Hence, the function $w=-\hspace{0.17em}\text{log}\hspace{0.17em}|\nabla u_{\infty }|$ satisfies $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=\mp \text{div}\left(\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}\right)=\pm \frac{{\nabla }^{\perp }{u}_{\infty }·\nabla |\nabla u_{\infty }|}{|\nabla u_{\infty }{|}^2}=|\nabla w|.$$

The equation (2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) has a geometric interpretation: it is the level set formulation of the inverse mean curvature flow.

The inverse mean curvature flow is an evolution equation for hypersurfaces. It is often studied on a Riemannian manifold, but we explain it here for an open set $\Omega \subseteq {\mathbb{R}}^n.$ Consider an oriented $(n-1)$-dimensional manifold N and a smooth map $\varphi :[0,T)\times N\to \Omega$ such that $N_t=\varphi (\{t\}\times N)$ is an immersed hypersurface for every $t\in [0,T).$ Suppose that $\nu :[0,T)\times N\to S^{n-1}$ is a smooth map such that $\nu (t, ·)$ is a normal vector field on $N_t\subseteq \Omega$ for every t, and let $H:[0,T)\times N\to \mathbb{R}$ be the function such that $H(t, ·)$ is the corresponding (scalar) mean curvature of N_t. We say that $\varphi$ is a classical solution of the inverse mean curvature flow if (3) $$\frac{\partial \varphi }{\partial t}=\frac{\nu }{H}$$(3) in $(0,t)\times N.$ This is a parabolic equation, so we may hope to solve it for a prescribed initial hypersurface N₀ under suitable boundary conditions. But this is not always possible, either because H has zeroes at t = 0 or because singularities develop in finite time. For this reason, a weak notion of solutions was proposed by Huisken and Ilmanen [12], based on a level set formulation. The underlying idea is to look for a function $w:\Omega \to \mathbb{R}$ such that $N_t=w^{-1}(\{t\}).$ As long as w is sufficiently smooth and $\nabla w\ne 0,$ Eq. (3)(3) $$\frac{\partial \varphi }{\partial t}=\frac{\nu }{H}$$(3) (3) $$\frac{\partial \varphi }{\partial t}=\frac{\nu }{H}$$(3) is equivalent to (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) .

Equation (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) (2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) , however, allows a weak interpretation as well. Huisken and Ilmanen use a variational principle for this purpose. In this paper, we use a different formulation, which is more convenient for our main results and perhaps more intuitive, too. We will see in Section 5, however, that the following condition implies that w is a weak solution in the sense of Huisken and Ilmanen, as long as we impose enough regularity such that the latter makes sense.

Definition 1.

A function $w\in W_{\text{loc}}^{1,1}(\Omega )$ is called a weak solution of (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) if there exists a measurable vector field $F:\Omega \to {\mathbb{R}}^n$ such that $\left|F\right|\le 1$ and $F·\nabla w=|\nabla w|$ almost everywhere in Ω, and $\text{div}F=|\nabla w|$ weakly in Ω.

By the last condition, we mean that $${\int }_{\Omega }(\nabla \eta ·F+\eta |\nabla w|) dx=0$$ for all $\eta \in C_0^{\infty }(\Omega ).$

We now restrict our attention to n = 2 again. For $w=-\hspace{0.17em}\text{log}\hspace{0.17em}|\nabla u_{\infty }|,$ Eq. (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) (2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) means that the level sets of $|\nabla u_{\infty }|$ move by the inverse mean curvature flow. Furthermore, if $u_{\infty }\in C^2(\Omega )$ with $\nabla u_{\infty }\ne 0$ and $\nabla |\nabla u_{\infty }|\ne 0,$ as assumed for the above calculations, then the level sets of $|\nabla u_{\infty }|$ are the streamlines of $u_{\infty }.$

We study the question to what extent these observations persist when we remove the regularity assumptions. It is not true in general that viscosity solutions of (1)(1) $${\left(\frac{\partial u_{\infty }}{\partial x_1}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_1^2}+2\frac{\partial u_{\infty }}{\partial x_1}\frac{\partial u_{\infty }}{\partial x_2}\frac{{\partial }^2{u}_{\infty }}{\partial x_1\partial x_2}+{\left(\frac{\partial u_{\infty }}{\partial x_2}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_2^2}=0.$$(1) give rise even to weak solutions of (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) . The function $u_{\infty }(x)=|x_1{|}^{4/3}-|x_2{|}^{4/3}$ provides a counterexample here, too. In this case, as $\nabla |\nabla u_{\infty }|\ne 0$ almost everywhere, there is only one possible choice for the vector field F from Definition 1. We can then check that (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) does not hold on the coordinate axes. By contrast, the function $u_{\infty }(x)=\xi ·x,$ for a constant $\xi \in {\mathbb{R}}^2\setminus \left\{0\right\},$ may appear an unlikely candidate for the inverse mean curvature flow. Its streamlines are straight lines and the curvature vanishes identically. But in the formulation of Definition 1 (and also in the formulation of Huisken and Ilmanen [12]), the inverse mean curvature flow can deal with this situation. It is easily seen that any constant function is a weak solution of (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) .

If $\nabla w=0,$ then the level sets are not hypersurfaces in general. We may, for example, have a certain value t₀ such that the sets $N_t=w^{-1}(\{t\})$ evolve smoothly for $t<t_0$ and for $t>t_0,$ but $N_{t_0}$ has a non-empty interior. Then we have a jump in N_t at the time t₀. This is indeed the typical way in which the weak inverse mean curvature flow resolves singularities.

In this paper, we will establish that $w=-\hspace{0.17em}\text{log}\hspace{0.17em}|\nabla u_{\infty }|$ does solve (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) weakly under an additional assumption, which is based on the idea that the function $|\nabla u_{\infty }|$ is monotone along the level sets of $u_{\infty }.$ This induces a sense of direction on the level sets and, by comparison with $\nabla u_{\infty },$ an orientation of ${\mathbb{R}}^2.$ Writing $B_r(x)$ for the open disc in ${\mathbb{R}}^2$ with centre x and radius r, we can formalise this notion as follows.

Definition 2.

Let $u_{\infty }\in C^1(\Omega )$ with $\nabla u_{\infty }\ne 0$ everywhere. For $G\subseteq \Omega ,$ an orientation of $u_{\infty }$ in G is a continuous function $\omega :G\to \{-1,1\}$ such that for any $x\in G$ there exists r > 0 with the following property: for all $y,z\in B_r(x)\cap G,$ if $u_{\infty }(y)=u_{\infty }(z)$ and $(z-y)·{\nabla }^{\perp }{u}_{\infty }(x)\ge 0,$ then $$\omega (x)|\nabla u_{\infty }(z)|\ge \omega (x)|\nabla u_{\infty }(y)|.$$

For example, the function $u_{\infty }(x)=|x_1{|}^{4/3}-|x_2{|}^{4/3}$ has no orientation in ${\mathbb{R}}^2,$ but does have an orientation in $\left\{x\in {\mathbb{R}}^2:x_1{x}_2\ne 0\right\},$ which is $\omega (x)=\frac{x_1{x}_2}{\left|x_1\right|\left|x_2\right|}.$ More generally, if $u_{\infty }\in C^2(\Omega )$ is ∞-harmonic with $\nabla u_{\infty }\ne 0$ and $\nabla |\nabla u_{\infty }|\ne 0,$ then $u_{\infty }$ has an orientation in Ω. In this case, we may consider the basis of ${\mathbb{R}}^2$ given by the pair of vectors $(\nabla u_{\infty }(x),\nabla |\nabla u_{\infty }|(x))$ at any point $x\in \Omega .$ We set $\omega (x)=1$ if this basis gives the standard orientation and $\omega (x)=-1$ otherwise. Then we can check that the conditions of the definition are satisfied.

It is also possible to have an orientation when $\nabla |\nabla u_{\infty }|=0.$ For example, if $|\nabla u_{\infty }|$ is constant, then we may choose ω constant (of either value). This applies, e.g., to the function $u_{\infty }(x)=\xi ·x,$ or to the function $u_{\infty }:{\mathbb{R}}^2\setminus \left\{0\right\}\to \mathbb{R}$ with $u_{\infty }(x)=\left|x\right|.$

Theorem 3.

Let $u_{\infty }\in C^1(\Omega )$ be an ∞-harmonic function with $\nabla u_{\infty }\ne 0$ in Ω. Suppose that ω is an orientation of $u_{\infty }$ in Ω. Then $|\nabla u_{\infty }|$ belongs to $W_{\text{loc}}^{1,q}(\Omega )$ for all $q<\infty$ and satisfies(4) $$|\nabla |\nabla u_{\infty }\left|\right|\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}=\omega \nabla |\nabla u_{\infty }|$$(4) almost everywhere and(5) $$\text{div}\left(\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}\right)=-\omega \frac{|\nabla |\nabla u_{\infty }\left|\right|}{|\nabla u_{\infty }|}$$(5) weakly in Ω. Hence the function $w=-\hspace{0.17em}\text{log}\hspace{0.17em}|\nabla u_{\infty }|$ is a weak solution of (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) .

Note that the condition $u_{\infty }\in C^1(\Omega )$ is satisfied automatically for viscosity solutions of (1)(1) $${\left(\frac{\partial u_{\infty }}{\partial x_1}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_1^2}+2\frac{\partial u_{\infty }}{\partial x_1}\frac{\partial u_{\infty }}{\partial x_2}\frac{{\partial }^2{u}_{\infty }}{\partial x_1\partial x_2}+{\left(\frac{\partial u_{\infty }}{\partial x_2}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_2^2}=0.$$(1) by the results of Evans and Savin [5]. The condition that $\nabla u_{\infty }\ne 0$ and the existence of an orientation, however, are additional assumptions.

Equation (4)(4) $$|\nabla |\nabla u_{\infty }\left|\right|\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}=\omega \nabla |\nabla u_{\infty }|$$(4) (4) $$|\nabla |\nabla u_{\infty }\left|\right|\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}=\omega \nabla |\nabla u_{\infty }|$$(4) may be regarded as another representation of the Aronsson equation (1)(1) $${\left(\frac{\partial u_{\infty }}{\partial x_1}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_1^2}+2\frac{\partial u_{\infty }}{\partial x_1}\frac{\partial u_{\infty }}{\partial x_2}\frac{{\partial }^2{u}_{\infty }}{\partial x_1\partial x_2}+{\left(\frac{\partial u_{\infty }}{\partial x_2}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_2^2}=0.$$(1) (1) $${\left(\frac{\partial u_{\infty }}{\partial x_1}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_1^2}+2\frac{\partial u_{\infty }}{\partial x_1}\frac{\partial u_{\infty }}{\partial x_2}\frac{{\partial }^2{u}_{\infty }}{\partial x_1\partial x_2}+{\left(\frac{\partial u_{\infty }}{\partial x_2}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_2^2}=0.$$(1) . Equation (5)(5) $$\text{div}\left(\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}\right)=-\omega \frac{|\nabla |\nabla u_{\infty }\left|\right|}{|\nabla u_{\infty }|}$$(5) (5) $$\text{div}\left(\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}\right)=-\omega \frac{|\nabla |\nabla u_{\infty }\left|\right|}{|\nabla u_{\infty }|}$$(5) , on the other hand, provides additional information about the behaviour of the solutions.

It is already known from work of Koch, Zhang, and Zhou [13] that $|\nabla u_{\infty }|\in W_{\text{loc}}^{1,2}(\Omega ).$ This result applies to all viscosity solutions of (1)(1) $${\left(\frac{\partial u_{\infty }}{\partial x_1}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_1^2}+2\frac{\partial u_{\infty }}{\partial x_1}\frac{\partial u_{\infty }}{\partial x_2}\frac{{\partial }^2{u}_{\infty }}{\partial x_1\partial x_2}+{\left(\frac{\partial u_{\infty }}{\partial x_2}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_2^2}=0.$$(1) and does not require any additional assumptions. Theorem 3 improves this regularity to $W_{\text{loc}}^{1,q}(\Omega )$ for any $q<\infty ,$ but only if $\nabla u_{\infty }\ne 0$ and if there is an orientation. As discussed in the aforementioned paper, so much regularity cannot be expected in general. A counterexample is given by the previously considered function $u_{\infty }(x)=|x_1{|}^{4/3}-|x_2{|}^{4/3}.$ Nevertheless, the regularity statement from Theorem 3 can be improved somewhat.

Theorem 4.

Let $u_{\infty }\in C^1(\Omega )$ be an ∞-harmonic function with $\nabla u_{\infty }\ne 0$ in Ω. Let $G\subseteq \Omega$ be an open set and $\Gamma \subset \partial G\cap \Omega$. Suppose that ω is an orientation of $u_{\infty }$ in $G\cup \Gamma$. Suppose further that for every $x\in \Gamma$ there exist r > 0 and a Lipschitz function $f:\mathbb{R}\to \mathbb{R}$ such that$$G\cap B_r(x)=\left\{y\in B_r(x):\omega (x)y·{\nabla }^{\perp }{u}_{\infty }(x)<f(y·\nabla u_{\infty }(x))\right\}.$$

Let $U\subseteq G$ be an open set with $\overline{U}\subseteq G\cup \Gamma$. Then $|\nabla u_{\infty }|\in W^{1,q}(U)$ for every $q<\infty .$

In less technical terms, we require that $|\nabla u_{\infty }|$ is non-decreasing if we travel along a level set of $u_{\infty }$ inside G towards Γ. There is no such restriction outside of Γ. In some cases, when $|\nabla u_{\infty }|$ has local maxima on Γ, it may be possible to apply the theorem on the other side of Γ as well with the opposite orientation. Even then, however, it does not follow that $-\hspace{0.17em}\text{log}\hspace{0.17em}|\nabla u_{\infty }|$ will satisfy Eq. (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) (2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) on Γ.

The assumption that f is Lipschitz continuous is stronger than necessary; we use it for the sake of a simpler statement. A weaker assumption is used in Proposition 5 below.

Clearly we need some prior information about the behaviour of $u_{\infty }$ before we can apply a result such as this. Such information is available, for example, for the ∞-harmonic functions studied by Lindgren and Lindqvist [8, 9]. Combining their results with Theorem 4, we see that under the assumptions of the second paper [9], we have local $W^{1,q}$-regularity of $|\nabla u_{\infty }|$ for all $q<\infty$ away from what Lindgren and Lindqvist call the attracting streamlines.

The main purpose of this paper, however, is not to provide regularity results, but to explore the relationship between ∞-harmonic functions and the inverse mean curvature flow. It seems that this has not been discussed in the literature before even in the smooth case, although some related calculations are present in the work of Aronsson [2] and Evans [14]. The proof of Theorem 3 shows that the connection is in fact deeper than the simple calculations at the beginning of the introduction suggest. The arguments are based on the following ideas, explained here for $u_{\infty }\in C^1(\overline{\Omega })$ when Ω is a simply connected domain with Lipschitz boundary.

According to the results of Jensen [4], an ∞-harmonic function can be approximated by p-harmonic functions. For $p\in [2,\infty ),$ let therefore u_p denote the unique minimiser of the functional $$E_p(u)={\left(\frac{1}{p}{\int }_{\Omega }|\nabla u{|}^p dx\right)}^{1/p}$$ in the space $u_{\infty }+W_0^{1,p}(\Omega ).$ Then it satisfies the p-Laplace equation $$\text{div}(|\nabla u_p{|}^{p-2}\nabla u_p)=0.$$

We use the notation ${\Delta }_{p}u=\text{div}(|\nabla u{|}^{p-2}\nabla u)$ for the p-Laplace operator; then we can write this equation in the form ${\Delta }_p{u}_p=0.$

We eventually consider the limit $p\to \infty ,$ but for the moment we fix $p<\infty .$ Let $p\prime$ be its conjugate exponent with $\frac{1}{p}+\frac{1}{p\prime }=1.$ Then we note that $\text{curl}(|\nabla u_p{|}^{p-2}{\nabla }^{\perp }{u}_p)={\Delta }_p{u}_p=0.$ Hence there exists $v_p\in W^{1,p\prime }(\Omega )$ satisfying $$\nabla v_p=\omega |\nabla u_p{|}^{p-2}{\nabla }^{\perp }{u}_p.$$

Then we compute $|\nabla v_p{|}^{p\prime -2}\nabla v_p=\omega {\nabla }^{\perp }{u}_p.$ Therefore, $${\Delta }_{p\prime }{v}_p=0.$$

Thus we have the same sort of equation, but we can now consider the limit $p\prime \to 1.$ The duality between these two problems has been exploited for different purposes before [15, 16], but the consequences for the limit behaviour have never been studied in detail, perhaps because swapping $p\to \infty$ for $p\prime \to 1$ does not seem helpful superficially. Here, however, is where the inverse mean curvature flow and its $p\prime$-approximation come into play.

Set $w_p=\frac{1}{1-p}\text{log}\hspace{0.17em}{v}_p.$ Then we compute $${\Delta }_{p\prime }{w}_p=|\nabla w_p{|}^{p\prime }.$$

Equation (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) (2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) arises as the formal limit as $p\prime \to 1.$ This connection between $p\prime$-harmonic functions and the inverse mean curvature flow has been used before to construct weak solutions of the latter [17–20]. In the context of ∞-harmonic functions, the advantage of this transformation is that it removes some of the degenerate behaviour that arises for v_p in the limit.

Next we use some tools developed for the inverse mean curvature flow [18, 20] to show that we have at least a sequence $p_k\to \infty$ such that $w_{p_k}$ converges weakly in $W_{\text{loc}}^{1,q}(\Omega ),$ for any $q<\infty ,$ to a weak solution w of (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) . Then we can reverse the above transformations to see what this means for $u_{\infty }.$ We compute $$|\nabla w_p{|}^{p^{\prime }-2}\nabla w_p=-{(p\prime -1)}^{p^{\prime }-1}\omega e^{w_p}{\nabla }^{\perp }{u}_p.$$

The left-hand side, at least if restricted to a certain subsequence, will converge weakly in $L_{\text{loc}}^q(\Omega ;R^2)$ for every $q<\infty$ to a vector field F, and we will eventually see that F satisfies the conditions from Definition 1. The right-hand side converges to $-\omega e^w{\nabla }^{\perp }{u}_{\infty }.$ At almost every point $x\in \Omega$ such that $\nabla w(x)\ne 0,$ we conclude that $$\frac{\nabla w(x)}{|\nabla w(x)|}=-\omega e^{w(x)}{\nabla }^{\perp }{u}_{\infty }(x),$$ and at such a point we therefore recover the relationship $w(x)=-\hspace{0.17em}\text{log}\hspace{0.17em}|\nabla u_{\infty }(x)|$ and also Eqs. (4)(4) $$|\nabla |\nabla u_{\infty }\left|\right|\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}=\omega \nabla |\nabla u_{\infty }|$$(4) (4) $$|\nabla |\nabla u_{\infty }\left|\right|\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}=\omega \nabla |\nabla u_{\infty }|$$(4) and (5)(5) $$\text{div}\left(\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}\right)=-\omega \frac{|\nabla |\nabla u_{\infty }\left|\right|}{|\nabla u_{\infty }|}$$(5) .

But it is possible that $\nabla w$ vanishes, and this is indeed expected for situations such as when $\nabla u_{\infty }$ is constant. In this case, we need much better information about the functions w_p, and this is the most intricate part of the proof. We do not go into the details here, but because of the technical difficulties arising when $\nabla w=0,$ we will first consider a small neighbourhood of a given point where $\nabla u_{\infty }$ is approximately constant. As a consequence, we need to show at the end of the proof that a local weak solution of the inverse mean curvature flow gives rise to a global weak solution. This is the main reason why we favour Definition 1 over the definition of Huisken and Ilmanen [12]. At least in the presence of Eqs. (4)(4) $$|\nabla |\nabla u_{\infty }\left|\right|\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}=\omega \nabla |\nabla u_{\infty }|$$(4) (4) $$|\nabla |\nabla u_{\infty }\left|\right|\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}=\omega \nabla |\nabla u_{\infty }|$$(4) and (5)(5) $$\text{div}\left(\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}\right)=-\omega \frac{|\nabla |\nabla u_{\infty }\left|\right|}{|\nabla u_{\infty }|}$$(5) , this step turns out to be quite straightforward.

This strategy resembles some arguments that have been used for several higher order variational problems related to the Aronsson equation [21–25]. These papers study minimisers of certain functionals involving the $L^{\infty }$-norm. They rely on the idea of approximating the $L^{\infty }$-norm by the L^p-norm for $p<\infty ,$ studying minimisers of the resulting functionals, and reformulating the Euler-Lagrange equation in a way that removes the expected degeneracy in the limit $p\to \infty ,$ so that conclusions about the original problem can be drawn. It is typically quite easy to find bounds for the relevant quantities in the appropriate spaces in this context, but it is necessary and difficult to show that they stay away from 0.

It may seem that the above observations are specific to two-dimensional domains, but they conceivably have a higher-dimensional generalisation—not for the Aronsson equation (1)(1) $${\left(\frac{\partial u_{\infty }}{\partial x_1}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_1^2}+2\frac{\partial u_{\infty }}{\partial x_1}\frac{\partial u_{\infty }}{\partial x_2}\frac{{\partial }^2{u}_{\infty }}{\partial x_1\partial x_2}+{\left(\frac{\partial u_{\infty }}{\partial x_2}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_2^2}=0.$$(1) (1) $${\left(\frac{\partial u_{\infty }}{\partial x_1}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_1^2}+2\frac{\partial u_{\infty }}{\partial x_1}\frac{\partial u_{\infty }}{\partial x_2}\frac{{\partial }^2{u}_{\infty }}{\partial x_1\partial x_2}+{\left(\frac{\partial u_{\infty }}{\partial x_2}\right)}^2\frac{{\partial }^2{u}_{\infty }}{\partial x_2^2}=0.$$(1) , but for an analogous problem involving differential forms. Indeed, the relationship between $p\prime$-harmonic functions and the $p\prime$-approximation of Eq. (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) (2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) exists for any dimension. If d denotes the exterior derivative and $d^{*}$ its formal L²-adjoint, then we may write the equation ${\Delta }_{p\prime }{v}_p=0$ in the form $d^{*}(|d{v}_p{|}^{p^{\prime }-2}d{v}_p)=0.$ Assuming that this is satisfied in a star-shaped domain $\Omega \subseteq {\mathbb{R}}^n,$ it implies that $|d{v}_p{|}^{p^{\prime }-2}d{v}_p=d^{*}{u}_p$ for some 2-form u_p on Ω. Moreover, u_p will satisfy $d(|d^{*}{u}_p{|}^{p-2}{d}^{*}{u}_p)=0,$ which is the Euler-Lagrange equation for the functional $$E_p(u)={\left(\frac{1}{p}{\int }_{\Omega }|d^{*}u{|}^p dx\right)}^{1/p}.$$

This suggests that we study the problem of minimising $\left|\right|d^{*}u|{|}_{L^{\infty }(\Omega )}$ if we wish to find a connection to the inverse mean curvature flow. (If n = 3, we may alternatively minimise $\left|\right|\text{curl}u|{|}_{L^{\infty }(\Omega )}$ for vector fields $u:\Omega \to {\mathbb{R}}^3.$) Indeed, formal calculations analogous to Aronsson’s [1] lead to the equation (6) $$d|d^{*}{u}_{\infty }{|}^2\wedge d^{*}{u}_{\infty }=0.$$(6)

(For n = 3, we alternatively have the equation $\nabla |\text{curl}\mathrm{ }{u}_{\infty }{|}^2\times \text{curl}\mathrm{ }{u}_{\infty }=0.$) But almost nothing is known about this equation; indeed, even the vector-valued optimal Lipschitz extension problem and the Aronsson equation for vector-valued functions $u:\Omega \to {\mathbb{R}}^N$ with $N\ge 2$ are poorly understood despite some existing work on the former by Sheffield and Smart [26] and a series of papers on the latter by Katzourakis [27–32]. In particular, several of the tools for the proof of Theorem 3 are missing in higher dimensions, and we have no results here apart from the following calculations for C²-solutions, which are completely analogous to the above calculations for n = 2.

Suppose that $u_{\infty }$ is a 2-form with coefficients in $C^2(\Omega ).$ If $u_{\infty }$ solves (6)(6) $$d|d^{*}{u}_{\infty }{|}^2\wedge d^{*}{u}_{\infty }=0.$$(6) and satisfies $d^{*}{u}_{\infty }\ne 0$ and $d\left|d^{*}{u}_{\infty }\right|\ne 0$ in Ω, then we conclude that $$\frac{d\left|d^{*}{u}_{\infty }\right|}{\left|d\right|d^{*}{u}_{\infty }\left|\right|}=\pm \frac{d^{*}{u}_{\infty }}{\left|d^{*}{u}_{\infty }\right|}.$$

Define $w=-\hspace{0.17em}\text{log}\hspace{0.17em}\left|d^{*}{u}_{\infty }\right|.$ Then $$-d^{*}\left(\frac{dw}{\left|dw\right|}\right)=\pm d^{*}\left(\frac{d^{*}{u}_{\infty }}{\left|d^{*}{u}_{\infty }\right|}\right)=\pm \frac{d\left|d^{*}{u}_{\infty }\right|·d^{*}{u}_{\infty }}{|d^{*}{u}_{\infty }{|}^2}=\frac{\left|d\right|d^{*}{u}_{\infty }\left|\right|}{\left|d^{*}{u}_{\infty }\right|}=\left|dw\right|.$$

As the operator $-d^{*}$ for 1-forms can be identified with the divergence for vector fields, this means that w solves Eq. (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) (2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) . Of course it is no longer appropriate to speak of streamlines here. Their higher-dimensional counterparts are the hypersurfaces characterised by the condition that their tangent vectors X satisfy $d^{*}{u}_{\infty }(X)=0,$ and using (6)(6) $$d|d^{*}{u}_{\infty }{|}^2\wedge d^{*}{u}_{\infty }=0.$$(6) we can check that they coincide with the level sets of $|\nabla u_{\infty }|.$

Sections 2–4 are devoted to the proofs of Theorem 3 and Theorem 4. Then, in Section 5, we prove that weak solutions of (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) in the sense of Definition 1 are also weak solutions in the sense of Huisken and Ilmanen [12]. This final section is not essential for the understanding of the main theorems, but it provides a connection with a larger body of literature on the inverse mean curvature flow.

2. Reduction to a local result

As discussed in the introduction, we first consider small neighbourhoods of a given point $x_0\in \Omega$ where $\nabla u_{\infty }$ is nearly constant. We may then rescale these neighbourhoods and thereby renormalise $\nabla u_{\infty }(x_0)$ to unit size, using the following observation: if $u_{\infty }$ is a given ∞-harmonic function, then for any $a\in \mathbb{R},$ r > 0, and $R\in \mathrm{O}(2),$ the rescaled function ${\tilde{u}}_{\infty }(x)=a{u}_{\infty }(\mathit{rRx}+x_0)$ is also ∞-harmonic. If $a\det R>0,$ then the transformation preserves the orientation, and if $a\det R<0,$ it reverses the orientation of $u_{\infty }.$ (We do not consider the case a = 0.)

The following result should be thought of as a statement about $u_{\infty }$ after such a rescaling, chosen such that $\nabla u_{\infty }(x_0)$ becomes the second standard basis vector and the orientation becomes negative. Here and throughout the rest of the paper, we use the notation (e₁, e₂) for the standard basis of ${\mathbb{R}}^2,$ and we also write $Q_r={(-r,r)}^2$ for r > 0.

Proposition 5.

There exists $\delta >0$ with the following property. Suppose that $u_{\infty }\in C^1(\overline{Q_1})$ is ∞-harmonic with $|\nabla u_{\infty }-e_2|\le \delta$ in $\overline{Q_1}$. For $t\in [-\frac{1}{2},\frac{1}{2}]$, let $L_t=\left\{x\in Q_1:u_{\infty }(x)=u_{\infty }(0,t)\right\}$, and suppose that the numbers $m_t\in [-1,1]$ satisfy the following condition: for all $x,y\in L_t$, if $x_1\le y_1\le m_t$, then $|\nabla u_{\infty }(x)|\le |\nabla u_{\infty }(y)|$. Let$$M=\underset{t\in [-\frac{1}{2},\frac{1}{2}]}{\cup }\left\{x\in L_t:x_1\le m_t\right\}.$$

Then there exists $w\in \underset{q<\infty }{\cap }{W}^{1,q}(Q_{1/4})$ such that

1. $w\le -\hspace{0.17em}\text{log}\hspace{0.17em}|\nabla u_{\infty }|$ and $e^w{\nabla }^{\perp }{u}_{\infty }·\nabla w=|\nabla w|$ almost everywhere in $Q_{1/4},$

2. $w=-\hspace{0.17em}\text{log}\hspace{0.17em}|\nabla u_{\infty }|$ in $Q_{1/4}\cap M$, and

3. the equation$$\text{div}(e^w{\nabla }^{\perp }{u}_{\infty })=|\nabla w|$$

   holds weakly in $Q_{1/4}.$

We give the proof of this result in Section 4 after some auxiliary results in Section 3. But first, we show how Theorem 3 and Theorem 4 follow from Proposition 5.

Proof of Theorem 3.

For any $x_0\in \Omega ,$ we can choose $a\in \mathbb{R},$ r > 0, and $R\in \text{SO}(2)$ such that Proposition 5 applies to $\tilde{u}(x)=a{u}_{\infty }(\mathit{rRx}+x_0).$ Since $|\nabla {\tilde{u}}_{\infty }|$ is monotone along the level sets of $u_{\infty },$ we may choose m_t = 1 for every $t\in [-\frac{1}{2},\frac{1}{2}].$ Hence we obtain a function $\tilde{w}\in \underset{q<\infty }{\cap }{W}^{1,q}(Q_{1/4})$ satisfying (a)-(c) in $Q_{1/4}.$ In particular $\tilde{w}=-\hspace{0.17em}\text{log}\hspace{0.17em}|\nabla {\tilde{u}}_{\infty }|,$ and it follows that $|\nabla {\tilde{u}}_{\infty }|\in W^{1,q}(Q_{1/4})$ for every $q<\infty .$

From the pointwise equations (a) and (b), we obtain $$\frac{{\nabla }^{\perp }{\tilde{u}}_{\infty }}{|\nabla {\tilde{u}}_{\infty }|}=-\frac{\nabla |\nabla {\tilde{u}}_{\infty }|}{|\nabla |\nabla {\tilde{u}}_{\infty }\left|\right|}$$ at almost every point where $\nabla |\nabla {\tilde{u}}_{\infty }|\ne 0.$ This amounts to Eq. (4)(4) $$|\nabla |\nabla u_{\infty }\left|\right|\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}=\omega \nabla |\nabla u_{\infty }|$$(4) (4) $$|\nabla |\nabla u_{\infty }\left|\right|\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}=\omega \nabla |\nabla u_{\infty }|$$(4) for ${\tilde{u}}_{\infty },$ which is trivially satisfied where the gradient vanishes. The combination of (b) and (c) gives (5)(5) $$\text{div}\left(\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}\right)=-\omega \frac{|\nabla |\nabla u_{\infty }\left|\right|}{|\nabla u_{\infty }|}$$(5) for ${\tilde{u}}_{\infty }.$ In terms of $u_{\infty },$ this means that there exists a neighbourhood U of x₀ such that $|\nabla u_{\infty }|\in W^{1,q}(U)$ for every $q<\infty$ and (4)(4) $$|\nabla |\nabla u_{\infty }\left|\right|\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}=\omega \nabla |\nabla u_{\infty }|$$(4) holds almost everywhere in U, while (5)(5) $$\text{div}\left(\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}\right)=-\omega \frac{|\nabla |\nabla u_{\infty }\left|\right|}{|\nabla u_{\infty }|}$$(5) holds weakly in U.

It follows that $|\nabla u_{\infty }|\in W_{\text{loc}}^{1,q}(\Omega )$ and the two equations hold in Ω. Let $$F=-\omega \frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}.$$

For the function $w=-\hspace{0.17em}\text{log}\hspace{0.17em}|\nabla u_{\infty }|,$ we then compute $$\nabla w·F=|\nabla w|$$ because of (4)(4) $$|\nabla |\nabla u_{\infty }\left|\right|\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}=\omega \nabla |\nabla u_{\infty }|$$(4) , and $$\text{div}F=|\nabla w|$$ because of (5)(5) $$\text{div}\left(\frac{{\nabla }^{\perp }{u}_{\infty }}{|\nabla u_{\infty }|}\right)=-\omega \frac{|\nabla |\nabla u_{\infty }\left|\right|}{|\nabla u_{\infty }|}$$(5) . Thus w is a weak solution of (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) . □

Proof of Theorem 4.

For points in G, the arguments in the proof of Theorem 3 apply. For $x_0\in \Gamma ,$ we can still argue similarly. Here we can still choose $a\in \mathbb{R},$ r > 0, and $R\in \text{SO}(2)$ such that the function $\tilde{u}(x)=a{u}_{\infty }(\mathit{rRx}+x_0)$ satisfies $|\nabla {\tilde{u}}_{\infty }-e_2|<\delta$ in $\overline{Q_1}$ and such that $$\frac{1}{r}{R}^{-1}(G-x_0)\cap Q_1=\left\{x\in Q_1:x_1<f(x_2)\right\}$$ for some Lipschitz function $f:[-1,1]\to \mathbb{R}$ with $f(0)=0,$ the Lipschitz constant of which is independent of the rescaling. We define L_t as in Proposition 5 for ${\tilde{u}}_{\infty }.$ If δ is sufficiently small, then each L_t will intersect the graph of f exactly once for every $t\in [-\frac{1}{2},\frac{1}{2}].$ We choose m_t such that the unique point $x\in L_t$ with $x_1=m_t$ is this intersection point.

The orientation in Theorem 4 is such that $|\nabla u_{\infty }|$ is non-decreasing if we approach Γ from inside G. In terms of ${\tilde{u}}_{\infty },$ this means that $|\nabla {\tilde{u}}_{\infty }|$ is non-decreasing when we travel along L_t from left to right up to m_t. Then the hypothesis of Proposition 5 is satisfied, so we infer $|\nabla {\tilde{u}}_{\infty }|\in W^{1,q}(M\cap Q_{1/4})$ for every $q<\infty .$ Since $M\cap Q_{1/4}$ corresponds to a neighbourhood of x₀ in $G\cup \Gamma ,$ a standard covering argument now implies the desired statement. □

3. Some estimates for p-harmonic functions

In this section we consider solutions of the equation ${\Delta }_{p}u=0.$ For the proofs of our main results, we require an $L^{\infty }$-estimate for the gradient away from the boundary. We have the following lemma, which is an easy consequence of estimates due to Bhattacharya, DiBenedetto, and Manfredi [3].

Lemma 6.

There exists a constant C > 0 with the following property. Suppose that $\Omega \subseteq {\mathbb{R}}^2$ is an open set. For $\varrho >0$, let ${\Omega }_{\varrho }=\left\{x\in \Omega :\text{dist}(x,\partial \Omega )>\varrho \right\}$. Let $p\ge 2$. Then for any p-harmonic function $u\in W^{1,p}(\Omega ),$ $$\left|\right|\nabla u|{|}_{L^{\infty }({\Omega }_{\varrho })}\le {\left(\frac{C}{{\varrho }^n}\right)}^{\frac{1}{p}}||\nabla u|{|}_{L^p(\Omega )},$$provided that $\text{diam} {\Omega }_{\varrho }\ge \varrho .$

Proof.

Bhattacharya, DiBenedetto, and Manfredi [3, Part III, Proposition 1.1] prove an inequality similar to this, the difference being that they consider a more general equation of the form $${\Delta }_{p}u=f(x,u,\nabla u)$$ (for a function f satisfying a certain growth condition) and that they consequently obtain $$\left|\right|\nabla u|{|}_{L^{\infty }({\Omega }_{\varrho })}\le {\left(\frac{C}{{\varrho }^n}\right)}^{\frac{1}{p}}{\left({\int }_{\Omega }(1+|\nabla u{|}^p) dx\right)}^{\frac{1}{p}}.$$

If we study the equation ${\Delta }_{p}u=0$ instead, then we can apply this result to $cu$ instead of u for any constant c > 0. Letting $c\to \infty ,$ we obtain the desired inequality. □

We require another estimate for p-harmonic functions. The following result gives an estimate from below for the partial derivative $\frac{\partial u}{\partial x_2},$ provided that we have a suitable domain and suitable boundary data.

Lemma 7.

Let $f,g:[-1,1]\to \mathbb{R}$ be two Lipschitz functions with f < g and $U=\left\{x\in (-1,1)\times \mathbb{R}:f(x_1)<x_2<g(x_1)\right\}$. Let $\varphi :\overline{U}\to \mathbb{R}$ be a Lipschitz function such that $\frac{\partial \varphi }{\partial x_2}\ge 1$ for almost all $x\in U$ and such that there are two numbers $a,b\in \mathbb{R}$ with $\varphi (x)=a+x_2$ when $x_2=f(x_1)$ and $\varphi (x)=b+x_2$ when $x_2=g(x_1)$. Then the solution $u:\overline{U}\to \mathbb{R}$ of the boundary value problem$$\begin{array}{c}{\Delta }_{p}u=0\hspace{1em}\text{in}\mathrm{ }U,\\ u=\varphi \text{on}\mathrm{ }\partial U,\end{array}$$satisfies $\frac{\partial u}{\partial x_2}\ge 1$ in U.

Proof.

Let $c=\max \left\{\right|\left|f\right|{|}_{L^{\infty }(-1,1)},\left|\right|g\left|{|}_{L^{\infty }(-1,1)}\right\}+2$ and write $U_0=(-1,1)\times (-c,c).$ Extend $\varphi$ to U₀ by $\varphi (x)=a+x_2$ when $x_2<f(x_1)$ and $\varphi (x)=b+x_2$ when $x_2>g(x_1).$ Choose a sequence of functions ${\varphi }_k\in C^{\infty }(\overline{U_0}),$ for $k\in \mathbb{N},$ such that ${\varphi }_k\to \varphi$ in $W^{1,p}(U_0)$ as $k\to \infty$ and such that

• ${\varphi }_k(x)=a+x_2$ when $x_2<f(x_1)-\frac{1}{k},$

• ${\varphi }_k(x)=b+x_2$ when $x_2>g(x_1)+\frac{1}{k},$ and

• $\frac{\partial {\varphi }_k}{\partial x_2}\ge 1.$

Now choose a sequence of domains $U_k\subseteq U_0$ with smooth boundaries, such that each U_k is of the form $U_k=\left\{x\in (-1,1)\times \mathbb{R}:f_k(x_1)<x_2<g_k(x_1)\right\}$ for some smooth functions $f_k,g_k:(-1,1)\to \mathbb{R}$ with $$f(x_1)-\frac{2}{k}<f_k(x_1)<f(x_1)-\frac{1}{k}<g(x_1)+\frac{1}{k}<g_k(x_1)<g(x_1)+\frac{2}{k}$$ for $-1<x_1<1.$

Let $u_k:U_k\to \mathbb{R}$ be the solution of $$\begin{array}{c}\text{div}\left({(|\nabla u_k{|}^2+k^{-2})}^{p/2-1}\nabla u_k\right)=0\hspace{1em}\text{in}\mathrm{ }{U}_k,\\ u_k={\varphi }_k\text{on}\mathrm{ }\partial U_k.\end{array}$$

Extend u_k to U₀ by $u_k(x)=a+x_2$ when $x_2<f_k(x_1)$ and $u_k(x)=b+x_2$ when $x_2>g_k(x_1).$ Then the sequence ${(u_k)}_{k\in \mathbb{N}}$ is clearly bounded in $W^{1,p}(U_0),$ and we may assume that it converges weakly in this space to a limit $\tilde{u}\in W^{1,p}(U_0).$ We claim that $\tilde{u}=u$ in U.

In order to prove this, note that $${\int }_{U_k}{\left(|\nabla u_k{|}^2+k^{-2}\right)}^{p/2} dx\le {\int }_{U_k}{\left(|\nabla ({\varphi }_k+w){|}^2+k^{-2}\right)}^{p/2} dx$$ for any $w\in W_0^{1,p}(U),$ because u_k minimises this quantity for its boundary data. Letting $k\to \infty ,$ we find that $$\begin{array}{cc}{\int }_U|\nabla \tilde{u}{|}^p dx\hfill & \le \underset{k\to \infty }{\lim \inf }{\int }_{U_k}{\left(|\nabla u_k{|}^2+k^{-2}\right)}^{p/2} dx\hfill \\ \hfill & \le \underset{k\to \infty }{\lim \inf }{\int }_{U_k}{\left(|\nabla ({\varphi }_k+w){|}^2+k^{-2}\right)}^{p/2} dx.\hfill \end{array}$$

Moreover, $$\begin{array}{l}{\int }_{U_k}{\left(|\nabla ({\varphi }_k+w){|}^2+k^{-2}\right)}^{p/2} dx\\ \begin{array}{cc}\hfill & ={\int }_{U_0}{\left(|\nabla ({\varphi }_k+w){|}^2+k^{-2}\right)}^{p/2} dx-{\int }_{U_0\setminus U_k}{\left(|\nabla \varphi {|}^2+k^{-2}\right)}^{p/2} dx\hfill \\ \hfill & \to {\int }_{U_0}|\nabla (\varphi +w){|}^p dx-{\int }_{U_0\setminus U}|\nabla \varphi {|}^p dx\hfill \\ \hfill & ={\int }_U|\nabla (\varphi +w){|}^p dx\hfill \end{array}\end{array}$$ as $k\to \infty .$ Therefore, $${\int }_U|\nabla \tilde{u}{|}^p dx\le {\int }_U|\nabla (\varphi +w){|}^p dx$$ for any $w\in W_0^{1,p}(U).$ Furthermore, it is clear that $\tilde{u}=\varphi$ on $\partial U.$ Since the functional $$v\mapsto {\int }_U|\nabla v{|}^p dx$$ has a unique minimiser in $\varphi +W_0^{1,p}(U),$ which is u, we conclude that $\tilde{u}=u.$

The regularity theory of Lieberman [33] shows that $u_k\in C^{\infty }(\overline{U_k}).$ Moreover, the function $\frac{\partial u_k}{\partial x_2}$ satisfies the equation $$\text{div}\left(A_k\nabla \frac{\partial u_k}{\partial x_2}\right)=0,$$ where $$A_k={\left(|\nabla u_k{|}^2+k^{-2}\right)}^{p/2-1}\left(I+(p-2)\frac{\nabla u_k\otimes \nabla u_k}{|\nabla u_k{|}^2+k^{-2}}\right)$$ and I is the identity matrix. This is a uniformly elliptic equation.

The comparison principle applies to u_k and implies that $a+x_2\le u_k(x)\le b+x_2$ for $x\in U_k.$ Hence $\frac{\partial u_k}{\partial x_2}\ge 1$ on $$\left\{x\in (-1,1)\times \mathbb{R}:x_2=f_k(x_1)\mathrm{ }\text{or}\mathrm{ }{x}_2=g_k(x_1)\right\}.$$

On the rest of $\partial U_k,$ the inequality is inherited directly from ${\varphi }_k.$ Applying the maximum principle to $\frac{\partial u_k}{\partial x_2},$ we prove that $\frac{\partial u_k}{\partial x_2}\ge 1$ in U_k. Then the desired inequality follows for u as well. □

4. Proof of Proposition 5

This section contains the key arguments of this paper.

We first fix $\delta \in (0,\frac{1}{16})$ and also fix a constant $\sigma \in [\frac{1}{2},1-8\delta ).$ The values of both will be determined later. (We will require that δ and $1-\sigma$ are sufficiently small.) We consider an ∞-harmonic function $u_{\infty }\in C^1(\overline{Q_1})$ that satisfies the hypotheses of Proposition 5. We wish to find a function $w\in \underset{q<\infty }{\cap }{W}^{1,q}(Q_{1/4})$ with the properties (a)-(c). In particular, we wish to show that w is a weak solution of the inverse mean curvature flow.

The function $\tilde{u}(x)=u_{\infty }(x)-\sigma x_2$ satisfies $$|\nabla \tilde{u}-(1-\sigma )e_2|\le \delta .$$

Hence the level sets of $\tilde{u}$ are Lipschitz graphs with Lipschitz constants bounded by $$\frac{\delta }{\sqrt{{(1-\sigma )}^2-{\delta }^2}}.$$

Let $a=\tilde{u}(0,-\frac{3}{4})$ and $b=\tilde{u}(0,\frac{3}{4}).$ Set $$U=\left\{x\in Q_1:a<\tilde{u}(x)<b\right\}.$$

Since $\delta \le \frac{1}{8}(1-\sigma )$ by the choice of σ, it follows that $[-1,1]\times [-\frac{1}{2},\frac{1}{2}]\subseteq \overline{U}\subseteq [-1,1]\times (-1,1).$

For $p\in [2,\infty ),$ let $u_p\in W^{1,p}(U)$ be the unique solutions of $$\begin{array}{c}{\Delta }_p{u}_p=0\hspace{1em}\text{in}\mathrm{ }U,\\ u=u_{\infty }\text{on}\mathrm{ }\partial U.\end{array}$$

Then we apply Lemma 7 to the functions $u_p/\sigma ,$ with boundary data given by $u_{\infty }/\sigma .$ This implies that (7) $$\frac{\partial u_p}{\partial x_2}\ge \sigma$$(7) in U. Moreover, (8) $${\int }_U|\nabla u_p{|}^p dx\le {\int }_U|\nabla u_{\infty }{|}^p dx\le 4{(1+\delta )}^p,$$(8) as u_p minimises this quantity among all functions with the same boundary data. Define $U_p=\left\{x\in U:\text{dist}(x,\partial U)>2^{-\sqrt{p}}\right\}.$ Then Lemma 6 implies that $$\left|\right|\nabla u_p|{|}_{L^{\infty }(U_p)}\le {\left(4^{1+\sqrt{p}}C\right)}^{\frac{1}{p}}(1+\delta ),$$ where C is the constant from Lemma 6, provided that p is sufficiently large. In particular, if p is sufficiently large, then (9) $$\left|\right|\nabla u_p|{|}_{L^{\infty }(U_p)}\le 1+2\delta .$$(9)

Inequalities (7(7) $$\frac{\partial u_p}{\partial x_2}\ge \sigma$$(7) ) and (9(9) $$\left|\right|\nabla u_p|{|}_{L^{\infty }(U_p)}\le 1+2\delta .$$(9) ) then imply that (10) $$\frac{\partial u_p}{\partial x_2}\ge \frac{\sigma |\nabla u_p|}{1+2\delta }.$$(10) in U_p.

By (8)(8) $${\int }_U|\nabla u_p{|}^p dx\le {\int }_U|\nabla u_{\infty }{|}^p dx\le 4{(1+\delta )}^p,$$(8) , we have uniform bounds for u_p in $W^{1,q}(U)$ for any $q<\infty .$ It therefore follows that a subsequence converges weakly in these spaces. The limit is ∞-harmonic and coincides with $u_{\infty }$ by the uniqueness result of Jensen [4, Corollary 3.14]. This implies that we have in fact the convergence $u_p\rightharpoonup u_{\infty }$ weakly in $W^{1,q}(U)$ for any $q<\infty .$ Clearly this convergence also holds uniformly in U. Moreover, we have the following variant of a result by Lindgren and Lindqvist [9].

Lemma 8.

For any precompact set $K⋐U$, the convergence $|\nabla u_p|\to |\nabla u_{\infty }|$ holds uniformly in K.

Proof.

The arguments of Lindgren and Lindqvist [9, Section 3] (which depend to some degree on the ideas of Koch, Zhang, and Zhou [13] and also use an inequality of Lebesgue [34]) can be used here. Although their paper deals with an annular domain and with specific boundary conditions, their reasoning applies more generally to p-harmonic functions in a domain $U\subseteq {\mathbb{R}}^2$ satisfying $|\nabla u_p|>0$ in U and $$\underset{p\to \infty }{\lim \sup }\left|\right|\nabla u_p|{|}_{L^{\infty }(K)}<\infty$$ for any $K⋐U.$ In our case, the first property follows from (7)(7) $$\frac{\partial u_p}{\partial x_2}\ge \sigma$$(7) and the second from (9)(9) $$\left|\right|\nabla u_p|{|}_{L^{\infty }(U_p)}\le 1+2\delta .$$(9) . □

Remark.

Lemma 8 does not imply that $\nabla u_p\to \nabla u_{\infty }.$ Nevertheless, the ideas of Koch, Zhang, and Zhou [13], as adapted by Lindgren and Lindqvist [9], do give convergence almost everywhere. But we do not need this information here.

For $p\in [2,\infty ),$ let $p\prime \in (1,2]$ denote the conjugate exponent with $\frac{1}{p}+\frac{1}{p\prime }=1.$ Since U is simply connected and since $\text{curl}(|\nabla u_p{|}^{p-2}{\nabla }^{\perp }{u}_p)={\Delta }_p{u}_p=0,$ there exists $v_p\in W^{1,p\prime }(U)$ satisfying $$\nabla v_p=-|\nabla u_p{|}^{p-2}{\nabla }^{\perp }{u}_p.$$

Then we compute $|\nabla v_p{|}^{p^{\prime }-2}\nabla v_p=-{\nabla }^{\perp }{u}_p.$ Therefore, $${\Delta }_{p\prime }{v}_p=0$$ in U. Note that the level sets of v_p are the streamlines of u_p. Moreover, (10)(10) $$\frac{\partial u_p}{\partial x_2}\ge \frac{\sigma |\nabla u_p|}{1+2\delta }.$$(10) implies that (11) $$\frac{\partial v_p}{\partial x_1}\ge \frac{\sigma |\nabla v_p|}{1+2\delta }$$(11) in U_p. That is, the angle between $\nabla v_p$ and e₁ is at most $$\arccos \left(\frac{\sigma }{1+2\delta }\right).$$

If δ and $1-\sigma$ are sufficiently small, then this means that for every $\beta \in \mathbb{R}$ there exists $\theta \in [-1,1]$ such that $$(-1,\theta )\times [-\frac{1}{2},\frac{1}{2}]\subseteq \left\{x\in (-1,1)\times [-\frac{1}{2},\frac{1}{2}]:v_p(x)<\beta \right\}\subseteq (-1,\theta +\frac{1}{16})\times [-\frac{1}{2},\frac{1}{2}]$$ for p large enough.

We can further prove the following inequalities for v_p. Here we use the notation $a_{+}=\max \{a,0\}$ for $a\in \mathbb{R}.$ For $\xi =({\xi }_1,{\xi }_2)\in {\mathbb{R}}^2,$ we write ${\xi }^{\perp }=(-{\xi }_2,{\xi }_1).$

Lemma 9.

Let $\lambda :[0,\mathcal{\ell }]\to U$ be a C¹-curve with $|\lambda \prime |\equiv 1$ and $|\lambda \prime -e_1|\le \delta$. Let $x=\lambda (0)$ and $y=\lambda (\mathcal{\ell })$. Then for any $\varrho >0,$ $$\underset{p\to \infty }{\lim \sup }{(\underset{B_{\varrho }(y)\cap U}{\inf }{v}_p-\underset{B_{\varrho }(x)\cap U}{\sup }{v}_p)}_{+}^{\frac{1}{p-1}}\le \underset{0\le s\le \mathcal{\ell }}{\sup }|\nabla u_{\infty }(\lambda (s))|$$and$$\underset{p\to \infty }{\lim \inf }{(\underset{B_{\varrho }(y)\cap U}{\sup }{v}_p-\underset{B_{\varrho }(x)\cap U}{\inf }{v}_p)}^{\frac{1}{p-1}}\ge {⨏}_0^{\mathcal{\ell }}{(\lambda \prime (s))}^{\perp }·\nabla u_{\infty }(\lambda (s)) ds.$$

Proof.

We may approximate λ in the C¹-topology with smooth curves. Therefore, we may assume without loss of generality that $\lambda \in C^{\infty }([0,\mathcal{\ell }];U).$ Let $ϵ>0.$ Define $\Phi :[0,\mathcal{\ell }]\times [-T,T]\to U$ by $\Phi (s,t)=\lambda (s)+t{(\lambda \prime (s))}^{\perp },$ where T > 0 is chosen so small that (12) $$|\nabla u_{\infty }(\Phi (s,t))-\nabla u_{\infty }(\lambda (s))|\le ϵ$$(12) for all $s\in [0,\mathcal{\ell }]$ and $t\in [-T,T].$ Since $\det D\Phi (s,t)=1+t\lambda \prime (s)·{(\lambda ″(s))}^{\perp },$ we may further choose T so small that $|\det D\Phi -1|<ϵ$ and $|{(\det D\Phi )}^{-1}-1|<ϵ$ in $[0,\mathcal{\ell }]\times [-T,T].$ We write $\Sigma =\Phi ([0,\mathcal{\ell }]\times [-T,T])$ and ${\Sigma }_t=\Phi ([0,\mathcal{\ell }]\times \{t\})$ for $-T\le t\le T,$ and we set $X=\frac{\partial \Phi }{\partial t}°{\Phi }^{-1}.$ (I.e., $X(\Phi (s,t))={(\lambda \prime (s))}^{\perp }.$)

Now note that $\Sigma \subseteq U_p$ for p sufficiently large and T sufficiently small. Hence $X·\nabla u_p\ge 0$ by (10)(10) $$\frac{\partial u_p}{\partial x_2}\ge \frac{\sigma |\nabla u_p|}{1+2\delta }.$$(10) and $X^{\perp }·\nabla v_p\le 0$ by (11)(11) $$\frac{\partial v_p}{\partial x_1}\ge \frac{\sigma |\nabla v_p|}{1+2\delta }$$(11) .

We write ${\mathcal{H}}^1$ for the 1-dimensional Hausdorff measure. Then (13) $$\begin{array}{cc}{⨏}_{-T}^T(v_p(\Phi (\mathcal{\ell },t))-v_p(\Phi (0,t))) dt\hfill & =-{⨏}_{-T}^T{\int }_{{\Sigma }_t}{X}^{\perp }·\nabla v_p d{\mathcal{H}}^1 dt\hfill \\ \hfill & =-\frac{1}{2T}{\int }_{\Sigma }{X}^{\perp }·\nabla v_p dx\hfill \\ \hfill & =\frac{1}{2T}{\int }_{\Sigma }|\nabla u_p{|}^{p-2}X·\nabla u_p dx\hfill \\ \hfill & \le \frac{\left|\Sigma \right|}{2T}\underset{\Sigma }{\sup }|\nabla u_p{|}^{p-1}.\hfill \end{array}$$(13)

By (12)(12) $$|\nabla u_{\infty }(\Phi (s,t))-\nabla u_{\infty }(\lambda (s))|\le ϵ$$(12) and Lemma 8, if p is sufficiently large, then $$\underset{\Sigma }{\sup }|\nabla u_p|\le \underset{0\le s\le \mathcal{\ell }}{\sup }|\nabla u_{\infty }(\lambda (s))|+2ϵ.$$

If $T<\varrho ,$ it follows that $${(\underset{B_{\varrho }(y)\cap U}{\inf }{v}_p-\underset{B_{\varrho }(x)\cap U}{\sup }{v}_p)}_{+}^{\frac{1}{p-1}}\le {\left(\frac{\left|\Sigma \right|}{2T}\right)}^{\frac{1}{p-1}}\left(\underset{0\le s\le \mathcal{\ell }}{\sup }|\nabla u_{\infty }(\lambda (s))|+2ϵ\right).$$

Letting $p\to \infty$ and $ϵ\to 0,$ we obtain the first inequality.

We can also estimate (14) $$\begin{array}{l}{\int }_0^{\mathcal{\ell }}(u_p(\Phi (s,T))-u_p(\Phi (s,-T))) ds\\ \begin{array}{cc}\hfill & ={\int }_0^{\mathcal{\ell }}{\int }_{-T}^T\frac{\partial \Phi }{\partial t}(s,t)·\nabla u_p(\Phi (s,t)) dt ds\hfill \\ \hfill & ={\int }_{\Sigma }X·\nabla u_p \left|\det D{\Phi }^{-1}\right| dx\hfill \\ \hfill & \le (1+ϵ){\int }_{\Sigma }X·\nabla u_p dx\hfill \\ \hfill & \le (1+ϵ)|\Sigma {|}^{\frac{p-2}{p-1}}{\left({\int }_{\Sigma }{(X·\nabla u_p)}^{p-1} dx\right)}^{\frac{1}{p-1}}\hfill \\ \hfill & \le (1+ϵ)|\Sigma {|}^{\frac{p-2}{p-1}}{\left({\int }_{\Sigma }|\nabla u_p{|}^{p-2}X·\nabla u_p dx\right)}^{\frac{1}{p-1}}\hfill \\ \hfill & =(1+ϵ)|\Sigma {|}^{\frac{p-2}{p-1}}{\left(-{\int }_{\Sigma }{X}^{\perp }·\nabla v_p dx\right)}^{\frac{1}{p-1}}.\hfill \end{array}\end{array}$$(14)

Since $u_p\to u_{\infty }$ uniformly, for p sufficiently large we have the inequality $$\begin{array}{l}{\int }_0^{\mathcal{\ell }}(u_p(\Phi (s,T))-u_p(\Phi (s,-T))) ds\\ \begin{array}{cc}\hfill & \ge (1-ϵ){\int }_0^{\mathcal{\ell }}(u_{\infty }(\Phi (s,T))-u_{\infty }(\Phi (s,-T))) ds\hfill \\ \hfill & =(1-ϵ){\int }_0^{\mathcal{\ell }}{\int }_{-T}^T\frac{\partial \Phi }{\partial t}(s,t)·\nabla u_{\infty }(\Phi (s,t)) dt ds.\hfill \end{array}\end{array}$$

Moreover, by (12)(12) $$|\nabla u_{\infty }(\Phi (s,t))-\nabla u_{\infty }(\lambda (s))|\le ϵ$$(12) and because ${(\lambda \prime (s))}^{\perp }·\nabla u_{\infty }(\lambda (s))\ge \frac{1}{2}$ by the assumptions on λ, we know that $\frac{\partial \Phi }{\partial t}(s,t)·\nabla u_{\infty }(\Phi (s,t))\ge (1-2ϵ){(\lambda \prime (s))}^{\perp }·\nabla u_{\infty }(\lambda (s)).$ Thus (15) $${\int }_0^{\mathcal{\ell }}(u_p(\Phi (s,T))-u_p(\Phi (s,-T))) ds\ge 2\mathcal{\ell }T(1-3ϵ){⨏}_0^{\mathcal{\ell }}{(\lambda \prime (s))}^{\perp }·\nabla u_{\infty }(\lambda (s)) ds.$$(15)

Recall that $${⨏}_{-T}^T(v_p(\Phi (\mathcal{\ell },t))-v_p(\Phi (0,t))) dt=-\frac{1}{2T}{\int }_{\Sigma }{X}^{\perp }·\nabla v_p dx$$ according to (13)(13) $$\begin{array}{cc}{⨏}_{-T}^T(v_p(\Phi (\mathcal{\ell },t))-v_p(\Phi (0,t))) dt\hfill & =-{⨏}_{-T}^T{\int }_{{\Sigma }_t}{X}^{\perp }·\nabla v_p d{\mathcal{H}}^1 dt\hfill \\ \hfill & =-\frac{1}{2T}{\int }_{\Sigma }{X}^{\perp }·\nabla v_p dx\hfill \\ \hfill & =\frac{1}{2T}{\int }_{\Sigma }|\nabla u_p{|}^{p-2}X·\nabla u_p dx\hfill \\ \hfill & \le \frac{\left|\Sigma \right|}{2T}\underset{\Sigma }{\sup }|\nabla u_p{|}^{p-1}.\hfill \end{array}$$(13) . We also know that $\left|\Sigma \right|\le 2(1+ϵ)\mathcal{\ell }T.$ Therefore, combining the above inequalities (14(14) $$\begin{array}{l}{\int }_0^{\mathcal{\ell }}(u_p(\Phi (s,T))-u_p(\Phi (s,-T))) ds\\ \begin{array}{cc}\hfill & ={\int }_0^{\mathcal{\ell }}{\int }_{-T}^T\frac{\partial \Phi }{\partial t}(s,t)·\nabla u_p(\Phi (s,t)) dt ds\hfill \\ \hfill & ={\int }_{\Sigma }X·\nabla u_p \left|\det D{\Phi }^{-1}\right| dx\hfill \\ \hfill & \le (1+ϵ){\int }_{\Sigma }X·\nabla u_p dx\hfill \\ \hfill & \le (1+ϵ)|\Sigma {|}^{\frac{p-2}{p-1}}{\left({\int }_{\Sigma }{(X·\nabla u_p)}^{p-1} dx\right)}^{\frac{1}{p-1}}\hfill \\ \hfill & \le (1+ϵ)|\Sigma {|}^{\frac{p-2}{p-1}}{\left({\int }_{\Sigma }|\nabla u_p{|}^{p-2}X·\nabla u_p dx\right)}^{\frac{1}{p-1}}\hfill \\ \hfill & =(1+ϵ)|\Sigma {|}^{\frac{p-2}{p-1}}{\left(-{\int }_{\Sigma }{X}^{\perp }·\nabla v_p dx\right)}^{\frac{1}{p-1}}.\hfill \end{array}\end{array}$$(14) ) and (15(15) $${\int }_0^{\mathcal{\ell }}(u_p(\Phi (s,T))-u_p(\Phi (s,-T))) ds\ge 2\mathcal{\ell }T(1-3ϵ){⨏}_0^{\mathcal{\ell }}{(\lambda \prime (s))}^{\perp }·\nabla u_{\infty }(\lambda (s)) ds.$$(15) ), we obtain $$\begin{array}{l}{⨏}_{-T}^T(v_p(\Phi (\mathcal{\ell },t))-v_p(\Phi (0,t))) dt\\ \ge \frac{{(1-3ϵ)}^{p-1}}{{(1+ϵ)}^{2p-3}}\mathcal{\ell }{\left({⨏}_0^{\mathcal{\ell }}{(\lambda \prime (s))}^{\perp }·\nabla u_{\infty }(\lambda (s)) ds\right)}^{p-1}.\end{array}$$

Thus if $T<\varrho ,$ then $${(\underset{B_{\varrho }(y)\cap U}{\sup }{v}_p-\underset{B_{\varrho }(x)\cap U}{\inf }{v}_p)}^{\frac{1}{p-1}}\ge \frac{1-3ϵ}{{(1+ϵ)}^2}{\mathcal{\ell }}^{\frac{1}{p-1}}{⨏}_0^{\mathcal{\ell }}{(\lambda \prime (s))}^{\perp }·\nabla u_{\infty }(\lambda (s)) ds.$$

Since $ϵ>0$ was chosen arbitrarily, the second inequality follows. □

Now fix a constant $\gamma \in (0,1-\delta ).$ Since we may add arbitrary constants to v_p without changing the properties used, we may assume that $v_p(-\frac{3}{4},0)={\gamma }^{p-1}$ for every $p\in [2,\infty ).$ In view of inequality (11(11) $$\frac{\partial v_p}{\partial x_1}\ge \frac{\sigma |\nabla v_p|}{1+2\delta }$$(11) ) and the considerations immediately following it, if δ and $1-\sigma$ are sufficiently small and p is sufficiently large, then $v_p\le {\gamma }^{p-1}$ in $(-1,-\frac{7}{8}]\times [-\frac{1}{2},\frac{1}{2}]$ and $v_p\ge {\gamma }^{p-1}$ in $[-\frac{5}{8},1)\times [-\frac{1}{2},\frac{1}{2}].$ In particular, the functions $$w_p=\frac{\hspace{0.17em}\text{log}\hspace{0.17em}{v}_p}{1-p}$$ are well-defined and satisfy $w_p\le -\hspace{0.17em}\text{log}\hspace{0.17em}\gamma$ at least in $[-\frac{5}{8},1)\times [-\frac{1}{2},\frac{1}{2}].$ We observe that (16) $${\Delta }_{p\prime }{w}_p=|\nabla w_p{|}^{p\prime }$$(16) wherever w_p is defined. Thus for any $\eta \in C_0^{\infty }(Q_{1/2})$ with $\eta \ge 0,$ $$\begin{array}{ll}{\int }_U{\eta }^{p\prime }|\nabla w_p{|}^{p\prime } dx\hfill & =-p\prime {\int }_U{\eta }^{p\prime -1}|\nabla w_p{|}^{p\prime -2}\nabla \eta ·\nabla w_p dx\hfill \\ \hfill & \le p\prime {({\int }_U|\nabla \eta {|}^{p\prime } dx)}^{\frac{1}{p^{\prime }}}{({\int }_U{\eta }^{p\prime }|\nabla w_p{|}^{p\prime } dx)}^{\frac{1}{p}}.\hfill \end{array}$$

Hence $${\int }_U{\eta }^{p\prime }|\nabla w_p{|}^{p\prime } dx\le {(p\prime )}^{p\prime }{\int }_U|\nabla \eta {|}^{p\prime } dx.$$

This means that $\left|\right|\nabla w_p|{|}_{L^1(K)}$ is uniformly bounded for any precompact set $K⋐Q_{1/2}.$

We apply Lemma 9 to $\lambda (s)=(s-\frac{15}{16},0)$ for $0\le s\le \frac{27}{16}.$ We already know that $$\underset{B_{1/16}(-15/16,0)}{\sup }{v}_p\le {\gamma }^{p-1}.$$

The first inequality in Lemma 9 then gives the estimate $$\underset{B_{1/16}(3/4,0)}{\inf }{v}_p\le {(1+2\delta )}^{p-1}+{\gamma }^{p-1}$$ for p sufficiently large. Using (11)(11) $$\frac{\partial v_p}{\partial x_1}\ge \frac{\sigma |\nabla v_p|}{1+2\delta }$$(11) again, we conclude that $$v_p\le {(1+2\delta )}^{p-1}+{\gamma }^{p-1}$$ in $Q_{1/2}.$ It follows that w_p is uniformly bounded in $L^{\infty }(Q_{1/2}).$

A result from the theory of the inverse mean curvature flow [20, Proposition 2.1] implies that for any $q\ge p\prime ,$ we have the inequality $${\int }_{Q_{1/2}}{\eta }^2{e}^{-2w_p}|\nabla w_p{|}^{q+2} dx\le \left(5+\frac{4q+8}{p\prime }\right){\int }_{Q_{1/2}}{e}^{-2w_p}|\nabla \eta {|}^2|\nabla w_p{|}^q dx$$ for any $\eta \in C_0^{\infty }(Q_{1/2}),$ provided that p is sufficiently large. Because w_p is uniformly bounded in $L^{\infty }(Q_{1/2}),$ the factor $e^{-2w_p}$ is also uniformly bounded and bounded away from 0. We can then iterate this inequality and obtain a uniform bound for $\left|\right|w_p|{|}_{W^{1,q}(K)}$ for any precompact $K⋐Q_{1/2}$ and any $q<\infty .$ Therefore, there exists a sequence $p_k\to \infty$ such that $w_{p_k}\rightharpoonup w$ weakly in $\underset{q<\infty }{\cap }{W}_{\text{loc}}^{1,q}(Q_{1/2})$ for some function $w:Q_{1/2}\to \mathbb{R}.$ By Morrey’s inequality and the Arzelà-Ascoli theorem, the convergence is also locally uniform and w is continuous.

Lemma 10.

Let $\lambda :[0,\mathcal{\ell }]\to U$ be a solution of the equation$$\lambda \prime (s)=-\frac{{\nabla }^{\perp }{u}_{\infty }(\lambda (s))}{|{\nabla }^{\perp }{u}_{\infty }(\lambda (s))|}$$for $s\in [0,\mathcal{\ell }]$. Let $x=\lambda (0)$ and $y=\lambda (\mathcal{\ell })$. If $x\in (-1,-\frac{15}{16}]\times [-\frac{3}{8},\frac{3}{8}]$ and $y\in Q_{1/4}$, then$$\underset{0\le s\le \mathcal{\ell }}{\inf }(-\hspace{0.17em}\text{log}\hspace{0.17em}|\nabla u_{\infty }(\lambda (s))|)\le w(y)\le -\hspace{0.17em}\text{log}\hspace{0.17em}|\nabla u_{\infty }(y)|.$$

Proof.

Set $A=\underset{0\le s\le \mathcal{\ell }}{\sup }|\nabla u_{\infty }(\lambda (s))|.$ We apply Lemma 9 with $\varrho \le \frac{1}{16}.$ Then $$\underset{B_{\varrho }(x)\cap U}{\sup }{v}_p\le {\gamma }^{p-1}.$$

Let $ϵ>0.$ It follows that for p sufficiently large, $$\underset{B_{\varrho }(y)}{\inf }{v}_p\le {(A+ϵ)}^{p-1}+{\gamma }^{p-1},$$ and therefore, $$\begin{array}{cc}\underset{B_{\varrho }(y)}{\sup }{w}_p\hfill & \ge \frac{1}{1-p}\text{log}\hspace{0.17em}\left({(A+ϵ)}^{p-1}+{\gamma }^{p-1}\right)\hfill \\ \hfill & =-\hspace{0.17em}\text{log}\hspace{0.17em}(A+ϵ)+\frac{1}{1-p}\text{log}\hspace{0.17em}\left(1+{\left(\frac{\gamma }{A+ϵ}\right)}^{p-1}\right).\hfill \end{array}$$

Since $\gamma <1-\delta \le A,$ it follows that $$\underset{B_{\varrho }(y)}{\sup }w\ge -\hspace{0.17em}\text{log}\hspace{0.17em}(A+ϵ).$$

Since w is continuous, we obtain the first estimate by letting $\varrho \to 0$ and $ϵ\to 0.$

For the proof of the second inequality, we apply Lemma 9 to the restriction of λ to $[k,\mathcal{\ell }],$ where $k\in [0,\mathcal{\ell })$ is chosen such that $\lambda (k)\in Q_{1/2}$ and $${⨏}_k^{\mathcal{\ell }}|\nabla u_{\infty }(\lambda (s))| ds\ge |\nabla u_{\infty }(y)|-ϵ.$$

Then $$\underset{p\to \infty }{\lim \inf }{\left(\underset{B_{\varrho }(y)}{\sup }{v}_p-\underset{B_{\varrho }(\lambda (k))}{\inf }{v}_p\right)}^{\frac{1}{p-1}}\ge |\nabla u_{\infty }(y)|-ϵ.$$

Note that $\underset{B_{\varrho }(\lambda (k))}{\inf }{v}_p\ge {\gamma }^{p-1}>0$ by the above observations. Hence for p large enough, $$\underset{B_{\varrho }(y)}{\sup }{v}_p\ge {(|\nabla u_{\infty }(y)|-2ϵ)}^{p-1}$$ and $$\underset{B_{\varrho }(y)}{\inf }{w}_p\le -\hspace{0.17em}\text{log}\hspace{0.17em}(|\nabla u_{\infty }(y)|-2ϵ).$$

The same inequality follows for w. Again we conclude the proof by letting $\varrho \to 0$ and $ϵ\to 0.$ □

Under the assumptions of Proposition 5, the level sets L_t of $u_{\infty }$ can be parametrised by curves λ as in Lemma 10. Considering the monotonicity of $|\nabla u_{\infty }|$ along these level sets, it follows that $$\underset{0\le s\le \mathcal{\ell }}{\sup }|\nabla u_{\infty }(\lambda (s))|=|\nabla u_{\infty }(\lambda (\mathcal{\ell }))|$$ if $\lambda (\mathcal{\ell })\in M.$ Furthermore, if δ is sufficiently small, then any L_t that intersects $Q_{1/4}$ will also intersect $(-1,-\frac{15}{16}]\times [-\frac{3}{8},\frac{3}{8}].$ Thus Lemma 10 implies that $w=-\hspace{0.17em}\text{log}\hspace{0.17em}|\nabla u_{\infty }|$ in $Q_{1/4}\cap M,$ which is statement (b).

Next we have a closer look at Eq. (16)(16) $${\Delta }_{p\prime }{w}_p=|\nabla w_p{|}^{p\prime }$$(16) (16) $${\Delta }_{p\prime }{w}_p=|\nabla w_p{|}^{p\prime }$$(16) and study what it means for the limit. We define $F_p=|\nabla w_p{|}^{p\prime -2}\nabla w_p,$ so we can write (17) $$\text{div}{F}_p=|\nabla w_p{|}^{p\prime }.$$(17)

For any $q<\infty$ and any precompact set $K⋐Q_{1/2},$ we have the inequality $${\left({\int }_K|F_p{|}^q dx\right)}^{\frac{1}{q}}\le |K{|}^{\frac{p-q}{pq}}{\left({\int }_K|F_p{|}^p dx\right)}^{\frac{1}{p}}=|K{|}^{\frac{p-q}{pq}}{\left({\int }_K|\nabla w_p{|}^{p\prime } dx\right)}^{\frac{1}{p}}.$$

Hence $$\underset{p\to \infty }{\lim \sup }{\left({\int }_K|F_p{|}^q dx\right)}^{\frac{1}{q}}\le |K{|}^{\frac{1}{q}}.$$

Therefore, we may assume that $F_{p_k}\rightharpoonup F$ weakly in $L_{\text{loc}}^q(Q_{1/2};{\mathbb{R}}^2)$ for every $q<\infty .$ Moreover, $$\left|\right|F|{|}_{L^{\infty }(K)}=\underset{q\to \infty }{\lim }|\left|F\right|{|}_{L^q(K)}\le 1.$$

That is, we have the inequality $\left|F\right|\le 1$ almost everywhere in $Q_{1/2}.$

We also observe that by the definition of w_p, $$F_p={(p-1)}^{-\frac{1}{p-1}}{e}^{w_p}{\nabla }^{\perp }{u}_p.$$

As $\nabla u_{p_k}\rightharpoonup \nabla u_{\infty }$ weakly in $L_{\text{loc}}^q(U)$ and $w_{p_k}\to w$ locally uniformly in $Q_{1/2},$ this implies that $$F=e^w{\nabla }^{\perp }{u}_{\infty }.$$

It follows that $w\le -\hspace{0.17em}\text{log}\hspace{0.17em}|\nabla u_{\infty }|.$

Lemma 11.

For any $\eta \in C_0^{\infty }(Q_{1/2}),$ $${\int }_U\eta |\nabla w_{p_k}{|}^{p_k^{\prime }} dx\to {\int }_U\eta |\nabla w| dx.$$

Proof.

The following arguments are known [18, p. 82], but are repeated here for the reader’s convenience.

We first show that for any compact set $K\subset Q_{1/2}$ and any function $f\in W^{1,p\prime }(Q_{1/2})$ with f = w_p in $Q_{1/2}\setminus K,$ the inequality (18) $${\int }_K\left(\frac{1}{p\prime }|\nabla w_p{|}^{p\prime }+w_p|\nabla w_p{|}^{p\prime }\right) dx\le {\int }_K\left(\frac{1}{p\prime }|\nabla f{|}^{p\prime }+f|\nabla w_p{|}^{p\prime }\right) dx$$(18) holds true. Indeed, using (16)(16) $${\Delta }_{p\prime }{w}_p=|\nabla w_p{|}^{p\prime }$$(16) , an integration by parts, and Young’s inequality, we compute $$\begin{array}{ll}{\int }_K(w_p-f)|\nabla w_p{|}^{p\prime } dx\hfill & =-{\int }_{Q_{1/2}}|\nabla w_p{|}^{p\prime -2}\nabla w_p·(\nabla w_p-\nabla f) dx\hfill \\ \hfill & \le \frac{1}{p\prime }{\int }_K(|\nabla f{|}^{p\prime }-|\nabla w_p{|}^{p\prime }) dx.\hfill \end{array}$$

Now given $\eta \in C_0^{\infty }(Q_{1/2})$ with $0\le \eta \le 1,$ we set $f=\eta w+(1-\eta )w_p$ and use (18)(18) $${\int }_K\left(\frac{1}{p\prime }|\nabla w_p{|}^{p\prime }+w_p|\nabla w_p{|}^{p\prime }\right) dx\le {\int }_K\left(\frac{1}{p\prime }|\nabla f{|}^{p\prime }+f|\nabla w_p{|}^{p\prime }\right) dx$$(18) . Then $$\begin{array}{l}{\int }_{Q_{1/2}}\left(\frac{1}{p\prime }|\nabla w_p{|}^{p\prime }+\eta (w_p-w)|\nabla w_p{|}^{p\prime }\right) dx\\ \begin{array}{cc}\hfill & \le \frac{1}{p\prime }{\int }_{Q_{1/2}}|\eta \nabla w+(1-\eta )\nabla w_p+(w-w_p)\nabla \eta {|}^{p\prime } dx\hfill \\ \hfill & \le \frac{3^{p\prime -1}}{p\prime }{\int }_{Q_{1/2}}({\eta }^p|\nabla w{|}^{p\prime }+{(1-\eta )}^{p\prime }|\nabla w_p{|}^{p\prime }+|w-w_p{|}^{p\prime }|\nabla \eta {|}^{p\prime }) dx.\hfill \end{array}\end{array}$$

(In the last step, we have used Hölder’s inequality for the counting measure.) Letting $p\to \infty$ (and therefore $p\prime \to 1$), we find that $$\underset{p\to \infty }{\lim \sup }{\int }_{Q_{1/2}}\eta |\nabla w_p{|}^{p\prime } dx\le {\int }_{Q_{1/2}}\eta |\nabla w| dx.$$

As we already know that $w_{p_k}\rightharpoonup w$ weakly in $W^{1,1}(\text{supp} \eta ),$ the desired convergence follows when $0\le \eta \le 1.$ It is then easy to prove in general. □

Using Lemma 11, we see that (17)(17) $$\text{div}{F}_p=|\nabla w_p{|}^{p\prime }.$$(17) gives rise to $$\text{div}F=|\nabla w|$$ in $Q_{1/2},$ which amounts to statement (c). Testing this equation with $\eta e^{-w},$ we find that $$\begin{array}{cc}{\int }_U\eta e^{-w}|\nabla w| dx\hfill & =-{\int }_U{e}^{-w}(\nabla \eta -\eta \nabla w)·F dx\hfill \\ \hfill & =-{\int }_U(\nabla \eta -\eta \nabla w)·{\nabla }^{\perp }{u}_{\infty } dx\hfill \\ \hfill & ={\int }_U\eta \nabla w·\nabla u_{\infty }^{\perp } dx\hfill \end{array}$$ for any $\eta \in C_0^{\infty }(Q_{1/2}).$ Therefore, $$e^{-w}|\nabla w|=\nabla w·{\nabla }^{\perp }{u}_{\infty }$$ almost everywhere in $Q_{1/2}.$ Thus we have proved statement (a) as well.

5. Weak solutions of the inverse mean curvature flow

The study of weak solutions of the inverse mean curvature flow goes back to a seminal paper of Huisken and Ilmanen [12], where they are defined in terms of a variational condition. Equation (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) (2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) is not actually variational, but Huisken and Ilmanen get around that problem by asking that a function w minimise a functional depending on w itself. They work with local Lipschitz functions, but the same ideas make sense for functions $w\in W_{\text{loc}}^{1,1}(\Omega )\cap L_{\text{loc}}^{\infty }(\Omega ).$ According to their definition, w is a weak solution of (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) if (19) $${\int }_K(1+w)|\nabla w| dx\le {\int }_K\left(|\nabla \tilde{w}|+\tilde{w}|\nabla w|\right) dx$$(19) for every precompact set $K⋐\Omega$ and every $\tilde{w}\in W_{\text{loc}}^{1,1}(\Omega )\cap L_{\text{loc}}^{\infty }(\Omega )$ with $\tilde{w}=w$ in $\Omega \setminus K.$

The same paper also proves a number of properties implied by this condition, including a comparison principle, a resulting uniqueness theorem, and a minimising hull property for the sublevel sets. Despite some technical differences, many of the underlying ideas will apply to the situation discussed in the introduction, too. It is therefore useful to know that the condition from Definition 1 implies inequality (19)(19) $${\int }_K(1+w)|\nabla w| dx\le {\int }_K\left(|\nabla \tilde{w}|+\tilde{w}|\nabla w|\right) dx$$(19) , provided that $w\in L_{\text{loc}}^{\infty }(\Omega ).$

Proposition 12.

Suppose that $w\in W_{\text{loc}}^{1,1}(\Omega )\cap L_{\text{loc}}^{\infty }(\Omega )$ is a weak solution of (2)(2) $$\text{div}\left(\frac{\nabla w}{|\nabla w|}\right)=|\nabla w|$$(2) in the sense of Definition 1. Then for every $K⋐\Omega$ and every $\tilde{w}\in W_{\text{loc}}^{1,1}(\Omega )\cap L^{\infty }(\Omega )$ with $\tilde{w}=w$ in $\Omega \setminus K$, inequality (19) holds true.

Proof.

Let $F\in L^{\infty }(\Omega ;{\mathbb{R}}^n)$ be a vector field satisfying $\left|F\right|\le 1$ and $F·\nabla w=|\nabla w|$ almost everywhere in Ω and $$\text{div}F=|\nabla w|$$ weakly. Then for almost every $x\in \Omega ,$ either $\nabla w(x)=0$ or $$F(x)=\frac{\nabla w(x)}{|\nabla w(x)|}.$$

Let $K⋐\Omega$ be a precompact set and let $\tilde{w}\in W_{\text{loc}}^{1,1}(\Omega )\cap L_{\text{loc}}^{\infty }(\Omega )$ with $\tilde{w}=w$ in $\Omega \setminus K.$ Then $$|\nabla \tilde{w}|\ge F·\nabla \tilde{w}=|\nabla w|+F·(\nabla \tilde{w}-\nabla w)$$ almost everywhere. It follows that $${\int }_K|\nabla \tilde{w}| dx\ge {\int }_K|\nabla w| dx-{\int }_K(\tilde{w}-w)|\nabla w| dx.$$

Rearranging the terms, we obtain inequality (19(19) $${\int }_K(1+w)|\nabla w| dx\le {\int }_K\left(|\nabla \tilde{w}|+\tilde{w}|\nabla w|\right) dx$$(19) ). □