Note on an eigenvalue problem with applications to a Minkowski type regularity problem in $\mathbb{R}$^n

TL;DR

This paper investigates the existence, uniqueness, and homogeneity of solutions to a class of p-Laplace PDEs within specific conical domains, with applications to Minkowski-type regularity problems in low-dimensional Euclidean spaces.

Contribution

It establishes conditions under which solutions are homogeneous of a specific degree, advancing understanding of PDE solutions in conical geometries and their geometric applications.

Findings

Solutions are homogeneous of degree 1 - (n-1)/p when p > n - 1.

Existence and uniqueness of solutions are proven for certain boundary conditions.

Applications to Minkowski regularity problems in 2D and 3D are demonstrated.

Abstract

We consider existence and uniqueness of homogeneous solutions $ u > 0 $ to certain PDE of $p$-Laplace type, $ p $ fixed, $ n - 1 <p< \infty, n \geq 2, $ when $ u $ is a solution in $K(\alpha)\subset\mathbb{R}^n$ where \[ K (\alpha) := \{ x = (x_1, \dots, x_n ): x_1 > \cos \alpha \, | x| \} \quad \mbox{for fixed}\, \, \alpha \in (0, \pi ], \] with continuous boundary value zero on $ \partial K ( \alpha ) \setminus \{0\}$. In our main result we show that if $ u $ has continuous boundary value $0$ on $ \partial K ( \pi )$ then $u$ is homogeneous of degree $ 1 - (n-1)/p $ when $ p > n - 1. $ Applications of this result are given to a Minkowski type regularity problem in $ \mathbb{R}^{n}$ when $n=2,3$.

Equations

$$K(\alpha ):=\{x=(x_1,\dots ,x_n):x_1>\cos \alpha |x|\}\text{mbox}forfixed\alpha \in (0,\pi ],$$

$$K(\alpha ):=\{x=(x_1,\dots ,x_n):x_1>\cos \alpha |x|\}\text{mbox}for\alpha \in (0,\pi ].$$

$$\displaystyle u(tx)=t^{\lambda}u(x)\,\,\mbox{for some real $\lambda$ whenever $t>0$ and $x\in K(\alpha)$.}$$

$$u(x)=u(r,\theta )=r^{\lambda }\varphi (\theta )\text{mbox}for0\le \theta <\alpha \text{mbox}andr>0$$

$$0=$$

$$+\lambda [\lambda (p-1)+(n-p)][{\lambda }^2{\varphi }^2(\theta )+({\varphi }^{\prime }{)}^2(\theta ){]}^{(p-2)/2}\varphi (\theta )(\sin \theta {)}^{n-2}.$$

$$\begin{array}{rl}0=& ((p-1){\psi }^2+{\lambda }^2){\psi }^{\prime }\\ & +({\lambda }^2+{\psi }^2)[(p-1){\psi }^2+(n-2)\cot \theta \psi +{\lambda }^2(p-1)+\lambda (n-p)].\end{array}$$

$$\frac{\lambda d\psi }{{\lambda }^2+{\psi }^2}-\frac{(\lambda -1)d\psi }{{\lambda }^2+{\psi }^2+\lambda (2-p)/(p-1)}+d\theta =0.$$

$$\pm 1-\frac{\lambda -1}{\sqrt{{\lambda }^2+\lambda (2-p)/(p-1)}}=\frac{2\alpha }{\pi }$$

$${\lambda }_1=1\text{mbox}and{\lambda }_2=\frac{p-3-2\sqrt{p^2-3p+3}}{3(p-1)}.$$

$${\lambda }_1=1-1/p\text{mbox}and{\lambda }_2=(1/16)\left(7p-16-\sqrt{81p^2-288p+288}\right)/(p-1).$$

$$-2{\lambda }_2(\pi )=\frac{p+1-n}{p-1}=\beta =\frac{n+1}{4n-5}$$

$$B(z,r)=\{y\in {\mathbb{R}}^n:|z-y|<r\}\text{mbox}wheneverz\in {\mathbb{R}}^n\text{mbox}andr>0.$$

$${\mathcal{H}}^{\gamma }(E)=\underset{\delta \to 0}{\lim }\inf \left\{\sum _j{r}_j^{\gamma };E\subset \bigcup _{j}B(x_j,r_j),r_j\le \delta \right\}$$

$$\Vert h{\Vert }_{1,q}=\Vert h{\Vert }_q+\Vert |\nabla h|{\Vert }_q$$

$$\displaystyle\begin{split}(a)&\,\,f(t\eta)=t^{p}f(\eta)\quad\mbox{when}\,\,t>0\,\,\mbox{and}\,\,\eta\in\mathbb{R}^{n}.\\ (b)&\,\,\mbox{There exists $\tilde{a}_{1}\geq 1$ such that if $\eta,\xi\in\mathbb{R}^{n}\setminus\{0\},$ then }\\ &\mbox{$\hskip 31.79897pt$}\tilde{a}_{1}^{-1}\,|\xi|^{2}|\eta|^{p-2}\,\leq\,\sum_{i,j=1}^{n}\frac{\partial^{2}f}{\partial\eta_{i}\partial\eta_{j}}(\eta)\,\xi_{i}\,\xi_{j}\leq\tilde{a}_{1}\,|\xi|^{2}|\eta|^{p-2}.\\ (c)&\,\,\mbox{There exists $\tilde{a}_{2}\geq 1$ such that for $\mathcal{H}^{n}$-almost every $\eta\in B(0,2)\setminus B(0,1/2),$ }\\ &\mbox{$\hskip 106.23698pt$}{\displaystyle\sum_{i,j,k=1}^{n}}\left|\frac{\partial^{3}f}{\partial\eta_{i}\partial\eta_{j}\partial\eta_{k}}(\eta)\right|\leq\tilde{a}_{2}.\end{split}$$

$$\int ⟨\mathcal{A}(\nabla v(y)),\nabla \theta (y)⟩dy=0\text{mbox}whenever\theta \in W_0^{1,p}(G).$$

$$\displaystyle(+)\,\,\,\,\,\,\,\,\quad\mbox{$u_{1}$ has continuous boundary value 0 on $\partial K(\alpha)$.}$$

$$\displaystyle(++)\,\,\quad\lim_{|x|\to\infty}u_{2}(x)=0\quad\mbox{and $u_{2}$ has continuous boundary value 0 on $\partial K(\alpha)\setminus\{0\}$}.$$

$${\lambda }_1(\alpha )-1+\frac{n-1}{p}\approx (\pi -\alpha {)}^{\frac{p+1-n}{p-1}}\text{mbox}as\alpha \to \pi .$$

$${\lambda }_1(\alpha )\approx \{\begin{array}{ll}(\pi -\alpha {)}^{\frac{n-1-p}{p-1}}& \text{mbox}for1<p<n-1\\ -\frac{1}{\log (\pi -\alpha )}& \text{mbox}forp=n-1\end{array}\text{mbox}as\alpha \to \pi .$$

$$\begin{array}{rl}(i)& {\int }_{{\mathbb{S}}^{n-1}}|⟨\theta ,\zeta ⟩|d\mu (\zeta )>0\text{mbox}forall\theta \in {\mathbb{S}}^{n-1},\\ (ii)& {\int }_{{\mathbb{S}}^{n-1}}\zeta d\mu (\zeta )=0.\end{array}$$

$$\displaystyle(a)\mbox{$\hskip 14.45377pt$}{\displaystyle\lim_{y\to x}}\nabla U(y)=\nabla U(x)\,\,\mbox{exists for $\mathcal{H}^{n-1}$-almost every $x\in\partial E$}$$

$$y\in {\mathbb{R}}^n\setminus E$$

$$\displaystyle(b)\mbox{$\hskip 14.45377pt$}{\displaystyle\int_{\partial E}f(\nabla U(x))\,d\mathcal{H}^{n-1}<\infty}.$$

$$\displaystyle(c)\mbox{$\hskip 14.45377pt$}{\displaystyle\int_{\mathbf{g}^{-1}(K,E)}f(\nabla U(x))\,d\mathcal{H}^{n-1}}=\mu(K)\quad\mbox{whenever }K\subset\mathbb{S}^{n-1}\,\,\mbox{is a Borel set}.$$

$$\displaystyle(d)\mbox{$\hskip 14.45377pt$}\mbox{$E$ is the unique set up to translation for which $(c)$ holds.}$$

$$\displaystyle(c^{\prime})\mbox{$\hskip 14.45377pt$}\mbox{ There exists $\tilde{b},0<\tilde{b}<\infty,$ with }$$

$$\displaystyle\mbox{$\hskip 35.41269pt$}{\displaystyle\tilde{b}\int_{\mathbf{g}^{-1}(K,E)}f(\nabla\tilde{U})\,d\mathcal{H}^{n-1}}=\,\mu(K)\mbox{ whenever $K\subset\mathbb{S}^{n-1}$ is a Borel set}.$$

$$\displaystyle(d^{\prime})\mbox{$\hskip 14.45377pt$}\mbox{$E$ is the unique set up to translation satisfying $(c^{\prime})$ with $\mathcal{A}$-capacity 1. }$$

$$\displaystyle\mathcal{H}^{n-1}(\mathbf{g}^{-1}(K,E))=\mu(K)\mbox{ whenever $K\subset\mathbb{S}^{n-1}$ is a Borel set.}$$

$$\bar{c}(\delta {)}^{-1}\le {\int }_{\partial K(\alpha )\cap \{x:x_1\ge -1+4\delta \}}\sin (\pi -\alpha )f(\nabla u(y,\alpha ))d{\mathcal{H}}^{n-1}\le \bar{c}(\delta ).$$

$$0<\pi-\alpha<<\delta<<1\mbox{ and $\delta$ is fixed.}$$

$$|\nabla u(\cdot ,\alpha )|\le c^{\prime }(\pi -\alpha {)}^{\frac{2-n}{p-1}}\text{mbox}on\partial K(\alpha )\cap [B(0,2)\setminus B(0,1/2)]$$

Full text

[TABLE]

with continuous boundary value zero on $\partial K(\alpha)\setminus\{0\}$. In our main result we show that if $u$ has continuous boundary value [math] on $\partial K(\pi)$ then $u$ is homogeneous of degree $1-(n-1)/p$ when $p>n-1.$ Applications of this result are given to a Minkowski type regularity problem in $\mathbb{R}^{n}$ when $n=2,3$.

1. 1 Introduction
2. 2 Basic estimates and definitions for $\mathcal{A}$-harmonic functions
3. 3 Boundary Harnack inequalities and uniqueness in Theorem A
4. 4 Proof of (1.7) in Theorem A
5. 5 Proof of Theorem B
6. 6 Closing Remarks

1. Introduction

Let $u>0$ be a homogeneous $p$-harmonic function in the cone $K(\alpha)\subset\mathbb{R}^{n}$, $n\geq 2$, with continuous boundary value 0 on $\partial K(\alpha)\setminus\{0\}$ where

[TABLE]

More specifically, for fixed $p,1<p<\infty$, $u$ is a weak solution to $\nabla\cdot\left(|\nabla u|^{(p-2)}\,\nabla u\right)=0$ in $K(\alpha)$ and

[TABLE]

Given $x\in\mathbb{R}^{n}\setminus\{0\}$, introduce spherical coordinates $r=|x|$ and $x_{1}=r\cos\theta$ for $0\leq\theta\leq\pi$. If $u$ as in (1.1) is $p$-harmonic in $K(\alpha)$ and $u(1,0,\dots,0)=1$ then using rotational invariance of the $p$-Laplace equation, it turns out that $u$ has additionally the following form

[TABLE]

with $\phi(0)=1$ and $\phi(\alpha)=0$ for some $\lambda(\alpha)=\lambda\in(-\infty,\infty)$ and $\phi\in C^{\infty}([0,\alpha])$.

It was first shown by Krol’ and Maz’ya in [KM72] that if $1<p\leq n-1$ and $\alpha\in(0,\pi)$, $\alpha$ is near enough $\pi$, then there exists a unique solution to (1.1) in $K(\alpha)$ of the special form (1.2) with $\lambda(\alpha)>0$. Tolksdorf in [Tol83] showed that given $\alpha\in(0,\pi)$, for $i=1,2$, there exist unique $\lambda_{i}$ with $\lambda_{2}<0<\lambda_{1}$ and $\phi_{i}$ where $\phi_{i}$ is infinitely differentiable on $[0,\alpha]$ satisfying $\phi_{i}(\alpha)=0$ and $\phi_{i}(0)=1$ and $u_{i}(r,\theta)=r^{\lambda_{i}}\phi_{i}(\theta)$ are solutions to the $p$- Laplace equation in $K(\alpha)$. Also Porretta and Véron gave another proof of Tolksdorf’s result in [PV09]. A similar study was made in more general Lipschitz cones by Gkikas and Véron in [GV18].

Next we discuss what is known about “eigenvalues” $\lambda$ in (1.2) for various $\alpha$ and $n$. Krol’ in [Kro73] (see also [Aro86]) used (1.2) and separation of variables to show for $u$ as in (1.2) that

[TABLE]

Letting $\psi=\phi^{\prime}/\phi$ in the above equation he obtained, the first order DE

[TABLE]

If $n=2$ the cotangent term in the above DE goes out and variables can be separated in (1.3) to get

[TABLE]

The boundary conditions imply that $\phi$ is decreasing on $(0,\alpha)$ so $\psi(\alpha)=-\infty$ and $\psi(0)=0$. Using this fact and integrating it follows that

[TABLE]

where $+1$ is taken if $\lambda>0$ and $-1$ if $\lambda<0.$ For later discussion we note that if $\alpha=\pi/2$, i.e., $K(\pi/2)$ is a half-space, then (1.4) gives

[TABLE]

We remark that $\lambda_{1}=\lambda_{1}(\pi/2)=1$ for $n\geq 2$ since $x_{1}=r\cos\theta$ is $p$-harmonic for $1<p<\infty$. Also if $\alpha=\pi$ and $n=2,$ i.e., $K(\pi)=\mathbb{R}^{2}\setminus(-\infty,0]$, then (1.4) yields

[TABLE]

For other values of $\lambda_{2}=\lambda_{2}(\alpha)$ when $n=2$, see [LV13]. For $n\geq 3$, $\alpha=\pi/2$, and $p=2$, one can use the Kelvin transformation to get $\lambda_{2}(\pi/2)=1-n$ while if $p=n,$ it follows from conformal invariance of the $n$-Laplacian that $\lambda_{2}(\pi/2)=-1$. Also if $p=(4n-2)/3$ then

[TABLE]

since $u(r,\theta)=r^{-\beta/2}(\cos(\theta/2))^{\beta}$ in (1.2) for $\alpha=\pi.$ DeBlassie and Smits in [DS16] obtained estimates on $-\lambda_{2}(\pi/2)$, $1<p\neq 2<\infty$, by leaving out the cotangent term in (1.3). In fact their solution to the DE in (1.3) with the cotangent term omitted leads to a supersolution of the form (1.2) for the $p$-Laplace equation, so leads to a lower estimate for $-\lambda_{2}(\pi/2)$ in (1.3). Upper and lower estimates for $\lambda_{2}(\alpha)$ for $\alpha\in(0,\pi/2]$ were also obtained by these authors in [DS18], by finding $p$-harmonic subsolution and supersolution of the form $r^{k}\tilde{\phi}(\theta)$ where $k<0$ and $\tilde{\phi}$ is the solution to (1.2) when $p=2$ in $K(\alpha)$. Sub and super $p$-harmonic solutions of the form $r^{k}\cos\theta$ were also found in $K(\pi/2)$ by Llorente, Manfredi, Troy, and Wu in [LMTW19]. These estimates were then used to find upper and lower bounds for $\lambda_{2}(\pi/2)$ in $K(\pi/2).$ In [LMTW19], the authors also use shooting methods to give a strictly ODE proof for existence of a solution to (1.3) on $[0,\pi/2]$ satisfying $\psi(0)=0$ and $\lim_{\theta\to\pi/2}\psi(\theta)=-\infty$ when $p$ and $n$ are fixed with $1<p<\infty$ and $n\geq 2$.

In this paper we consider problems similar to the above for certain PDEs of $p$-Laplace type. Our results, when specialized to the $p$-Laplace equation for fixed $p>n-1,$ give a unique solution $u$ to (1.2) in $K(\pi)$ with continuous boundary value [math] on $\partial K(\pi)$ and $\lambda=\lambda_{1}(\pi)=1-(n-1)/p$ when $n\geq 3$ (compare with Krol’s $n=2$ and $\alpha=\pi$ result). To be more specific we need some notation. Put

[TABLE]

Let $\langle\cdot,\cdot\rangle$ denote the standard inner product on $\mathbb{R}^{n}$ and let $|y|=\langle y,y\rangle^{1/2}$ be the Euclidean norm of $y.$ Let $dy$ denote $n$-dimensional Lebesgue measure on $\mathbb{R}^{n}$ and let $\mathcal{H}^{\gamma}$, $0<\gamma\leq n,$ denote $\gamma$-dimensional Hausdorff measure on $\mathbb{R}^{n}$ defined by

[TABLE]

where the infimum is taken over all possible $\delta$-covering $\{B(x_{j},r_{j})\}$ of $E$. If $O\subset\mathbb{R}^{n}$ is open and $1\leq q\leq\infty,$ then by $W^{1,q}(O)$ we denote the space of equivalence classes of functions $h$ with distributional gradient $\nabla h=(h_{y_{1}},\dots,h_{y_{n}}),$ both of which are $q$-th power integrable on $O.$ Let

[TABLE]

be the norm in $W^{1,q}(O)$ where $\|\cdot\|_{q}$ is the usual Lebesgue $q$ norm of functions in the Lebesgue space $L^{q}(O).$ Next let $C^{\infty}_{0}(O)$ be the set of infinitely differentiable functions with compact support in $O$ and let $W^{1,q}_{0}(O)$ be the closure of $C^{\infty}_{0}(O)$ in the norm of $W^{1,q}(O).$ Given $p,1<p<\infty,$ suppose f:\mathbb{R}^{n}\mbox{\rightarrow}[0,\infty) satisfies:

[TABLE]

Note that our assumptions in (1.5) imply that second derivatives of $f$ are Lipschitz and homogeneous of degree $p-2$ on $\mathbb{R}^{n}\setminus\{0\}.$ To conform with the notation in [AGH*+*17] and [ALSV18] we put $\mathcal{A}=\nabla f$ for fixed $p$, $1<p<\infty$, and given an open set $O$ we say that $v$ is $\mathcal{A}$-harmonic in $O$ provided $v\in W^{1,p}(G)$ for each open $G$ with $\bar{G}\subset O$ and

[TABLE]

As a short notation for (1.6) we write $\nabla\cdot\mathcal{A}(\nabla v)=0$ in $O$. Note that if $f(\eta)=p^{-1}|\eta|^{p}$ then $v$ as in (1.6) is $p$-harmonic in $O.$ The definition of $\mathcal{A}$-capacity, a $\mathcal{A}$-capacitary function, and of the $\mathcal{A}$-harmonic Green’s function with pole at $\infty$ are given in section 2.

In this article, we first prove

Theorem A**.**

Fix $f$ as in (1.5), $n\geq 2,\alpha\in(0,\pi],$ and suppose $1<p<\infty$ when $\alpha\in(0,\pi),$ while $p>n-1$ when $\alpha=\pi.$ For $i=1,2,$ there exists a unique $\mathcal{A}$-harmonic function $u_{i}>0$ in $K(\alpha)$ with $u_{i}(1,0,\dots,0)=1$ satisfying

[TABLE]

Moreover, (1.1) holds with $\lambda=\lambda_{i}(\alpha)$, for $i=1,2$, where $\lambda_{2}(\alpha)<0<\lambda_{1}(\alpha)$ with the property that $|\lambda_{i}(\alpha)|$ is decreasing on $(0,\pi)$. Finally, $\lambda_{1}(\pi)=1-(n-1)/p$ for $p>n-1$ and

[TABLE]

Remark 1.1**.**

We remark that if $1<p\leq n-1$ then a slit has $p$-capacity zero in $\mathbb{R}^{n}$ for $n\geq 3$ and so one can show (see [HKM06, chapter 2]) that there are no solutions to (1.3). In fact, Krol’ and Maz’ya in the paper mentioned earlier obtained that

[TABLE]

Here and in (1.7), $\approx$ means the ratio of the two functions is bounded above and below by positive constants depending only on $p,n,$ and possibly $\tilde{a}_{1}$, $\tilde{a}_{2}$ in (1.5). We regard (1.7) as our main contribution in Theorem 1.2. For an outline of our efforts in trying to prove this equality we refer the reader to [ALV19]. As mentioned above, our proof of existence and uniqueness in Theorem A for $p$-harmonic functions when $0<\alpha<\pi$ is considerably less general than the proof in [PV09] given for “Lipschitz cones”. Our proof, however, differs somewhat from the proof of these authors (even for $p$-harmonic functions). We include a proof in our setting mainly to facilitate the proof of (1.7) but also for completeness.

In order to give an application of Theorem A we need some background material. Let $E\subset\mathbb{R}^{n}$ be a convex set with nonempty interior. Then for $\mathcal{H}^{n-1}$ almost every $x\in\partial E,$ there is a well defined outer unit normal, $\mathbf{g}(x,E)$ to $\partial E.$ The function $\mathbf{g}(\cdot,E):\partial E\mapsto\mathbb{S}^{n-1}$ (whenever defined) is called the Gauss map for $\partial E.$ Let $\mu$ be a finite positive Borel measure on $\mathbb{S}^{n-1}=\{x\in\mathbb{R}^{n}:|x|=1\}$ satisfying

[TABLE]

Then in [ALSV18], it was shown that

Theorem 1.2**.**

Let $\mu$ be as in (1.8), $f$ as in (1.5), and $p$ fixed, $n\leq p<\infty$. Then there exists a compact convex set $E$ with non-empty interior and an $\mathcal{A}$-harmonic Green’s function $U$ for $\mathbb{R}^{n}\setminus E$ with pole at infinity satisfying

[TABLE]

Also in [AGH*+*17] the authors proved

Theorem 1.3**.**

Let $\mu$ be as in (1.8) and $f$ be as in (1.5). Then for fixed $p$ with $1<p\not=n-1<n$, there exists a compact convex set $E$ with non-empty interior and an $\mathcal{A}$-capacitary function, $\tilde{U}$ for $E$, satisfying $(a)-(d)$ of Theorem 1.2 with $U=\tilde{U}$. If $p=n-1,$ then there exists a compact convex set $E$ with non-empty interior having $\mathcal{A}$-capacity $1$, and a corresponding $\mathcal{A}$-capacitary function $\tilde{U}$ for $E$ satisfying $(a)$ and $(b)$ of Theorem 1.2 with $U=\tilde{U},$ as well as,

[TABLE]

As an application of Theorem A when $n=2,3,$ we prove the regularity of the Minkowski problem.

Theorem B**.**

Let $\mu$ be as in (1.8) and $f$ as in (1.5). Suppose also that $\hat{\alpha}\in(0,1),k$ is a non-negative integer, and $d\mu=\Theta\,d\mathcal{H}^{n-1}$ on $\mathbb{S}^{n-1}$ for some $0<\Theta\in C^{k,\hat{\alpha}}(\mathbb{S}^{n-1}).$ If $k\geq 1,$ assume $f\in C^{k+2,\hat{\alpha}}(\mathbb{R}^{n}\setminus\{0\}).$ Let $E$ be the compact convex set with non-empty interior in Theorem 1.2 or Theorem 1.3 corresponding to $\mu.$ If either $n=2,3,$ and $1<p<\infty,$ or $n\geq 4$ and $1<p\leq 2,$ then $\partial E$ is locally the graph of a $C^{k+2,\hat{\alpha}}(\mathbb{R}^{n-1})$ function.

Remark 1.4**.**

Theorems 1.2, 1.3, and B are generalizations of existence, uniqueness, and regularity for the classical Minkowski Problem. To give a little history, the classical Minkowski existence and uniqueness theorem states that if $\mu$ is as in (1.8) , then there exists a unique compact convex set $E$ (up to translation) with non-empty interior such that

[TABLE]

When $E$ is a polyhedron, the measure is a sum of point masses at the normals to each of the faces, and the coefficient at a normal is the surface area of that face.

The analogue of Theorem B concerning regularity in the Minkowski problem was studied by Pogorelov in [Pog78], Nirenberg in [Nir53], Cheng and Yau in [CY76], and Caffarelli in [Caf90b, Caf91, Caf90a, Caf89]. See also recent work of Savin in [Sav13] and De Philippis and Figalli in [DPF13]. In all papers regularity of $\partial E$ reduces to a corresponding regularity problem for the graph of a convex solution to a certain Monge-Ampère equation with [math] boundary values. A more thorough discussion of this reduction is given in section 5.

Theorems 1.3 and B were first proved by Jerison in [Jer96] for Laplace’s equation (i.e., when $f(\eta)=|\eta|^{2}/2$) and after that generalized to $p$-harmonic functions when $1<p<2$ in [CNS*+*15] for $n>2$. It will turn out that it suffices to assume that $\Theta$ is bounded above and below on $\mathbb{S}^{n-1}$ in order to conclude $\partial E$ is strictly convex and locally the graph of a $C^{1,\epsilon}$ function where $\epsilon>0$ depends on $\tilde{a}_{1},\tilde{a}_{2},p,n,$ the eccentricity of $E,$ and the bounds for $\Theta$.

1.1. Outline of the proof of Theorems A and B

Existence in Theorem A for $\alpha\in(0,\pi)$ follows easily from interior regularity results and Wiener type estimates for $\mathcal{A}$-harmonic functions listed in section 2. Uniqueness in Theorem A for $\alpha\in(0,\pi)$ follows from boundary Harnack inequalities, originally proved for positive $p$-harmonic functions vanishing on a portion of a Lipschitz domain in [LN07, LN10]. These inequalities were updated to $\mathcal{A}$-harmonic functions for fixed $p$ with $1<p<n$ in [AGH*+*17] and for $p\geq n$ in [ALSV18]. Uniqueness in the case $\alpha=\pi$ is somewhat more involved (since $K(\pi)\cap B(0,\rho)$ is not a Lipschitz domain), using not only the above boundary Harnack inequalities but also arguments from [LLN08] and [LN18]. To outline the proof of (1.7) we now write $u(\cdot,\alpha)$ and $\lambda(\alpha)$ for $u_{1}$ and $\lambda_{1}$ in Theorem A relative to $K(\alpha).$ First it follows easily from our existence and uniqueness results that $\lambda$ is continuous and decreasing as a function of $\alpha$ on $(0,\pi)$ with ${\displaystyle\lim_{\alpha\to\pi}}\lambda(\alpha)=\lambda(\pi).$ From boundary Harnack inequalities for $\mathcal{A}$-harmonic functions, as well as an integral identity proved in [AGH*+*17] for $n-1<p<n$ and in [ALSV18] for $p\geq n,$ we eventually obtain

[TABLE]

in (4.10) where

[TABLE]

Also $c(\delta)\geq 1$ is a positive constant depending only on $p,n,$ and $\tilde{a}_{1},\tilde{a}_{2}$ in (1.5). To estimate the integral in (1.10) we use a boundary Harnack inequality for $\mathcal{A}$-harmonic functions on lower dimensional sets from [LN18] to essentially obtain

[TABLE]

where $c^{\prime}$ depends on $p,n,$ and $\tilde{a}_{1},\tilde{a}_{2}$ in (1.5). From (1.10), (1.11), and homogeneity of $u(\cdot,\alpha)$ we finally get

[TABLE]

where $c(\delta)$ has the same dependence as $\bar{c}(\delta)$ above and we have also used the fact that an element of surface area on $\partial K(\alpha)$ is of the form $[\sin(\pi-\alpha)]^{n-2}r^{n-2}dr$. From (1.12) and some arithmetic we conclude

[TABLE]

for some $c^{*}=c^{*}(p,n,\tilde{a}_{1},\tilde{a}_{2})\geq 1$ and so get the desired upper estimate for $\lambda_{1}(\alpha)$ in Theorem A. The lower estimate is similar. We note that a slightly different proof of Theorem A for $p$-harmonic functions when $n-1<p<n$ (with more details) is outlined in [ALV19].

As for the proof of Theorem B, armed with Theorems A, 1.2, and 1.3., we can follow closely the proof in [CNS*+*15], who in turn followed closely the proof in [Jer96]. Indeed, Jerison in [Jer96], first converts Theorem B into a regularity statement for the solution, say $\hat{u}$ to a Monge Ampère equation whose right-hand side corresponds to a measure $\hat{\mu}$ on $\mathbb{S}^{n-1}$. To show regularity of $\hat{u},$ he first generalized the Alexandrov-Bakelman inequality (see [Jer96, Lemma 7.3]) and then used this generalization to prove a certain integral inequality for $\hat{\mu}$ in Theorem 6.5 of [Jer96]. This inequality was then used to show that arguments in [Caf89, Caf90b, Caf91, Caf90a] could be used to eventually obtain Theorem B (see also [GH00]). Theorem A is used in Theorem B to prove the analogue of Theorem 6.5 in [Jer96] when $n=2,3$ and $p>2$. In fact, Theorem A is used only in the proof of Lemma 5.8. Unfortunately this lemma is not strong enough to be used in the rest of Jerison’s proof when $p>2,$ unless $n=2,3.$

As for the plan of this paper, in section 2, we state some basic properties of $\mathcal{A}$-harmonic functions, give the definitions mentioned after Theorem 1.3, and prove existence in Theorem A. In section 3, we state several boundary Harnack inequalities and then apply these inequalities to prove uniqueness in Theorem A. In section 4 we state integral identities from [AGH*+*17, ALSV18] and then use these identities to prove Theorem A. Theorem B is proved in section 5. In section 6 we make closing remarks concerning generalizations of Theorems A and B.

2. Basic estimates and definitions for $\mathcal{A}$-harmonic functions

In this section we first introduce some notation and then state some fundamental estimates for $\mathcal{\tilde{A}}=\nabla\tilde{f}$-harmonic functions when $p$ is fixed, $1<p<\infty,$ and $\tilde{f}$ satisfies (1.5) with $f=\tilde{f}$. Second, we define the $\mathcal{A}$-capacitary function when $1<p<n$ and $\mathcal{A}$-harmonic Green’s function with pole at $\infty$ when $p\geq n$ of a compact convex set $E.$ Third, we show existence of $u_{i}$ for $i=1,2$, in Theorem A relative to $K(\alpha)$ when $\alpha\in(0,\pi).$ Concerning constants, unless otherwise stated, in this section, and throughout the paper, $c$ will denote a positive constant $\geq 1$, not necessarily the same at each occurrence, depending at most on $p,n,\tilde{a}_{1},\tilde{a}_{2},$ which sometimes we refer to as depending on the data. In general, $c(t_{1},\dots,t_{m})$ denotes a positive constant $\geq 1,$ which may depend at most on $p,n,\tilde{a}_{1},\tilde{a}_{2}$ and $t_{1},\dots,t_{m},$ not necessarily the same at each occurrence. Also, as in the introduction, if $B\approx C$ then $B/C$ is bounded from above and below by constants which, unless otherwise stated, depend at most on the data. Let $e_{k}$ be the $n$ tuple with one in the $k$th position and zeros elsewhere. Let $d(E_{1},E_{2})$ denote the distance between the sets $E_{1}$ and $E_{2}.$ For short we write $d(x,E_{2})$ for $d(\{x\},E_{2}).$ Also put $E_{1}+E_{2}=\{x+y:x\in E_{1},y\in E_{2}\}$ and $\lambda E=\{\lambda x:x\in E\}$ for $\lambda>0$. Let $\mbox{diam}(E)$, $\bar{E}$, and $\partial E$ denote the diameter, closure, and boundary of $E$ respectively. We write ${\displaystyle\max_{E}\tilde{u},\,\min_{E}\tilde{u}}$ to denote the essential supremum and infimum of $\tilde{u}$ on $E$ whenever $E\subset\mathbb{R}^{n}$ and $\tilde{u}$ is defined on $E$.

Lemma 2.1**.**

Given $p,1<p<\infty,n\geq 2,$ and $\tilde{f}$ as in (1.5), let $\tilde{u}$ be a positive $\tilde{\mathcal{A}}=\nabla\tilde{f}$-harmonic function in $B(w,4r)$ for $r>0$.Then

[TABLE]

Furthermore, there exists $\tilde{\sigma}=\tilde{\sigma}(p,n,\tilde{a}_{1},\tilde{a}_{2})\in(0,1)$ such that if $x,y\in B(w,r)$, then

[TABLE]

Proof.

A proof of this lemma can be found in [Ser64]. ∎

Lemma 2.2**.**

Let $p,n,\tilde{f},\tilde{\mathcal{A}},\tilde{u},w,r,$ be as in Lemma 2.1. Then $\tilde{u}$ has a representative locally in $W^{1,p}(B(w,4r)),$ with Hölder continuous partial derivatives in $B(w,4r)$ (also denoted $\tilde{u}$), and there exist $\tilde{\beta}\in(0,1]$ and $c\geq 1$, depending only on $p,n,\tilde{a}_{1},\tilde{a}_{2},$ such that if $x,y\in B(w,r),$ then

[TABLE]

Proof.

A proof of Lemma 2.2 can be found in [Tol84]. ∎

Definition 2.3**.**

Fix $p,1<p<\infty$ and let $\tilde{f}$ be as in (1.5) with $f=\tilde{f}.$ If $\tilde{K}$ is a compact subset of the connected open set $D,$ define the $\tilde{\mathcal{A}}=\nabla\tilde{f}$-capacity of $\tilde{K}$ relative to $D$ by

[TABLE]

In case $\tilde{f}(\eta)=p^{-1}|\eta|^{p}$ for $\eta\in\mathbb{R}^{n},$ we write $\mbox{Cap}_{p}(\tilde{K},D)$ instead of $\mbox{Cap}_{\mathcal{\tilde{A}}}(\tilde{K},D).$ If $D=\mathbb{R}^{n}$ we also write $\mbox{Cap}_{\mathcal{\tilde{A}}}(\tilde{K})$ and $\mbox{Cap}_{p}(\tilde{K})$ for short. We note from (1.5) that

[TABLE]

for $\tau>0$ and $x_{0}\in\mathbb{R}^{n}.$ Ratio constants depend only on the data. If $n\leq p<\infty$ then $\mbox{Cap}_{\mathcal{\tilde{A}}}(\tilde{K})\equiv 0$ (see [HKM06, Chapter 2]).

Definition 2.4**.**

Let $p,\tilde{f},\tilde{A},$ be as in Definition 2.3. A closed set $\tilde{K}\subset\mathbb{R}^{n}$ is called uniformly $(r_{0},p)$-fat if there exists $\hat{c}\geq 1$ such that

[TABLE]

for all $0<r\leq r_{0}$ and $w\in\tilde{K}.$ The largest such $\hat{c}^{-1}$ is called the uniform $(r_{0},p)$-fatness constant of $\tilde{K}$.

Lemma 2.5**.**

Let $p,\tilde{f},\mathcal{\tilde{A}},$ be as in Definition 2.4 and suppose that $\tilde{K}$ is a uniformly $(r_{0},p)$-fat compact set with $\tilde{K}\cap B(z,3\rho)\not=\emptyset,$ where $r_{0}=\mbox{diam}(\tilde{K})$. Let $\zeta\in C_{0}^{\infty}(B(z,4\rho))$ with $\zeta\equiv 1$ on $B(z,3\rho).$ If $0\leq\tilde{u}$ is $\tilde{\mathcal{A}}$-harmonic in $B(z,4\rho)\setminus\tilde{K},$ and $\tilde{u}\zeta\in W_{0}^{1,p}(B(z,4\rho)\setminus\tilde{K}),$ then $\tilde{u}$ has a continuous extension to $B(z,3\rho)$ obtained by putting $\tilde{u}\equiv 0$ on $\tilde{K}\cap B(z,3\rho)$. Moreover, if $0<r<\min\{r_{0},\rho\}$ and $w\in\tilde{K}\cap B(z,2\rho),$ then

[TABLE]

where $c_{1}$ depends only on $p,n,\tilde{a}_{1},\tilde{a}_{2},$ and the uniform $(r_{0},p)$-fatness constant for $\tilde{K}$. Furthermore, there exist $\hat{\sigma}\in(0,1)$ and $c_{2}\geq 1$, having the same dependence as $c_{1}$, such that

[TABLE]

whenever $x,y\in B(w,r/2)$ and $0<r<\min\{r_{0},\rho\}$.

Proof.

Here $(i)$ in (2.4) is a standard Caccioppoli inequality and $(ii)$ for $y\in\tilde{K}$ follows from uniform $(r_{0},p)$-fatness of $\tilde{K}$ and essentially Theorem 6.18 in [HKM06]. Combining this fact with (2.1) $(iii)$ we obtain $(ii).$ ∎

Lemma 2.6**.**

Let ${\mathcal{\tilde{A}}},p,\tilde{f},\tilde{K},r_{0},z,\rho,\tilde{u}$ be as in Lemma 2.5. Then there exists a unique finite positive Borel measure $\tilde{\nu}$ with support contained in $\tilde{K}\cap B(z,3\rho)$ such that

[TABLE]

Moreover, there exists $\bar{c}\geq 1,$ with the same dependence as $c_{1}$ in Lemma 2.5, for which

[TABLE]

whenever $0<r<\min\{r_{0},\rho\}$ and $w\in\tilde{K}\cap B(z,\rho)$. Furthermore, suppose for some constant $\Lambda\geq 1$ that if $w\in\tilde{K}\cap B(z,\rho),$ and $0<s<r,$ there exists $a_{s}(w)\in B(w,r)\setminus\tilde{K}$ with

[TABLE]

Suppose also that whenever $w_{1},w_{2}\in B(z,2r)\setminus\tilde{K}$ and $0<r\leq\rho/\Lambda,$ there exists a rectifiable curve \tau:[0,1]\mbox{\rightarrow}B(z,2\rho)\setminus\tilde{K} with $\tau(0)=w_{1}$ and $\tau(1)=w_{2},$ and such that

[TABLE]

If $w\in B(z,r/2)\cap\tilde{K}$ then

[TABLE]

Ratio constants depend only on the data, the uniform $(r_{0},p)$-fatness constant for $\tilde{K},$ and $\Lambda.$

Proof.

For the proof of (2.5), see [HKM06, Theorem 21.2] The left-hand inequality in (2.6) follows from (2.5), (1.5), and Hölder’s inequality, using a test function, $\phi,$ with $\phi\equiv 1$ on $\bar{B}(w,r/2).$ The proof of the right-hand inequality in (2.6) follows from [KZ03] (see also [EL91]). Here (2.7) is equivalent to a Harnack chain condition used in the definition of an non-tangentially accessible domain (see [JK82]). The proof of the middle inequality in (2.8) follows from an argument often attributed to Carleson (see [AS05]) and just uses (2.4) $(ii),$ (2.1) $(ii),$ and (2.6). The first and last inequalities in (2.8) give the “doubling property” of $\nu$ measure. ∎

Remark 2.7**.**

Uniform $(r_{0},p)$-fatness of $\mathbb{R}^{n}\setminus D$ for some $r_{0}>0$ is a sufficient condition for solvability of the Dirichlet problem for $\mathcal{\tilde{A}}$-harmonic PDEs in a bounded domain $D$ in the sense that if $\phi$ is a continuous function on $\partial D,$ then there exists an $\mathcal{\tilde{A}}$-harmonic function $\Phi$ in $D$ with continuous boundary values equal to $\phi$ on $\partial D.$ In fact, if $\mathbb{R}^{n}\setminus D$ is uniformly $(r_{0},p)$-fat then for every $w\in\mathbb{R}^{n}\setminus D$ and $0<r<r_{0}$

[TABLE]

That is, uniform $(r_{0},p)$-fatness implies Wiener regularity (see [HKM06, Theorem 6.33]). We also remark that if $E\subset B(0,\rho)$ is a closed convex set with $\mbox{diam}(E)=1$ and $\mathcal{H}^{k}(E)>0$ for some positive integer $k>n-p$ then $E$ is $(1,p)$-uniformly fat and $\mbox{Cap}_{\mathcal{\tilde{A}}}(E,B(0,2\rho))\approx 1$ with ratio constants depending only on the data when $1<p<n$ while for $p\geq n$ these constants depend on the data and also $\rho.$ On the other hand, if $\mathcal{H}^{k}(E)<\infty$ for some positive integer $k\leq n-p$ then $\mbox{Cap}_{\mathcal{\tilde{A}}}(E)=0$ (see [HKM06, Chapter 2]).

2.1. Definition of $\mathcal{A}$-capacitary and $\mathcal{A}$-harmonic Green’s functions

Definition 2.8**.**

Let $1<p<n$ and $f$ be as in (1.5) and let $E$ be a compact convex set with $\mbox{Cap}_{\mathcal{A}}(E)>0$. Then the $\mathcal{A}$-capacitary function of $E$, say $\tilde{U},$ is the unique continuous function $\tilde{U}\not\equiv 1,0<\tilde{U}\leq 1,$ on $\mathbb{R}^{n}$ satisfying

[TABLE]

For existence and uniqueness of $\tilde{U}$ see Lemma 4.1 in [AGH*+*17]. We note that if $\tilde{\nu}$ denotes the measure associated with $\tilde{U}$ as in Lemma 2.6 then $\tilde{\nu}(E)=\mbox{Cap}_{\mathcal{A}}(E)$ (see [AGH*+*17, Lemma 4.2]). Therefore, if $E\subset B(0,1)$ with $\mbox{diam}(E)\geq 1/2$ and $n-1<p<n$ then from (2.8) and Remark 2.7 we have

[TABLE]

where $c$ depends only on the data.

In order to define an $\mathcal{A}$-harmonic Green’s function with pole at $\infty$ when $p\geq n,$ we first have to define a fundamental solution, say $F,$ with pole at [math] in $\mathbb{R}^{n}$ when $p\geq n.$ Definitions for $p=n$ and $n<p<\infty$ are different and we start with $p=n$.

Definition 2.9**.**

If $p=n$ we say that $F$ is a fundamental solution to $\nabla\cdot\mathcal{A}(\nabla F)=0$ in $\mathbb{R}^{n}$ with pole at [math] if

[TABLE]

If $p>n$ we say that $F$ is a fundamental solution to $\nabla\cdot\mathcal{A}(\nabla F)=0$ in $\mathbb{R}^{n}$ with pole at [math] if

[TABLE]

Existence and uniqueness of $F$ in (2.11) and (2.12) are proved in Lemma 4.4 and Lemma 4.6 of [ALSV18], respectively.

Definition 2.10**.**

Let $p\geq n$ and for a given compact, convex set $E\subset\mathbb{R}^{n}$ with $0\in E$ we say that $U$ is the $\mathcal{A}$-harmonic Green’s function for $\mathbb{R}^{n}\setminus E$ with pole at $\infty,$ if $U:\mathbb{R}^{n}\setminus E\to(0,\infty)$ has continuous boundary value [math] on $\partial E$, $U$ is $\mathcal{A}$-harmonic in $\mathbb{R}^{n}\setminus E$, and $U(x)=F(x)+k(x)$ where $k(x)$ is a bounded function in a neighbourhood of $\infty$ and $F$ is the fundamental solution as in Definition 2.9.

Remark 2.11**.**

In [ALSV18] the authors show that $U$ exists and is unique if and only if the convex compact set $E$ is either $(a)$ non-empty when $p>n$ or $(b)$ contains at least two points when $p=n.$ If $U$ exists then it was also shown that $k\leq 0$ in $\mathbb{R}^{n}\setminus E$ and $k$ is Hölder continuous in a neighbourhood of $\infty$ with ${\displaystyle\lim_{x\to\infty}}k(x)=k(\infty).$ They then define

[TABLE]

If $E$ is a single point and $p=n$ (so $U$ does not exist), set $\mathcal{C}_{\mathcal{A}}(E):=0.$ Here $\gamma$ is a constant depending only on the data which occurs in the asymptotic expansion of $F(x)$ as $x\to\infty.$ From the definition of $\mathcal{C}_{\mathcal{A}}(E)$ and translation, dilation invariance of $\mathcal{A}$-harmonic functions it follows as in (2.3) that if $x_{0}\in\mathbb{R}^{n}$, $r>0$, and $E$ is a convex compact set then

[TABLE]

Also if $\nu$ is the measure associated with $U$ as in Lemma 2.6 then (see Lemmas 5.2, 5.3 in [ALSV18]), $\nu(E)=1$. Hence if $E\subset B(0,1)$ with $1/2\leq\mbox{diam}(E)$ it follows from (2.8) that

[TABLE]

Finally, if $E_{1}\subset E_{2}$ are compact convex sets and $U_{1}$ and $U_{2}$ the corresponding $\mathcal{A}$-harmonic Green’s functions with pole at $\infty$ then

[TABLE]

2.2. Existence in Theorem A

To show existence and uniqueness for $u_{1}$ and $u_{2}$ in Theorem A we shall also need the following lemma.

Lemma 2.12**.**

Fix $p$ with $1<p<\infty$ and $\alpha\in(0,\pi)$ and suppose $0<r\leq R/10$. Let $v$ be the $\mathcal{A}$-harmonic function in $D=[K(\alpha)\setminus\bar{B}(re_{1},\frac{r\alpha}{100})]\cap B(0,R)$ with continuous boundary values $v\equiv 1$ on $\partial B(re_{1},\frac{r\alpha}{100})$ and $v\equiv 0$ on $[\partial B(0,R)\cap K(\alpha)]\cup[\partial K(\alpha)\cap B(0,R)].$ Then there exists $c\geq 1$ such that

[TABLE]

Here $c$ depends on the data and $\alpha$ if $1<p\leq n-1,$ while $c$ depends only on the data if $p>n-1$.

Proof.

Let $\hat{D}=\{y:y+re_{1}\in D\}$ and define $\hat{v}$ on $\hat{D}$ by $\hat{v}(y)=v(y+re_{1})$ for $y\in\hat{D}$ (see Figure 1). Given $\lambda$ with $1<\lambda<1001/1000$, set $\hat{D}(\lambda)=\{y\in\hat{D}:\lambda y\in\hat{D}\}.$

From the definition of $D,\hat{D},v,\hat{v},$ and translation and dilation invariance of $\mathcal{A}$-harmonic functions we see that $y\mapsto\hat{v}(y)$ and $y\mapsto\hat{v}(\lambda y)$ are both $\mathcal{A}$-harmonic in $\hat{D}(\lambda)$. If

[TABLE]

we claim that

[TABLE]

where $\breve{c}\geq 1$ has the same dependence as $c$ in Lemma 2.12. Using the boundary maximum principle for $\mathcal{A}$-harmonic functions and continuity of $h$ and $\hat{v}$ we see that it suffices to prove (2.17) when $y\in\partial\hat{D}(\lambda).$ To do this we note from the definition of $\hat{D}(\lambda)$ that if $y\in\partial\hat{D}(\lambda),$ then either $y=z/\lambda$ for some $z\in\partial\hat{D}$ with $\hat{v}(z)=0$ or $y\in\partial\hat{D}$ and $\hat{v}(y)=1.$ In the first case we see that $\hat{v}(\lambda y)=0$ so (2.17) is trivially true. In the second case let $\tilde{f}(\eta)=f(-\eta),$ and note that $1-\hat{v}$ is $\nabla\tilde{f}=\mathcal{\tilde{A}}$-harmonic in $\hat{D}.$ Using this note, uniform fatness of $K(\alpha)\cap B(0,R),$ the definition of $\hat{D},$ (2.4) $(ii)$ for $\hat{v},$ and Harnack’s inequality we deduce that if $r^{\prime}=\frac{2\alpha r}{100},$ then $1-\hat{v}\geq c_{*}^{-1}$ on $\partial B(0,r^{\prime})$ for some $c_{*}\geq 1$ with the same dependence as $c$ in the statement of Lemma 2.12. Thus $c_{*}(1-\hat{v})\geq 1$ on $\partial B(0,r^{\prime})$ and $(1-\hat{v})\equiv 0$ on $\partial B(0,r^{\prime}/2).$ Also this function is $\nabla\tilde{f}=\mathcal{\tilde{A}}$-harmonic in $T=B(0,r^{\prime})\setminus\bar{B}(0,r^{\prime}/2).$

Using these facts and a barrier type argument as in [AGH*+*17, section 7] or [ALSV18, (4.6)-(4.9)], it follows (since $|y|=r^{\prime}/2$) that

[TABLE]

where $\bar{c}\geq 1$ depends only on the data. From (2.18) we conclude that (2.17) also holds in the second case when $y\in\partial\hat{D}(\lambda).$ Thus (2.17) holds on $\partial\hat{D}(\lambda)$ so by the above maximum principle is valid in $\hat{D}(\lambda).$ Letting $\lambda\to 1$ in (2.17) and using (2.2) $(\hat{a}),$ as well as the chain rule, we get

[TABLE]

for $y\in\hat{D}.$ Clearly this inequality implies (2.16). ∎

To begin the proof of existence in Theorem A for $u_{1},$ let $v$ and $D$ be as in Lemma 2.12 and put $R=l,r=l/10$, for sufficiently large positive integer $l$ (say $l\geq 100$). Set $v_{l}=M_{l}\,v$ where $M_{l}>0$ is chosen so that $v_{l}(e_{1})=1.$ Extend $v_{l}$ to a continuous function in $\bar{B}(0,l)$ by defining $v_{l}\equiv 0$ on $[\bar{B}(0,l)\setminus K(\alpha)]\cup\partial B(0,l)$ while $v_{l}\equiv M_{l}$ on $\bar{B}(\frac{le_{1}}{10},\frac{l\alpha}{1000}).$ Using Lemmas 2.1, 2.2, 2.5 and letting l\mbox{\rightarrow}\infty it follows from Ascoli’s theorem that a subsequence of $(v_{l}),$ also denoted $(v_{l})$, converges uniformly to $u_{1},$ an $\mathcal{A}$-harmonic function in $K(\alpha)$ that is also Hölder continuous in $\mathbb{R}^{n}$ with $u_{1}\equiv 0$ on $\mathbb{R}^{n}\setminus K(\alpha).$

To construct $u_{2}$, we let $r=1/l$, $R=l$, and let $v_{l}=\hat{M}_{l}\,v$ for $l=2,3,\ldots,$ where $\hat{M}_{l}$ is chosen so that $v_{l}(e_{1})=1.$ Extend $v_{l}$ to a continuous function on $\bar{B}(0,l)$ by putting $v_{l}\equiv 0$ on $[B(0,l)\setminus K(\alpha)]\cup\partial B(0,l)$ and $v_{l}\equiv\hat{M}_{l}$ on $\bar{B}(e_{1}/l,\frac{\alpha}{100l}).$ Also from Lemmas 2.1, 2.2, 2.5 and (2.8) we deduce for $l>\rho>2/l,$ that there exists $c\geq 1$ and $\breve{\beta}\in(0,1)$ such that

[TABLE]

Here $c$ and $\breve{\beta}$ depend on the data and $\alpha$ if $1\leq p\leq n-1$ while these constants depend only on the data if $p>n-1$. Letting $l\to\infty,$ it follows from the above lemmas, and Ascoli’s theorem that a subsequence of $(v_{l}),$ also denoted $(v_{l}),$ converges uniformly to $u_{2},$ an $\mathcal{A}$-harmonic function in $K(\alpha)$ that is locally Hölder continuous in $\mathbb{R}^{n}\setminus\{0\}$ with $u_{2}\equiv 0$ on $\mathbb{R}^{n}\setminus(K(\alpha)\cup\{0\}).$ Moreover, (2.19) holds with $v_{l}$ replaced by $u_{2}$ and from (2.16) we have

[TABLE]

3. Boundary Harnack inequalities and uniqueness in Theorem A

To prove that $u_{1}$ and $u_{2}$ are unique and satisfy (1.1) in Theorem A we use a variety of boundary Harnack inequalities, mostly in Lipschitz domains. To set the stage for these inequalities, let $K\subset\mathbb{R}^{n-1},n\geq 2,$ be a non-empty compact set and recall that $\phi:K\to\mathbb{R}$ is said to be Lipschitz on $K$ provided there exists $\hat{b},0<\hat{b}<\infty,$ such that

[TABLE]

The infimum of all $\hat{b}$ such that (3.1) holds is called the Lipschitz norm of $\phi$ on $K,$ denoted by $\|\phi\hat{\|}_{K}$. It is well-known that if $K\subset\mathbb{R}^{n-1}$ is compact, then $\phi$ has an extension to $\mathbb{R}^{n-1}$ (also denoted by $\phi$) which is differentiable almost everywhere in $\mathbb{R}^{n-1}$ and

[TABLE]

In fact, one can take $c=1$ (see [Fed69, Section 2.10.43]). Now suppose that $D$ is an open set, $w\in\partial D,\hat{r}>0,$ and

[TABLE]

in an appropriate coordinate system for some Lipschitz function $\phi$ on $\mathbb{R}^{n-1}$ with $\phi(w^{\prime})=w_{n}.$ Note from elementary geometry that if $\zeta\in\partial D\cap B(w,2\hat{r})$ and $0<s<\hat{r}$, we can find points

[TABLE]

for a constant $c$ depending on $\|\phi\hat{\|}$. In the following, we let $a_{s}(\zeta)$ denote one such point. Also let $\Delta(w,r)=\partial D\cap B(w,r),r>0,$ and if $\zeta\in\Delta(w,2\hat{r})$ and $t>1$ let

[TABLE]

Unless otherwise stated we always assume that $t$ is fixed and so large that $\Gamma(\zeta)$ contains the inside of a truncated cone with vertex at $\zeta,$ height $\hat{r},$ axis along the positive $e_{n}$ axis, and of angle opening $\theta=\theta(t)>0.$ We note for $D,\hat{r}$, and $w$ as above that $\mathbb{R}^{n}\setminus(D\cap B(w,\hat{r}))$ is uniformly $(\hat{r},p)$-fat for $1<p<\infty.$ Thus, if $v$ satisfies the same hypotheses as $\tilde{u}$ in Lemmas 2.5 and 2.6, then these Lemmas are valid with $\tilde{u}$ replaced by $v$ in the above $D.$ It follows that (see [ALSV18, Section 8] and [AGH*+*17, section 10] there exists $\bar{c}\geq 1,$ depending only on the data and $\|\phi\hat{\|},$ such that if $0<r\leq\hat{r}$ and $\bar{r}=r/\bar{c},$ then

[TABLE]

where $w$ is as in (3.2) and $a_{\bar{r}}(w)$ is a point associated to $w$ as in the display below (3.2). Moreover, there exists $\hat{\sigma}\in(0,1),$ depending only on the data and $\|\phi\hat{\|}$, such that

[TABLE]

Finally, there exists a unique finite positive Borel measure $\nu$ on $\mathbb{R}^{n}$, with support contained in $\bar{\Delta}(w,r)$, such that

[TABLE]

Also in [AGH*+*17, section 10] for $1<p<n$ and in [ALSV18, section 8] for $p\geq n$ we updated to $\mathcal{A}$-harmonic functions the following Lemmas proved in [LN07], [LN10], for $p$-harmonic functions when $1<p<\infty.$

Lemma 3.1**.**

Let $D,\hat{r},w,\phi$ be as in (3.2), $p$ fixed, $1<p<\infty,$ and $0<r\leq\hat{r}.$ Also let $v$ be a positive $\nabla f=\mathcal{A}$-harmonic in $D\cap B(w,r)$ and continuous in $B(w,r)$ with $v\equiv 0$ on $B(w,r)\setminus D.$ There exists $c_{\star}\geq 1,$ depending only on the data and $\|\phi\hat{\|}$, such that if $4\tilde{r}=r/c_{\star}$ and $x\in B(w,\tilde{r})\cap D,$ then

[TABLE]

Moreover, $\Delta(w,\tilde{r})$ has a tangent plane for $\mathcal{H}^{n-1}$-almost every $y\in\Delta(w,\tilde{r})$. If $\mathbf{n}(y)$ denotes the unit normal to this tangent plane pointing into $D\cap B(w,2\tilde{r}),$ then

[TABLE]

and

[TABLE]

Finally, there exists $q>p/(p-1)$ and $c_{\star\star}$ with the same dependence as $c_{\star}$ such that

[TABLE]

To prove uniqueness for $u_{1}$ in Theorem A we need the following boundary Harnack inequality.

Lemma 3.2**.**

Let $D,\hat{r},w,\phi,p,$ be as in Lemma 3.1 and $0<r\leq\hat{r}.$ Also let $v_{i}$, for $i=1,2$ be positive $\nabla f=\mathcal{A}$-harmonic functions in $D\cap B(w,r)$ and continuous in $B(w,r)$ with $v_{1}\equiv v_{2}\equiv 0$ on $B(w,r)\setminus D.$ Then there exist $\beta_{+}\in(0,1)$ and $c_{+}\geq 1$, depending only on the data and $\|\phi\hat{\|}$, such that if $r^{+}=r/c^{+}$ then

[TABLE]

whenever $x,y\in D\cap B(w,r^{+})$.

3.1. Uniqueness in Theorem A for $0<\alpha<\pi$

To prove uniqueness for $u_{1}$ when $p,1<p<\infty,$ and $\alpha\in(0,\pi)$ are fixed, suppose $\hat{u}>0$ in $K(\alpha)$ and is also $\mathcal{A}$-harmonic as well as continuous in $\mathbb{R}^{n}$ with $\hat{u}\equiv 0$ on $\mathbb{R}^{n}\setminus K(\alpha)$ and $\hat{u}(e_{1})=1.$ Using Lemma 3.2 with $D=B(0,R)\cap K(\alpha),v_{1}=u_{1},v_{2}=\hat{u},$ and $w=0,\hat{r}=R/2,$ we find that

[TABLE]

in $B(0,\frac{R}{2c_{+}})$ for some $c_{+}$ and $\beta_{+}$ depending only on the data and the Lipschitz constant of $\partial K(\alpha)$. Fixing $x,y,$ and letting $R\to\infty$ it follows that $u_{1}=\hat{u}.$ To show that $u_{1}$ has the form (1.2) observe that for fixed $t>0,$ the function $x\mapsto u_{1}(tx)$ for $x\in K(\alpha)$ is positive, $\mathcal{A}$-harmonic, and has boundary value 0 on $\partial K(\alpha),$ so by uniqueness of $u_{1},$ we have

[TABLE]

Differentiating (3.12) with respect to $t$ (permissible by Lemma 2.2) and evaluating at $t=1$ we see that

[TABLE]

If we put $\rho=|x|,x/|x|=\omega\in\mathbb{S}^{n-1},$ in this identity we obtain that

[TABLE]

Dividing this equality by $\rho u_{1}(\rho\omega),$ integrating with respect to $\rho,$ and exponentiating, we find that $u_{1}(r\omega)=r^{\lambda_{1}}u_{1}(\omega)$ whenever $\omega\in\mathbb{S}^{n-1}$ where $\lambda_{1}=\langle e_{1},\nabla u_{1}(e_{1})\rangle$.

To prove uniqueness for $u_{2}$ in $K(\alpha)$ with $p$ and $\alpha$ fixed with $0<\alpha<\pi,1<p<\infty,$ we let $0<\hat{u}$ be $\mathcal{A}$-harmonic in $K(\alpha)$ with continuous boundary value 0 on $\partial K(\alpha)\setminus\{0\},\hat{u}(e_{1})=1,$ and

[TABLE]

From Lemma 3.2 we see that if $w\in\partial K(\alpha)\setminus\{0\}$, $r=|w|/4$, $v_{1}\not=v_{2}$ with $v_{2}=\hat{u}$ or $v_{2}=u_{2}$, then (3.10) in Lemma 3.2 is valid for both $\hat{u}/u_{2}$ and $u_{2}/\hat{u}$. Now (3.10) for $\hat{u},u_{2}$, (3.13), (2.19) for $u_{2}$, Harnack’s inequality and the maximum principle for $\mathcal{A}$-harmonic functions yield that

[TABLE]

where $c\geq 1$ depends only on the data. Indeed, if for example

[TABLE]

then the above program first gives $u_{2}(x)/\hat{u}(x)\to 0$ as $x\to 0$ in $K(\alpha)$ and second that $u_{2}\equiv 0,$ clearly a contradiction.

Now (3.14), (3.6) $(a)$ for $\hat{u}$ and $u_{2}$ when $w\in\partial K(\alpha)\setminus\{0\}$ and $r=|w|/4$, and (2.20) imply that there exist $c_{*}\geq 1$ and $\hat{\beta}\in(0,1)$, depending only on the data and $\alpha$, such that

[TABLE]

whenever $0<\rho<1/c_{*}$ and $u^{\prime}\not=u^{\prime\prime}\in\{\hat{u},u_{2}\}.$ Fixing $x,y,$ and letting $\rho\to 0$ we conclude that $\hat{u}=u_{2}$. The proof of (3.15) is quite similar to the proof of (3.10) (given the above assumptions) only arguments are made in $\mathbb{R}^{n}\setminus B(0,\rho)$ rather than $B(0,r).$ For the proof of a somewhat stronger inequality than (3.15) when $\hat{u}$ and $u_{2}$ are $p$-harmonic functions, see the proof of Theorem 3 and Corollary 5.25 in [LN10]. The proof of (3.15) when $\hat{u}$ and $u_{2}$ are $\mathcal{A}$-harmonic is essentially unchanged, so we omit the details. Homogeneity of $u_{2},$ i.e., (1.1), assuming uniqueness, is proved in the same way as for $u_{1}$ when $\alpha\in(0,\pi).$

3.2. Existence and uniqueness in Theorem A for $\alpha=\pi$

It remains to show existence and uniqueness in Theorem A when $\alpha=\pi$ and $p>n-1$. To do this, for $i=1,2,$ we temporarily write

[TABLE]

for the functions in Theorem A corresponding to $K(\alpha).$ From the maximum principle for $\mathcal{A}$-harmonic functions it follows that if $0<\alpha_{1}<\alpha_{2}<\pi,$ then $u_{1}(\cdot,\alpha_{1})\leq\bar{c}\,u_{1}(\cdot,\alpha_{2})$ in $K(\alpha_{1})\cap B(0,1)$ so necessarily

[TABLE]

Also strict inequality must hold since otherwise from (1.1) it would follow that $u_{1}(\cdot,\alpha_{1})/u_{1}(\cdot,\alpha_{2})$ has an absolute maximum in $K(\alpha_{1})$ which again leads to a contradiction by way of the maximum principle for $\mathcal{A}$-harmonic functions. Similarly , if $0<\alpha_{1}<\alpha_{2}<\pi,$ then $u_{2}(\cdot,\alpha_{1})\leq\bar{c}\,u_{2}(\cdot,\alpha_{2})$ in $K(\alpha_{1})\setminus B(0,1)$ and $\lambda_{2}(\alpha)<0$ for $\alpha\in(0,\pi),$ thanks to (2.19) for $u_{2}(\cdot,\alpha).$ Thus

[TABLE]

Moreover, strict inequality holds in this equation since otherwise we could get a contradiction by the same argument as above. We conclude from our considerations for $i=1,2,$ that

[TABLE]

For $i=1,2$, let

[TABLE]

We note that if $\alpha\in(0,\pi]$ and $n-1<p<\infty$ then Lemmas 2.1, 2.2, 2.5, and (2.8) are valid for $u_{1}$ in $K(\alpha)\cap B(0,\rho)$ with constants depending only on the data as follows from uniform $(\rho,p)$-fatness of $(\mathbb{R}^{n}\setminus K(\alpha))\cap B(0,\rho)$ when $n-1<p<\infty.$ Using these facts and Ascoli’s theorem we find that as $m\to\infty$, a subsequence of $\{u_{1}(\cdot,\pi-1/m)\},$ converges uniformly on compact subsets of $\mathbb{R}^{n}$ to $u_{1}(\cdot,\pi),$ a Hölder continuous function on $\mathbb{R}^{n}$ which is $\mathcal{A}$-harmonic in $K(\pi)$ with $u_{1}\equiv 0$ on $\partial K(\pi).$ Similarly, Lemmas 2.1, 2.2, 2.5, (2.8), (2.20), and (2.19) (with $v_{l}$ replaced by $u_{2}$) are valid for $u_{2}(\cdot,\alpha)$ in $K(\alpha)\cap B(w,\rho)$ whenever $w\in\partial K(\alpha)\setminus\{0\}$ and $\rho<|w|/4.$ All constants depend only on the data for $n-1<p<\infty.$ Using these facts as above, we obtain $u_{2}(\cdot,\pi),$ a uniform limit on compact subsets of $\mathbb{R}^{n}\setminus\{0\},$ of a subsequence of $(u_{2}(\cdot,\pi-1/m))$ as $m\to\infty.$ Also $u_{2}(\cdot,\pi)$ is $\mathcal{A}$-harmonic in $K(\alpha)\setminus\{0\}$ and locally Hölder continuous on $\mathbb{R}^{n}\setminus\{0\}$ . Moreover, (2.19), (2.20) hold with $v_{l},u_{2},$ replaced by $u_{2}(\cdot,\pi).$ From (1.1) for $\alpha\in(0,\pi)$ and (3.16) we deduce for $i=1,2,$ that

[TABLE]

To prove uniqueness of $u_{i}(\cdot,\pi)$ for $i=1,2,$ we need several Lemmas analogous to Lemmas 3.1 and 3.2 for Lipschitz domains.

Lemma 3.3**.**

Fix $p$ with $n-1<p<\infty$, $n>2$, $t>0$, and let $\breve{I}$ be the line segment with endpoints $-3te_{1}/2$ and $-te_{1}/2$. Let $0<v$ be $\mathcal{A}=\nabla f$-harmonic in $B(-te_{1},t/2)\setminus\breve{I}$ with continuous boundary value 0 on $\breve{I}.$ Then there exists $c\geq 4,$ depending only on the data, such that

[TABLE]

for $x\in B(-te_{1},t/c)\setminus\breve{I}.$

Proof.

See Lemma 7.1 in [LN18]. ∎

Lemma 3.4**.**

Let $p,n,f,t,\breve{I},$ be as in Lemma 3.3. For fixed $\rho,0<\rho<t/2,$ let $0<v_{i},\,i=1,2,$ be $\mathcal{A}=\nabla f$-harmonic in $B(-te_{1},\rho)\setminus\breve{I}.$ There exist $c_{*}\geq 1$ and $\beta_{*}\in(0,1),$ depending only on the data, such that

[TABLE]

whenever $x,y\in B(-te_{1},\rho/c_{*})\setminus\breve{I}.$

Proof.

See Lemma 6.2 in [LN18]. ∎

Proof of uniqueness of $u_{1}(\cdot,\pi)$.

We now prove uniqueness of $u_{1}(\cdot,\pi)$ when $p>n-1$ and $n\geq 2.$ Suppose $0\leq\hat{u}$ is also $\mathcal{A}$-harmonic in $K(\pi)$ with continuous boundary value 0 on $\partial K(\pi)$ and $\hat{u}(e_{1})=1.$ Then from (3.19), Harnack’s inequality, and the maximum principle for $\mathcal{A}$-harmonic functions we deduce for $n\geq 3$ as in (3.14) that

[TABLE]

where $\tilde{c}$ depends only on the data. To prove (3.20) for $n=2$ we note that both components of $B(-te_{1},\rho)\setminus\breve{I}$ are Lipschitz domains so we can use the boundary Harnack inequality for Lipschitz domains (Lemma 3.2) to estimate the ratio of $u_{1}(\cdot,\pi)/\hat{u}$ in $B(-te_{1},t/c)\setminus\breve{I}.$ Doing this and using Harnack’s inequality, the maximum principle for $\mathcal{A}$-harmonic functions, once again, it follows that Lemma 3.4 and (3.20) are also valid when $n=2.$ Next observe from homogeneity of $u_{1},u_{1}\geq 0,$ and Lemmas 2.1, 2.2, that given $0<\delta<\pi$ there exists $c(\delta)\geq 1$, depending only on the data and $\delta$, such that

[TABLE]

for $x\in K(\pi-\delta)$. Using Lemma 3.3 for $u_{1}(\cdot,\pi)$ when $n>2$ and (3.1) $(a)$ on both sides of $\partial K(\pi)$ when $n=2,$ we deduce for fixed $\delta=\delta_{0}$ near enough $\pi$ that (3.21) is valid when $x\in K(\pi)$ for some $c(\delta_{0}),$ depending only on the data. Finally (3.20), (3.21), and Lemmas 3.3, 3.4, can be used for $n\geq 3$ as in [LN18, subsection 4.2, Assumption 1] and for $n=2$ as in [LLN08] to show first that (3.21) with $u_{1}$ replaced by $\hat{u}$ holds when $x\in K(\pi)$ for some $0<\delta=\delta_{1}<\delta_{0}$. Second that there exists, $c_{**}\geq 1,\beta_{**}\in(0,1),$ depending only on the data with

[TABLE]

whenever $\rho>0$ and $x,y$ in $K(\pi)\cap B(0,\rho).$ Letting $\rho\to\infty$ in this inequality it follows that $\hat{u}=u_{1}(\cdot,\pi)$ so $u_{1}$ is unique. ∎

Sketch of Proof of (3.22).

To briefly outline the strategy in the proof of (3.22), assuming (3.21) for $\hat{u},u_{1},$ in $\mathbb{R}^{n}\setminus\partial K(\pi),$ when $n\geq 3,$ suppose $a,b\in(0,\infty).$ Then using Lemmas 2.1, 2.2, and (3.21), one can show that $\chi(x)=(a\,|\nabla\hat{u}(x)|+b\,|\nabla u_{1}(x)|)^{p-2}$ is an $A_{2}$ weight on $\mathbb{R}^{n}$ with $A_{2}$ constant $\leq c$ where $c$ depends only on the data. That is,

[TABLE]

Also $\zeta=a\,u_{1}-b\,\hat{u}$ is a weak solution to the degenerate elliptic divergence form PDE,

[TABLE]

where

[TABLE]

whenever $x\in K(\pi).$ Moreover, for some $c\geq 1$ depending only on the data,

[TABLE]

Using (3.23)-(3.25), one can then use the boundary Harnack inequalities from divergence form linear degenerate elliptic PDE whose degeneracy is given in terms of an $A_{2}$ weight to get (3.22) (see section 4 in [LN18]) . (3.21) for $\hat{u}$ is proven by a perturbation type argument as in (4.42)-(4.45) of [LN18]. ∎

Uniqueness of $u_{2}(\cdot,\pi)$.

Uniqueness of $u_{2}(\cdot,\pi)$ is proved similarly. Indeed suppose $\hat{u}$ is also $\mathcal{A}$-harmonic in $K(\pi)$ with continuous boundary value 0 on $\partial K(\pi)\setminus\{0\},$ and $\lim_{|x|\to\infty}\hat{u}(x)=0.$ Then (3.20) and (3.21) in $K(\pi)$ are valid with $u_{1}(\cdot,\pi)$ replaced by $u_{2}(\cdot,\pi)$ by the same argument as the one we gave for $u_{1}(\cdot,\pi).$ These inequalities can then be used as outlined above to show that for some $\bar{c}^{*}\geq 1$ and $\bar{\beta}^{*}\in(0,1),$ depending only on the data, that

[TABLE]

whenever $|x|,|y|\geq 2\rho.$ Letting $\rho\to 0$ we then get $u_{2}(\cdot,\pi)=\hat{u}.$ This completes the proof of uniqueness for $u_{1}(\cdot,\pi)$ and $u_{2}(\cdot,\pi).$ ∎

4. Proof of (1.7) in Theorem A

To show $\lambda(\pi)=1-(n-1)/p$ for fixed $p>n-1,n\geq 2,$ and $f$ as in (1.5), we let $0<\delta<10^{-100}$ be a small but fixed number. Also $\epsilon>0,$ $0<\epsilon<<\delta^{1000}$ is allowed to vary. Put

[TABLE]

Given $\eta\in\mathbb{R}^{n}\setminus\{0\},$ let $\hat{f}(\eta)=f(-\eta)$ when $n-1<p<n.$ If $n-1<p<n$ let $U=1-\hat{U}$ where $\hat{U}$ is the $\hat{\mathcal{A}}=\nabla\hat{f}$-capacitary function for $E$ as in Theorem 1.3, so $U$ is $\mathcal{A}=\nabla f$-harmonic. If $n\leq p<\infty$ let $U$ be the $\mathcal{A}=\nabla f$-harmonic Green’s function for $E$ as in Theorem 1.2. We also write $u$ and $\lambda$ for $u_{1}(\cdot,\pi-\epsilon)$ and $\lambda_{1}(\pi-\epsilon)$ in Theorem A when there is no chance of confusion. We shall need the following lemma (see Definition 2.8, Remark 2.11 for notation).

Lemma 4.1**.**

We have

[TABLE]

where $\mathbf{n}(x)$ denotes the outer unit normal at $x\in\partial E$ and $c$ depends only on the data.

Proof.

Lemma 4.1 is proved in [ALV19] (see Remark 11.3) for $p\geq n$ and in [AGH*+*17] (see Remark 13.4) for $n-1<p<n$, using the Hadamard variational formula. The integral in these remarks is defined in terms of a measure on $\mathbb{S}^{n-1}$ obtained by way of the Gauss map, so for example as in $(c)$ of Theorem 1.2 for $p\geq n,$ and the support function of a convex set relative to zero rather than $-e_{1}.$ However, using (1.8) $(ii)$ and the definition of a support function it is easily seen that both integrals are equal. ∎

To obtain estimates on $U$ near $\partial E$ we note that in [LN18, Lemma 5.3], it was shown that for fixed $p$ with $n-1<p<\infty$ and $n\geq 3,$ a continuous function $w$ on $\mathbb{R}^{n}$ exists with $w\equiv 0$ on $T$ where

[TABLE]

Also $w$ is $\mathcal{A}$-harmonic in $\mathbb{R}^{n}\setminus T$ and for $x\in\mathbb{R}^{n},$

[TABLE]

Ratio constants depend only on the data. We use (4.1) to show that there exists $\tilde{c}_{1}\geq 1$ depending only on the data with

[TABLE]

To prove (4.2) observe from Lemma 4.1, (2.10) with $\mathcal{A}$ replaced by $\mathcal{\hat{A}}$ for $n-1<p<n,$ and (2.14) when $p\geq n$ that $w\leq c^{\prime}U$ on $\partial B(0,2).$ Using (4.1) and the boundary maximum principle for $\mathcal{A}$-harmonic functions it follows that for some $c^{\prime\prime}\geq 1,$

[TABLE]

where constants depend only on the data. Using (4.1) in (4.3) we see for $\epsilon>0,$ sufficiently small, that (4.2) is valid. Next we show for some $\tilde{c}_{2}\geq 4,$ depending only on the data that

[TABLE]

To prove (4.4) let $v_{1}$ be the $\mathcal{A}$-harmonic function in $B(-\frac{1}{2}e_{1},\frac{1}{4})\setminus T$ with continuous boundary values $v_{1}=u$ on $\partial B(-\frac{1}{2}e_{1},\frac{1}{4})$ and $v_{1}\equiv 0$ on $T\cap B(-\frac{1}{2}e_{1},\frac{1}{4}).$ Comparing boundary values of $u$ and $v_{1}$, we see from the maximum principle for $\mathcal{A}$-harmonic functions that $u\leq v_{1}$ in $B(-\frac{1}{2}e_{1},\frac{1}{4})\setminus T.$ This inequality, (4.1), and Lemma 3.4 with $v_{2}=w,$ give (4.4) since $u(-\frac{1}{2}e_{1}+\frac{1}{4}e_{n})\approx w(-\frac{1}{2}e_{1}+\frac{1}{4}e_{n}).$

Let $S=E\cap\{y:y_{1}\geq-1+4\delta\}$ and let $V$ be the $\mathcal{A}$-harmonic Green’s function for the complement of $S_{1}=E\cap\{x:x_{1}\leq-1+4\delta\}$ (see Figure 2) with a pole at infinity when $p\geq n$ while $V=1-\hat{V}$ where $\hat{V}$ is the $\mathcal{\hat{A}}$-capacitary function for $S_{1}$ if $n-1<p<n.$

We note from (2.15) that $V\geq U$ in $\mathbb{R}^{n}$ when $p\geq n$. Using this note, (3.6) $(b)$, (3.7), and the Hopf boundary maximum principle we deduce for $n<p<\infty$ that

[TABLE]

thanks to Lemma 4.1 with $E$ replaced by $S_{1}$ and (2.13), where $c$ depends only on the data. If $1<p<n$ we see from (2.9) $(b)$ that U(x),V(x)\mbox{\rightarrow}1 as |x|\mbox{\rightarrow}\infty so $U\leq V$ in $\mathbb{R}^{n},$ by the maximum principle for $\mathcal{A}$-harmonic functions. In view of this fact and (2.3) we conclude that (4.5) remains valid when $1<p<n$ if $\delta^{\frac{|p-n|}{p-1}}$ is replaced by $\delta^{|p-n|}$. If $p=n$ it follows from (2.4) $(ii),$ (2.8), for $U$ with $w=-e_{1},$ and (2.14), (2.8), dilation invariance and Harnack’s inequality for $\mathcal{A}$-harmonic functions, as applied to $V,$ that for some $c\geq 1,\hat{\sigma}\in(0,1),$ depending only on the data,

[TABLE]

Then by the boundary maximum principle for $\mathcal{A}$-harmonic functions,

[TABLE]

Using (4.6) and arguing as above it follows for some $c\geq 1$ that

[TABLE]

From (4.5), (4.7), Lemma 4.1, we see for $\delta>0$ sufficiently small that

[TABLE]

where constants depend only on the data.

Finally, we claim for some $c(\delta)\geq 1,$ depending only on the data and $\delta$ that

[TABLE]

Once (4.9) is proved we get Theorem A as follows. Note that $S\subset B(0,1-2\delta)$ and in Lemma 4.1, $\langle x+e_{1},\mathbf{n}(x)\rangle=\sin\epsilon$ when $x\not=0$ and $x\in\partial S\cap\partial E.$ Using this note, Lemma 4.1, (4.8). (4.9), and the Hopf boundary maximum principle we find that for some $\bar{c}(\delta)\geq 1,$ depending only on the data and $\delta,$

[TABLE]

We also note that $\partial E\cap B(-1/2,1/4)$ is Lipschitz on a scale of $\epsilon/100.$ That is, if $z\in\partial E\cap\bar{B}(-1/2,1/4),$ there exists \phi:\mathbb{R}^{n-1}\mbox{\rightarrow}\mathbb{R} satisfying $\|\phi\hat{\|}\leq 100$ such that after a possible rotation of coordinates,

[TABLE]

From (4.11), (3.8), (1.5) $(a)$, (3.9) with $q$ replaced by $p/(p-1)$ (permissible by Hölder’s inequality), (3.5) $(b)$, and Harnack’s inequality all applied to $u$ and $U$ we see that

[TABLE]

where ratio constants depend only on the data. Using this inequality, (4.9), (4.2), (4.4), (4.1), and the Hopf boundary maximum principle once again, we obtain that

[TABLE]

where ratio constants depend on the data and $\delta.$ Integrating (4.13) over $z\in\partial E\cap B(-1/2,1/4),$ and interchanging the order of integration or giving a covering argument, we conclude after some arithmetic that

[TABLE]

Using (4.14), $(\lambda(\pi-\epsilon)-1)p$-homogeneity of $f(\nabla u),$ and $0<\lambda(\pi-\epsilon)<1$ in (4.10) we arrive at

[TABLE]

where for brevity we have written $\lambda$ for $\lambda(\pi-\epsilon).$ Also ratio constants depend only on $\delta$ and the data. Clearly (4.15) implies that

[TABLE]

So if $\alpha=\pi-\epsilon$ and we use the notation in Theorem A it follows from this inequality that there exist $\delta_{0}$ and $\epsilon_{0}$ with $0<\epsilon_{0}<<\delta_{0},$ and a positive constant $k\geq 1$ depending on $\delta_{0}$ and the data such that if $\pi-\epsilon_{0}<\alpha<\pi,$ then

[TABLE]

Thus Theorem A is true once we prove claim (4.9).

Claim (4.9) is easily proved for $n=2$ using (3.10) on both sides of $\partial K(\pi-\epsilon)$ in each of the Lipschitz domains obtained from removing the positive $x_{1}$ axis from $B(0,1-2\delta)\cap\partial K(\pi-\epsilon)$ (see Figure 2) as well as Harnack’ s inequality and $u(e_{1})\approx U(e_{1})\approx 1.$

Thus we assume $n>2.$ In this case we give an argument which was first used in [BL05, Lemma 2.16] and later in [LN18, section 6.1]. To begin note that (4.9) on $\partial B(0,1-2\delta)\cap K(\pi-\epsilon)$ follows from

[TABLE]

for some $c\geq 1,$ depending only on the data, as we see from $0<\epsilon<<\delta<<1,$ $U(e_{1})\approx u(e_{1})\approx 1,$ and Harnack’s inequality for $\mathcal{A}$-harmonic functions. Then (4.9) in $B(0,1-2\delta)\cap K(\pi-\epsilon)$ follows from the boundary maximum principle for $\mathcal{A}$-harmonic functions.

To prove the right-hand inequality in (4.17) let

[TABLE]

when $t\in[\delta/200,\delta/10]$ and suppose that $u/U>\zeta$ at some point in $\partial C_{\delta/200}\setminus E.$ Given $t\in(\delta/200,\delta/100)$ observe from Harnack’s inequality and the maximum principle for $\mathcal{A}$-harmonic functions that either we have $u(y)/U(y)>\zeta$ at some $y$ in $\partial C_{t}\setminus E$ with $y_{1}=\delta\pm t-1$ or the right-hand inequality in (4.17) holds. If $\zeta$ is large enough this observation implies that there exists $I=[\delta/200,\delta/100]$ or $I=[-\delta/100,-\delta/200]$ such that for all $t\in I$ there is $y^{\prime\prime}=y^{\prime\prime}(t)$ with $0<|y^{\prime\prime}|<\delta/1000$ and

[TABLE]

If for example there exists $t^{\prime}\in[-\delta/100,-\delta/200]$ such that for all $y=(\delta+t^{\prime}-1,y^{\prime\prime})$ in $\partial C_{-t^{\prime}}\setminus E$ we have $u(y)/U(y)\leq\zeta,$ then we can apply the above analysis in

[TABLE]

whenever $t\in[\delta/200,\delta/100]$ to conclude the existence of $I=[\delta/200,\delta/100].$

Let $\nu$ and $\tau$ denote the measures associated with $u$ and $U$ restricted to $C_{\delta/10}.$ We observe from (3.5) $(b)$ that $\nu$ and $\tau$ are doubling measures in the sense that if $z\in C_{\delta/100}\cap\partial E$ and $0<s<\delta/200,$ then

[TABLE]

for $\theta\in\{\nu,\tau\}$ and some $c\geq 1$ depending only on the data.

Given $t\in I,$ choose $y^{\prime\prime}(t)$ as in (4.18). If $|y^{\prime\prime}(t)|>4\epsilon$ we put $\rho(t)=|y^{\prime\prime}(t)|.$ Otherwise since as noted earlier $\partial E\cap B(-1,1/4)$ is Lipschitz on a scale of $\epsilon/100,$ we deduce from (3.10) of Lemma 3.2 that there exists $\hat{y}^{\prime\prime}=\hat{y}^{\prime\prime}(t)$ with $|\hat{y}^{\prime\prime}|=4\epsilon$ and

[TABLE]

In this case we put $\rho(t)=4\epsilon.$ Set $\tilde{y}^{\prime\prime}(t)=y^{\prime\prime}(t)$ when $|y^{\prime\prime}(t)|>4\epsilon$ while $\tilde{y}^{\prime\prime}(t)=\hat{y}^{\prime\prime}(t),$ otherwise. Using (4.19) and (3.5) $(b)$ once again it follows that

[TABLE]

Next using a standard covering lemma we see there exists $\{t_{j}\},t_{j}\in I,$ for which (4.20) holds with $t,\tilde{y}^{\prime\prime}(t),\rho(t),$ replaced by $t_{j},\tilde{y}^{\prime\prime}(t_{j}),\rho(t_{j}).$ Also if $\kappa(t_{j})=(\delta+t_{j}-1)e_{1},$ then

[TABLE]

From (4.19), (4.20), (4.21), (3.5), and Harnack’s inequality it follows, for some $c\geq 1,$ depending only on the data, that

[TABLE]

Also from (2.8) and Harnack’s inequality we see that

[TABLE]

where ratio constants depend only on the data. Using these inequalities in (4.22) we find that

[TABLE]

The right-hand inequality in (4.17) follows from this display and Harnack’s inequality for $\mathcal{A}$-harmonic functions with $2\zeta=\max(u/U)$ on $\partial C_{\delta/200}\setminus E.$ Interchanging the roles of $u$ and $U$ in this argument we get the left-hand inequality in (4.17). This completes the proof of claim (4.9) and so also of Theorem A.

5. Proof of Theorem B

We begin this section with a discussion of some familiar concepts from convex geometry which were used in [Jer96, CNS*+*15] to prove analogues of Theorem B. Let $E\subset\mathbb{R}^{n}$ be a compact convex set with non-empty interior. Translating and dilating $E$ if necessary we may assume that $\bar{B}(0,1)\subset E$ is a ball with largest radius contained in $E$ while $\bar{B}(0,\tilde{R}_{0})$ is the ball with smallest radius and center at the origin containing $E$ for some $\tilde{R}_{0}>0$. Then $\tilde{e}=1/\tilde{R}_{0}$ is called the eccentricity of $E.$ From basic geometry one sees that if $w\in\partial E$ there exists $\hat{c}=\hat{c}(n)\geq 1,$ depending only on $n$ such that $\partial E$ can be covered by at most $N=\hat{c}^{2}(\tilde{e})^{1-n}$ balls, $B(w,\tilde{r}),$ with $w\in\partial E,\tilde{r}\geq 1/8,$ and the property that after a possible change of coordinates there exists a real valued convex function $\phi$ on

[TABLE]

which extends to a Lipschitz function on $\mathbb{R}^{n-1}$ with $\|\phi\hat{\|}\leq\hat{c}/\tilde{e}$. Moreover, if we let $w=(w^{\prime},w_{n})$ and $\phi(w^{\prime})=w_{n},$ after a possible change of coordinates, we also have

[TABLE]

Definition 5.1**.**

Let $\psi$ be a real valued convex function on a bounded convex open set $\Omega\subset\mathbb{R}^{n-1}.$ If $x^{\prime}\in\Omega$ we write $\theta=(\theta_{1},\dots,\theta_{n-1})\in\partial\psi(x^{\prime})$ provided $\psi(y^{\prime})\geq\psi(x^{\prime})+\langle\theta,y^{\prime}-x^{\prime}\rangle$ whenever $y^{\prime}\in\Omega.$ If $\tau$ is a finite positive Borel measure on $\Omega$ then $\psi$ is said to be a solution to the Monge-Ampère equation

[TABLE]

in the sense of Alexandrov provided that

[TABLE]

Let $\mathbf{g}(\cdot,E)$ denote the Gauss function for $\partial E,$ suppose (5.1) is valid, and set $\Omega=B^{\prime}(w^{\prime},\tilde{r})$ and $\phi=\psi$. If $x^{\prime}\in\Omega$ then one can define

[TABLE]

We note that the mapping $x^{\prime}\mapsto\frac{\langle x^{\prime},-1\rangle}{(1+|x^{\prime}|^{2})^{1/2}}=\xi$ is one to one from $\mathbb{R}^{n-1}$ onto $\mathbb{S}^{n-1}\cap\{\xi:\,\xi_{n}<0\}.$ Moreover, the inverse of this mapping has Jacobian $|\xi_{n}|^{-n}$ at $\xi$ with $|\xi|=1$ and $\xi_{n}<0$. Using this fact, it follows from (5.2), (5.3), and (5.4) that if $K\subset B(w^{\prime},\tilde{r})$ is a Borel set and $\tilde{K}:=\{(x^{\prime},\phi(x^{\prime})):x^{\prime}\in K\}$ then

[TABLE]

in the sense of Alexandrov. Next suppose for fixed $p$ with $1<p<\infty$ that $U$, $E$, and $\mu$ are as in Theorem 1.2 or $U=1-\tilde{U}$ where $\tilde{U}$ is as in Theorem 1.3. Then for $\mathcal{H}^{n-1}$-almost every $y\in\partial E$ we see from Theorems 1.2 and 1.3 that

[TABLE]

Thus $\mathbf{g}(\cdot,E)$ is well defined by (5.6) on a Borel set $E_{1}\subset E$ with $\mathcal{H}^{n-1}(E\setminus E_{1})=0.$ If also $d\mu=\Theta d\mathcal{H}^{n-1}|_{\mathbb{S}^{n-1}}$ and there exists $\tilde{a}_{3}\geq 1$ such that

[TABLE]

then from $\||\nabla\phi|\|_{\infty}\leq c/\tilde{e}<\infty,$ finiteness and positivity of $\mu,$ as well as the Radon-Nikodym theorem we conclude for $\tau$ and $K$ as in (5.5) that

[TABLE]

Thus to prove regularity of $\partial E,$ we study the Monge-Ampère equation in domains of the form $\Omega=B^{\prime}(w^{\prime},\tilde{r})$ with measure as in (5.8). To outline some of the work of previous authors on the Monge-Ampère equation we need several definitions.

Definition 5.2**.**

Given $\psi$, $\Omega$ as in Definition 5.1 and $x^{\prime}\in\Omega$, $t>0$, $\theta\in\partial\psi(x^{\prime})$, we put

[TABLE]

and call $S=S(x^{\prime},\theta,t)$ a cross section of $\psi.$ Define the reduced distance $\delta(\cdot,S)$ on $S(x^{\prime},\theta,t)$ by

[TABLE]

Note from convexity of $\psi$ that $S(x^{\prime},\theta,t)$ is a convex set. Let $\bar{x}^{\prime}$ denote the centroid of $S$ and for $0<\lambda<1$, set

[TABLE]

For ease of writing, for $y^{\prime}\in\Omega$, we put

[TABLE]

when $x^{\prime}$, $\theta$, $t$ are understood. Then from a theorem of John (see [Fig17, A.3.2]) it follows that there exists a unique ellipsoid, $\mathcal{E}$ of maximum volume with $\mathcal{E}\subset S\subset(n-1)\mathcal{E}.$ Using this fact and basic geometry we deduce the existence of a positive constant $\beta(n)$ and an affine mapping of the form $Tz^{\prime}=A(z^{\prime}-\bar{x}^{\prime})$ for $z^{\prime}\in\mathbb{R}^{n-1}$ where $A$ is an $n-1\times n-1$ nonsingular matrix with $T(\bar{x}^{\prime})=0$ and

[TABLE]

Here $T(S)$ is said to be a normalization of $S$. Note that $\delta(z^{\prime},T(S))=\delta(T^{-1}z^{\prime},S)$ and if $\Psi(z^{\prime})=\psi(T^{-1}z^{\prime})$ for $z^{\prime}\in T(S)$ then $\Psi$ is convex and

[TABLE]

where $A^{t}$ is the transpose of $A.$ Also $\Psi$ is a solution to the Monge-Ampère equation in $T(S)$ with measure $\mathcal{T}$ where

[TABLE]

whenever $K\subset S$ is a Borel set. Finally, let $\tilde{\Psi}(Ty^{\prime})=\tilde{\psi}(y^{\prime})$ for $y^{\prime}\in\Omega.$

Using the above normalizations it was shown in [Jer96, Lemma 7.3] that

Lemma 5.3**.**

Let $\Omega,\psi,\tau,t,x^{\prime},\theta,$ be as in Definitions 5.1, 5.2, and suppose $\bar{S}(x^{\prime},\theta,t)\subset\Omega.$ Then given $0<\epsilon\leq 1,$ there is a positive constant $C(\epsilon,n)$ such that

[TABLE]

whenever $z^{\prime}\in T(S)$.

Proof.

See Lemma 7.3 in [Jer96]. ∎

Our goal is to show for $\psi=\phi$, $\Omega=B^{\prime}(w^{\prime},\tilde{r}),$ and $\Theta,\tilde{a}_{3}$ as in (5.7), (5.8), that there exists $\epsilon_{0}\in(0,1]$ for which

[TABLE]

whenever $\bar{S}=\bar{S}(x^{\prime},\theta,t)\subset\Omega$ where $\epsilon_{0}$ and $\hat{C}$ depend on the data, $\tilde{e},$ and $\tilde{a}_{3}$ in (5.7).

Before proving (5.12) we show as in [Jer96] and [GH00], how (5.12) can be used to prove Theorem B. Indeed, normalizing this problem we deduce first that if $\tilde{\Psi}(Tz^{\prime})=\tilde{\psi}(z^{\prime}),$ then from (5.12) it follows as in Proposition 2.10 of [GH00] that

[TABLE]

Using this inequality in (5.11) of Lemma 5.3 with $\epsilon=\epsilon_{0}$ and $z^{\prime}=T(x^{\prime}),$ $\tilde{\Psi}(T(x^{\prime}))=-t,$ we deduce

[TABLE]

where ratio constants have the same dependence as $\hat{C}$ in (5.12). Using (5.12)-(5.14), it follows that

Lemma 5.4**.**

Let $\psi$ be a real valued convex function on the convex open set $\Omega,$ and continuous on $\bar{\Omega}.$ If $\psi\geq l$ on $\partial\Omega$ where $l$ is an affine function and $\psi(z^{\prime})=l(z^{\prime})$ at some point $z^{\prime}\in\Omega,$ then either $\{y^{\prime}:\psi(y^{\prime})=l(y^{\prime})\}=\{z^{\prime}\}$ or this set has no extremal points in $\Omega.$

Proof.

See Theorem 1 in [Caf91] or Theorem 4.1 in [GH00]. The proof in either paper is by contradiction and uses invariance of (5.11) and (5.12) under affine mappings as well as the following result. Suppose that $\hat{\psi}_{j}$ for $j=1,2,\dots,$ are convex functions and solutions to the Monge-Ampère equation with measures $\hat{\tau}_{j}$ in an open set $\hat{\Omega}$. If $(\hat{\psi}_{j})$ converges uniformly on compact subsets of $\hat{\Omega}$ to $\hat{\psi},$ a solution to the Monge-Ampère equation in $\hat{\Omega}$ with measure $\hat{\tau},$ then $\hat{\tau}_{j}\rightharpoonup\hat{\tau}$ weakly in $\hat{\Omega}$ (see [Gut01, Lemma 1.2.2]). ∎

Applying Lemma 5.4 with $\psi=\phi$ and $\Omega=B^{\prime}(w^{\prime},\tilde{r})$ as in (5.1) we see that $\partial E$ is strictly convex since otherwise it would follow from repeated application of Lemma 5.4 to balls (as in (5.1)), with non-empty intersection, that $\partial E$ contains a line segment of infinite length. From this contradiction we conclude that $\partial E$ is strictly convex. Now given $w=(w^{\prime},\phi(w^{\prime}))\in\partial E,$ with $x^{\prime}\in\bar{B}(w^{\prime},\tilde{r}/4),$ where $\phi,\tilde{r}$ are as in (5.1), we choose $t$ so that for $S(x^{\prime},\theta,t)$ as in (5.9) we have

[TABLE]

Geometrically this means there is a point $z=(z^{\prime},z_{n})\in\partial E$ with $z^{\prime}\in\bar{S}(x^{\prime},\theta,t)\cap\partial B^{\prime}(x^{\prime},\tilde{r}/2)$ which lies at most $t$ distance from the support plane $y_{n}=l_{x^{\prime},\theta}(y^{\prime})$ to $\partial E$ at $(x^{\prime},\phi(x^{\prime})).$ We claim that $t\geq t_{0}>0,$ where $t_{0}$ has the same dependence as $\hat{C}$ in (5.12). Indeed, otherwise using a compactness argument, the above convergence result, and Lemma 5.4 we could obtain a contradiction to the strict convexity of $\partial E.$ Finally, we observe from Lipschitzness of $\phi$ as in (5.1) that there exists $r_{1}\geq r_{0}>0$ where $r_{0}$ has the same dependence as $t_{0}$ with $B^{\prime}(x^{\prime},r_{1})\subset S.$ Next if $\tilde{\phi}$ is as in (5.10) with $\phi=\psi,$ we claim there is a $\sigma>2,$ with the same dependence as $t_{0},r_{0},$ satisfying

[TABLE]

Here $\hat{S}_{\lambda}$ is defined in the same way as $S_{\lambda}$ in (5.10) only with $\bar{x}^{\prime}$ replaced by $x^{\prime}.$ Indeed, this inequality holds for $l=1$ since otherwise we could use Lemma 5.4 and a compactness argument, as above, to contradict the strict convexity of $\partial E.$ Iterating this inequality we obtain (5.15). From (5.15) and arbitrariness of $x^{\prime}\in B^{\prime}(w^{\prime},\tilde{r}/4),$ we get first that

[TABLE]

whenever $y^{\prime}\in B^{\prime}(w^{\prime},\tilde{r}/4),$ where $\hat{C}_{1}\geq 1,\alpha^{\prime}\in(0,1),$ depend on the data, $\tilde{e},$ and $\tilde{a}_{3}$ in (5.7). Also from convexity and uniform Lipschitzness of $\phi$ we deduce the existence of $\hat{C}_{2}\geq 1,$ having the same dependence as $\hat{C}_{1}$ for which

[TABLE]

whenever $x^{\prime},y^{\prime}\in B^{\prime}(w^{\prime},\tilde{r}/4)$. Combining (5.16), (5.17), and using the triangle inequality, we find that

[TABLE]

Using (5.18) and results from [Lie88] we see that $\nabla U$ when $1<p<n$ or $\nabla\tilde{U}$ when $n\leq p<\infty$ has a $C^{1,\beta^{\prime}}$ extension to $\partial E$ for some $\beta^{\prime}\in(0,1)$ having the same dependence as $\alpha^{\prime}.$ Also from [Lie88] or (5.23) (to be proved) we have $\min\{|\nabla U|,|\nabla\tilde{U}|\}>0$ on $\partial E$ where constants depend only on the data and $\tilde{e}.$ In view of this information and (5.2), (5.5), (5.8), we find that if $0<\Theta\in C^{0,\hat{\alpha}}(\mathbb{S}^{n-1}),$ then for some $0<s_{1},s_{2},\alpha_{*},$ having the same dependence as $\alpha^{\prime},$

[TABLE]

From the above remarks, (5.19), and [Caf89, Caf90b, Caf90a], we conclude that $\phi\in C^{2,\hat{\alpha}}(\Omega).$ Further applications of [Caf89, Caf90b, Caf90a] also give the higher order smoothness results in Theorem B.

It remains to prove (5.12) in order to complete the proof of Theorem B. Throughout the proof of this inequality we let $C\geq 1$ be a positive constant which may depend only on the data, $\tilde{e},$ and $\tilde{a}_{3},$ not necessarily the same at each occurrence. Also if $A\approx B,$ proportionality constants may depend on the data, $\tilde{e},$ and $\tilde{a}_{3}.$ Let $\hat{f}(\eta)=f(-\eta)$ when $1<p<n$ and $\hat{f}(\eta)=f(\eta)$ when $n\leq p<\infty$. Also set $\hat{U}=1-\tilde{U}$ when $1<p<n$ and $\hat{U}=U$ when $n\leq p<\infty.$ Then $\hat{U}$ is $\mathcal{\hat{A}}=\nabla\hat{f}$-harmonic in $\mathbb{R}^{n}\setminus E$ with continuous boundary value 0 on $\partial E.$ Observe from the discussion above (5.1), (5.7), and (5.8) that if $K\subset\Omega$ is a compact set then

[TABLE]

Thus we only prove (5.12) for $\chi(\cdot)$. Recall that $\bar{S}=\bar{S}(x^{\prime},\theta,t)\subset\Omega=B^{\prime}(w^{\prime},\tilde{r}).$ We see that

[TABLE]

is the part of $\partial E$ that lies below or on the plane

[TABLE]

and above or on the support plane $\{(y^{\prime},y_{n}):y_{n}-\phi(x^{\prime})-\langle\theta,y^{\prime}-x^{\prime}\rangle=0\}$ to $\partial E$ at $x=(x^{\prime},\phi(x^{\prime})).$ Then $S$ can be viewed as the projection of $F$ onto the plane $y_{n}=0$ by lines parallel to $e_{n}$ or the $y_{n}$ axis. To simplify the geometry in what follows and for use in adapting the work in [Jer96] to our situation we also project $F$ onto $\Sigma_{1}$ by lines parallel to $e_{n}.$ More specifically, given $y=(y^{\prime},\phi(y^{\prime}))\in F,$ let $\pi(y)\in\Sigma_{1}$ be that point with

[TABLE]

Let $\tilde{S}=\pi(F),$ and note that $\tilde{S}$ is convex. Define the reduced distance $\delta(\cdot,\tilde{S})$ as in Definition 5.2 with $S$ and $\mathbb{R}^{n-1},$ replaced by $\tilde{S}$ and $\Sigma_{1}$ respectively. From (5.1) and the discussion above this display we deduce that

[TABLE]

where ratio constants depend only on $n,\tilde{e}.$ Let $\delta(x,F)=\delta(\pi(x),\tilde{S})$ when $x\in F.$ Then from (5.20) and (5.21) we conclude that to prove (5.12) it suffices to show,

[TABLE]

Remark 5.5**.**

We first remark that (5.22) is the exact counterpart of Theorem 6.5 in [Jer96]. We also note that in [Jer96, section 6], the analogue of $F$ is projected onto $\Sigma_{1}$ by rays through the origin. If $P(y)$ denotes this radial projection of $y\in F$ onto $\Sigma_{1},$ then in [Jer96] the reduced distance of $y\in F$ is defined to be equal to $\delta(P(y),\tilde{S}).$ Using the definition of reduced distance and (5.1) it is easily verified as in (5.21) that $\delta(P(y),\tilde{S})\approx\delta(\pi(y),\tilde{S})$ where proportionality constants depend only on $n$ and $\tilde{e}.$ Thus (5.22) implies the corresponding inequality in [Jer96] and vice-versa.

To prove (5.22) we shall require the following lemma.

Lemma 5.6**.**

Let $w$, $E$, and $\tilde{r}$ be as in (5.1). There exists $C\geq 1,$ depending only on the data and $\tilde{e}$, such that if $0<r\leq\tilde{r}/C$ then

[TABLE]

Proof.

Let $H$ be an open half-space with $H\cap E=\emptyset$ and $\partial H$ a support plane for $\partial E$ at $w.$ Let $\xi$ denote a unit normal pointing into $H$ and let $v$ be the $\mathcal{\hat{A}}$-harmonic function in $\mathbb{G}=H\cap B(w,r)\setminus\bar{B}(w+r\xi/2,r/8)$ with continuous boundary values, $v\equiv 0$ on $\partial(H\cap B(w,r))$ while $v\equiv\hat{U}(w+r\xi/2)$ on $\partial B(w+r\xi/2,r/8).$ Comparing boundary values and using Harnack’s inequality for $\mathcal{\hat{A}}$-harmonic functions we see that $v\leq c\hat{U}$ in $\mathbb{G}$. Also using the boundary Harnack inequality in Lemma 3.2 and comparing $v$ to a linear function, say $l,$ which vanishes on $\partial H$ with $l(w+r\xi/2)=\hat{U}(w+r\xi/2)$ we arrive at

[TABLE]

whenever $0<s\leq r/C^{\prime}$ where $C^{\prime}\geq 1$ depends only on the data. Letting $s\to 0$ in this display we get from Lemma 3.1 for $\mathcal{H}^{n-1}$-almost every $w\in\partial E$ that

[TABLE]

Next observe from (3.9) with $q=(p-1)/p$, (3.8) of Lemma 3.1, and $\eqref{3.5}(b)$ that there exists $C\geq 1$ depending only on the data and $\tilde{e}$ such that for $0<r\leq\tilde{r}/C,$

[TABLE]

Combining (5.25), (5.24), and using arbitrariness of $w,$ Harnack’ s inequality for $\mathcal{\hat{A}}$-harmonic functions, we conclude the validity of Lemma 5.6. ∎

Note from Lemma 5.6 that

[TABLE]

Following [Jer96, Lemma 6.7] we first note from (5.1) that if $\zeta\in F$ and $b$ denotes the radius of the largest $n-1$ dimensional ball contained in $\tilde{S}$ (the so called inradius of $\tilde{S}$) then

[TABLE]

for some $C_{+}\geq 1,$ depending only on the data and $\tilde{e}.$ Second we state

Lemma 5.7**.**

If $y,z\in F$ and $\delta(y,F)\approx 1,$ then

[TABLE]

Proof.

The analogue of Lemma 5.7 in [Jer96] is Lemma 6.8. Given Lemma 5.6 and (5.27) we can essentially copy the clever geometric argument in [Jer96], so we refer to this paper for details. ∎

Lemma 5.8**.**

There exists $\epsilon_{1}\in(0,1]$ and $C\geq 2$ depending only on the data, $\tilde{e},$ such that $\epsilon_{1}\geq C^{-1}$ when $1<p\leq n-1$ while $\epsilon_{1}\geq 1+(1-n)/p+C^{-1}$ when $p>n-1.$ Moreover, if $\hat{x}\in F,$ then

[TABLE]

Proof.

As in Lemma 6.13 of [Jer96] we note that if $\delta(\hat{x},F)\approx 1$ then   (5.29) follows from Lemmas 5.6 and 5.7. Thus we assume that $\delta(\hat{x},F)<<1$ and choose $y,z\in F$ so that $\delta(y,F)\approx 1$ and $\pi(z)=z\in F\cap\tilde{S}$ with $\pi(\hat{x})$ lying on the line segment from $\pi(z)$ to $\pi(y).$ Let $\rho=|\pi(z)-\pi(y)|.$ We note that if $\rho<100\,b,$ then (5.29) follows from Lemmas 5.7, 5.6 with $w=\hat{x},y,$ (5.27), and Harnack’s inequality for $\mathcal{\hat{A}}$-harmonic functions with $\epsilon_{1}=1.$ Thus we assume $\rho\geq 100\,b.$ Then from the John ellipsoid theorem mentioned below (5.10) we deduce

[TABLE]

so we assume, as we may, that $|\pi(\hat{x})-\pi(z)|<\rho/100.$ Next we define the cone:

[TABLE]

From convexity of $E$ we see that $E\cap B(y,\rho/2),$ contains a ball of radius $\rho/C^{\prime\prime},$ where $C^{\prime\prime}$ depends only on the data and $\tilde{e}.$ Let $X$ denote a point that lies $\rho$ distance from $\Gamma\cup E$ and at most $2\rho$ from $y.$ As in the proof of Theorem A we first construct $V$ a positive $\mathcal{\hat{A}}$-harmonic function in $\mathbb{R}^{n}\setminus\Gamma$ which is continuous in $\mathbb{R}^{n}$ with $V\equiv 0$ on $\Gamma$ and $V(X)=\hat{U}(X).$ Second we use the fact that $\pi(z)=z$ and the boundary Harnack inequality in Lemma 3.2 as in the proof of Theorem A to deduce that $V$ is unique, homogeneous, and

[TABLE]

From the discussion below the definition of $\Gamma$ we observe that $\mathbb{R}^{n}\setminus\Gamma$ is contained in a translation and rotation of $K(\alpha)$ for some $\alpha\in(0,\pi)$ with $\pi-\alpha\geq\bar{C}^{-1}$. Using this fact, Lemma 3.2, and Theorem A we see that if $p>n-1,$ $\epsilon_{1}-1+(n-1)/p\geq C^{-1}.$ If $1<p\leq n-1,$ one can use a compactness argument or an argument as in [KM72] to show that $\epsilon_{1}\geq C^{-1}.$ Let $\Gamma_{1}$ be the convex hull of $E\cap\bar{B}(y,\rho/2)$ and $z.$ Also let $V^{*}$ be the $\mathcal{A}$-capacitary function for $\mathbb{R}^{n}\setminus\Gamma_{1}$ when $1<p<n$ while $V^{*}$ is the $\mathcal{A}$-harmonic Green’s function for $\mathbb{R}^{n}\setminus\Gamma_{1}$ when $p\geq n.$ Define $\hat{f}$ and $\mathcal{\hat{A}}$ as above (5.20) and observe that $\hat{V}^{*}=1-V^{*}$ is $\mathcal{\hat{A}}$-harmonic when $1<p<n$ while $\hat{V}^{*}=V^{*}$ is $\mathcal{\hat{A}}$-harmonic when $p\geq n$ with continuous boundary value [math] on $\Gamma_{1}.$ We first let

[TABLE]

and claim that

[TABLE]

To prove (5.31) $(a)$ observe from (2.8) that

[TABLE]

This inequality, $\Gamma_{1}\subset E,$ and the boundary maximum principle for $\mathcal{\hat{A}}$-harmonic functions give (5.31) $(a)$. On the other hand, (5.31) $(b)$ follows from (2.8), Harnack’s inequality for $\mathcal{\hat{A}}$-harmonic functions, Lemma 3.2, and the fact that $\Gamma_{1}\cap\bar{B}(z,\rho/2)=\Gamma\cap\bar{B}(z,\rho/2)$. Finally, (5.31) $(c)$ follows from these inequalities and the fact that

[TABLE]

We conclude from (5.31) that

[TABLE]

If $\tilde{C}\geq 1$ is large enough depending on $\tilde{e}$ and the data, then from (5.32), the fact that $\rho\geq 100\,b,$ (5.27), Harnack’s inequality for $\mathcal{\hat{A}}$-harmonic functions, and Lemma 5.6 we deduce that

[TABLE]

and

[TABLE]

Next we draw the line segment $\hat{l}$ from $z$ to $\pi(y)-\tilde{C}be_{n}.$ From similar triangles and the definition of $\bar{\delta}$ below (5.29), we see that \pi(\hat{x})-\bar{\delta}\tilde{C}be_{n}\mbox{ lies on \hat{l}. } From this observation and homogeneity of $V$ we get

[TABLE]

Now since $\Gamma$ is convex we can repeat the argument given in Lemma 5.6 with $\hat{U}$ replaced by $V$ to get (5.23) with $\hat{U}$ replaced by $V,$ $w$ by $\pi(\hat{x})$, and $r$ by $\bar{\delta}\tilde{C}b,\tilde{C}b,$ provided $\tilde{C}$ is large enough. We obtain

[TABLE]

From Lemma 5.6, (5.33)-(5.36), Harnack’s inequality for $\mathcal{\hat{A}}$-harmonic functions, we see that

[TABLE]

where $C,C^{\prime}$ depend only on the data and $\tilde{e}.$ Thus Lemma 5.8 is valid. ∎

To complete the proof of Theorem B we need Lemma 6.16 from [Jer96] which in our situation can be stated as following lemma.

Lemma 5.9**.**

With the same notation as in Lemma 5.8 choose a coordinate system with axes parallel to the axes of an optimal inscribed ellipsoid contained in $\tilde{S}.$ Let $\mathcal{Q}$ be a tiling of $\tilde{S}$ by closed cubes $\subset\sum_{1}$ and of side-length $s\leq b$ with sides parallel to the coordinate axes. If $Q\in\mathcal{Q},$ let $Q^{*}$ be the cube concentric to $Q$ with side-length $10((n-1)!)^{2}s$ and let

[TABLE]

There exists $c(n)\geq 1$ such that

[TABLE]

where $C$ depends only on the data and $\tilde{e}.$

Let $\epsilon_{1}$ be as in Lemma 5.8 and put $\epsilon_{0}=\epsilon_{1}$ if $1<p\leq n-1$ while $\epsilon_{0}=\epsilon_{1}-1+(n-1)/p$ if $p>n-1$. To prove (5.22) and thus complete the proof of Theorem B we first note from Lemma 5.9 that if $\epsilon\in(0,1),$

[TABLE]

as follows from summing separately over cubes $Q\in\mathcal{Q}$ with $\delta^{*}(Q)\leq 2^{-k}s,k=0,1,2,\dots$ Second from Lemmas 5.6, 5.8, we deduce that if $\hat{y},\hat{z}\in F$ and $\pi(\hat{y}),\pi(\hat{z})\in Q^{*}$, then

[TABLE]

Hence,

[TABLE]

Using (5.39) with $s=b/2,$ and the above inequality we conclude that

[TABLE]

as we obtain from (5.39) if $1<p\leq 2$ or $n=2,3,$ and $p>2.$ Indeed , $\delta^{*}(Q)^{1-\epsilon_{0}-p+p\epsilon_{1}}\leq\delta^{*}(Q)^{-1+(p-1)\epsilon_{0}}$ if $1<p\leq 2$ while $\delta^{*}(Q)^{1-\epsilon_{0}-p+p\epsilon_{1}}=\delta^{*}(Q)^{2-n+(p-1)\epsilon_{0}}\leq\delta^{*}(Q)^{-1+(p-1)\epsilon_{0}}$ if $n=2,3$ and $p>2.$ Thus (5.22) is valid and the proof of Theorem B is now complete.

6. Closing Remarks

Here we discuss possible generalizations of Theorems A and B. First can any of the hypotheses on $f$ in (1.5) $(a),(b),(c)$ be weakened or even removed ? For example can $(c)$ be replaced by the assumption that $f$ is $C^{2}$ in $\mathbb{R}^{n}\setminus\{0\}?$ Does one need ${\mathcal{A}}=\nabla f$ or is it enough to assume {\mathcal{A}}:\mathbb{R}^{n}\setminus\{0\}\mbox{\rightarrow}\mathbb{R}^{n} is a homogeneous $p-1$ vector field with continuous first partials satisfying structure conditions similar to 1.5 $(b)$? Does one really need uniform ellipticity in $(b)?$ We cannot give a quick answer to any of these questions, still we note that existence and uniqueness for $u$ as in Theorem B made important use of boundary Harnack inequalities from [AGH*+*17] and [LN18]. In both references, theorems are stated for an $f$ satisfying (1.5). However [AGH*+*17] is concerned with proving boundary Harnack inequalities for much more general Lipschitz domains. Using smoothness of $\partial K(\alpha),$ it appears likely that at least for $0<\alpha<\pi,$ it would be enough to assume $f$ has continuous second partials, rather than Lipschitz second partials in (1.5) $(c)$. Also in [LN18] the emphasis was on domains, a portion of whose boundaries, are $k$ Reifenberg flat, $1\leq k<n-1.$ If $2\leq k<n-1,$ the authors of this paper needed an assumption similar to (1.5) in order to construct a lower dimensional barrier, which ultimately provided a lower bound for a certain boundary Harnack inequality. if $k=1$ though these considerations can be avoided and one can use the same argument as in the proof of (4.17) to show for example that (3.20) holds. Moreover, this argument is valid for more general $\mathcal{A}$ vector fields and corresponding $\mathcal{A}$-harmonic functions as outlined above. The proof that $\lambda_{1}(\pi)=1-\frac{(n-1)}{p}$ when $p>n-1,$ made important use of Lemma 4.1. This Lemma was proved in [AGH*+*17] for $n-1<p<n$ and in [ALSV18] for $n\leq p<\infty.$ In both papers it was assumed that (1.5) held for $f$ primarily in order to prove uniqueness in certain Brunn-Minkowski type inequalities for $\mathcal{A}$-capacity and in the proof of Theorems 1.2 and 1.3. The proof of Lemma 4.1 in either paper follows from a Rellich type inequality, which could easily hold if $f$ has continuous second partials and perhaps also is true for more general $\mathcal{A}$ than when $\mathcal{A}=\nabla f$.

In the proof of Theorem B we first show in (5.18) that $\partial E$ is locally $C^{1,\alpha^{\prime}}$ when $\Theta$ is bounded above and below on $\mathbb{S}^{n-1}$. We then assumed $\Theta\in C^{0,\hat{\alpha}}({\mathbb{S}^{n-1}})$ in order to complete the proof of Theorem B when $k=0.$ If instead of $\Theta\in C^{0,\hat{\alpha}}({\mathbb{S}^{n-1}}),$ one assumes that $\Theta$ is only continuous on ${\mathbb{S}^{n-1}},$ then one can use results from [Caf90a] to conclude that $\partial E$ is locally $W^{2,q}$ for $1<q<\infty.$ As for possible generalizations of Theorem B, recall that Theorem A was used only to prove Lemma 5.8, which was then used in (5.40). In (5.40) one needs to estimate the second line of this display from above by the left-hand side of (5.39), which is only possible for the values of $p$ in Theorem B, as explained below (5.40). Moreover, (5.40) was the final step in the proof of (5.22) which was the key inequality needed to eventually conclude Theorem B. Examples from Section 8 of [Jer96], show that for $p=2$, the exponent $\epsilon_{0}$ in (5.22) is dependent on the eccentricity of $E.$ Using Theorem A and proceeding operationally it appears likely that the same example implies when $n\geq 4$ and $p\geq 1-(n-1)/p$ that

[TABLE]

where $0\in\partial E$ and $F=\{x\in\partial{E}:x_{n}<t\}.$ To briefly outline this example, let $x^{\prime}=(x_{1},\dots,x_{n-1})$ and let

[TABLE]

where $0<t_{1}<<s<<1.$ Let $E$ be the convex hull of $Q\cup I.$ Repeating the argument in [Jer96] through display (8.3) with $h=|\nabla\hat{U}|$ one uses Theorem A to get

[TABLE]

for $x_{1}\in I$ where $\eta>0$ can be arbitrarily small and $c,$ may depend on $s$, $t_{1},$ and the data. Now

[TABLE]

Using this observation and (6.42) it follows that

[TABLE]

Moreover, the argument in [Jer96] can be used to get

[TABLE]

where $\eta_{1}>0$ can also be arbitrarily small and $\tilde{c},c_{+},$ have the same dependence as $c$ in (6.42). Choosing $\eta,\eta_{1},$ small enough and then fixing $s,t_{1},$ we deduce (6.41) from (6.43), (6.44).

We note, however that $\mu,$ as defined in Theorems 1.2, 1.3, corresponding to $E$ in the example above, does not satisfy the hypotheses of Theorem B since for example $\mu(\{e_{n}\})>0.$ Thus it is an interesting open question whether Theorem B remains true when $2<p<\infty$ and $n\geq 4.$ Perhaps one should first try to answer this question under the additional assumption that $\|\Theta-1\|_{\infty}\leq\epsilon,\epsilon>0,$ small, since Theorems 1.2, 1.3, give $E=$ ball when $\Theta\equiv 1.$ Somewhat similar questions have recently been considered in generalizations of the work of Caffarelli in [Caf90a] on the Monge-Ampère equation (see Theorems 3.13 and 3.14 and Corollary 3.15 in [Fig18]).

Acknowledgement

Part of this research was done while the second author was visiting TIFR in Bangalore India. The second author thanks TIFR for their gracious hospitality.