# PDE Comparison Principles for Robin Problems

Jeffrey J. Langford  Department of Mathematics, Bucknell University,
Lewisburg, Pennsylvania 17837<EMAIL_ADDRESS>
###### Abstract.

We compare the solutions of two Poisson problems in a spherical shell with
Robin boundary conditions, one with given data, and one where the data has
been cap symmetrized. When the Robin parameters are nonnegative, we show that
the solution to the symmetrized problem has larger (increasing) convex means.
We prove similar results on balls. We also prove a comparison principle on
generalized cylinders with mixed boundary conditions (Neumann and Robin).

##  1\. Introduction

Comparison principles in elliptic pde began with the work of Talenti [33]. In
his seminal work, Talenti compared the solutions of two Poisson problems with
Dirichlet boundary conditions:

$\begin{array}[]{rclccccrclcc}-\Delta u&=&f&\text{in}&\Omega,&&&-\Delta
v&=&f^{\\#}&\text{in}&\Omega^{\\#},\\\
u&=&0&\text{on}&\partial\Omega,&&&v&=&0&\text{on}&\partial\Omega^{\\#}.\end{array}$

In the first problem, the given data $f\geq 0$ belongs to $L^{2}(\Omega)$ and
$\Omega\subseteq\mathbb{R}^{n}$ denotes a bounded Lipschitz domain. In the
second, symmetrized problem, $\Omega^{\\#}\subseteq\mathbb{R}^{n}$ denotes a
centered open ball with $|\Omega|=|\Omega^{\\#}|$ (here $|\cdot|$ denotes
Lebesgue measure), and $f^{\\#}\in L^{2}(\Omega^{\\#})$ denotes the Schwarz
rearrangement, a radially decreasing function on $\Omega^{\\#}$ whose upper
level sets have the same measure as those of $f$, i.e.
$|\\{x\in\Omega:f(x)>t\\}|=|\\{x\in\Omega^{\\#}:f^{\\#}(x)>t\\}|$ for each
$t\in\mathbb{R}$. Talenti showed that the solutions $u$ and $v$ compare
through their Schwarz rearrangements:

(1) $u^{\\#}\leq v\quad\textup{in}\quad\Omega^{\\#}.$

This conclusion implies, for instance,

(2) $\int_{\Omega}\phi(u)\,dx\leq\int_{\Omega^{\\#}}\phi(v)\,dx$

for each increasing convex function $\phi:\mathbb{R}\to\mathbb{R}$. By the
Minimum Principle, $u$ and $v$ are nonnegative, so choosing
$\phi(x)=\chi_{[0,\infty)}(x)\cdot x^{p}$ gives

(3) $\|u\|_{L^{p}(\Omega)}\leq\|v\|_{L^{p}(\Omega^{\\#})},\qquad 1\leq
p\leq\infty.$

The novelty of Talenti’s result is that it allows us to estimate properties of
the solution to a given pde in terms of the solution of a related problem that
is, in general, much easier to solve. In particular, notice that the solution
$v$ to the symmetrized problem is purely radial.

Papers on comparison principles related to Talenti’s original work fill many
journal pages. We direct the interested reader to the recent survey article,
and references therein, by Talenti [34]. This literature is rich with pde
comparison principles for problems with Dirichlet boundary conditions. We
recall two results most relevant to our paper. In [2], Alvino, Diaz, Lions,
and Trombetti prove a comparison principle for Steiner symmetrization. To
Steiner symmetrize a domain, we intersect the given domain with hyperplanes,
and replace those intersections with centered balls of equal volume. To
Steiner symmetrize a function, we apply the Schwarz rearrangement to each
slice function. In this Steiner regime, the authors show that the solutions of
the given and symmetrized problems compare via their convex means as in (2).
In [35], Weitsman establishes a comparison principle for cap symmetrization.
Under cap symmetrization, we intersect a given domain with spheres, and
replace those intersections with centered spherical caps of equal surface
measure. Likewise, we cap symmetrize functions by applying the spherical
symmetrization (the analogue of the Schwarz rearrangement for functions
defined on spheres) to each radial slice function. In the cap regime, Weitsman
shows that the solutions of the given and symmetrized problems compare as in
(1), with $\\#$ denoting cap symmetrization.

In contrast to the Dirichlet setting, there are few Neumann results that
compare solutions to Poisson problems of the form

$\begin{array}[]{rclccccrclcc}-\Delta u&=&f&\text{in}&\Omega,&&&-\Delta
v&=&f^{\\#}&\text{in}&\Omega^{\\#},\\\ \frac{\partial
u}{\partial\nu}&=&0&\text{on}&\partial\Omega,&&&\frac{\partial
v}{\partial\nu}&=&0&\text{on}&\partial\Omega^{\\#},\end{array}$

where $\frac{\partial u}{\partial\nu}$ is the outer normal derivative and
$\\#$ denotes a prescribed symmetrization. The Neumann problem presents a
slight technicality in that the solutions $u$ and $v$ are not unique, so one
must impose an appropriate normalization, say, that $u$ and $v$ have zero
mean. For all known comparison results, the initial domain $\Omega$ and the
symmetrized domain $\Omega^{\\#}$ are equal. Thus, in moving from the first
problem to the symmetrized one, it is only the input function $f$ that is
rearranged. For example, in [26], the author compares $u$ and $v$ when
$\Omega$ is a spherical shell (the region between two concentric spheres) and
$\\#$ is the cap symmetrization. The author shows that the solutions compare
through their convex means as in (2). Baernstein, Laugesen, and the author
prove a generalized version of this result (see Theorem 10.20) in [11]. See
also [27] for a number of related comparison results on spheres and two-
dimensional consequences in the plane via projection mappings. In [28], the
author compares $u$ and $v$ when $\Omega=D\times(0,\ell)$ is a generalized
cylinder and $\\#$ denotes the monotone decreasing rearrangement in the final
variable (a one-dimensional Steiner symmetrization in one direction). Again,
the author shows that the solutions compare via their convex means as in (2).
In [13], Brock considers similar cylinder-like sets and proves a related
comparison principle with an eye towards comparing the extreme values ($\sup$
and $\inf$) of $u$ and $v$.

Likely driven by the recent surge of interest in Robin problems in spectral
theory (see any of [4], [5], [14], [15], [16], [17], [18], [19], [20], [21],
[23]), in [3], Alvino, Nitsch, and Trombetti establish the first Robin
comparison principle in the spirit of Talenti. The authors compare the
solutions to the Poisson problems

$\begin{array}[]{rclccccrclcc}-\Delta u&=&f&\text{in}&\Omega,&&&-\Delta
v&=&f^{\\#}&\text{in}&\Omega^{\\#},\\\ \frac{\partial u}{\partial\nu}+\alpha
u&=&0&\text{on}&\partial\Omega,&&&\frac{\partial v}{\partial\nu}+\alpha
v&=&0&\text{on}&\partial\Omega^{\\#},\end{array}$

where $f\geq 0$, $\alpha>0$, $\Omega^{\\#}$ is a centered ball whose volume
equals that of $\Omega$, and $\\#$ denotes the Schwarz rearrangement. The
authors show that $u$ and $v$ compare through their Lorentz norms; when the
dimension $n=2$, it follows that

$\|u\|_{L^{1}(\Omega)}\leq\|v\|_{L^{1}(\Omega^{\\#})}\qquad\textup{and}\qquad\|u\|_{L^{2}(\Omega)}\leq\|v\|_{L^{2}(\Omega^{\\#})}.$

Also in dimension $n=2$, when $f=1$, the authors show that $u$ and $v$ compare
as in (1). They therefore deduce a two-dimensional isoperimetric inequality
for Robin torsional rigidity. See also the related paper [1]. But beyond these
two papers, the author is not aware of any others that discuss pde comparison
principles for Robin problems. In this paper, we prove several such results.

To state our results we require some notation. For $0<a<b<\infty$, let

$A=A(a,b)=\\{x\in\mathbb{R}^{n}:a<|x|<b\\}$

denote a spherical shell with inner radius $a$ and outer radius $b$. For a
function $f\in L^{1}(A)$, let $f^{\\#}$ denote the cap symmetrization (for a
precise definition, see Definitions 2.2, 2.7, and 2.8). Denote

$A^{\bigstar}=(a,b)\times(0,\pi)$

for the polar rectangle associated to $A$ and define functions
$Jf,f^{\bigstar}:A^{\bigstar}\to\mathbb{R}$ a.e. by

$\displaystyle Jf(r,\theta)$
$\displaystyle=\int_{K(\theta)}f(r\xi)\,d\sigma(\xi),$ $\displaystyle
f^{\bigstar}(r,\theta)$
$\displaystyle=\int_{K(\theta)}f^{\\#}(r\xi)\,d\sigma(\xi),$

where $\sigma$ denotes standard surface measure on $\mathbb{S}^{n-1}$ and

$K(\theta)=\\{\xi\in\mathbb{S}^{n-1}:d(e_{1},\xi)<\theta\\}$

denotes the open polar cap on $\mathbb{S}^{n-1}$ centered at $e_{1}$ with
radius $\theta$, computed in the spherical distance. Our first main result is
a Robin comparison principle for the cap symmetrization in shells. Notice that
our result allows for the possibility of mixed boundary conditions.

######  Theorem 1.1 (Robin Comparison Principle on Shells).

Say $0<a<b<\infty$ and $A=A(a,b)\subseteq\mathbb{R}^{n}$ is a spherical shell
with inner radius $a$ and outer radius $b$. Let $\alpha_{1},\alpha_{2}\geq 0$
and $f\in L^{2}(A)$; when $\alpha_{1}=\alpha_{2}=0$, we also assume
$\int_{A}f\,dx=0$. Say $u$ and $v$ solve the Poisson problems

$\begin{array}[]{rclccccrclcc}-\Delta u&=&f&\text{in}&A,&&&-\Delta
v&=&f^{\\#}&\text{in}&A,\\\ \frac{\partial
u}{\partial\nu}+\alpha_{1}u&=&0&\text{on}&\\{|x|=a\\},&&&\frac{\partial
v}{\partial\nu}+\alpha_{1}v&=&0&\text{on}&\\{|x|=a\\},\\\ \frac{\partial
u}{\partial\nu}+\alpha_{2}u&=&0&\text{on}&\\{|x|=b\\},&&&\frac{\partial
v}{\partial\nu}+\alpha_{2}v&=&0&\text{on}&\\{|x|=b\\},\end{array}$

where $f^{\\#}$ denotes the cap symmetrization of $f$; when
$\alpha_{1}=\alpha_{2}=0$, we also assume that $u$ and $v$ are normalized so
that $\int_{A}u\,dx=\int_{A}v\,dx=0$. Then for almost every $r\in(a,b)$, the
inequality

$u^{\bigstar}(r,\theta)\leq Jv(r,\theta)$

holds for each $\theta\in(0,\pi)$. In particular, for almost every $r\in(a,b)$
and for each convex increasing function $\phi:\mathbb{R}\to\mathbb{R}$, we
have

$\int_{\\{|x|=r\\}}\phi(u)\,dS\leq\int_{\\{|x|=r\\}}\phi(v)\,dS;$

when $\alpha_{1}=\alpha_{2}=0$, the above inequality holds for all convex
functions.

We also prove an analogous comparison principle on balls that requires a bit
more notation. For $0<b<\infty$, let

$B=B(0,b)=\\{x\in\mathbb{R}^{n}:|x|<b\\}.$

Denote

$B^{\bigstar}=(0,b)\times(0,\pi)$

for the polar rectangle associated to $B$ and define functions
$Jf,f^{\bigstar}:B^{\bigstar}\to\mathbb{R}$ a.e. by

$\displaystyle Jf(r,\theta)$
$\displaystyle=\int_{K(\theta)}f(r\xi)\,d\sigma(\xi),$ $\displaystyle
f^{\bigstar}(r,\theta)$
$\displaystyle=\int_{K(\theta)}f^{\\#}(r\xi)\,d\sigma(\xi),$

with $f^{\\#}$ the cap symmetrization of $f$ (see also the discussion
preceding the proof of Theorem 1.2 in Section 3). Notice that the conclusion
of the ball result below is slightly stronger than that of the shell result,
in that the convex means comparison between $u$ and $v$ now holds for all
convex functions, not just the increasing ones. This stronger conclusion is a
consequence of our proof, which shows that for almost every $r\in(0,b)$ the
solutions $u$ and $v$ have the same mean on the sphere $\\{|x|=r\\}$.

######  Theorem 1.2 (Robin Comparison Principle on Balls).

Say $0<b<\infty$ and let $B\subseteq\mathbb{R}^{n}$ denote an open ball
centered at $0$ with radius $b$. Let $\alpha\geq 0$ and $f\in L^{2}(A)$; when
$\alpha=0$, we also assume $\int_{B}f\,dx=0$. Say $u$ and $v$ solve the
Poisson problems

$\begin{array}[]{rclccccrclcc}-\Delta u&=&f&\text{in}&B,&&&-\Delta
v&=&f^{\\#}&\text{in}&B,\\\ \frac{\partial u}{\partial\nu}+\alpha
u&=&0&\text{on}&\\{|x|=b\\},&&&\frac{\partial v}{\partial\nu}+\alpha
v&=&0&\text{on}&\\{|x|=b\\},\end{array}$

where $f^{\\#}$ denotes the cap symmetrization of $f$; when $\alpha=0$, we
also assume that $u$ and $v$ are normalized so that
$\int_{B}u\,dx=\int_{B}v\,dx=0$. Then for almost every $r\in(a,b)$, the
inequality

$u^{\bigstar}(r,\theta)\leq Jv(r,\theta)$

holds for each $\theta\in(0,\pi)$. In particular, for almost every $r\in(a,b)$
and for each convex function $\phi:\mathbb{R}\to\mathbb{R}$, we have

$\int_{\\{|x|=r\\}}\phi(u)\,dS\leq\int_{\\{|x|=r\\}}\phi(v)\,dS.$

To state our final main result, we also introduce some additional notation.
Let $D\subseteq\mathbb{R}^{n-1}$ denote a bounded $C^{\infty}$ domain and say
$\ell>0$. Define a generalized cylinder by

$\Omega=D\times(0,\ell).$

Points of $\Omega$ are denoted by pairs $(x,y)$ with $x\in D$ and
$y\in(0,\ell)$. Given a function $f\in L^{1}(\Omega)$, we can hold almost
every $x\in D$ fixed, and take a decreasing rearrangement in the variable $y$.
Doing so yields the monotone decreasing rearrangement of $f$ in the
$y$-direction (for the precise definition, see Definitions 2.2 and 2.13).
Define functions $Jf,f^{\bigstar}:\Omega\to\mathbb{R}$ a.e. by

$\displaystyle Jf(x,y)$ $\displaystyle=\int_{0}^{y}f(x,t)\,dt,$ $\displaystyle
f^{\bigstar}(x,y)$ $\displaystyle=\int_{0}^{y}f^{\\#}(x,t)\,dt,$

where $f^{\\#}$ is the monotone decreasing rearrangement of $f$ in the
$y$-direction. We are now prepared to state our last result, a comparison
principle on cylinders with mixed boundary conditions.

######  Theorem 1.3 (Mixed Robin/Neumann Comparison Principle on Cylinders).

Let $\Omega=D\times(0,\ell)$ denote a cylinder whose base
$D\subset\mathbb{R}^{n-1}$ is a bounded $C^{\infty}$ domain and say $f\in
L^{2}(\Omega)$. We divide the boundary $\partial\Omega$ into two pieces,
$\partial\Omega_{1}=(D\times\\{0\\})\cup(D\times\\{\ell\\})$ and
$\partial\Omega_{2}=\partial D\times[0,\ell]$. Let $u$ and $v$ be solutions to
the Poisson problems

$\begin{array}[]{rclccccrclcc}-\Delta u&=&f&\text{in}&\Omega,&&&-\Delta
v&=&f^{\\#}&\text{in}&\Omega,\\\ \frac{\partial
u}{\partial\nu}&=&0&\text{on}&\partial\Omega_{1}&&&\frac{\partial
v}{\partial\nu}&=&0&\text{on}&\partial\Omega_{1},\\\ \frac{\partial
u}{\partial\nu}+\alpha u&=&0&\text{on}&\partial\Omega_{2},&&&\frac{\partial
v}{\partial\nu}+\alpha v&=&0&\text{on}&\partial\Omega_{2},\end{array}$

where $f^{\\#}$ denotes the decreasing rearrangement of $f$ in the
$y$-direction; when $\alpha=0$, we assume $f$, $u$, and $v$ have zero mean on
$\Omega$. Assume that $f$ and $f^{\\#}$ are continuous on $\overline{\Omega}$
and are sufficiently regular to ensure that the solutions $u,v$ belong to
$C^{2}(\overline{\Omega})$. If $\alpha\geq 0$, then for each $x\in D$

$u^{\bigstar}(x,y)\leq Jv(x,y)$

for every $0\leq y\leq\ell$. In particular,

$\int_{0}^{\ell}\phi(u(x,y))\,dy\leq\int_{0}^{\ell}\phi(v(x,y))\,dy$

for each convex function $\phi:\mathbb{R}\rightarrow\mathbb{R}$.

Note that in [13], Brock proves a result related to Theorem 1.3 using Green’s
functions. Again, his conclusions are weaker than those obtained here, as it
appears his main goal is to compare the extreme values of $u$ and $v$ rather
than their convex means or $L^{p}$-norms. Indeed, all of our comparison
principles have corollaries comparing the $L^{p}$-norms of $u$ and $v$ (see
Corollaries 3.2, 3.4, and 3.6).

The techniques used in this paper are similar to those used by Baernstein in
[9], the author in [26] and [28] to prove Neumann versions of Theorems 1.1,
1.2, and 1.3, and Baernstein, Laugesen, and the author in Chapter 10 of [11].
Indeed, the techniques employed here are drastically different from those used
by Talenti to prove his original comparison result (and also those used in
[3]). Whereas Talenti’s work relies on the classic isoperimetric inequality,
the coarea formula, and a detailed analysis of a function’s (in particular
$u$’s) level sets, our work relies on $\bigstar$-functions, so-called
“subharmonicity” results (certain weak differential inequalities), and a weak
Maximum Principle. In a certain sense, the machinery used here is robust, as
it can be used to prove Dirichlet, Neumann, and now Robin comparison
principles.

The remainder of the paper is outlined as follows. In Section 2, we present
three subsections. In the first, we introduce star functions on general
measure spaces; in the second subsection, we lay out all of the background
necessary to prove Theorems 1.1 and 1.2, including relevant rearrangements and
subharmonicity results; in the third subsection, we present background for
Theorem 1.3, introducing relevant rearrangements and subharmonicity results.
In Section 3, we prove our main results and related consequences.

##  2\. Background

###  2.1. Robin Problems.

We start by collecting basic facts about solutions to Poisson problems with
Robin boundary conditions. Say $\Omega\subseteq\mathbb{R}^{n}$ is a bounded
Lipschitz domain, $\alpha\in L^{\infty}(\partial\Omega)$ is nonnegative and
not identically zero, and $f\in L^{2}(\Omega)$. We consider the Robin Poisson
problem

(4) $\displaystyle-\Delta u$ $\displaystyle=$ $\displaystyle
f\quad\text{in}\quad\Omega,$ (5) $\displaystyle\frac{\partial u}{\partial
n}+\alpha u$ $\displaystyle=$ $\displaystyle
0\quad\text{on}\quad\partial\Omega.$

A function $u\in H^{1}(\Omega)$ is said to be a weak solution provided

$\int_{\Omega}\nabla u\cdot\nabla v\,dx+\int_{\partial\Omega}\alpha
uv\,dS=\int_{\Omega}fv\,dx,\qquad v\in H^{1}(\Omega),$

where the boundary integral is interpreted in the trace sense.

The bilinear form associated to the Robin problem is

$B[v,w]=\int_{\Omega}\nabla v\cdot\nabla w\,dx+\int_{\partial\Omega}\alpha
vw\,dS,\qquad v,w\in H^{1}(\Omega).$

The boundedness of $\alpha$ on $\partial\Omega$ and the Trace Theorem give
that $B$ is bounded, i.e. there exists a positive constant $c_{1}$ where

$|B[v,w]|\leq c_{1}\|v\|_{H^{1}(\Omega)}\|w\|_{H^{1}(\Omega)},\qquad v,w\in
H^{1}(\Omega).$

The theorems of Banach-Alaoglu and Rellich-Kondrachov, and the assumption that
$\alpha$ is positive on a set of positive measure on $\partial\Omega$ give
that $B$ is coercive. So, there exists a positive constant $c_{2}$ with

$B[v,v]\geq c_{2}\|v\|_{H^{1}(\Omega)}^{2},\qquad v\in H^{1}(\Omega).$

The Lax-Milgram Theorem now implies that for each $f\in L^{2}(\Omega)$, there
exists a unique $u\in H^{1}(\Omega)$ where

$B[u,v]=\int_{\Omega}fv\,dx,\qquad v\in H^{1}(\Omega).$

In other words, the Robin problem (4), (5) with data $f$ has a unique solution
$u$. Moreover, by coercivity, the solution depends continuously on the data:

(6) $\|u\|_{H^{1}(\Omega)}\leq\frac{1}{c_{2}}\|f\|_{L^{2}(\Omega)}.$

###  2.2. Star Functions on General Measure Spaces.

The goal of our paper is to compare the solutions of two Robin problems as in
$\eqref{eq:RPpde},\eqref{eq:RPbc}$, one with initial data, and one where the
data has been rearranged, or symmetrized. As we shall see, the two solutions
compare via their star functions. So in this subsection, we collect basic
facts about star functions on general measure spaces to be used throughout the
paper. Much of the background presented in this subsection also appears
Section 2.1 of [26]. For more on applications of the star function, we direct
the reader to the work of Baernstein [6], [7], [8], [10].

Unless specified otherwise, $(X,\mu)$ denotes a finite measure space. We start
by precisely defining what it means for two functions to have the same size.

######  Definition 2.1 (Rearrangements).

Suppose that $f\in L^{1}(X)$ and $g\in L^{1}(Y)$ are defined on finite measure
spaces $(X,\mu)$ and $(Y,\nu)$. We say $f$ and $g$ are rearrangements of each
other provided

$\mu\left(\\{x\in X:t<f(x)\\}\right)=\nu\left(\\{y\in
Y:t<g(y)\\}\right),\qquad t\in\mathbb{R}.$

For a comprehensive treatment of rearrangements, we direct the interested
reader to the recently published manuscript of Baernstein [11]. See also [24]
and [29].

Given an integrable function, we can always build a decreasing function on an
interval that rearranges the given function’s values. This function is called
the decreasing rearrangement and it plays a central role in defining other
rearrangements (symmetrizations) throughout our paper.

######  Definition 2.2 (Decreasing Rearrangement).

Let $f\in L^{1}(X)$. Define $f^{\ast}:[0,\mu(X)]\rightarrow[-\infty,\infty]$
via

$f^{\ast}(t)=\begin{cases}\underset{X}{\textup{ess\ sup}}\ f&\textup{if}\
t=0,\\\ \inf\\{s:\mu(\\{x:s<f(x)\\})\leq t\\}&\textup{if}\ t\in(0,\mu(X)),\\\
\underset{X}{\textup{ess\ inf}}\ f&\textup{if}\ t=\mu(X).\end{cases}$

We call $f^{\ast}$ the decreasing rearrangement of $f$.

We are now prepared to define star functions on general measure spaces.

######  Definition 2.3 (Star Function for a General Measure Space).

Let $f\in L^{1}(X)$. We define the star function of $f$ on the interval
$[0,\mu(X)]$ by the formula

$f^{\bigstar}(t)=\underset{\mu(E)=t}{\sup}\ {\displaystyle\int_{E}f\,d\mu},$

where the $\sup$ is taken over all measurable subsets $E\subseteq X$ with
$\mu(E)=t$.

Our next proposition collects two important facts about the star function.
First, on nonatomic measure spaces, the $\sup$ defining $f^{\bigstar}$ is
always achieved by some measurable subset $E$. Second, there is a simple
connection between star function and the decreasing rearrangement. The result
below appears as Proposition 9.2 of [11].

######  Proposition 2.4.

Assume $f\in L^{1}(X)$ with $(X,\mu)$ a finite nonatomic measure space. Then
for each $t\in[0,\mu(X)]$, there exists a measurable subset $E\subseteq X$
with $\mu(E)=t$ such that

$f^{\bigstar}(t)=\int_{E}f\,d\mu.$

Moreover,

$f^{\bigstar}(t)=\int_{0}^{t}f^{\ast}(s)\,ds,$

where $f^{\ast}$ is the decreasing rearrangement of $f$.

The next result allows us to interpret star function inequalities in terms of
convex means. See Propositions 10.1 and 10.3 of [11].

######  Proposition 2.5 (Majorization).

Assume $(X,\mu)$ is a finite measure space and $u,v\in L^{1}(X)$. Then

$u^{\bigstar}\leq v^{\bigstar}$

on $[0,\mu(X)]$ if and only if

$\int_{X}\phi(u)\,d\mu\leq\int_{X}\phi(v)\,d\mu$

for every increasing convex function $\phi:\mathbb{R}\rightarrow\mathbb{R}$.
If $\int_{X}u\,d\mu=\int_{X}v\,d\mu$, then the word “increasing” may be
removed from the previous statement.

The last result in this subsection gives further consequences of star function
inequalities when additional information about the functions is known. The
following result combines Corollary 10.2 and Proposition 10.3 of [11].

######  Corollary 2.6.

Say $u,v\in L^{1}(X)$ where $(X,\mu)$ is a finite measure space and assume
$u^{\bigstar}\leq v^{\bigstar}$ on $[0,\mu(X)]$. If either $u,v\geq 0$ or
$\int_{X}u\,d\mu=\int_{X}v\,d\mu$, then

$\|u\|_{L^{p}(X)}\leq\|v\|_{L^{p}(X)},\quad 1\leq p\leq\infty.$

If $\int_{X}u\,d\mu=\int_{X}v\,d\mu$, moreover

$\displaystyle\underset{X}{\operatorname{\textup{ess\ sup}}}\ u$
$\displaystyle\leq\underset{X}{\operatorname{\textup{ess\ sup}}}\ v,$
$\displaystyle\underset{X}{\operatorname{\textup{ess\ inf}}}\ u$
$\displaystyle\geq\underset{X}{\operatorname{\textup{ess\ inf}}}\ v,$
$\displaystyle\underset{X}{\operatorname{\textup{osc}}}\ u$
$\displaystyle\leq\underset{X}{\operatorname{\textup{osc}}}\ v,$

where $\operatorname{\textup{osc}}=\operatorname{\textup{ess\
sup}}-\operatorname{\textup{ess\ inf}}$.

###  2.3. Spherical Shells: Symmetrization, Star Functions, and
Subharmonicity.

In this subsection, we collect several key definitions and results for
functions defined on spherical shells. These results drive the proof of
Theorem 1.1. Much of the information in this subsection also appears in
Section 2.2 of [26].

We first discuss rearrangements of functions defined on spheres. Denote

$\mathbb{S}^{n}=\\{(\xi_{1},\xi_{2},\ldots,\xi_{n+1})\in\mathbb{R}^{n+1}:\xi_{1}^{2}+\xi_{2}^{2}+\cdots+\xi_{n+1}^{2}=1\\}$

for the standard unit sphere. We write $\sigma$ for standard surface measure
on $\mathbb{S}^{n}$ and $d$ for the spherical distance, computed using lengths
of arcs of great circles. Let

$K(\theta)=\\{\xi\in\mathbb{S}^{n}:d(\xi,e_{1})<\theta\\}$

denote the open polar cap, or geodesic ball, on $\mathbb{S}^{n}$ centered at
$e_{1}=(1,0,\ldots,0)$ with radius $\theta$.

We next define the spherical rearrangement. Notice that this definition is the
analogue of the Schwarz rearrangement in the spherical setting.

######  Definition 2.7 (Spherical Rearrangement).

Given $F\in L^{1}(\mathbb{S}^{n})$, we define
$F^{\\#}:\mathbb{S}^{n}\rightarrow[-\infty,\infty]$ by the formula

$\displaystyle F^{\\#}(\xi)$ $\displaystyle=$ $\displaystyle
F^{\ast}\left(\sigma\left(K(\theta)\right)\right),$

where $\theta$ is the spherical distance between the point $\xi$ and $e_{1}$,
and $F^{\ast}$ is the decreasing rearrangement of $F$. We call $F^{\\#}$ the
spherical rearrangement of $F$.

For $0<a<b<\infty$, we write

$A=A(a,b)=\\{x\in\mathbb{R}^{n}:a<|x|<b\\}$

for the spherical shell with inner radius $a$ and outer radius $b$. If $f\in
L^{1}(A)$, the cap symmetrization of $f$ is obtained by spherically
rearranging $f$’s radial slice functions. More precisely, we have the
following definition.

######  Definition 2.8 (Cap Symmetrization).

Suppose $f\in L^{1}(A)$. We define $f^{\\#}:A\rightarrow[-\infty,\infty]$ as
follows. By Fubini’s Theorem, the slice function
$f^{r}:\mathbb{S}^{n}\rightarrow\mathbb{R}$ defined by $f^{r}(\xi)=f(r\xi)$
belongs to $L^{1}(\mathbb{S}^{n-1})$ for almost every $r\in(a,b)$. For such
$r$, the cap symmetrization of $f$ on $\\{|x|=r\\}$ is defined by

$\displaystyle f^{\\#}(r\xi)$ $\displaystyle=$
$\displaystyle(f^{r})^{\\#}(\xi),$

where $(f^{r})^{\\#}$ denotes the spherical rearrangement of the slice
function $f^{r}$. We leave $f^{\\#}$ undefined on those spheres $\\{|x|=r\\}$
for which $f^{r}\not\in L^{1}(\mathbb{S}^{n-1})$.

Given a function defined on a spherical shell $A(a,b)$, its star function is
defined in the polar rectangle

$A^{\bigstar}=\\{(r,\theta)\in\mathbb{R}^{2}:a<r<b\text{\ and\
}0<\theta<\pi\\}.$

We have the following definition.

######  Definition 2.9 (Star Functions on Spherical Shells).

If $f\in L^{1}(A)$, define $f^{\bigstar}:A^{\bigstar}\rightarrow\mathbb{R}$
a.e. by the formula

$f^{\bigstar}(r,\theta)=\underset{\sigma(E)=\sigma(K(\theta))}{\max}\
{\displaystyle\int_{E}f(r\xi)\
\text{d}\sigma(\xi)}=\int_{K(\theta)}f^{\\#}(r\xi)\ \text{d}\sigma(\xi),$

where the $\max$ is taken over all measurable subsets $E$ of
$\mathbb{S}^{n-1}$ with the same surface measure as $K(\theta)$ and $f^{\\#}$
denotes the cap symmetrization of $f$.

Several remarks are in order. First, notice that the formula for
$f^{\bigstar}(r,\theta)$ only makes sense if the radial slice function
$f^{r}\in L^{1}(\mathbb{S}^{n-1})$. By Fubini’s Theorem,
$f^{\bigstar}(r,\theta)$ is well-defined for almost every $r\in(a,b)$ and for
each $\theta\in(0,\pi)$. Second, the definition of $f^{\bigstar}$ given above
above is obtained by applying the star function of Definition 2.3 to each of
$f$’s slice functions, up to a change of variable. Moreover, since
$(\mathbb{S}^{n},\sigma)$ is nonatomic, Proposition 2.4 tells us we are
justified in using either $\sup$ or $\max$ to define $f^{\bigstar}$, since the
$\max$ is obtained by some appropriate subset.

We close this subsection by stating “commutativity” and “subharmonicity”
results for cap symmetrization. With $A$ as above, suppose $u\in L^{1}(A)$.
Define $Ju:A^{\bigstar}\rightarrow\mathbb{R}$ a.e. by

(7) $Ju(r,\theta)=\int_{K(\theta)}u(r\xi)\,d\sigma(\xi).$

If $u^{\\#}$ is the cap symmetrization of $u$, we therefore have

(8)
$u^{\bigstar}(r,\theta)=\int_{K(\theta)}u^{\\#}(r\xi)\,d\sigma(\xi)=Ju^{\\#}(r,\theta).$

Define operators $\Delta^{\bigstar}$ and $\Delta^{\bigstar t}$ acting on $F\in
C^{2}(A^{\bigstar})$ via the formulas

(9) $\displaystyle\Delta^{\bigstar}F$
$\displaystyle=\partial_{rr}F+\frac{n-1}{r}\partial_{r}F+r^{-2}[\partial_{\theta\theta}F-(n-2)(\cot\theta)\partial_{\theta}F],$
$\displaystyle\Delta^{\bigstar t}F$
$\displaystyle=\partial_{rr}F-\frac{n-1}{r}\partial_{r}F+r^{-2}(n-1)F$
$\displaystyle\qquad+r^{-2}[\partial_{\theta\theta}F+(n-2)(\cot\theta)\partial_{\theta}F-(n-2)(\csc^{2}\theta)F].$

Then for $F\in C^{2}(A^{\bigstar})$ and $G\in C_{c}^{2}(A^{\bigstar})$, we
have

$\int_{A^{\bigstar}}(\Delta^{\bigstar}F)G\,dr\,d\theta=\int_{A^{\bigstar}}F\Delta^{\bigstar
t}G\,dr\,d\theta.$

In other words, $\Delta^{\bigstar t}$ is formally the adjoint of
$\Delta^{\bigstar}$ when $L^{2}(A^{\bigstar})$ is given the measure
$dr\,d\theta$.

The following results appear as Theorems 2.18 and 2.20 in [26].

######  Theorem 2.10 (Commutativity Relation for Cap Symmetrization).

If $u\in C^{2}(A)$, then

$J\Delta u=\Delta^{\bigstar}Ju$

on $A^{\bigstar}$.

Our last result requires the following definition:

######  Definition 2.11.

For $u\in C^{2}(A)$ and $f\in L_{loc}^{1}(A)$, we say that
$-\Delta^{\bigstar}u^{\bigstar}\leq f^{\bigstar}$ holds in the weak sense if

$-\int_{A^{\bigstar}}u^{\bigstar}\Delta^{\bigstar
t}G\,dr\,d\theta\leq\int_{A^{\bigstar}}f^{\bigstar}G\,dr\,d\theta$

for every nonnegative $G\in C_{c}^{2}(A^{\bigstar})$.

We finally have the following result, the major tool in proving Theorem 1.1.

######  Theorem 2.12 (Subharmonicity for Cap Symmetrization).

Suppose $u\in C^{2}(A)$ satisfies $-\Delta u=f$. Then

$-\Delta^{\bigstar}u^{\bigstar}\leq f^{\bigstar}$

in the weak sense.

###  2.4. Generalized Cylinders: Symmetrization, Star Functions, and
Subharmonicity.

The structure of this subsection mimics the structure of the last one, as our
goal is to gather all of the machinery necessary to prove our third main
result, Theorem 1.3. Much of the information gathered here also appears in
Sections 2 and 3 of [28]. The major change in this subsection is that here, we
consider cylindrical domains rather than shell domains. To start, let
$D\subseteq\mathbb{R}^{n-1}$ denote a bounded $C^{\infty}$ domain and say
$\ell>0$. Define

(10) $\Omega=D\times(0,\ell).$

Points in $\Omega$ will be denoted by pairs $(x,y)$ with $x\in D$ and
$y\in(0,\ell)$. Given a function $f$ on $\Omega$, we can perform a decreasing
rearrangement in the $y$-variable, holding each $x\in D$ fixed. Doing so gives
the decreasing rearrangement of $f$ in the $y$-direction. More precisely, we
have:

######  Definition 2.13 (Decreasing Rearrangement in the $y$-direction).

Let $f\in L^{1}(\Omega)$ with $\Omega$ as in (10). Define a function $f^{\\#}$
on $\Omega$ by

$f^{\\#}(x,y)=(f^{x})^{\ast}(y),$

where $f^{x}$ denotes the slice function that maps $y\in(0,\ell)$ to $f(x,y)$
and $\ast$ denotes the decreasing rearrangement. We call $f^{\\#}$ the
decreasing rearrangement of $f$ in the y-direction.

Given $u\in L^{1}({\Omega})$, Fubini’s Theorem guarantees that for almost
every $x\in D$, the slice function $f^{x}\in L^{1}(0,\ell)$. We are therefore
led to define $Ju$ and $u^{\bigstar}$ a.e. on $\Omega$ via

$Ju(x,y)=\int_{0}^{y}u(x,t)\,dt$

and

(11) $u^{\bigstar}(x,y)=Ju^{\\#}(x,y)=\int_{0}^{y}u^{\\#}(x,t)\,dt$

where $u^{\\#}$ denotes the monotone decreasing rearrangement of $u$ in the
$y$-direction. By Proposition 2.4,

$u^{\bigstar}(x,y)=\underset{\mathcal{L}^{1}(E)=y}{\max}\int_{E}u(x,t)\,dt,$

where the $\max$ is taken over all measurable subsets $E$ of $(0,\ell)$ with
one-dimensional Lebesgue measure (length) equal to $y$.

We are now prepared to state our commutativity result for the cylindrical
setting. See Proposition 2 of [27].

######  Proposition 2.14 (Commutativity for Decreasing Rearrangement in
$y$-direction).

Let $\Omega$ be as in (10). If $u\in C^{2}(\overline{\Omega})$ satisfies
$u_{y}(x,0)=0$ for $x\in D$, then

$J\Delta u=\Delta Ju$

in $\Omega$.

We finally have the following subharmonicity result in the cylindrical
setting. See Theorem 1 of [27].

######  Theorem 2.15 (Subharmonicity for Decreasing Rearrangement in
$y$-direction).

Let $\Omega$ be as in (10). Say $u\in C^{2}(\overline{\Omega})$ and
$u_{y}(x,0)=u_{y}(x,\ell)=0$ for $x\in D$. If $-\Delta u=f$ in $\Omega$, then

$-\Delta u^{\bigstar}\leq f^{\bigstar}$

in the weak sense in $\Omega$. That is,

$-\int_{\Omega}u^{\bigstar}\Delta
G\,dx\,dy\leq\int_{\Omega}f^{\bigstar}G\,dx\,dy$

for nonnegative $G\in C_{c}^{2}(\Omega$).

##  3\. Proofs of Main Results

We start this section with a proof of our paper’s first main result.

###### Proof of Theorem 1.1.

The proof is long, and so we break it down into several steps. In Steps 1
through 4, we show that for almost every $r\in(a,b)$,
$u^{\bigstar}(r,\theta)\leq Jv(r,\theta)$ for each $\theta\in(0,\pi)$. In Step
5, we deduce the inequalities about convex means.

Step 1: Maximum Principle. First suppose that $f\in C_{c}^{\infty}(A)$ is a
test function. In particular, $f$ is Lipschitz continuous on $\mathbb{R}^{n}$.
Then $f^{\\#}$ is also Lipschitz continuous on $\mathbb{R}^{n}$; see Section
7.5 of [11] or Lemma 6.7 in [32]. By standard regularity results, the
solutions $u,v$ belong to $C^{2}(\overline{A})$. See Theorem 6.31 in [22] and
the discussion following its proof on Fredholm alternatives. See also [31] for
an explicit discussion of the Neumann problem.

Put

$w(r,\theta)=u^{\bigstar}(r,\theta)-Jv(r,\theta)-\varepsilon Q(r,\theta),$

for $(r,\theta)\in A^{\bigstar}$, where

$Q(r,\theta)=(r-a)(r-b)+C\theta(\pi-\theta).$

A direct computation shows

$\Delta^{\bigstar}Q(r,\theta)=2+\frac{n-1}{r}(2r-a-b)-\frac{C}{r^{2}}\left(2+(n-2)(\pi-2\theta)\cot\theta\right).$

Since $(\pi-2\theta)\cot\theta\geq 0$ on $(0,\pi)$, we may choose $C>0$
sufficiently large to guarantee $\Delta^{\bigstar}Q\leq 0$ on $A^{\bigstar}$.
We then compute in the weak sense

$\displaystyle-\Delta^{\bigstar}w$
$\displaystyle=-\Delta^{\bigstar}u^{\bigstar}+\Delta^{\bigstar}Jv+\varepsilon\Delta^{\bigstar}Q$
$\displaystyle\leq f^{\bigstar}+J\Delta v+\varepsilon\Delta^{\bigstar}Q$
$\displaystyle=f^{\bigstar}-f^{\bigstar}+\varepsilon\Delta^{\bigstar}Q$
$\displaystyle\leq 0,$

where the first inequality follows from Theorems 2.10 and 2.12, the second
equality follows from (8), and the second inequality follows from our choice
of $C$. Since the operator $\Delta^{\bigstar}$ is uniformly elliptic in
$A^{\bigstar}$, the Maximum Principle for weakly $\Delta^{\bigstar}$
subharmonic functions (see Theorem 1 of [30], for example) implies

(12) $\sup_{A^{\bigstar}}w\leq\max_{\partial A^{\bigstar}}w.$

Step 2: Boundary Analysis on Vertical Sides of $\partial A^{\bigstar}$. We
first claim that $\displaystyle\max_{\partial A^{\bigstar}}w$ is not achieved
at a point where $r=a$ or $r=b$, i.e. on a vertical side of $\partial
A^{\bigstar}$. For $\theta\in[0,\pi]$ and $h>0$ small, let
$E(a+h,\theta)\subseteq\mathbb{S}^{n-1}$ denote a measurable subset with
$\sigma(E(a+h,\theta))=\sigma(K(\theta))$ for which the $\sup$ defining
$u^{\bigstar}(a+h,\theta)$ is achieved. We estimate

$\displaystyle\frac{e^{-\alpha_{1}(a+h)}w(a+h,\theta)-e^{-\alpha_{1}a}w(a,\theta)}{h}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad=e^{-\alpha_{1}(a+h)}\int_{E(a+h,\theta)}\frac{u((a+h)\xi)}{h}\,d\sigma(\xi)-e^{-\alpha_{1}a}\int_{E(a,\theta)}\frac{u(a\xi)}{h}\,d\sigma(\xi)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-e^{-\alpha_{1}(a+h)}\int_{K(\theta)}\frac{v((a+h)\xi)}{h}\,d\sigma(\xi)+e^{-\alpha_{1}a}\int_{K(\theta)}\frac{v(a\xi)}{h}\,d\sigma(\xi)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-\varepsilon\frac{e^{-\alpha_{1}(a+h)}Q(a+h,\theta)-e^{-\alpha_{1}a}Q(a,\theta)}{h}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\geq\int_{E(a,\theta)}\frac{e^{-\alpha_{1}(a+h)}u((a+h)\xi)-e^{-\alpha_{1}a}u(a\xi)}{h}\,d\sigma(\xi)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-\int_{K(\theta)}\frac{e^{-\alpha_{1}(a+h)}v((a+h)\xi)-e^{-\alpha_{1}a}v(a\xi)}{h}\,d\sigma(\xi)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-\varepsilon\frac{e^{-\alpha_{1}(a+h)}Q(a+h,\theta)-e^{-\alpha_{1}a}Q(a,\theta)}{h}.$
.

From the Dominated Convergence Theorem we deduce

$\displaystyle\liminf_{h\to
0^{+}}\frac{e^{-\alpha_{1}(a+h)}w(a+h,\theta)-e^{-\alpha_{1}a}w(a,\theta)}{h}$
$\displaystyle\geq-e^{-\alpha_{1}a}\int_{E(a,\theta)}(-u_{r}(a\xi)+\alpha_{1}u(a\xi))\,d\sigma(\xi)$
$\displaystyle\qquad+e^{-\alpha_{1}a}\int_{K(\theta)}(-v_{r}(a\xi)+\alpha_{1}v(a\xi))\,d\sigma(\xi)$
$\displaystyle\qquad+\varepsilon\left(e^{-\alpha_{1}a}(b-a)+\alpha_{1}e^{-\alpha_{1}a}C\theta(\pi-\theta)\right)$
$\displaystyle=\varepsilon\left(e^{-\alpha_{1}a}(b-a)+\alpha_{1}e^{-\alpha_{1}a}C\theta(\pi-\theta)\right),$

where the equality holds courtesy of the Robin boundary conditions. It follows
that for all $h>0$ sufficiently small,

(13)
$w(a+h,\theta)>e^{\alpha_{1}h}w(a,\theta)+\frac{e^{\alpha_{1}h}h\varepsilon(b-a)}{2}>w(a,\theta),$

where the last inequality uses $\alpha_{1}\geq 0$. When $\theta\in(0,\pi)$,
(13) says that $\displaystyle\max_{\partial A^{\bigstar}}w$ is not achieved at
a point with $r=a$, since otherwise the Maximum Principle (12) would be
violated. Likewise, when $\theta=0$ or $\theta=\pi$,
$\displaystyle\max_{\partial A^{\bigstar}}w$ is not achieved at a point with
$r=a$ since according to (13), $w$ assumes strictly larger values at other
points $(r,\theta)\in\partial A^{\bigstar}$ with $r\in(a,b)$.

We next do a similar analysis of the behavior of $w$ at points along the
boundary $\partial A^{\bigstar}$ with $r=b$. To this end, say
$\theta\in[0,\pi]$ and $h>0$ is small. We estimate

$\displaystyle\frac{e^{\alpha_{2}(b-h)}w(b-h,\theta)-e^{\alpha_{2}b}w(b,\theta)}{h}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad=e^{\alpha_{2}(b-h)}\int_{E(b-h,\theta)}\frac{u((b-h)\xi)}{h}\,d\sigma(\xi)-e^{\alpha_{2}b}\int_{E(b,\theta)}\frac{u(b\xi)}{h}\,d\sigma(\xi)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad-e^{\alpha_{2}(b-h)}\int_{K(\theta)}\frac{v((b-h)\xi)}{h}\,d\sigma(\xi)+e^{\alpha_{2}b}\int_{K(\theta)}\frac{v(b\xi)}{h}\,d\sigma(\xi)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad-\varepsilon\frac{e^{\alpha_{2}(b-h)}Q(b-h,\theta)-e^{\alpha_{2}b}Q(b,\theta)}{h}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\geq\int_{E(b,\theta)}\frac{e^{\alpha_{2}(b-h)}u((b-h)\xi)-e^{\alpha_{2}b}u(b\xi)}{h}\,d\sigma(\xi)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-\int_{K(\theta)}\frac{e^{\alpha_{2}(b-h)}v((b-h)\xi)-e^{\alpha_{2}b}v(b\xi)}{h}\,d\sigma(\xi)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-\varepsilon\frac{e^{\alpha_{2}(b-h)}Q(b-h,\theta)-e^{\alpha_{2}b}Q(b,\theta)}{h}.$

As before, we use the Robin boundary conditions to deduce

$\displaystyle\liminf_{h\to
0^{+}}\frac{e^{\alpha_{2}(b-h)}w(b-h,\theta)-e^{\alpha_{2}b}w(b,\theta)}{h}$
$\displaystyle\geq-e^{\alpha_{2}b}\int_{E(b,\theta)}(u_{r}(b\xi)+\alpha_{2}u(b\xi))\,d\sigma(\xi)$
$\displaystyle\qquad+e^{\alpha_{2}b}\int_{K(\theta)}(v_{r}(b\xi)+\alpha_{2}v(b\xi))\,d\sigma(\xi)$
$\displaystyle\qquad+\varepsilon\left(e^{\alpha_{2}b}(b-a)+\alpha_{2}e^{\alpha_{2}b}C\theta(\pi-\theta)\right)$
$\displaystyle=\varepsilon\left(e^{\alpha_{2}b}(b-a)+\alpha_{2}e^{\alpha_{2}b}C\theta(\pi-\theta)\right).$

For all $h>0$ sufficiently small, we therefore have

$w(b-h,\theta)>e^{\alpha_{2}h}w(b,\theta)+\frac{e^{\alpha_{2}h}h\varepsilon(b-a)}{2}>w(b,\theta),$

as $\alpha_{2}\geq 0$. By the same argument as before, the string of strict
inequalities immediately above shows that $\displaystyle\max_{\partial
A^{\bigstar}}w$ cannot be achieved at a point with $r=b$.

Step 3: Boundary Analysis on Horizontal Sides of $\partial A^{\bigstar}$. We
start by analyzing the behavior of $u^{\bigstar}(r,\pi)-Jv(r,\pi)$ in $r$.
Define

$\psi(r)=u^{\bigstar}(r,\pi)-Jv(r,\pi)=\int_{\mathbb{S}^{n-1}}(u(r\xi)-v(r\xi))\,d\sigma(\xi),\quad
r\in[a,b].$

For $a\leq r_{1}<r_{2}\leq b$, note that

$\displaystyle
r_{2}^{n-1}\psi^{\prime}(r_{2})-r_{1}^{n-1}\psi^{\prime}(r_{1})$
$\displaystyle=r_{2}^{n-1}\int_{\mathbb{S}^{n-1}}\left(u_{r}(r_{2}\xi)-v_{r}(r_{2}\xi)\right)\,d\sigma(\xi)$
$\displaystyle\qquad-
r_{1}^{n-1}\int_{\mathbb{S}^{n-1}}\left(u_{r}(r_{1}\xi)-v_{r}(r_{1}\xi)\right)\,d\sigma(\xi)$
$\displaystyle=\int_{\partial A(r_{1},r_{2})}\left(\frac{\partial
u}{\partial\nu}-\frac{\partial v}{\partial\nu}\right)\,dS$
$\displaystyle=\int_{A(r_{1},r_{2})}\left(\Delta u-\Delta v\right)\,dx$
$\displaystyle=\int_{A(r_{1},r_{2})}\left(-f+f^{\\#}\right)\,dx$
$\displaystyle=0,$

since $f$ and $f^{\\#}$ are rearrangements of each other on each sphere
$\\{|x|=r\\}$. It follows that the function $r^{n-1}\psi^{\prime}(r)=k_{1}$ is
constant on $[a,b]$, and so $\psi^{\prime}(r)$ does not change sign on this
interval.

If $k_{1}\neq 0$, note that

$\frac{\partial}{\partial r}w(r,\pi)=k_{1}r^{1-n}-\varepsilon(2r-a-b).$

By taking $\varepsilon$ sufficiently small, we may assume that
$\frac{\partial}{\partial r}w(r,\pi)$ has the same sign as $k_{1}$ on $[a,b]$,
and this makes $w(r,\pi)$ strictly monotone on $[a,b]$. But our work in Step 2
shows that $\displaystyle\max_{\partial A^{\bigstar}}w$ cannot be achieved at
a point on $(\\{a\\}\times[0,\pi])\cup(\\{b\\}\times[0,\pi])$, and so
$\displaystyle\max_{\partial A^{\bigstar}}w$ is achieved at at point where
$\theta=0$. Since $u^{\bigstar}-Jv=0$ when $\theta=0$, (12) implies

$\sup_{A^{\bigstar}}(u^{\bigstar}-Jv)\leq\max_{r\in[a,b]}\left(-\varepsilon(r-a)(r-b)\right)+\varepsilon\|Q\|_{L^{\infty}(A^{\bigstar})}=\frac{\varepsilon}{4}(b-a)^{2}+\varepsilon\|Q\|_{L^{\infty}(A^{\bigstar})}.$

Sending $\varepsilon\to 0^{+}$ gives $u^{\bigstar}-Jv\leq 0$ in
$A^{\bigstar}$.

If $k_{1}=0$, then

(14) $\int_{\mathbb{S}^{n-1}}\left(u(r\xi)-v(r\xi)\right)\,d\sigma(\xi)=k_{2}$

is constant for $r\in[a,b]$. It follows that

$\displaystyle
a^{n-1}\alpha_{1}\int_{\mathbb{S}^{n-1}}\left(u(a\xi)-v(a\xi)\right)\,d\sigma(\xi)$
$\displaystyle=a^{n-1}\alpha_{1}k_{2},$ $\displaystyle
a^{n-1}\int_{\mathbb{S}^{n-1}}\left(u_{r}(a\xi)-v_{r}(a\xi)\right)\,d\sigma(\xi)$
$\displaystyle=0.$

Subtracting these equations and using the Robin boundary conditions yields

$0=\int_{\\{|x|=a\\}}\left(\left(\alpha_{1}u+\frac{\partial
u}{\partial\nu}\right)-\left(\alpha_{1}v+\frac{\partial
v}{\partial\nu}\right)\right)\,dS=a^{n-1}\alpha_{1}k_{2}.$

An identical argument yields the equation

$0=b^{n-1}\alpha_{2}k_{2}.$

If either $\alpha_{1}$ or $\alpha_{2}$ is nonzero, then $k_{2}=0$ and so

(15)
$\int_{\mathbb{S}^{n-1}}\left(u(r\xi)-v(r\xi)\right)\,d\sigma(\xi)=0,\quad
r\in[a,b].$

If $\alpha_{1}=\alpha_{2}=0$, we integrate (14) and use the normalization
assumptions on $u,v$:

$0=\int_{A}(u-v)\,dx=k_{2}\int_{a}^{b}r^{n-1}\,dr$

which implies that $k_{2}=0$, and so equation (15) holds in this case too.
When $k_{1}=0$, we therefore deduce

$w(r,0)=w(r,\pi)=-\varepsilon(r-a)(r-b).$

Using (12), our work in Step 2 implies

$\sup_{A^{\bigstar}}(u^{\bigstar}-Jv)\leq\max_{r\in[a,b]}(-\varepsilon(r-a)(r-b))+\varepsilon\|Q\|_{L^{\infty}(A^{\bigstar})}.$

As before, sending $\varepsilon\to 0$ gives $u^{\bigstar}-Jv\leq 0$ in
$A^{\bigstar}$.

Step 4: Approximation for the General Result. Now say $f\in L^{2}(A)$ is a
general function. Let $u,v$ be as in the statement of Theorem 1.1. Choose a
sequence of test functions $f_{k}\in C^{\infty}_{c}(A)$ where $f_{k}\to f$ in
$L^{2}(A)$. If $\alpha_{1}=\alpha_{2}=0$, we assume $f,f_{k}$ have zero mean.
Since cap symmetrization is a contraction in the $L^{2}$-distance (see Theorem
3 of [9], for example), we also have $f_{k}^{\\#}\to f^{\\#}$ in $L^{2}(A)$.
Let $u_{k}$ and $v_{k}$ be solutions to

$\begin{array}[]{rclccccrclcc}-\Delta u_{k}&=&f_{k}&\text{in}&A,&&&-\Delta
v_{k}&=&f_{k}^{\\#}&\text{in}&A,\\\ \frac{\partial
u_{k}}{\partial\nu}+\alpha_{1}u_{k}&=&0&\text{on}&\\{|x|=a\\},&&&\frac{\partial
v_{k}}{\partial\nu}+\alpha_{1}v_{k}&=&0&\text{on}&\\{|x|=a\\},\\\
\frac{\partial
u_{k}}{\partial\nu}+\alpha_{2}u_{k}&=&0&\text{on}&\\{|x|=b\\},&&&\frac{\partial
v_{k}}{\partial\nu}+\alpha_{2}v_{k}&=&0&\text{on}&\\{|x|=b\\}.\end{array}$

If $\alpha_{1}=\alpha_{2}=0$, we also assume $u_{k},v_{k}$ have zero mean. By
our work in Steps 1-3, for each $k$,

(16) $u_{k}^{\bigstar}(r,\theta)\leq Jv_{k}(r,\theta)$

for $(r,\theta)\in A^{\bigstar}$. If either $\alpha_{1}$ or $\alpha_{2}$ is
nonzero, then (6) implies $u_{k}\to u$ in $L^{2}(A)$. If
$\alpha_{1}=\alpha_{2}=0$, we also have convergence in $L^{2}(A)$; see
Corollary 1.29 of [25], for instance. In particular, we have convergence in
$L^{1}(A)$. Spelled out, we have

$\lim_{k\to\infty}\int_{a}^{b}\int_{\mathbb{S}^{n-1}}\big{|}u_{k}(r\xi)-u(r\xi)\big{|}\,d\sigma(\xi)\
r^{n-1}\,dr=0.$

Thus, by passing to a subsequence, we may assume

(17)
$\lim_{k\to\infty}\int_{\mathbb{S}^{n-1}}\big{|}u_{k}(r\xi)-u(r\xi)\big{|}\,d\sigma(\xi)=0$

for almost every $r\in(a,b)$. Since cap symmetrization is a contraction in the
$L^{1}$-distance (again, see Theorem 3 of [9]), this last equality implies
that for almost every $r\in(a,b)$,

(18) $\lim_{k\to\infty}u_{k}^{\bigstar}(r,\theta)=u^{\bigstar}(r,\theta)$

for each $\theta\in(0,\pi)$. By an identical argument, we may pass to another
subsequence and assume that for almost every $r\in(a,b)$

(19) $\lim_{k\to\infty}Jv_{k}(r,\theta)=Jv(r,\theta)$

for each $\theta\in(0,\pi)$. Finally, sending $k\to\infty$ in (16) and using
(18), (19) gives the theorem’s first conclusion.

Step 5: Conclusions about Convex Means. Note that $Jv\leq v^{\bigstar}$ by
definition. Thus, the theorem’s conclusion about increasing convex means
follows from Proposition 2.5. It remains to show that all convex means of $u$
are dominated by those of $v$, not just the increasing ones, when
$\alpha_{1}=\alpha_{2}=0$.

So suppose $\alpha_{1}=\alpha_{2}=0$, and again say $f\in C_{c}^{\infty}(A)$
is a test function with zero mean. By our work in Step 3,

$r^{n-1}\int_{\mathbb{S}^{n-1}}(u_{r}(r\xi)-v_{r}(r\xi))\,d\sigma(\xi)=k_{1}$

is constant when $r\in[a,b]$. Taking $r=a$ and using the Neumann boundary
conditions shows $k_{1}=0$. By (15), $u$ and $v$ have the same mean on each
sphere $\\{|x|=r\\}$ for $r\in[a,b]$. Approximating a general $f\in L^{2}(A)$
with test functions $f_{k}\in C_{c}^{\infty}(A)$, all with zero mean, we again
let $u_{k}$ and $v_{k}$ solve the Poisson problems from Step 4 normalized to
have zero mean. Then

$\int_{\mathbb{S}^{n-1}}\left(u_{k}(r\xi)-v_{k}(r\xi)\right)\,d\sigma(\xi)=0,\quad
r\in[a,b].$

Since $u_{k}\to u$ and $v_{k}\to v$ in $L^{2}(A)$, for almost every
$r\in(a,b)$, we have

$\displaystyle\lim_{k\to\infty}\int_{\mathbb{S}^{n-1}}u_{k}(r\xi)\,d\sigma(\xi)$
$\displaystyle=\int_{\mathbb{S}^{n-1}}u(r\xi)\,d\sigma(\xi),$
$\displaystyle\lim_{k\to\infty}\int_{\mathbb{S}^{n-1}}v_{k}(r\xi)\,d\sigma(\xi)$
$\displaystyle=\int_{\mathbb{S}^{n-1}}v(r\xi)\,d\sigma(\xi),$

and for such $r$

$\int_{\mathbb{S}^{n-1}}\left(u(r\xi)-v(r\xi)\right)\,d\sigma(\xi)=0.$

Thus in this case, all convex means of $u$ are dominated by those of $v$ for
almost every $r\in(a,b)$ again by Proposition 2.5.

∎

As a consequence of Theorem 1.1, we see that the solution to the symmetrized
problem is itself symmetrized.

######  Corollary 3.1.

The solution $v$ in Theorem 1.1 satisfies $v=v^{\\#}$ a.e.

###### Proof.

First suppose that $f\in C_{c}^{\infty}(A)$ is a test function. Then as in the
proof of Theorem 1.1, $f^{\\#}$ is Lipschitz continuous and $v\in
C^{2}(\overline{A})$. Letting $f^{\\#}$ play the role of $f$, the proof of
Theorem 1.1 gives that $v^{\bigstar}(r,\theta)\leq Jv(r,\theta)$ on
$A^{\bigstar}$. The reverse inequality holds by definition of the star
function, and so $v^{\bigstar}=Jv$ on $A^{\bigstar}$. When the dimension
$n=2$, we use complex notation to write this equality as

$\int_{-\theta}^{\theta}v(re^{i\phi})\,d\phi=\int_{-\theta}^{\theta}v^{\\#}(re^{i\phi})\,d\phi,$

for each $r\in(a,b)$ and $\theta\in(0,\pi)$. Differentiating this equation
with respect to $\theta$ gives

$v(re^{i\theta})+v(re^{-i\theta})=2v^{\\#}(re^{i\theta}).$

We claim that $v(re^{i\theta})=v(re^{-i\theta})$. If this were not the case,
then say $v(re^{i\theta})>v(re^{-i\theta})$ for some $\theta$. We can then
find $\varepsilon>0$ sufficiently small so that

$\int_{-\theta}^{\theta}v(re^{i\phi})\,d\phi<\int_{-\theta+\varepsilon}^{\theta+\varepsilon}v(re^{i\phi})\,d\phi,$

and this would contradict the fact that
$v^{\bigstar}(r,\theta)=\int_{-\theta}^{\theta}v(re^{i\phi})\,d\phi$. Having
verified the claim, we conclude $v(re^{i\theta})=v^{\\#}(re^{i\theta})$.

If $n\geq 3$, we write out the equation $v^{\bigstar}=Jv$ on $A^{\bigstar}$
using spherical coordinates. We have

$\int_{0}^{\theta}\int_{\mathbb{S}^{n-2}}v(ry\sin t,r\cos
t)\,d\sigma(y)\sin^{n-2}t\,dt=\int_{0}^{\theta}\int_{\mathbb{S}^{n-2}}v^{\\#}(ry\sin
t,r\cos t)\,d\sigma(y)\sin^{n-2}t\,dt$

for each $r\in(a,b)$ and $\theta\in(0,\pi)$. Differentiating with respect to
$\theta$ and dividing by $\sin^{n-2}\theta$ gives

(20)
$\int_{\mathbb{S}^{n-2}}v(ry\sin\theta,r\cos\theta)\,d\sigma(y)=\int_{\mathbb{S}^{n-2}}v^{\\#}(ry\sin\theta,r\cos\theta)\,d\sigma(y)$

for each $r\in(a,b)$ and $\theta\in(0,\pi)$. Note that
$v^{\\#}(ry\sin\theta,r\cos\theta)$ is constant in $y\in\mathbb{S}^{n-2}$. We
claim that $v(ry\sin\theta,r\cos\theta)$ is also constant for
$y\in\mathbb{S}^{n-2}$. Say this is not the case, and
$v(ry_{1}\sin\theta,r\cos\theta)>v(ry_{2}\sin\theta,r\cos\theta)$ for
$y_{1},y_{2}\in\mathbb{S}^{n-2}$. Then we can find positive numbers
$\varepsilon_{1},\varepsilon_{2}>0$ where

(21) $\inf_{z_{1}\in B_{1}}v(rz_{1})>\sup_{z_{2}\in B_{2}}v(rz_{2});$

here $B_{1}=B((y_{1}\sin\theta,\cos\theta),\varepsilon_{1})$ and
$B_{2}=B((y_{2}\sin\theta,\cos\theta),\varepsilon_{2})$, where
$B((y_{1}\sin\theta,\cos\theta),\varepsilon_{1})$ denotes the spherical cap
(or ball) on $\mathbb{S}^{n-1}$ centered at $(y_{1}\sin\theta,\cos\theta)$
with radius $\varepsilon_{1}$, and similarly for $B_{2}$. We additionally
assume

$\sigma(B_{1}\setminus K(\theta))=\sigma(B_{2}\cap K(\theta)).$

Write

$K_{1}=(K(\theta)\setminus B_{2}))\cup(B_{1}\setminus K(\theta)).$

Then $\sigma(K_{1})=\sigma(K(\theta))$, and using (21), we have

$\displaystyle\int_{K_{1}}v(r\xi)\,d\sigma(\xi)$
$\displaystyle=\int_{K(\theta)\setminus
B_{2}}v(r\xi)\,d\sigma(\xi)+\int_{B_{1}\setminus
K(\theta)}v(r\xi)\,d\sigma(\xi)$ $\displaystyle>\int_{K(\theta)\setminus
B_{2}}v(r\xi)\,d\sigma(\xi)+\int_{B_{2}\cap K(\theta)}v(r\xi)\,d\sigma(\xi)$
$\displaystyle=\int_{K(\theta)}v(r\xi)\,d\sigma(\xi),$

which contradicts $v^{\bigstar}=Jv$. We conclude that
$v(ry\sin\theta,r\cos\theta)$ is constant for $y\in\mathbb{S}^{n-2}$. Equation
(20) implies

$v(ry\sin\theta,r\cos\theta)=v^{\\#}(ry\sin\theta,r\cos\theta)$

for each $y\in\mathbb{S}^{n-2}$, $r\in(a,b)$, and $\theta\in(0,\pi)$. That is,
$v=v^{\\#}$.

If $f\in L^{2}(A)$ is a general function, pick a sequence of test functions
$f_{k}\in C_{c}^{\infty}(A)$ where $f_{k}\to f$ in $L^{2}(A)$. If
$\alpha_{1}=\alpha_{2}=0$, we assume each $f_{k}$ has zero mean. As cap
symmetrization is a contraction in the $L^{2}$-distance, $f_{k}^{\\#}\to
f^{\\#}$ in $L^{2}(A)$. As before, each $f_{k}^{\\#}$ is Lipschitz continuous
on $\overline{A}$. Let $v_{k}$ solve

$\begin{array}[]{rclcc}-\Delta v_{k}&=&f_{k}^{\\#}&\text{in}&A,\\\
\frac{\partial
v_{k}}{\partial\nu}+\alpha_{1}v_{k}&=&0&\text{on}&\\{|x|=a\\},\\\
\frac{\partial
v_{k}}{\partial\nu}+\alpha_{2}v_{k}&=&0&\text{on}&\\{|x|=b\\}.\end{array}$

If $\alpha_{1}=\alpha_{2}=0$, we assume each $v_{k}$ has zero mean. As in the
proof of Theorem 1.1, $v_{k}\to v$ in $L^{2}(A)$. By the work above,
$v_{k}=v_{k}^{\\#}$ for each $k$. Again, cap symmetrization is a contraction
in the $L^{2}$-distance, and so $v_{k}^{\\#}\to v^{\\#}$ in $L^{2}(A)$. We
deduce that $v=v^{\\#}$ a.e., as desired. ∎

######  Corollary 3.2.

Let $u$ and $v$ be as in Theorem 1.1. If $\alpha_{1},\alpha_{2}>0$ and $f\geq
0$ or if $\alpha_{1}=\alpha_{2}=0$, then for almost every $r\in(a,b)$, the
radial slice functions $u(r\ \\!\cdot)$ and $v(r\ \\!\cdot)$ satisfy

$\|u(r\ \\!\cdot)\|_{L^{p}(\mathbb{S}^{n-1})}\leq\|v(r\
\\!\cdot)\|_{L^{p}(\mathbb{S}^{n-1})},\quad 1\leq p\leq\infty$

and consequently

$\|u\|_{L^{p}(A)}\leq\|v\|_{L^{p}(A)},\quad 1\leq p\leq\infty.$

If $\alpha_{1}=\alpha_{2}=0$, then for almost every $r\in(a,b)$,

$\displaystyle\underset{\mathbb{S}^{n-1}}{\operatorname{\textup{ess\ sup}}}\
u(r\ \\!\cdot)$
$\displaystyle\leq\underset{\mathbb{S}^{n-1}}{\operatorname{\textup{ess\
sup}}}\ v(r\ \\!\cdot),$
$\displaystyle\underset{\mathbb{S}^{n-1}}{\operatorname{\textup{ess\ inf}}}\
u(r\ \\!\cdot)$
$\displaystyle\geq\underset{\mathbb{S}^{n-1}}{\operatorname{\textup{ess\
inf}}}\ v(r\ \\!\cdot),$
$\displaystyle\underset{\mathbb{S}^{n-1}}{\operatorname{\textup{osc}}}\ u(r\
\\!\cdot)$
$\displaystyle\leq\underset{\mathbb{S}^{n-1}}{\operatorname{\textup{osc}}}\
v(r\ \\!\cdot).$

Consequently

$\displaystyle\underset{A}{\operatorname{\textup{ess\ sup}}}\ u$
$\displaystyle\leq\underset{A}{\operatorname{\textup{ess\ sup}}}\ v,$
$\displaystyle\underset{A}{\operatorname{\textup{ess\ inf}}}\ u$
$\displaystyle\geq\underset{A}{\operatorname{\textup{ess\ inf}}}\ v,$
$\displaystyle\underset{A}{\operatorname{\textup{osc}}}\ u$
$\displaystyle\leq\underset{A}{\operatorname{\textup{osc}}}\ v.$

###### Proof.

First suppose that $\alpha_{1},\alpha_{2}>0$ and $f\geq 0$. If $f\in
C^{\infty}_{c}(A)\geq 0$ is a test function, then as in the proof of Theorem
1.1, $u,v\in C^{2}(\overline{A})$. In this case, $u$ and $v$ are superharmonic
functions and hence obey a Minimum Principle. That is, $u$ and $v$ are
minimized along the boundary $\partial A$. Suppose that $u$ is minimized at
the point $x_{0}\in\partial A$ and say $|x_{0}|=a$. Then by the Robin boundary
condition along $\\{|x|=a\\}$,

$\frac{\partial u}{\partial\nu}(x_{0})=-\alpha_{1}u(x_{0}).$

This equation implies that $u(x_{0})\geq 0$, since otherwise, $u$ would assume
a value strictly smaller than $u(x_{0})$ in the interior of $A$. A similar
consideration applies if $|x_{0}|=b$ and also to the function $v$. We deduce
$u,v\geq 0$ on $A$. If $0\leq f\in L^{2}(A)$ is a general function, we may
choose $0\leq f_{k}\in C_{c}^{\infty}(A)$ test functions where $f_{k}\to f$ in
$L^{2}(A)$. Letting $u_{k}$ and $v_{k}$ solve the Poisson problems from Step 4
of the proof of Theorem 1.1, we have $u_{k},v_{k}\geq 0$ for each $k$. By
passing to an appropriate subsequence, we may assume $u_{k}\to u$ and
$v_{k}\to v$ pointwise a.e. on $A$. We conclude that $u,v\geq 0$ a.e. on $A$.
In particular, $u,v\geq 0$ a.e on the sphere $\\{|x|=r\\}$ for a.e.
$r\in(a,b)$. The corollary’s conclusions in this case about $L^{p}$-norms of
slice functions follow from Theorem 1.1 and Corollary 2.6, and the conclusions
on shells clearly follow.

If $\alpha_{1}=\alpha_{2}=0$ and $f\in L^{2}(A)$ has zero mean, the proof of
Theorem 1.1 shows that for almost every $r\in(a,b)$,

$\int_{\mathbb{S}^{n-1}}u(r\xi)\,d\sigma(\xi)=\int_{\mathbb{S}^{n-1}}v(r\xi)\,d\sigma(\xi).$

The equality above together with Theorem 1.1 and Corollary 2.6 give the
claimed inequalities about $L^{p}$-norms, $\operatorname{\textup{ess\ sup}}$,
$\operatorname{\textup{ess\ inf}}$, and $\operatorname{\textup{osc}}$. The
remaining conclusions clearly follow. ∎

We next prove our paper’s second main result. But before we do so, we briefly
recall how cap symmetrization is defined for functions on balls, as the
definition has one technicality that does not arise in the shell setting. In
our result, recall that we take $B=\\{x\in\mathbb{R}^{n}:|x|<b\\}$ with
$0<b<\infty$. The cap symmetrization of a function $f\in L^{1}(B)$ is defined
as follows. When $0<r<b$ and the slice function
$f^{r}:\mathbb{S}^{n-1}\rightarrow\mathbb{R}$ belongs to
$L^{1}(\mathbb{S}^{n-1})$, we define

$f^{\\#}(r\xi)=(f^{r})^{\\#}(\xi),$

where $(f^{r})^{\\#}$ denotes the spherical rearrangement of the slice
function $f^{r}$. If $f^{r}\notin L^{1}(\mathbb{S}^{n-1})$, we leave $f^{\\#}$
undefined on the sphere $\\{|x|=r\\}$. If $f(0)$ is defined, we also define
$f^{\\#}(0)=f(0)$. When $f(0)$ is not defined, $f^{\\#}(0)$ is likewise left
undefined. Note that the definitions of $Jf$ and $f^{\bigstar}$ in this
setting (as defined in the Introduction), agree with Definition 2.9 and
equation (7) with $a=0$. Here now is our second main result.

###### Proof of Theorem 1.2..

First say $f\in C_{c}^{\infty}(B)$ is a test function. As before, $u,v\in
C^{2}(\overline{B})$. Say $\delta,\epsilon>0$ and let $A_{\delta}=A(\delta,b)$
denote a spherical shell with inner radius $\delta$ and outer radius $b$. Let
$w$ be defined as in the proof of Theorem 1.1:

$w(r,\theta)=u^{\bigstar}(r,\theta)-Jv(r,\theta)-\varepsilon Q(r,\theta),$

where

$Q(r,\theta)=r(r-b)+C\theta(\pi-\theta),$

and again $C$ is chosen so that

$\Delta^{\bigstar}Q(r,\theta)=2+\frac{n-1}{r}(2r-b)-\frac{C}{r^{2}}\left(2+(n-2)(\pi-2\theta)\cot\theta\right)\leq
0$

on $B^{\bigstar}$. As before, we have

$-\Delta^{\bigstar}w\leq 0$

weakly in $A_{\delta}^{\bigstar}$. By the Maximum Principle,

(22) $\sup_{A_{\delta}^{\bigstar}}w\leq\max_{\partial
A_{\delta}^{\bigstar}}w.$

Defining

$\psi(r)=\int_{\mathbb{S}^{n-1}}(u(r\xi)-v(r\xi))\,d\sigma(\xi),\qquad
r\in(0,b),$

the proof of Theorem 1.1 shows that $r^{n-1}\psi^{\prime}(r)=k_{1}$ is
constant on $(0,b)$. Letting $r\to 0^{+}$ and using $u,v\in
C^{2}(\overline{B})$, we conclude $k_{1}=0$. Thus

(23) $\int_{\mathbb{S}^{n-1}}(u(r\xi)-v(r\xi))\,d\sigma(\xi)=k_{2}$

is constant on $(0,b)$. By continuity, this equation holds on $[0,b]$. We
therefore have

$\displaystyle
b^{n-1}\alpha\int_{\mathbb{S}^{n-1}}\left(u(b\xi)-v(b\xi)\right)\,d\sigma(\xi)$
$\displaystyle=b^{n-1}\alpha k_{2},$ $\displaystyle
b^{n-1}\int_{\mathbb{S}^{n-1}}\left(u_{r}(b\xi)-v_{r}(b\xi)\right)\,d\sigma(\xi)$
$\displaystyle=0.$

Adding these equations and using the Robin boundary conditions shows

$0=\int_{\\{|x|=b\\}}\left(\left(\alpha u+\frac{\partial
u}{\partial\nu}\right)-\left(\alpha v+\frac{\partial
v}{\partial\nu}\right)\right)\,dS=b^{n-1}\alpha k_{2}.$

If $\alpha\neq 0$, we conclude $k_{2}=0$, and so

(24) $\int_{\mathbb{S}^{n-1}}(u(r\xi)-v(r\xi))\,d\sigma(\xi)=0,\qquad
r\in[0,b].$

If $\alpha=0$, we integrate (23) and use our normalization assumptions on
$u,v$:

$0=\int_{B}(u-v)\,dx=k_{2}\int_{0}^{b}r^{n-1}\,dr,$

so $k_{2}=0$ and again (24) holds.

Step 2 in the proof of Theorem 1.1 shows that $\displaystyle\max_{\partial
A_{\delta}^{\bigstar}}w$ cannot be achieved at a point with $r=b$. Writing

$T_{\delta}=(\\{\delta\\}\times[0,\pi])\cup((\delta,b)\times\\{0\\})\cup((\delta,b)\times\\{\pi\\}),$

the Maximum Principle (22) becomes

$\sup_{A_{\delta}^{\bigstar}}w\leq\sup_{T_{\delta}}w.$

Letting $\delta\to 0^{+}$, we see

(25) $\sup_{B^{\bigstar}}w\leq\sup_{T_{0}}w.$

Now $u^{\bigstar}-Jv=0$ by definition when $\theta=0$ and (24) implies that
$u^{\bigstar}-Jv=0$ when $\theta=\pi$. Moreover, (24) implies $u(0)=v(0)$, and
so $u^{\bigstar}-Jv=0$ when $r=0$. Thus, (25) implies

$\sup_{B^{\bigstar}}(u^{\bigstar}-Jv)\leq\sup_{T_{0}}(-\varepsilon
Q)+\varepsilon\|Q\|_{L^{\infty}(B^{\bigstar})}.$

Sending $\varepsilon\to 0^{+}$ shows $u^{\bigstar}-Jv\leq 0$ in
$B^{\bigstar}$. The approximation argument from the proof of Theorem 1.1
applies almost verbatim to our new setting, and shows that for a given
function $f\in L^{2}(B)$, for almost every $r\in(0,b)$, the inequality
$u^{\bigstar}(r,\theta)\leq Jv(r,\theta)$ holds for each $\theta\in(0,\pi)$.
Equation (24) and the approximation argument in the proof of Theorem 1.1 also
shows that for general $f\in L^{2}(B)$,

(26) $\int_{\mathbb{S}^{n-1}}(u(r\xi)-v(r\xi))\,d\sigma(\xi)=0$

for almost every $r\in(0,b)$. Thus, the theorem’s conclusion about convex
means follows from Proposition 2.5. ∎

The proof of Corollary 3.1 similarly carries over verbatim to give the
following result.

######  Corollary 3.3.

The solution $v$ in Theorem 1.2 satisfies $v=v^{\\#}$ a.e.

Equation (26), Theorem 1.2, and Corollary 2.6 give the following analogue of
Corollary 3.2 for the ball setting.

######  Corollary 3.4.

Let $u$ and $v$ be as in Theorem 1.2. Then for almost every $r\in(0,b)$, we
have

$\|u(r\ \\!\cdot)\|_{L^{p}(\mathbb{S}^{n-1})}\leq\|v(r\
\\!\cdot)\|_{L^{p}(\mathbb{S}^{n-1})},\quad 1\leq p\leq\infty,$

and

$\displaystyle\underset{\mathbb{S}^{n-1}}{\operatorname{\textup{ess\ sup}}}\
u(r\ \\!\cdot)$
$\displaystyle\leq\underset{\mathbb{S}^{n-1}}{\operatorname{\textup{ess\
sup}}}\ v(r\ \\!\cdot),$
$\displaystyle\underset{\mathbb{S}^{n-1}}{\operatorname{\textup{ess\ inf}}}\
u(r\ \\!\cdot)$
$\displaystyle\geq\underset{\mathbb{S}^{n-1}}{\operatorname{\textup{ess\
inf}}}\ v(r\ \\!\cdot),$
$\displaystyle\underset{\mathbb{S}^{n-1}}{\operatorname{\textup{osc}}}\ u(r\
\\!\cdot)$
$\displaystyle\leq\underset{\mathbb{S}^{n-1}}{\operatorname{\textup{osc}}}\
v(r\ \\!\cdot).$

Consequently,

$\|u\|_{L^{p}(B)}\leq\|v\|_{L^{p}(B)},\quad 1\leq p\leq\infty,$

and

$\displaystyle\underset{B}{\operatorname{\textup{ess\ sup}}}\ u$
$\displaystyle\leq\underset{B}{\operatorname{\textup{ess\ sup}}}\ v,$
$\displaystyle\underset{B}{\operatorname{\textup{ess\ inf}}}\ u$
$\displaystyle\geq\underset{B}{\operatorname{\textup{ess\ inf}}}\ v,$
$\displaystyle\underset{B}{\operatorname{\textup{osc}}}\ u$
$\displaystyle\leq\underset{B}{\operatorname{\textup{osc}}}\ v.$

We next prove our paper’s third main result.

###### Proof of Theorem 1.3.

Again, the proof is rather long, so we break it down into manageable steps.

Step 1: Maximum Principle. If $\alpha>0$, let $Q_{1}$ denote the solution to
the Robin problem

$\begin{array}[]{rclcc}\Delta Q_{1}&=&0&\text{in}&D,\\\ \frac{\partial
Q_{1}}{\partial\nu}+\alpha Q_{1}&=&1&\text{on}&\partial D.\end{array}$

If $\alpha=0$, let $Q_{1}$ be the solution to the Neumann problem

$\begin{array}[]{rclcc}\Delta Q_{1}&=&\frac{\mathcal{H}^{n-2}(\partial
D)}{\mathcal{L}^{n-1}(D)}&\text{in}&D,\\\ \frac{\partial
Q_{1}}{\partial\nu}&=&1&\text{on}&\partial D,\end{array}$

where $Q_{1}$ is additively normalized to have zero mean. Here we have written
$\mathcal{H}^{n-2}$ for $(n-2)$-dimensional Hausdorff measure and
$\mathcal{L}^{n-1}$ for $(n-1)$-dimensional Lebesgue measure. By Theorem 6.31
in [22] and [31], $Q_{1}$ belongs to $C^{2}(\overline{D})$. We are led to
define a function $Q$ by

$Q(x,y)=Q_{1}(x)+Cy(\ell-y),$

for $(x,y)\in\Omega$, where $C>0$ is chosen large enough to guarantee

$\Delta Q=\Delta_{x}Q_{1}-2C\leq 0$

on $\Omega$. Define for $\varepsilon>0$,

$w(x,y)=u^{\bigstar}(x,y)-Jv(x,y)-\varepsilon Q(x,y)$

for $(x,y)\in\Omega$.

In the weak sense, we have

$\displaystyle-\Delta w$ $\displaystyle=-\Delta u^{\bigstar}+\Delta
Jv+\varepsilon\Delta Q$ $\displaystyle\leq f^{\bigstar}+J\Delta
v+\varepsilon\Delta Q$
$\displaystyle=f^{\bigstar}-f^{\bigstar}+\varepsilon\Delta Q$
$\displaystyle\leq 0,$

where the first inequality follows from Proposition 2.14 and Theorem 2.15, the
second equality follows from (11), and the second inequality follows from our
choice of $C$. Thus $w$ is weakly subharmonic in $\Omega$ so by the Maximum
Principle (Theorem 1 in [30])

(27) $\sup_{\Omega}w\leq\max_{\partial\Omega}w.$

Step 2: Boundary Analysis on $\partial\Omega_{2}$. We first show that
$\displaystyle\max_{\partial\Omega}w$ is not achieved at a point of
$\partial\Omega_{2}$. Fix $(x,y)\in\partial\Omega_{2}$, so that $x\in\partial
D$ and $y\in[0,\ell]$. Let $n=n(x)$ denote the inner normal vector at $x$ to
$\partial D$. Let $E(x)$ denote a measurable subset of $(0,\ell)$ of length
$y$ for which the $\sup$ defining $u^{\bigstar}(x,y)$ is achieved. By
translating, we assume $0\in D$. Write $r=|x|$. We then have for $h>0$,

$\displaystyle\frac{e^{-\alpha(r+h)}w(x+hn,y)-e^{-\alpha
r}w(x,y)}{h}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad=e^{-\alpha(r+h)}\int_{E(x+hn)}\frac{u(x+hn,t)}{h}\,dt-e^{-\alpha
r}\int_{E(x)}\frac{u(x,t)}{h}\,dt$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-e^{-\alpha(r+h)}\int_{0}^{y}\frac{v(x+hn,t)}{h}\,dt+e^{-\alpha
r}\int_{0}^{y}\frac{v(x,t)}{h}\,dt$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\varepsilon\frac{e^{-\alpha(r+h)}Q(x+hn,y)-e^{-\alpha
r}Q(x,y)}{h}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\geq\int_{E(x)}\frac{e^{-\alpha(r+h)}u(x+hn,t)-e^{-\alpha
r}u(x,t)}{h}\,dt$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\int_{0}^{y}\frac{e^{-\alpha(r+h)}v(x+hn,t)-e^{-\alpha
r}v(x,t)}{h}\,dt$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\varepsilon\frac{e^{-\alpha(r+h)}Q(x+hn,y)-e^{-\alpha
r}Q(x,y)}{h}.$

Using the Dominated Convergence Theorem, we therefore have

$\displaystyle\underset{h\rightarrow 0^{+}}{\textup{lim
inf}}\frac{e^{-\alpha(r+h)}w(x+hn,y)-e^{-\alpha r}w(x,y)}{h}$
$\displaystyle\geq-e^{-\alpha r}\int_{E(x)}\left(\frac{\partial
u}{\partial\nu}(x,t)+\alpha u(x,t)\right)\,dt$ $\displaystyle\qquad+e^{-\alpha
r}\int_{0}^{y}\left(\frac{\partial v}{\partial\nu}(x,t)+\alpha
v(x,t)\right)\,dt$ $\displaystyle\qquad+\varepsilon e^{-\alpha
r}\left(1+\alpha Cy(\ell-y)\right)$ $\displaystyle=\varepsilon e^{-\alpha
r}\left(1+\alpha Cy(\ell-y)\right),$

where the last equality holds courtesy of the Robin boundary conditions on
$\partial\Omega_{2}$. For all $h>0$ sufficiently small, we therefore have

(28) $w(x+hn,y)>e^{\alpha h}w(x,y)+\frac{e^{\alpha h}h\varepsilon}{2}>w(x,y),$

as $\alpha\geq 0$. When $y\in(0,\ell)$, inequality (28) says that
$\displaystyle\max_{\partial\Omega}w$ is not achieved at a point
$(x,y)\in\partial\Omega_{2}$, since otherwise the Maximum Principle (27) would
be violated. Similarly, when $y=0$ or $y=\ell$,
$\displaystyle\max_{\partial\Omega}w$ is not achieved at a point
$(x,y)\in\partial\Omega_{2}$ since by (28), $w$ assumes strictly larger values
at points of $\partial\Omega_{1}$.

Step 3: Boundary Analysis on $\partial\Omega_{1}$. We start by defining

$\psi(x)=\int_{0}^{\ell}(u(x,t)-v(x,t))\,dt,\qquad x\in D.$

Using the Dominated Convergence Theorem and our assumption that $u,v\in
C^{2}(\overline{\Omega})$, we see that $\psi$ is harmonic in $D$:

$\displaystyle\Delta\psi(x)$
$\displaystyle=\int_{0}^{\ell}(\Delta_{x}u(x,t)-\Delta_{x}v(x,t))\,dt$
$\displaystyle=\int_{0}^{\ell}(-u_{yy}(x,t)-f(x,t)+v_{yy}(x,t)+f^{\\#}(x,t))\,dt$
$\displaystyle=\int_{0}^{\ell}(-f(x,t)+f^{\\#}(x,t))\,dt-
u_{y}(x,\ell)+u_{y}(x,0)+v_{y}(x,\ell)-v_{y}(x,0)$ $\displaystyle=0.$

This last equality holds by the Neumann boundary conditions of $u,v$ along
$\partial\Omega_{1}$ and since $f$ and $f^{\\#}$ are rearrangements of each
other.

Taking $x\in\partial D$, we next investigate the Robin boundary condition of
$\psi$:

$\frac{\partial\psi}{\partial\nu}(x)+\alpha\psi(x)=\int_{0}^{\ell}\left(\left(\frac{\partial
u}{\partial\nu}(x,t)+\alpha u(x,t)\right)-\left(\frac{\partial
v}{\partial\nu}(x,t)+\alpha v(x,t)\right)\right)\,dt=0,$

since $u$ and $v$ themselves satisfy Robin boundary conditions on
$\partial\Omega_{2}$. If $\alpha=0$, note that $\psi$ has zero mean by our
normalization assumption on $u,v$:

$\int_{D}\psi(x)\,dx=\int_{\Omega}(u-v)\,dx\,dy=0.$

The function $\psi$ therefore solves the problem

$\begin{array}[]{rclcc}\Delta\psi&=&0&\text{in}&D,\\\
\frac{\partial\psi}{\partial\nu}+\alpha\psi&=&0&\text{on}&\partial
D,\end{array}$

and when $\alpha=0$, $\int_{D}\psi\,dx=0$. Integrating by parts, $\psi$
satisfies the equation

$\int_{D}|\nabla\psi|^{2}\,dx=-\alpha\int_{\partial D}\psi^{2}\,dS,$

so we deduce $\psi\equiv 0$ in $D$. Since $u^{\bigstar}-Jv=0$ when $y=0$, we
conclude that

$w(x,0)=w(x,\ell)=-\varepsilon Q_{1}(x)$

for $x\in D$. Our work from Steps 2 and 3 and the Maximum Principle (27)
together give

$\sup_{\Omega}(u^{\bigstar}-Jv)\leq\varepsilon\|Q\|_{L^{\infty}(\Omega)}+\sup_{D}(-\varepsilon
Q_{1}).$

Sending $\varepsilon\to 0^{+}$ gives $u^{\bigstar}-Jv\leq 0$ in $\Omega$. Note
that since $\psi(x)\equiv 0$ on $D$, for each $x\in D$, the slice functions
$u(x,\cdot)$ and $v(x,\cdot)$ have the same mean on $(0,\ell)$. Thus
Proposition 2.5 gives the theorem’s claims on the convex means of $u$ and $v$.
∎

As before, we easily deduce that the solution of the symmetrized problem is
itself symmetrized.

######  Corollary 3.5.

Let $v$ be as in Theorem 1.3. Then $v=v^{\\#}$ on $\Omega$.

###### Proof.

Letting $f=f^{\\#}$, Theorem 1.3 gives that $v^{\bigstar}\leq Jv$ on $\Omega$.
Since the reverse inequality holds by definition, we have $v^{\bigstar}=Jv$ on
$\Omega$. Spelled out, this means that for each $x\in D$,

$\int_{0}^{y}v^{\\#}(x,t)\,dt=\int_{0}^{y}v(x,t)\,dt$

for each $y\in(0,\ell)$. Differentiating the above equation with respect to
$y$ shows that $v(x,y)=v^{\\#}(x,y)$ on $\Omega$, as desired. ∎

Finally, we have an analogue of Corollaries 3.2 and 3.4 for the cylindrical
setting.

######  Corollary 3.6.

Let $u$ and $v$ be as in Theorem 1.3. Then for each $x\in D$, the slice
functions $u(x,\cdot)$ and $v(x,\cdot)$ satisfy

$\|u(x,\cdot)\|_{L^{p}(0,\ell)}\leq\|v(x,\cdot)\|_{L^{p}(0,\ell)},\quad 1\leq
p\leq\infty.$

Moreover, for each $x\in D$,

$\displaystyle\underset{(0,\ell)}{\sup}\ u(x,\cdot)$
$\displaystyle\leq\underset{(0,\ell)}{\sup}\ v(x,\cdot),$
$\displaystyle\underset{(0,\ell)}{\inf}\ u(x,\cdot)$
$\displaystyle\geq\underset{(0,\ell)}{\inf}\ v(x,\cdot),$
$\displaystyle\underset{(0,\ell)}{\operatorname{\textup{osc}}}\ u(x,\cdot)$
$\displaystyle\leq\underset{(0,\ell)}{\operatorname{\textup{osc}}}\
v(x,\cdot).$

Consequently,

$\|u\|_{L^{p}(\Omega)}\leq\|v\|_{L^{p}(\Omega)},\quad 1\leq p\leq\infty,$

and

$\displaystyle\underset{\Omega}{\sup}\ u$
$\displaystyle\leq\underset{\Omega}{\sup}\ v,$
$\displaystyle\underset{\Omega}{\inf}\ u$
$\displaystyle\geq\underset{\Omega}{\inf}\ v,$
$\displaystyle\underset{\Omega}{\operatorname{\textup{osc}}}\ u$
$\displaystyle\leq\underset{\Omega}{\operatorname{\textup{osc}}}\ v.$

###### Proof.

The proof of Theorem 1.3 shows that for each $x\in D$,

$\int_{0}^{\ell}u(x,t)\,dt=\int_{0}^{\ell}v(x,t)\,dt.$

This equality together with Theorem 1.3 and Corollary 2.6 give the claimed
inequalities on the slice functions of $u$ and $v$, and as before, the
subsequent inequalities follow. ∎

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