# Minimality of vortex solutions to Ginzburg–Landau type systems for gradient
fields in the unit ball in dimension $N\geq 4$

Radu Ignat , Mickael Nahon and Luc Nguyen Institut de Mathématiques de
Toulouse, UMR 5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse
Cedex 9, France. Email<EMAIL_ADDRESS>Jean
Kuntzmann, UMR 5224, Université Grenoble Alpes, 700 avenue centrale, 38401
Saint Martin d’Hères. Email<EMAIL_ADDRESS>Institute and St Edmund Hall, University of Oxford, Andrew Wiles Building,
Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom.
Email<EMAIL_ADDRESS>
###### Abstract

We prove that the degree-one vortex solution is the unique minimizer for the
Ginzburg–Landau functional for gradient fields (that is, the Aviles–Giga
model) in the unit ball $B^{N}$ in dimension $N\geq 4$ and with respect to its
boundary value. A similar result is also proved for $\mathbb{S}^{N}$-valued
maps in the theory of micromagnetics. Two methods are presented. The first
method is an extension of the analogous technique previously used to treat the
unconstrained Ginzburg–Landau functional in dimension $N\geq 7$. The second
method uses a symmetrization procedure for gradient fields such that the
$L^{2}$-norm is invariant while the $L^{p}$-norm, $2<p<\infty$, and the
$H^{1}$-norm are lowered.

Keywords: Minimality, vortex solutions, gradient fields, Ginzburg–Landau,
Aviles–Giga, Hardy’s inequality, symmetrization.

MSC 2020: 26D10, 35J20, 35J50, 35Q56.

###### Contents

  1. 1 Introduction
    1. 1.1 The Aviles–Giga model
    2. 1.2 The $\mathbb{S}^{N}$-valued Ginzburg–Landau model
    3. 1.3 The extended model
    4. 1.4 Ideas of the proofs
  2. 2 First proof of main results
    1. 2.1 Proof of Theorem 3
    2. 2.2 Proof of Theorems 1 and 2
  3. 3 Symmetrization and second proof of main results in dimension $N\geq 5$
    1. 3.1 A symmetrization of scalar functions
    2. 3.2 A symmetrization of gradient fields and proof of Theorem 5
    3. 3.3 Second proof of Theorems 1, 2 and 3 in dimension $N\geq 5$
    4. 3.4 Symmetry for solutions to a nonlinear eigenvalue problem
  4. A The negativity of $F_{\epsilon}$ in dimension $N\in\\{2,3\\}$

##  1 Introduction

Let $B^{N}$ be the unit ball in $\mathbb{R}^{N}$. Consider the Ginzburg–Landau
(GL) functional

$E^{GL}_{\epsilon}[U]=\int_{B^{N}}\Big{[}\frac{1}{2}|\nabla
U|^{2}+\frac{1}{2\epsilon^{2}}W(1-|U|^{2})\Big{]}\,dx,$

where $\epsilon>0$, $W(t)=\frac{t^{2}}{2}$ and $U$ belongs to the set

$\mathcal{A}^{GL}=\\{U\in H^{1}(B^{N},\mathbb{R}^{N}):U(x)=x\text{ on
}\partial B^{N}\\}.$

The functional $E^{GL}_{\epsilon}$ has a unique radially symmetric critical
point in $\mathcal{A}^{GL}$ of the form

$U_{\epsilon}(x)=f_{\epsilon}(r)\frac{x}{r}\in\mathcal{A}^{GL},\quad r=|x|,$
(1.1)

where the profile $f_{\epsilon}$ is the unique solution to the ODE (see e.g.
[24, 28])

$\begin{cases}-f_{\epsilon}^{\prime\prime}(r)-\frac{N-1}{r}f_{\epsilon}^{\prime}(r)+\frac{N-1}{r^{2}}f_{\epsilon}(r)=\frac{1}{\epsilon^{2}}f_{\epsilon}(r)W^{\prime}(1-f_{\epsilon}(r)^{2}),\\\
f_{\epsilon}(0)=0,\ f_{\epsilon}(1)=1.\end{cases}$ (1.2)

Moreover $f_{\epsilon}>0$ and $f_{\epsilon}^{\prime}>0$ in $(0,1)$.

The map $U_{\epsilon}$ in (1.1), called the _( $\mathbb{R}^{N}$-valued)
Ginzburg–Landau vortex solution_ of topological degree one, can be considered
as a regularization of the singular harmonic map
$n:B^{N}\rightarrow\mathbb{S}^{N-1}$ given by $n(x)=\frac{x}{|x|}$ for every
$x\in B^{N}$, which is the unique minimizing ${\mathbb{S}}^{N-1}$-valued
harmonic map for $N\geq 3$ with respect to the boundary condition $n(x)=x$ on
$\partial B^{N}$ (see Brezis, Coron and Lieb [9] and Lin [39]). The question
about the minimality of $U_{\epsilon}$ for any $\epsilon>0$ was raised in
dimension $N=2$ in Bethuel, Brezis and Hélein [6, Problem 10, page 139], and
in higher dimensions in Brezis [8, Section 2]. It is not hard to see that,
when $\epsilon$ is sufficiently large, $E^{GL}_{\epsilon}$ is strictly convex
and so $U_{\epsilon}$ is the unique bounded critical point of
$E^{GL}_{\epsilon}$ in $\mathcal{A}^{GL}$ for every $N\geq 2$ (see e.g. [6] or
[32, Remark 3.3]). In dimension $N=2$, Pacard and Rivière showed in [47] that,
for small $\epsilon>0$, $U_{\epsilon}$ is the unique critical point of
$E^{GL}_{\epsilon}$ in $\mathcal{A}^{GL}$; however, whether $u_{\epsilon}$ is
the unique minimizer of $E_{\epsilon}^{GL}$ for all $\epsilon>0$ remains an
open question. In dimensions $N\geq 7$, this question was answered positively
in recent works of Ignat, Nguyen, Slastikov and Zarnescu [31, 32]:
$U_{\epsilon}$ is the unique minimizer of $E^{GL}_{\epsilon}$ in
$\mathcal{A}^{GL}$ _for every_ $\epsilon>0$. It is not known whether
$U_{\epsilon}$ minimizes $E^{GL}_{\epsilon}$ in $\mathcal{A}^{GL}$ in
dimensions $3\leq N\leq 6$ when $\epsilon$ is small. However, it is known that
for every $\epsilon>0$, $U_{\epsilon}$ is a local minimizer of
$E^{GL}_{\epsilon}$ in $\mathcal{A}^{GL}$ – for dimension $N=2$, see Mironescu
[43] and also Lieb and Loss [38]; for dimension $3\leq N\leq 6$, see Ignat and
Nguyen [26].

We note also that, when the domain is the whole space $\mathbb{R}^{N}$
instead, the minimality (in the sense of De Giorgi) of the vortex solution is
available: see Mironescu [44], Millot and Pisante [42] and Pisante [48]. See
also [12, 21, 22, 46] for studies on stability issues.

The main aim of this paper is to show that in dimensions $4\leq N\leq 6$ and
_for every_ $\epsilon>0$, $U_{\epsilon}$ is the unique minimizer of
$E^{GL}_{\epsilon}$ relative to the set of gradient field configurations in
$\mathcal{A}^{GL}$ (this is often referred to as the Aviles–Giga model).

###  1.1 The Aviles–Giga model

Consider a general non-negative convex $C^{2}$ potential
$W:(-\infty,1]\to[0,\infty)$ such that $W(0)=0$ and for every $\epsilon>0$,
the Ginzburg–Landau energy $E^{GL}_{\epsilon}(U)$ restricted to gradient
fields

$U=\nabla u\in H^{1}(B^{N},\mathbb{R}^{N})\quad\textrm{ such that }\quad
U|_{\partial B^{N}}=Id.$

Within a suitable rescaling (i.e., $\epsilon E^{GL}_{\epsilon}(\nabla u)$),
this is the so-called Aviles–Giga model (introduced with the standard
potential $W(t)=t^{2}/2$).

Note that the ($\mathbb{R}^{N}$-valued) Ginzburg–Landau vortex solution
$U_{\epsilon}$ introduced in (1.1) is a gradient field $U_{\epsilon}=\nabla
u_{\epsilon}$ for some radial function $u_{\epsilon}=u_{\epsilon}(r)$
determined (up to a constant) by $u^{\prime}_{\epsilon}=f_{\epsilon}$ in
$(0,1)$ where $f_{\epsilon}$ is the unique solution in (1.2).

We prove the following result:

######  Theorem 1.

Assume that $4\leq N\leq 6$ and $W:(-\infty,1]\to[0,\infty)$ is a $C^{2}$ non-
negative convex function such that $W(0)=0$. For every $\epsilon>0$, the
radially symmetric vortex solution $U_{\epsilon}$ in (1.1) is the unique
minimizer of $E^{GL}_{\epsilon}$ over the set of gradient fields $\\{U=\nabla
u\in\mathcal{A}^{GL}\\}$.

Note that the above result holds in dimension $N\geq 7$ as a consequence of
[31, 32]. We expect the result holds also in dimension $N\in\\{2,3\\}$. We
mention here the work Lorent [40, 41] and Lamy and Marconi [35] on stability
of the vortex solution in dimension $N=2$ and in the limit
$\epsilon\rightarrow 0$ (for the Aviles–Giga model as well as other
micromagnetic models).

###  1.2 The $\mathbb{S}^{N}$-valued Ginzburg–Landau model

We consider the following model:

$E^{MM}_{\eta}[M]=\int_{B^{N}}\Big{[}\frac{1}{2}|\nabla
M|^{2}+\frac{1}{2\eta^{2}}\tilde{W}(M_{N+1}^{2})\Big{]}\,dx$

where $\eta>0$ and $M=(\nabla m,M_{N+1})$ is a unit-length vector field that
is a gradient field in the first $N$ components belonging to

$\mathcal{A}^{MM}=\\{M=(\nabla m,M_{N+1})\in
H^{1}(B^{N},\mathbb{S}^{N})\,:\,M(x)=(x,0)\textrm{ on }\partial B^{N}\\}.$

The non-negative potential $\tilde{W}:[0,\infty)\rightarrow[0,\infty)$ is a
$C^{2}$ convex function such that $\tilde{W}(0)=0$.

This model comes from micromagnetics where the order parameter $M$ stands for
the magnetization in ferromagnetic materials (see [20])111In dimension $N=2$,
$E_{\eta}^{MM}$ is the reduced energy functional in a certain thin-film
ferromagnetic regime (see e.g. [13, Section 4.5] or [25, Section 7]) where,
after a rotation by $\frac{\pi}{2}$ in the first two components of $M$, the
condition $\nabla\times(M_{1},M_{2})=0$ is imposed in the space of admissible
configurations in $\mathcal{A}^{MM}$. , and also the Oseen-Frank theory for
nematic liquid crystals (see [1]). Considering radially symmetric critical
points of $E^{MM}_{\eta}$ in $\mathcal{A}^{MM}$, one is led to

$M_{\eta}(x)=(\tilde{f}_{\eta}(r)\frac{x}{r},g_{\eta}(r))\in\mathcal{A}^{MM}$
(1.3)

where the radial profiles $\tilde{f}_{\eta}$ and $g_{\eta}$ satisfy

$\tilde{f}_{\eta}^{2}+g_{\eta}^{2}=1\quad\textrm{in}\quad(0,1),$ (1.4)

and the system of ODEs:

$\displaystyle-\tilde{f}_{\eta}^{\prime\prime}-\frac{N-1}{r}\tilde{f}_{\eta}^{\prime}+\frac{N-1}{r^{2}}\tilde{f}_{\eta}$
$\displaystyle=\lambda(r)\tilde{f}_{\eta}\quad\textrm{in}\quad(0,1),$ (1.5)
$\displaystyle-g_{\eta}^{\prime\prime}-\frac{N-1}{r}g_{\eta}^{\prime}$
$\displaystyle=-\frac{1}{\eta^{2}}\tilde{W}^{\prime}(g_{\eta}^{2})g_{\eta}+\lambda(r)g_{\eta}\quad\textrm{in}\quad(0,1),$
(1.6) $\displaystyle\tilde{f}_{\eta}(1)$ $\displaystyle=1\text{ and
}g_{\eta}(1)=0,$ (1.7)

where

$\lambda(r)=(\tilde{f}_{\eta}^{\prime})^{2}+\frac{N-1}{r^{2}}\tilde{f}_{\eta}^{2}+(g_{\eta}^{\prime})^{2}+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(g_{\eta}^{2})g_{\eta}^{2}$
(1.8)

is the Lagrange multiplier due to the unit length constraint in
$\mathcal{A}^{MM}$. Note that indeed the vortex solution $M_{\eta}$ in (1.3)
is of the form $M_{\eta}=(\nabla m_{\eta},M_{\eta,N+1})\in\mathcal{A}^{MM}$
for some radial function $m_{\eta}=m_{\eta}(r)$ determined (up to a constant)
by $m^{\prime}_{\eta}=\tilde{f}_{\eta}$ in $(0,1)$.

As proved in [26], the solutions to (1.3)–(1.7) satisfy the dichotomy: either
$\tilde{f}_{\eta}(0)=0$ or $\tilde{f}_{\eta}(0)=1$. Furthermore, in the latter
case, it holds that $N\geq 3$ and $(\tilde{f}_{\eta}=1,g_{\eta}=0)$ in
$(0,1)$, which corresponds to the _equator map_

$\bar{M}(x):=(\frac{x}{r},0).$

In dimension $N\geq 7$, $\bar{M}$ is the unique minimizing harmonic map from
$B^{N}$ into $\mathbb{S}^{N}$ in $H^{1}(B^{N},\mathbb{S}^{N})$ with with
boundary condition $(Id,0)$ on $\partial B^{N}$ (Jäger and Kaul [34]; see also
Sandier and Shafrir [49] and [32, Example 1.6]); so $\bar{M}$ is the unique
minimizer of $E^{MM}_{\eta}$ in $\mathcal{A}^{MM}$ _for every_ $\eta>0$.
Therefore, in the following, we focus on dimensions $2\leq N\leq 6$ and on
escaping $\mathbb{S}^{N}$-valued radially symmetric vortex solutions

$M_{\eta}^{\pm}(x)=(\tilde{f}_{\eta}(r)\frac{x}{r},\pm
g_{\eta}(r))\quad\textrm{with}\quad g_{\eta}>0\textrm{ in }(0,1).$

It was proved in Hang and Lin [23] in dimension $N=2$ and [26] in dimension
$3\leq N\leq 6$ that, for any $\eta>0$, (1.3)–(1.7) has a unique escaping
solution $(\tilde{f}_{\eta},g_{\eta})$ with $g_{\eta}>0$ and $M_{\eta}^{\pm}$
are locally minimizers for $E_{\eta}^{MM}$. Moreover, $\tilde{f}_{\eta}(0)=0$,
$\tilde{f}_{\eta}>0$, $\tilde{f}_{\eta}^{\prime}>0$ and $g_{\eta}^{\prime}<0$
in $(0,1)$. (See also [37] for a related work in the context of micromagnetic
skyrmions in $\mathbb{R}^{2}$.)

We prove the following result:

######  Theorem 2.

Assume $4\leq N\leq 6$ and $\tilde{W}:[0,\infty)\rightarrow[0,\infty)$ is a
$C^{2}$ non-negative convex function such that $\tilde{W}(0)=0$. For every
$\eta>0$, $E^{MM}_{\eta}$ has exactly two minimizers over the set $\\{(\nabla
m,M_{N+1})\in\mathcal{A}^{MM}\\}$ and they are given by the escaping vortex
solutions $M_{\eta}^{\pm}(x)=(\tilde{f}_{\eta}(r)\frac{x}{r},\pm g_{\eta}(r))$
with $g_{\eta}>0$ in $(0,1)$. In particular, minimizers of $E^{MM}_{\eta}$ in
$\mathcal{A}^{MM}$ are radially symmetric for every $\eta>0$.

As in the case of the Aviles–Giga model, we expect the above result holds also
in dimension $N\in\\{2,3\\}$.

###  1.3 The extended model

More generally, we consider a family of extended energy functionals
$E_{\epsilon,\eta}$ depending on two positive parameters $\epsilon,\eta$ of
which $E_{\epsilon}^{GL}$ and $E_{\eta}^{MM}$ are limiting cases when
$\eta\rightarrow 0$ and $\epsilon\rightarrow 0$, respectively:

$E_{\epsilon,\eta}[U]=\int_{B^{N}}\Big{[}\frac{1}{2}|\nabla
U|^{2}+\frac{1}{2\epsilon^{2}}W(1-|U|^{2})+\frac{1}{2\eta^{2}}\tilde{W}(U_{N+1}^{2})\Big{]}\,dx,\quad\epsilon,\eta>0,$
(1.9)

where $U=(\nabla u,U_{N+1}):B^{N}\to\mathbb{R}^{N+1}$ is a gradient field in
the first $N$ components and belongs to

$\mathcal{A}=\\{U=(\nabla u,U_{N+1})\in
H^{1}(B^{N},\mathbb{R}^{N+1}):U(x)=(x,0)\text{ on }\partial B^{N}\\}.$

Here, $W:(-\infty,1]\rightarrow[0,\infty)$ and
$\tilde{W}:[0,\infty)\rightarrow[0,\infty)$ are non-negative $C^{2}$ convex
potentials such that $W(0)=\tilde{W}(0)=0$. We point out that these imply that
$W^{\prime}(0)=0$, $tW^{\prime}(t)\geq 0$ in $(-\infty,1]\setminus\\{0\\}$,
and $\tilde{W}^{\prime}(t)\geq 0$ in $[0,\infty)$. However, we allow the
possibility that $W$ or $\tilde{W}$ can be zero in a neighborhood of the
origin.

Radially symmetric critical points of $E_{\epsilon,\eta}$ in $\mathcal{A}$
take the form

$U_{\epsilon,\eta}=(f_{\epsilon,\eta}(r)\frac{x}{r},g_{\epsilon,\eta}(r))\in\mathcal{A},$
(1.10)

where $(f_{\epsilon,\eta},g_{\epsilon,\eta})$ satisfies the system of ODEs

$\displaystyle-
f_{\epsilon,\eta}^{\prime\prime}-\frac{N-1}{r}f_{\epsilon,\eta}^{\prime}+\frac{N-1}{r^{2}}f_{\epsilon,\eta}=\frac{1}{\epsilon^{2}}W^{\prime}(1-f_{\epsilon,\eta}^{2}-g_{\epsilon,\eta}^{2})f_{\epsilon,\eta},$
(1.11) $\displaystyle-
g_{\epsilon,\eta}^{\prime\prime}-\frac{N-1}{r}g_{\epsilon,\eta}^{\prime}=\frac{1}{\epsilon^{2}}W^{\prime}(1-f_{\epsilon,\eta}^{2}-g_{\epsilon,\eta}^{2})g_{\epsilon,\eta}-\frac{1}{\eta^{2}}\tilde{W}^{\prime}(g_{\epsilon,\eta}^{2})g_{\epsilon,\eta},$
(1.12) $\displaystyle f_{\epsilon,\eta}(1)=1\text{ and
}g_{\epsilon,\eta}(1)=0.$ (1.13)

Note that the above implies $f_{\epsilon,\eta}(0)=0$ and
$g_{\epsilon,\eta}^{\prime}(0)=0$ (see [26, Lemma A.5]). Also, note that the
first $N$ components of $U_{\epsilon,\eta}(r)$ is a gradient field
$\nabla\varphi_{\epsilon,\eta}$ for some radial function
$\varphi_{\epsilon,\eta}(r)$ determined (up to a constant) by
$\varphi_{\epsilon,\eta}^{\prime}=f_{\epsilon,\eta}$ in $(0,1)$.

In dimensions $N\geq 7$, it follows from [31, 32] that the _non-escaping_
vortex solution

$\bar{U}_{\epsilon}(x)=(f_{\epsilon}(r)\frac{x}{r},0)$

is the unique global minimizer of $E_{\epsilon,\eta}$ in $\mathcal{A}$ for
every $\epsilon>0$ and $\eta>0$. Therefore, in the following, we focus on
dimensions $2\leq N\leq 6$; we will analyse escaping radially symmetric vortex
solutions

$U_{\epsilon,\eta}^{\pm}=(f_{\epsilon,\eta}(r)\frac{x}{r},\pm
g_{\epsilon,\eta}(r)),\quad g_{\epsilon,\eta}>0\textrm{ in }(0,1).$

It is shown by [26] that such an escaping radially symmetric critical point
$U_{\epsilon,\eta}$ with $g_{\epsilon,\eta}>0$ exists if and only if $2\leq
N\leq 6$, $W^{\prime}(1)>0$, $0<\epsilon<\epsilon_{0}$ and
$\eta>\eta_{0}(\epsilon)$ for some $\epsilon_{0}\in(0,\infty)$ and a
continuous non-decreasing function
$\eta_{0}:[0,\epsilon_{0})\rightarrow[0,\infty)$ with $\eta_{0}(0)=0$. In this
case, it is the unique escaping solution of (1.10)–(1.13) with
$g_{\epsilon,\eta}>0$ in $(0,1)$; moreover, we have $f_{\epsilon,\eta}(0)=0$,
$f_{\epsilon,\eta}^{2}+g_{\epsilon,\eta}^{2}<1$, $f_{\epsilon,\eta}>0$,
$f_{\epsilon,\eta}^{\prime}>0$, $g_{\epsilon,\eta}^{\prime}<0$ in $(0,1)$. See
Section 1.4 and Figure 1 for more information.

We prove the following theorem:

######  Theorem 3.

Suppose $4\leq N\leq 6$ and $W:(-\infty,1]\rightarrow[0,\infty)$ and
$\tilde{W}:[0,\infty)\rightarrow[0,\infty)$ are $C^{2}$ non-negative convex
functions satisfying $W(0)=\tilde{W}(0)=0$. For every $\epsilon>0,\eta>0$, we
have the following dichotomy:

  * •

Either the escaping radially symmetric vortex solutions
$U_{\epsilon,\eta}^{\pm}$ exist and they are the only two minimizers of
$E_{\epsilon,\eta}$ in $\mathcal{A}$,

  * •

Or the escaping radially symmetric vortex solutions $U_{\epsilon,\eta}^{\pm}$
do not exist and the non-escaping vortex solution $\bar{U}_{\epsilon}$ is the
unique minimizer of $E_{\epsilon,\eta}$ in $\mathcal{A}$.

In particular, minimizers of $E_{\epsilon,\eta}$ in $\mathcal{A}$ are always
radially symmetric for every $\epsilon,\eta>0$.

To complete the picture, we recall facts from [26] on the escaping vs. non-
escaping phenomena. The escaping phenomenon is related to the loss of
stability of the non-escaping vortex solution $\bar{U}_{\epsilon}$. More
precisely, consider the stability operator
$\frac{\delta^{2}E_{\epsilon,\eta}}{\delta U_{N+1}^{2}}$ at
$\bar{U}_{\epsilon}$ along the $N+1$ direction:

$\bar{T}_{\epsilon,\eta}=-\Delta-\frac{1}{\epsilon^{2}}W^{\prime}(1-f_{\epsilon}^{2})+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(0).$

The first eigenvalue of $\bar{T}_{\epsilon,\eta}$ on
$H^{1}_{0}(B^{N},\mathbb{R})$ takes the form
$\ell(\epsilon)+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(0)$ where
$\ell(\epsilon)$ is the first eigenvalue of

$L_{\epsilon}:=-\Delta-\frac{1}{\epsilon^{2}}W^{\prime}(1-f_{\epsilon}^{2}).$
(1.14)

Then the

escaping vortex solutions $U_{\epsilon,\eta}^{\pm}$ with $g_{\epsilon,\eta}>0$
exists if and only if
$\ell(\epsilon)+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(0)<0$.

When $N\geq 7$ or $W^{\prime}(1)=0$, it holds always that $\ell(\epsilon)>0$,
hence escaping vortex solutions do not exist. When $2\leq N\leq 6$ and
$W^{\prime}(1)>0$,

there exists $\epsilon_{0}>0$ such that $\ell(\epsilon)>0$ for
$\epsilon>\epsilon_{0}$ and $\ell(\epsilon)<0$ for $0<\epsilon<\epsilon_{0}$.

Thus, in this case, the function $\eta_{0}(\epsilon)$ mentioned above (so that
escaping vortex solutions exist if and only if $0<\epsilon<\epsilon_{0}$ and
$\eta>\eta_{0}(\epsilon)$) is given by

$\eta_{0}(\epsilon)=\sqrt{\frac{\tilde{W}^{\prime}(0)}{|\ell(\epsilon)|}}\text{
for }0<\epsilon<\epsilon_{0}.$

In Figure 1, we describe the dichotomy of escaping and non-escaping phenomena
for minimizers222Recall from [26] that solutions of (1.11)-(1.13) satisfying
$g_{\epsilon,\eta}>0$, when they exist, are minimizing for $E_{\epsilon,\eta}$
relatively to the set of radially symmetric configurations. of
$E_{\epsilon,\eta}$ in radial symmetry in dimension $2\leq N\leq 6$. Theorem 3
asserts that, in dimension $4\leq N\leq 6$, this picture remains valid in the
larger set $\mathcal{A}$ of gradient field configurations in the first $N$
components.

Figure 1: Escaping vs. Non-escaping phenomenon in dimension $2\leq N\leq 6$.  $\epsilon$$\eta$$\epsilon_{0}$ non-escaping region escaping region | $\epsilon$$\eta$$\epsilon_{0}$ non-escaping region escaping region | $\epsilon$$\eta$ always non-escaping
---|---|---
$W^{\prime}(1)>0$ and $\tilde{W}^{\prime}(0)>0$. |  $W^{\prime}(1)>0$ and $\tilde{W}^{\prime}(0)=0$. |  $W^{\prime}(1)=0$.

For the case $\eta=\infty$ (that is the $\mathbb{R}^{N+1}$-valued
Ginzburg–Landau model), we refer the reader to the recent article Ignat and
Rus [33]. For a similar bifurcation from non-escaping to escaping phenomenon,
see Bethuel, Brezis, Coleman and Hélein [5].

###  1.4 Ideas of the proofs

Theorems 1 and 2 will be obtained from Theorem 3 by taking the limits $\eta\to
0$ or $\epsilon\to 0$, respectively. For simplicity, instead of describing the
proof of Theorem 3 (which is the main result), we explain instead the strategy
of the proof in the case $\eta=0$, i.e. Theorem 1 for the Aviles–Giga model.
We will present two methods of proof. Roughly speaking, the first method
follows the strategy in [31, 32] adapted to the situation of gradient field
configurations, and the second method uses a symmetrization procedure.

#### Method 1

It was observed in [31, 32] that the convexity of the potential $W$ implies
the inequality

$E_{\epsilon}^{GL}[U_{\epsilon}+V]-E_{\epsilon}^{GL}[U_{\epsilon}]\geq\frac{1}{2}\int_{B^{N}}L_{\epsilon}V\cdot
V\,dx=:F_{\epsilon}[V]\text{ for }V\in H_{0}^{1}(B^{N},\mathbb{R}^{N}),$
(1.15)

where $L_{\epsilon}$ is the operator defined in (1.14). Recall that in
dimension $N\geq 7$, the first eigenvalue $\ell(\epsilon)$ of $L_{\epsilon}$
is positive, $F_{\epsilon}[V]>0$ for $V\in
H_{0}^{1}(B^{N},\mathbb{R}^{N})\setminus\\{0\\}$, and hence, $U_{\epsilon}$ is
the unique minimizer of $E_{\epsilon}^{GL}$ in $\mathcal{A}^{GL}$. In
dimension $2\leq N\leq 6$, one has $\ell(\epsilon)<0$ for
$\epsilon<\epsilon_{0}$, and so it is not clear from the above argument if
$U_{\epsilon}$ is a minimizer of $E_{\epsilon}^{GL}$ in $\mathcal{A}^{GL}$.
However, in the current case of gradient field configurations (i.e. $V=\nabla
v$), we are able to conclude in dimension $N\geq 4$.

To appreciate the idea, consider the limit $\epsilon\rightarrow 0$ where
$L_{\epsilon}\rightarrow-\Delta-\frac{N-1}{r^{2}}=:L_{*}$ as bounded linear
operators from $H_{0}^{1}(B^{N},\mathbb{R}^{N})$ into
$H^{-1}(B^{N},\mathbb{R}^{N})$. Although $L_{*}$ is not positive definite when
$N\leq 6$, we have the following inequality333Here
$H_{0}^{2}(B^{N},\mathbb{R})$ is the closure of
$\mathcal{C}_{c}^{\infty}(B^{N},\mathbb{R})$ in $H^{2}(B^{N},\mathbb{R})$. In
particular, if $v\in H_{0}^{2}(B^{N},\mathbb{R})$, then $v$ and $\nabla v$
have zero trace on the boundary. for gradient fields in dimension $N\geq 4$:

$\int_{B^{N}}L_{*}(\nabla v)\cdot(\nabla v)\,dx=\int_{B^{N}}\Big{(}(\Delta
v)^{2}-\frac{N-1}{r^{2}}|\nabla v|^{2}\Big{)}\,dx\geq 0\text{ for }v\in
H_{0}^{2}(B^{N},\mathbb{R}),$

i.e. $L_{*}$ is positive definite on the subspace of gradient fields in
$H^{1}_{0}(B^{N},\mathbb{R}^{N})$. This is a consequence of the sharp Hardy
inequality for gradient fields (see e.g. [2, 10, 19, 51]):

$\int_{\mathbb{R}^{N}}(\Delta v)^{2}\,dx\geq
c_{N}\int_{\mathbb{R}^{N}}\frac{|\nabla v|^{2}}{r^{2}}\,dx,\text{ for }v\in
H_{0}^{2}(B^{N},\mathbb{R})\text{ where
}c_{N}:=\begin{cases}\frac{N^{2}}{4}&\text{ if }N\geq 5\\\ 3&\text{ if }N=4\\\
\frac{25}{36}&\text{ if }N=3,\\\ 0&\text{ if }N=2.\end{cases}$ (1.16)

For the general case $\epsilon>0$, we combine the above idea with the
machineries in [31, 32] and [26], based on the Hardy decomposition method.

Unfortunately, the above strategy does not work in dimension $N=\\{2,3\\}$
(for the proof, see Appendix A):

######  Proposition 4.

In dimension $N\in\\{2,3\\}$, there exists a function
$v\in\mathcal{C}_{c}^{2}(B^{N}\setminus\\{0\\})\subset H_{0}^{2}(B^{N})$ such
that $F_{\epsilon}(\nabla v)<0$ when $\epsilon$ is sufficiently small.

#### Method 2

As mentioned above, the second method of proof uses a symmetrization
procedure. For that, we use the spherical coordinates: for every $x\in B^{N}$,
we write $x=r\theta$ with $r=|x|$ and $\theta\in{\mathbb{S}^{N-1}}$. For $v\in
H^{1}(B^{N},\mathbb{R})$, we associate the radial function
$\check{v}=\check{v}(r)$ given by the formula

$\check{v}(r)=-\int_{r}^{1}\Big{(}\fint_{\mathbb{S}^{N-1}}|\nabla
v(s\theta)|^{2}d\sigma(\theta)\Big{)}^{1/2}\,ds\leq 0,\quad r\in(0,1).$ (1.17)

One can think of this as a kind of rearrangement in the spherical harmonic
decomposition of $v$ (see Section 3.2 for more detailed discussion).

We prove the following.

######  Theorem 5.

Let $N\geq 2$, $v\in H^{1}(B^{N},\mathbb{R})$ and $\check{v}$ be associated to
$v$ by (1.17). The following conclusions hold.

  1. (i)

The map $v\mapsto\check{v}$ is a Lipschitz continuous map from
$H^{1}(B^{N},\mathbb{R})$ into $H^{1}_{0}(B^{N},\mathbb{R})$. Moreover,

$\int_{\mathbb{S}^{N-1}}|\nabla\check{v}(r\theta)|^{2}\,d\sigma(\theta)=\int_{\mathbb{S}^{N-1}}|\nabla
v(r\theta)|^{2}\,d\sigma(\theta)\text{ for a.e. }r\in(0,1).$

  2. (ii)

Let $G:[0,\infty)\times[0,\infty)\rightarrow[0,\infty)$ be continuous. If $G$
is convex in the second variable, then

$\displaystyle\int_{B^{N}}G(r,|\nabla\check{v}|^{2})\,dx\leq\int_{B^{N}}G(r,|\nabla
v|^{2})\,dx.$

In particular, for any $2<p<\infty$,

$\int_{B^{N}}|\nabla\check{v}|^{p}\,dx\leq\int_{B^{N}}|\nabla v|^{p}\,dx.$

  3. (iii)

If $v\in H_{0}^{1}(B^{N})$, i.e. if $v=0$ on $\partial B^{N}$ and $1\leq p\leq
2$, then

$\int_{{\mathbb{S}^{N-1}}}|\check{v}(r\theta)|^{p}\,d\sigma(\theta)\geq\int_{{\mathbb{S}^{N-1}}}|v(r\theta)|^{p}\,d\sigma(\theta)\text{
for a.e. }r\in(0,1).$

  4. (iv)

Assume in addition that $v\in H^{2}(B^{N},\mathbb{R})$ with the boundary
condition $\nabla v(x)=cx$ on $\partial B^{N}$ for some constant
$c\in\mathbb{R}$. Then $\check{v}\in H^{2}(B^{N},\mathbb{R})$ and
$\nabla\check{v}(x)=|c|x$ on $\partial B^{N}$. If $N\geq 5$, then

$\int_{B^{N}}(\Delta\check{v})^{2}\,dx\leq\int_{B^{N}}(\Delta v)^{2}\,dx.$
(1.18)

If $N\in\\{2,3,4\\}$, (1.18) continues to hold provided that
$\int_{\mathbb{S}^{N-1}}v(r\theta)\theta\,d\sigma(\theta)=0$ for a.e.
$r\in(0,1)$. In either case, equality is attained if and only if $v$ is
radially symmetric and $|v^{\prime}|=\check{v}^{\prime}$ in $(0,1)$.

To apply Theorem 5 to prove Theorem 1, we only need to note that for $\nabla
u\in\mathcal{A}^{GL}$, by integrating by parts using $\nabla u(x)=x$ on
$\partial B^{N}$,

$\displaystyle\int_{B^{N}}|\nabla^{2}u|^{2}\,dx$
$\displaystyle=\int_{B^{N}}(\Delta
u)^{2}\,dx-\int_{\mathbb{S}^{N-1}}\underbrace{\sum_{i,j=1}^{N}\partial_{i}\partial_{j}u(\theta)(\delta_{ij}-\theta_{i}\theta_{j})}_{\nabla^{2}u:(I_{N}-\theta\otimes\theta)}\,d\sigma(\theta)$
$\displaystyle=\int_{B^{N}}(\Delta
u)^{2}\,dx-\int_{\mathbb{S}^{N-1}}\big{[}(N-1)\partial_{r}u(\theta)+\Delta_{{\mathbb{S}^{N-1}}}u(\theta)\big{]}\,d\sigma(\theta)$
$\displaystyle=\int_{B^{N}}(\Delta u)^{2}\,dx-(N-1)|\mathbb{S}^{N-1}|.$

(Here we have used the fact that $I_{N}-\theta\otimes\theta$ is the projection
onto the tangent hyperplane $T_{\theta}{\mathbb{S}^{N-1}}$.) Therefore, in
dimension $N\geq 5$, Theorem 5 immediately implies that minimizers of
$E_{\epsilon}^{GL}$ in $\\{U=\nabla u\in\mathcal{A}^{GL}\\}$ are radially
symmetric. Thanks to the characterization of radially symmetric critical
points in [26], Theorem 1 follows. Theorem 2 also follows in a similar manner.
For Theorem 3, we need an extra symmetrization for the $U_{N+1}$ component;
see Section 3.1.

In Theorem 5(iv), the requirement
$\int_{\mathbb{S}^{N-1}}v(r\theta)\theta\,d\sigma(\theta)=0$ in dimension
$N\in\\{2,3,4\\}$ cannot simply be dropped due to existence of counter-
examples. (For examples of symmetry breaking phenomena in the context of
Hardy’s inequality for gradient fields in dimension $N\in\\{3,4\\}$, see e.g.
[10].)

Our rearrangement is related to a vectorial rearrangement in Lieb and Loss
[38]. For $V\in H^{1}(B^{N},\mathbb{R}^{N})$, one associates the radially
symmetric vector field $\check{V}$ defined by

$\check{V}(x)=\left(\fint_{\mathbb{S}^{N-1}}|V(r\theta)|^{2}d\sigma(\theta)\right)^{\frac{1}{2}}\frac{x}{r}.$
(1.19)

It was shown in [38] that, provided $V(x)=x$ on $\partial B^{N}$ and
$\int_{\mathbb{S}^{N-1}}V(r\theta)d\sigma(\theta)=0$ for a.e. $r\in(0,1)$,

$\int_{B^{N}}|\nabla\check{V}|^{2}\,dx\leq\int_{B^{N}}|\nabla V|^{2}\,dx.$

It is readily seen that if $V=\nabla v$, then $\check{V}=\nabla\check{v}$.
Thus, when $N\in\\{2,3,4\\}$, the conclusion (1.18) in Theorem 5 can be
deduced from the above result in [38].

Fewer rearrangement methods are known to prove symmetry of solutions of higher
order elliptic equations than for second order ones. This can be partly
explained by the absence of a maximum principle for higher order elliptic
equations or systems, which makes Schwarz symmetrization methods inapplicable
in general. There are some exceptions, see for instance the two papers of
Nadirashvili [45] and Talenti [50] where it is shown by rearrangement
arguments that minimizers of $\frac{|\\{u\neq
0\\}|^{2}\int_{\mathbb{R}^{2}}(\Delta u)^{2}}{\int_{\mathbb{R}^{2}}u^{2}}$ are
radially symmetric.444For other results on symmetry of solutions of higher
order elliptic equations or systems which do not use rearrangement
inequalities, see e.g. [3, 11, 15, 18] and the references therein.

More recently, a rearrangement principle developed in Lenzmann and Sok [36]
deals with the radial symmetry of optimizers of Gagliardo–Nirenberg type
inequalities of arbitrarily high orders, as well as ground states of higher
order non-linear Schrödinger equations of the form

$Lv+\omega v=v|v|^{p-2}\text{ in }\mathbb{R}^{N}$

where $L$ is a certain pseudodifferential operator, $\omega>0$ and
$p\in(2,p^{*})$ for some critical exponent $p^{*}>2$ depending only on the
dimension $N$ and the operator $L$. The rearrangement principle here is based
on Schwarz rearrangement of the Fourier transform: any function
$v:\mathbb{R}^{N}\to\mathbb{C}$ is symmetrized as
$v^{\sharp}=\mathcal{F}^{-1}\left[|\mathcal{F}[v]|^{*}\right]$, where
$\mathcal{F}$ is the Fourier transform and $w^{*}$ designates the radially
decreasing Schwarz rearrangement of $w$.

We make a comparison between the rearrangement $\check{v}$ in Theorem 5 and
the rearrangement $v^{\sharp}$ of [36] in the following table.

|  $\check{v}$ on $B^{N}$ |  $v^{\sharp}$ on $\mathbb{R}^{N}$
---|---|---
$L^{2}$-norm | $\|\check{v}\|_{L^{2}}\geq\|v\|_{L^{2}}$ | $\|v^{\sharp}\|_{L^{2}}=\|v\|_{L^{2}}$
$L^{p}$-norm, $1\leq p<2$ | $\|\check{v}\|_{L^{p}}\geq\|v\|_{L^{p}}$ | ?
$L^{p}$-norm, even integer $p>2$ | ? | $\|v^{\sharp}\|_{L^{p}}\geq\|v\|_{L^{p}}$
$\dot{H}^{1}$-norm | $\|\nabla\check{v}\|_{L^{2}}=\|\nabla v\|_{L^{2}}$ | $\|\nabla v^{\sharp}\|_{L^{2}}\leq\|\nabla v\|_{L^{2}}$
$\dot{W}^{1,p}$-norm, $p>2$ | $\|\nabla\check{v}\|_{L^{p}}\leq\|\nabla v\|_{L^{p}}$ | ?
$\dot{H}^{2}$-norm | $\|\Delta\check{v}\|_{L^{2}}\leq\|\Delta v\|_{L^{2}}$ | $\|\Delta v^{\sharp}\|_{L^{2}}\leq\|\Delta v\|_{L^{2}}$
$\dot{H}^{s}$-norm, $s>0$ | ? | $\|v^{\sharp}\|_{\dot{H}^{s}}\leq\|v\|_{\dot{H}^{s}}$

As an application Theorem 5, we consider the radial symmetry of ‘ground state’
solutions to the nonlinear eigenvalue problem

$\begin{cases}\Delta^{2}v=\lambda v+|v|^{p-2}v\text{ on }B^{N},\\\
v=\partial_{r}v=0\text{ on }\partial B^{N},\end{cases}$ (1.20)

where $1\leq p<2$. See Section 3.4.

Acknowledgments. R.I. is partially supported by the ANR projects
ANR-21-CE40-0004 and ANR-22-CE40-0006-01. L.N. was partially supported by the
ANR LabEx CIMI (grant ANR-11-LABX-0040) within the French State Programme
“Investissements d’Avenir” during his visit to Université Toulouse III – Paul
Sabatier. Part of this work was done while all authors visited the Hausdorff
Research Institute for Mathematics in Bonn, funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy – EXC-2047/1 – 390685813, as part of the Trimester Program
on “Mathematics of Complex Materials”.

##  2 First proof of main results

###  2.1 Proof of Theorem 3

In this section we give the first proof of Theorem 3 based on the strategy in
[31, 32] and exploiting the additional structure of a gradient field for the
first $N$-components of the current admissible configurations.

Recall that $\ell(\epsilon)$ is the first eigenvalue of the operator
$L_{\epsilon}=-\Delta-\frac{1}{\epsilon^{2}}W^{\prime}(1-f_{\epsilon}^{2})$
and that the escaping radially symmetric critical points
$U_{\epsilon,\eta}^{\pm}$ with $g_{\epsilon,\eta}>0$ exist if and only if
$W^{\prime}(1)>0$ and
$\ell(\epsilon)+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(0)<0$ (equivalently
$0<\epsilon<\epsilon_{0}$ and $\eta>\eta_{0}(\epsilon)$). For fixed
$\epsilon>0,\eta>0$, we let

$\Phi=\begin{cases}U_{\epsilon,\eta}^{+}&\text{ if }W^{\prime}(1)>0\text{ and
}\ell(\epsilon)+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(0)<0\text{ (i.e. there is
an escaping solution),}\\\ \bar{U}_{\epsilon}&\text{ otherwise}\text{ (i.e.
there is no escaping solution),}\end{cases}$

and

$(f,g)=\begin{cases}(f_{\epsilon,\eta},g_{\epsilon,\eta})&\text{ if
}W^{\prime}(1)>0\text{ and
}\ell(\epsilon)+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(0)<0,\\\
(f_{\epsilon},0)&\text{ otherwise},\end{cases}$

so that $\Phi(x)=(f(r)\frac{x}{r},g(r))$.

We consider the differential operators $L_{\epsilon,\eta}$ and
$T_{\epsilon,\eta}$:

$L_{\epsilon,\eta}=-\Delta-\frac{1}{\epsilon^{2}}W^{\prime}(1-f^{2}-g^{2}),\quad
T_{\epsilon,\eta}=-\Delta-\frac{1}{\epsilon^{2}}W^{\prime}(1-f^{2}-g^{2})+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(g^{2}).$

For any $v\in H^{2}_{0}(B^{N},\mathbb{R})$, we let

$F_{\epsilon,\eta}[\nabla v]=\int_{B^{N}}\left((\Delta
v)^{2}-\frac{1}{\epsilon^{2}}W^{\prime}(1-f^{2}-g^{2})|\nabla
v|^{2}\right)\,dx=\int_{B^{N}}L_{\epsilon,\eta}(\nabla v)\colon(\nabla
v)\,dx.$ (2.1)

Note that $\int_{B^{N}}|\nabla^{2}v|^{2}\,dx=\int_{B^{N}}(\Delta v)^{2}\,dx$
since $v\in H^{2}_{0}(B^{N})$. Note also that, in the non-escaping case,
$\Phi=\bar{U}_{\epsilon}$, $L_{\epsilon,\eta}=L_{\epsilon}$,
$T_{\epsilon,\eta}=\bar{T}_{\epsilon,\eta}$ and
$F_{\epsilon,\eta}=F_{\epsilon}$ introduced in sections 1.3 and 1.4.

As in [31, 32], the starting point of the proof is the following consequence
of the convexity of $W$ and $\tilde{W}$:

######  Lemma 6.

For any $v\in H^{2}_{0}(B^{N},\mathbb{R})$ and $p\in
H^{1}_{0}(B^{N},\mathbb{R})$,

$E_{\epsilon,\eta}\big{[}\Phi+(\nabla
v,p)\big{]}-E_{\epsilon,\eta}[\Phi]\geq\frac{1}{2}F_{\epsilon,\eta}[\nabla
v]+\frac{1}{2}\int_{B^{N}}T_{\epsilon,\eta}p\cdot p\,dx.$

###### Proof.

We have

$\displaystyle E_{\epsilon,\eta}\big{[}\Phi+(\nabla
v,p)\big{]}-E_{\epsilon,\eta}[\Phi]$
$\displaystyle=\frac{1}{2}\int_{B^{N}}\Big{\\{}2\nabla\Phi:\nabla(\nabla
v,p)+|\nabla^{2}v|^{2}+|\nabla p|^{2}$
$\displaystyle\qquad+\frac{1}{\epsilon^{2}}[W(1-|\Phi+(\nabla
v,p)|^{2})-W(1-|\Phi|^{2})]$
$\displaystyle\qquad+\frac{1}{\eta^{2}}[\tilde{W}((g+p)^{2})-\tilde{W}(g^{2})]\Big{\\}}\,dx.$
(2.2)

By the convexity of $W$ and $\tilde{W}$, we have

$\displaystyle W(1-|\Phi+(\nabla v,p)|^{2})-W(1-|\Phi|^{2})$
$\displaystyle\geq W^{\prime}(1-|\Phi|^{2})\left(|\Phi|^{2}-|\Phi+(\nabla
v,p)|^{2}\right)$
$\displaystyle=-W^{\prime}(1-|\Phi|^{2})\left(2\Phi\cdot(\nabla v,p)+|\nabla
v|^{2}+p^{2}\right)$ $\displaystyle\tilde{W}((g+p)^{2})-\tilde{W}(g^{2})$
$\displaystyle\geq\tilde{W}^{\prime}(g^{2})\left((g+p)^{2}-g^{2}\right)$
$\displaystyle=\tilde{W}^{\prime}(g^{2})\left(2gp+p^{2}\right).$

Since $\Phi$ is a critical point of $E$, we also have

$\int_{B^{N}}\Big{\\{}\nabla\Phi:\nabla(\nabla
v,p)-\frac{1}{\epsilon^{2}}W^{\prime}(1-|\Phi|^{2})\Phi\cdot(\nabla v,p)\\\
+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(g^{2})gp\Big{\\}}\,dx=0.$

Inserting the last two estimates into (2.2) we arrive at

$\displaystyle E_{\epsilon,\eta}\left[\Phi+(\nabla
v,p)\right]-E_{\epsilon,\eta}[\Phi]$
$\displaystyle\qquad\geq\frac{1}{2}\int_{B^{N}}\left(|\nabla^{2}v|^{2}+|\nabla
p|^{2}-\frac{1}{\epsilon^{2}}W^{\prime}(1-f^{2}-g^{2})\left(|\nabla
v|^{2}+p^{2}\right)+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(g^{2})p^{2}\right)\,dx,$

which is precisely the conclusion. ∎

We will frequently make use of the following Hardy decomposition:

######  Lemma 7 ([29, Lemma A.1]).

Let $A:B^{N}\rightarrow\mathbb{R}^{N\times N}$ be a $C^{1}$ non-negative semi-
definite symmetric form, i.e. $A(x)\xi\cdot\xi\geq 0$ for every $x\in B^{N}$
and $\xi\in\mathbb{R}^{N}$. We define the operator

$L:=-\nabla\cdot(A\nabla)$

and consider a smooth positive function $\psi:B^{N}\rightarrow\mathbb{R}$.
Then for every $u\in\mathcal{C}_{c}^{\infty}(B^{N},\mathbb{R})$, we have the
following Hardy decomposition:

$\int_{B^{N}}Lu\cdot
u\,dx=\int_{B^{N}}\psi^{2}A(x)\nabla(\frac{u}{\psi})\cdot\nabla(\frac{u}{\psi})\,dx+\int_{B^{N}}\frac{u^{2}}{\psi^{2}}L\psi\cdot\psi\,dx.$

Before moving on with the proof, let us make a simple observation on the non-
negativity of $T_{\epsilon,\eta}$.

######  Lemma 8.

The first eigenvalue of $T_{\epsilon,\eta}$ on $H^{1}_{0}(B^{N},\mathbb{R})$
is $\left(\ell(\epsilon)+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(0)\right)_{+}$
and the corresponding first eigenspace of $T_{\epsilon,\eta}$

  * •

coincides with the first eigenspace of $L_{\epsilon}$ when
$\ell(\epsilon)+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(0)\geq 0$ (i.e. when
$g\equiv 0$), and

  * •

is generated by $g$ when
$\ell(\epsilon)+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(0)<0$ (i.e. when $g>0$).

In particular $T_{\epsilon,\eta}$ is non-negative semi-definite on
$H^{1}_{0}(B^{N},\mathbb{R})$ and

$\int_{B^{N}}T_{\epsilon,\eta}p\cdot
p\,dx\geq\int_{B^{N}}\Big{[}h^{2}\left|\nabla(\frac{p}{h})\right|^{2}+\left(\ell(\epsilon)+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(0)\right)_{+}p^{2}\Big{]}\,dx\geq
0,$

where $h$ is any first eigenfunction of $T_{\epsilon,\eta}$.

###### Proof.

Recall that, by [26, Theorem 2.4] on escaping and non-escaping critical points
of $E_{\epsilon,\eta}$, when $g\equiv 0$, we have
$\ell(\epsilon)+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(0)\geq 0$, while, when
$g>0$, $\ell(\epsilon)+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(0)<0$. The first
bullet point is then clear as
$T_{\epsilon,\eta}=L_{\epsilon}+\frac{1}{\eta^{2}}\tilde{W}^{\prime}(0)$ and
the first eigenvalue of $L_{\epsilon}$ is $\ell(\epsilon)$. When $g>0$, we
have

$T_{\epsilon,\eta}g=0,$

and so $g$ must be a first eigenfunction of $T_{\epsilon,\eta}$ and the first
eigenvalue of $T_{\epsilon,\eta}$ must be zero. The second bullet point
follows. The last assertion follows from the Hardy decomposition Lemma 7 with
the decomposition $p=h\frac{p}{h}$. ∎

The last ingredient for the proof of Theorem 3 is:

######  Proposition 9.

Suppose $N\geq 4$. For any $v\in H^{2}_{0}(B^{N},\mathbb{R})$ we have

$F_{\epsilon,\eta}[\nabla
v]\geq\frac{(N-2)^{2}}{4}\int_{B^{N}}\frac{(\partial_{r}v)^{2}}{r^{2}}\,dx+\left(\frac{N^{2}}{2}-2N\right)\int_{B^{N}}\frac{|\nabla
v|^{2}-(\partial_{r}v)^{2}}{r^{2}}\,dx\geq 0.$

######  Remark 10.

Note that for general $V\in H_{0}^{1}(B^{N},\mathbb{R}^{N})$ which is not
necessarily a gradient field, it was shown in [31, 32] in dimension $N\geq 7$
that $F_{\epsilon,\eta}[V]=F_{\epsilon}[V]\geq 0$.

Before giving the proof of the above proposition, let us prove Theorem 3.

###### First proof of Theorem 3.

Indeed, as $N\geq 4$, we have by Proposition 9 that $F_{\epsilon,\eta}[\nabla
v]\geq 0$ for every $v\in H^{2}_{0}(B^{N},\mathbb{R})$ with equality if and
only if $\partial_{r}v=0$ a.e., which implies $v=0$. Therefore, by Lemmas 6
and 8, $\Phi$ is a minimizer of our problem. If $\tilde{\Phi}$ is another
minimizer of $E_{\epsilon,\eta}$, then
$E_{\epsilon,\eta}[\tilde{\Phi}]=E_{\epsilon,\eta}[\Phi]$. By Lemmas 6, 8 and
Proposition 9, this is possible only if $\tilde{\Phi}-\Phi=(0,h)$ for some $h$
in the first eigenspace of $T_{\epsilon,\eta}$, which is radially symmetric
(because $T_{\epsilon,\eta}$ is radially symmetric). We thus have that
$\tilde{\Phi}$ is a radially symmetric minimizer of $E_{\epsilon,\eta}$. [26,
Therorem 2.4] then gives the desired uniqueness for minimizer(s). ∎

###### Proof of Proposition 9.

It is enough to prove the estimate for
$v\in\mathcal{C}^{\infty}_{c}(B^{N}\setminus\\{0\\},\mathbb{R})$. The general
case follows from Fatou’s lemma and the density of
$\mathcal{C}^{\infty}_{c}(B^{N}\setminus\\{0\\},\mathbb{R})$ in
$H^{2}_{0}(B^{N},\mathbb{R})$ (note $N\geq 4$).

We denote by $(\phi_{k})_{k\in\mathbb{N}}$ an orthonormal basis of
$L^{2}({\mathbb{S}^{N-1}})$ given by eigenfunctions of the Laplace-Beltrami
operator on the unit sphere, meaning that for any $k\in\mathbb{N}$ we have

$-\Delta_{{\mathbb{S}^{N-1}}}\phi_{k}=\lambda_{k}\phi_{k}$

where
$0=\lambda_{0}<N-1=\lambda_{1}=\ldots=\lambda_{N}<2N=\lambda_{N+1}\leq\cdots\longrightarrow+\infty$.
In particular we have

$\int_{{\mathbb{S}^{N-1}}}\phi_{k}\phi_{l}\,d\sigma(\theta)=\delta_{kl}\quad\text{
and
}\quad\int_{{\mathbb{S}^{N-1}}}\nabla_{{\mathbb{S}^{N-1}}}\phi_{k}\cdot\nabla_{{\mathbb{S}^{N-1}}}\phi_{l}\,d\sigma(\theta)=\lambda_{k}\delta_{kl}.$
(2.3)

Consider the decomposition of $v$ in spherical harmonics: we write

$v(r\theta)=\sum_{k\geq 0}v_{k}(r)\phi_{k}(\theta)\text{ for
}r\in(0,1),\theta\in{\mathbb{S}^{N-1}}$

where $v_{k}\in\mathcal{C}^{\infty}_{c}((0,1),\mathbb{R})$. We have

$\nabla v=\sum_{k\geq
0}\left(v_{k}^{\prime}\phi_{k}\frac{x}{r}+\frac{1}{r}v_{k}\nabla_{\mathbb{S}^{N-1}}\phi_{k}\right),\
\Delta v=\sum_{k\geq
0}\left(v_{k}^{\prime\prime}+\frac{N-1}{r}v_{k}^{\prime}-\frac{\lambda_{k}}{r^{2}}v_{k}\right)\phi_{k}.$

Using the orthogonality relations (2.3) and the identities

$\displaystyle\int_{0}^{1}r^{N-2}v_{k}^{\prime\prime}v_{k}^{\prime}\,dr$
$\displaystyle=-\frac{N-2}{2}\int_{0}^{1}r^{N-3}(v_{k}^{\prime})^{2}\,dr,$
$\displaystyle\int_{0}^{1}r^{N-4}v_{k}^{\prime}v_{k}\,dr$
$\displaystyle=-\frac{N-4}{2}\int_{0}^{1}r^{N-5}v_{k}^{2}\,dr\text{ for }k\geq
1,$ $\displaystyle\int_{0}^{1}r^{N-3}v_{k}^{\prime\prime}v_{k}\,dr$
$\displaystyle=\int_{0}^{1}\Big{[}-r^{N-3}(v_{k}^{\prime})^{2}+\frac{(N-3)(N-4)}{2}r^{N-5}v_{k}^{2}\Big{]}\,dr\text{
for }k\geq 1,$

we get

$\begin{split}&\int_{B^{N}}(\Delta v)^{2}dx=\sum_{k\geq
0}\int_{B^{N}}\left(v_{k}^{\prime\prime}+\frac{N-1}{r}v_{k}^{\prime}-\frac{\lambda_{k}}{r^{2}}v_{k}\right)^{2}\phi_{k}^{2}dx\\\
&\quad=\sum_{k\geq
0}\int_{0}^{1}\left(r^{N-1}(v_{k}^{\prime\prime})^{2}+(N-1+2\lambda_{k})r^{N-3}(v_{k}^{\prime})^{2}+\lambda_{k}(\lambda_{k}+2N-8))r^{N-5}v_{k}^{2}\right)dr,\end{split}$
(2.4)

and

$\displaystyle\int_{B^{N}}W^{\prime}(1-f^{2}-g^{2})|\nabla v|^{2}dx$
$\displaystyle=\sum_{k\geq
0}\int_{0}^{1}W^{\prime}(1-f^{2}-g^{2})\left(r^{N-1}(v_{k}^{\prime})^{2}+\lambda_{k}r^{N-3}v_{k}^{2}\right)dr.$

Inserting these into (2.1), we split $F_{\epsilon,\eta}$ into three terms as
follows:

$\displaystyle F_{\epsilon,\eta}[\nabla v]=$ $\displaystyle\sum_{k\geq
0}\Big{\\{}\underbrace{\int_{0}^{1}r^{N-1}\left((v_{k}^{\prime\prime})^{2}-\frac{1}{\epsilon^{2}}W^{\prime}(1-f^{2}-g^{2})(v_{k}^{\prime})^{2}\right)\,dr}_{\text{I}_{k}}$
$\displaystyle\qquad+\underbrace{\int_{0}^{1}\lambda_{k}r^{N-1}\left(r^{-2}(v_{k}^{\prime})^{2}-\frac{1}{\epsilon^{2}}W^{\prime}(1-f^{2}-g^{2})r^{-2}v_{k}^{2}\right)\,dr}_{\text{II}_{k}}$
$\displaystyle\qquad+\underbrace{\int_{0}^{1}\left((N-1+\lambda_{k})r^{N-3}(v_{k}^{\prime})^{2}+\lambda_{k}(\lambda_{k}+2N-8)r^{N-5}v_{k}^{2}\right)\,dr}_{\text{III}_{k}}\Big{\\}}.$

For terms $\text{I}_{k}$ and $\text{II}_{k}$ we will apply the Hardy
decomposition Lemma 7 using

$L_{\epsilon,\eta}f=-\frac{N-1}{r^{2}}f.$

More precisely, for any function
$w\in\mathcal{C}^{\infty}_{c}(B^{N},\mathbb{R})$ we have the identity

$\displaystyle\int_{B^{N}}L_{\epsilon,\eta}(fw)\cdot(fw)\,dx$
$\displaystyle=\int_{B^{N}}\big{(}f^{2}|\nabla
w|^{2}+w^{2}L_{\epsilon,\eta}f\cdot f\big{)}\,dx$
$\displaystyle=\int_{B^{N}}f^{2}\left(|\nabla
w|^{2}-\frac{N-1}{r^{2}}w^{2}\right)\,dx.$ (2.5)

  * •

Estimate of $\text{I}_{k}$: For the first term we use the decomposition
$v_{k}^{\prime}=f\frac{v_{k}^{\prime}}{f}$, i.e.
$w=\frac{v_{k}^{\prime}}{f}\in\mathcal{C}^{\infty}_{c}(B^{N}\setminus\\{0\\},\mathbb{R})$
in (2.5):

$\text{I}_{k}=\int_{0}^{1}r^{N-1}L_{\epsilon,\eta}(v_{k}^{\prime})\cdot(v_{k}^{\prime})dr=\int_{0}^{1}\Big{[}r^{N-1}f^{2}\left|\left(\frac{v_{k}^{\prime}}{f}\right)^{\prime}\right|^{2}-(N-1)r^{N-3}(v_{k}^{\prime})^{2}\Big{]}dr.$

We let $\zeta(r)=r^{-\frac{N-2}{2}}$ so that, when seen as a radial function
in $\mathbb{R}^{N}\setminus\\{0\\}$, $\zeta$ verifies

$-\nabla\cdot(f^{2}\nabla\zeta)=-f^{2}\Delta\zeta-2ff^{\prime}\zeta^{\prime}=\frac{(N-2)^{2}}{4r^{2}}f^{2}\zeta-2ff^{\prime}\zeta^{\prime}\geq\frac{(N-2)^{2}}{4r^{2}}f^{2}\zeta,$

since $\zeta^{\prime}<0$ and $f,f^{\prime}>0$ in $(0,1)$. By the Hardy
decomposition Lemma 7 for the operator $\nabla\cdot(f^{2}\nabla)$ and the
decomposition $\frac{v_{k}^{\prime}}{f}=\zeta\frac{v_{k}^{\prime}}{f\zeta}$,
we thus have

$\begin{split}\text{I}_{k}&\geq\int_{0}^{1}r^{N-1}\left(f^{2}\zeta^{2}\left|\left(\frac{v_{k}^{\prime}}{f\zeta}\right)^{\prime}\right|^{2}+\left(\frac{(N-2)^{2}}{4}-(N-1)\right)r^{N-3}(v_{k}^{\prime})^{2}\right)dr.\end{split}$
(2.6)

  * •

Estimate of $\text{II}_{k}$: First notice the elementary identity

$\displaystyle\int_{0}^{1}r^{N-3}(v_{k}^{\prime})^{2}dr$
$\displaystyle=\int_{0}^{1}\left(r^{N-1}\Big{(}\left(\frac{v_{k}}{r}\right)^{\prime}\Big{)}^{2}+2r^{N-4}v_{k}v_{k}^{\prime}-r^{N-5}v_{k}^{2}\right)dr$
$\displaystyle=\int_{0}^{1}\left(r^{N-1}\Big{(}\left(\frac{v_{k}}{r}\right)^{\prime}\Big{)}^{2}-(N-3)r^{N-5}v_{k}^{2}\right)dr$

so

$\text{II}_{k}=\lambda_{k}\int_{0}^{1}\left(r^{N-1}L_{\epsilon,\eta}(\frac{v_{k}}{r})\cdot(\frac{v_{k}}{r})-(N-3)r^{N-5}v_{k}^{2}\right)dr.$

This time we use the decomposition $\frac{v_{k}}{r}=f\frac{v_{k}}{rf}$ (i.e.
$w=\frac{v_{k}}{rf}$ in (2.5)) to obtain

$\text{II}_{k}=\lambda_{k}\int_{0}^{1}\left(r^{N-1}f^{2}\left|\left(\frac{v_{k}}{rf}\right)^{\prime}\right|^{2}-2(N-2)r^{N-5}v_{k}^{2}\right)dr.$

By the Hardy decomposition Lemma 7 for the operator $\nabla\cdot(f^{2}\nabla)$
and the decomposition $\frac{v_{k}}{rf}=\zeta\frac{v_{k}}{rf\zeta}$ as above
we get the estimate

$\begin{split}\text{II}_{k}&\geq\lambda_{k}\int_{0}^{1}\left(r^{N-1}f^{2}\zeta^{2}\left|\left(\frac{v_{k}}{rf\zeta}\right)^{\prime}\right|^{2}+\left(\frac{(N-2)^{2}}{4}-2(N-2)\right)r^{N-5}v_{k}^{2}\right)dr.\end{split}$
(2.7)

  * •

Estimate of $\text{III}_{k}$: For the last term we simply apply the Hardy
inequality once: for any $v\in\mathcal{C}^{\infty}_{c}((0,1),\mathbb{R})$,
$\int_{0}^{1}r^{N-3}(v^{\prime})^{2}dr\geq\frac{(N-4)^{2}}{4}\int_{0}^{1}r^{N-5}v^{2}dr$.
This gives

$\text{III}_{k}\geq\int_{0}^{1}\left((N-1)r^{N-3}(v_{k}^{\prime})^{2}+\lambda_{k}\left(\lambda_{k}+2N-8+\frac{(N-4)^{2}}{4}\right)r^{N-5}v_{k}^{2}\right)dr.$
(2.8)

Summing the estimates (2.6), (2.7), (2.8) we get

$\displaystyle F_{\epsilon,\eta}[\nabla v]$ $\displaystyle\geq\sum_{k\geq
0}\int_{0}^{1}\left(\frac{(N-2)^{2}}{4}r^{N-3}(v_{k}^{\prime})^{2}+\lambda_{k}\left(\frac{N^{2}}{2}-3N+1+\lambda_{k}\right)r^{N-5}v_{k}^{2}\right)dr$
$\displaystyle\geq\sum_{k\geq
0}\int_{0}^{1}\left(\frac{(N-2)^{2}}{4}r^{N-3}(v_{k}^{\prime})^{2}+\lambda_{k}\left(\frac{N^{2}}{2}-2N\right)r^{N-5}v_{k}^{2}\right)dr\text{
since }\lambda_{k}^{2}\geq(N-1)\lambda_{k}$
$\displaystyle=\frac{(N-2)^{2}}{4}\int_{B^{N}}\frac{(\partial_{r}v)^{2}}{r^{2}}dx+\left(\frac{N^{2}}{2}-2N\right)\int_{B^{N}}\frac{|\nabla
v|^{2}-(\partial_{r}v)^{2}}{r^{2}}dx.$

The result is proved. ∎

###  2.2 Proof of Theorems 1 and 2

Theorem 1 for the Aviles–Giga model is a simple consequence of Theorem 3 for
the extended model.

###### Proof of Theorem 1.

Fix $\epsilon>0$. Pick any convex $C^{2}$ function
$\tilde{W}:[0,\infty)\rightarrow[0,\infty)$ with $\tilde{W}(0)=0$ and
$\tilde{W}^{\prime}(0)>0$, e.g. $\tilde{W}(t)=t$. By [26], there exists a
small $\eta>0$ such that $E_{\epsilon,\eta}$ has no escaping radially
symmetric critical points. By Theorem 3, $\bar{U}_{\epsilon}=(U_{\epsilon},0)$
is the unique minimizer of $E_{\epsilon,\eta}$ in $\mathcal{A}$. It follows
that

$E_{\epsilon}^{GL}[\nabla u]=E_{\epsilon,\eta}[(\nabla u,0)]\geq
E_{\epsilon,\eta}[\bar{U}_{\epsilon}]=E_{\epsilon}[U_{\epsilon}]\text{ for all
}\nabla u\in\mathcal{A}^{GL}.$

This means that $U_{\epsilon}$ is a minimizer of $E_{\epsilon}^{GL}$ in
$\\{\nabla u\in\mathcal{A}^{GL}\\}$. Conversely, if $\nabla\tilde{u}$ is a
minimizer of $E_{\epsilon}^{GL}$ in $\\{\nabla u\in\mathcal{A}^{GL}\\}$, then

$E_{\epsilon,\eta}[(\nabla\tilde{u},0)]=E_{\epsilon}^{GL}[\nabla\tilde{u}]=E_{\epsilon}^{GL}[U_{\epsilon}]=E_{\epsilon,\eta}[\bar{U}_{\epsilon}],$

i.e. $(\nabla\tilde{u},0)$ is also a minimizer of $E_{\epsilon,\eta}$ in
$\mathcal{A}$. By Theorem 3, $\nabla\tilde{u}=U_{\epsilon}$ as desired. ∎

We next prove Theorem 2 for the $\mathbb{S}^{N}$-valued Ginzburg–Landau model.

###### Proof of Theorem 2.

Set $W(t)=t^{2}$ and fix some $\eta>0$. As $4\leq N\leq 6$ and
$W^{\prime}(1)>0$, we know by [26] that for $\epsilon>0$ small enough, there
exists a unique escaping radially symmetric critical point of the form

$U_{\epsilon,\eta}=(f_{\epsilon,\eta}(r)\frac{x}{r},g_{\epsilon,\eta}(r))\in\mathcal{A},\quad
g_{\epsilon,\eta}>0\textrm{ in }(0,1)$

of the energy $E_{\epsilon,\eta}$. Pick an arbitrary $M=(\nabla
m,M_{N+1})\in\mathcal{A}^{MM}$ (in particular, $|M|=1$) and set

$(\nabla v_{\epsilon,\eta},p_{\epsilon,\eta}):=M-U_{\epsilon,\eta}.$

Then by Section 2, we know that

$\displaystyle E^{MM}_{\eta}[M]$
$\displaystyle=E_{\epsilon,\eta}\big{[}U_{\epsilon,\eta}+(\nabla
v_{\epsilon,\eta},p_{\epsilon,\eta})\big{]}$ $\displaystyle\geq
E_{\epsilon,\eta}[U_{\epsilon,\eta}]+\frac{1}{2}F_{\epsilon,\eta}[\nabla
v_{\epsilon,\eta}]+\frac{1}{2}\int_{B^{N}}T_{\epsilon,\eta}p_{\epsilon,\eta}\cdot
p_{\epsilon,\eta}\,dx$

with

$\displaystyle F_{\epsilon,\eta}[\nabla v_{\epsilon,\eta}]$
$\displaystyle\geq\frac{(N-2)^{2}}{4}\int_{B^{N}}\frac{(\partial_{r}v_{\epsilon,\eta})^{2}}{r^{2}}\,dx+\left(\frac{N^{2}}{2}-2N\right)\int_{B^{N}}\frac{|\nabla_{{\mathbb{S}^{N-1}}}v_{\epsilon,\eta}|^{2}}{r^{4}}\,dx,$
$\displaystyle\int_{B^{N}}T_{\epsilon,\eta}p_{\epsilon,\eta}\cdot
p_{\epsilon,\eta}\,dx$
$\displaystyle\geq\int_{B^{N}}g^{2}_{\epsilon,\eta}\Big{|}\nabla\big{(}\frac{p_{\epsilon,\eta}}{g_{\epsilon,\eta}}\big{)}\Big{|}^{2}\,dx.$

By [26, Remark 2.17], for a subsequence $\epsilon\to 0$, we have that
$U_{\epsilon,\eta}\to M_{\eta}^{+}$ in $H^{1}(B^{N})$ (in particular, $\nabla
U_{\epsilon,\eta}\to\nabla M_{\eta}^{+}$ and $U_{\epsilon,\eta}\to
M_{\eta}^{+}$ a.e. in $B^{N}$) and $E_{\epsilon,\eta}(U_{\epsilon,\eta})\to
E^{MM}_{\eta}[M_{\eta}^{+}]$ where
$M_{\eta}^{+}=(\tilde{f}_{\eta}\frac{x}{r},g_{\eta})$ is the unique escaping
radially symmetric critical point of $E^{MM}_{\eta}$ with $g_{\eta}>0$ in
$(0,1)$. Therefore,

$(\nabla v_{\epsilon,\eta},p_{\epsilon,\eta})\to
M-M_{\eta}^{+}=:(\nabla\tilde{v}_{\eta},\tilde{p}_{\eta})$

in $H^{1}(B^{N})$ and a.e. in $B^{N}$ as well as $\nabla(\nabla
v_{\epsilon,\eta},p_{\epsilon,\eta})\to\nabla(\nabla\tilde{v}_{\eta},\tilde{p}_{\eta})$
a.e. in $B^{N}$ for a subsequence $\epsilon\to 0$. By Fatou’s lemma, it
follows for a subsequence $\epsilon\to 0$:

$\displaystyle E^{MM}_{\eta}[M]$
$\displaystyle=E^{MM}_{\eta}[M_{\eta}^{+}+(\nabla\tilde{v}_{\eta},\tilde{p}_{\eta})]$
$\displaystyle\geq
E^{MM}_{\eta}[M_{\eta}^{+}]+\frac{1}{2}\int_{B^{N}}g^{2}_{\eta}\Big{|}\nabla\big{(}\frac{\tilde{p}_{\eta}}{g_{\eta}}\big{)}\Big{|}^{2}\,dx$
$\displaystyle\quad+\frac{(N-2)^{2}}{8}\int_{B^{N}}\frac{(\partial_{r}\tilde{v}_{\eta})^{2}}{r^{2}}+\frac{1}{2}\left(\frac{N^{2}}{2}-2N\right)\int_{B^{N}}\frac{|\nabla_{{\mathbb{S}^{N-1}}}\tilde{v}_{\eta}|^{2}}{r^{4}}.$

We conclude to the minimality of $M_{\eta}^{+}$. If $M$ is another minimizer,
within the above notations, then
$E_{\eta}^{MM}[M]=E_{\eta}^{MM}[M_{\eta}^{+}]$ and so
$\partial_{r}\tilde{v}_{\eta}=0$ in $B^{N}$ yielding $\tilde{v}_{\eta}=0$ (as
$\tilde{v}_{\eta}=0$ on $\partial B^{N}$); also, $\tilde{p}_{\eta}=\alpha
g_{\eta}$ for some constant $\alpha\in\mathbb{R}$. Since $|M|=1$ and
$M=(0,\tilde{p}_{\eta})+M_{\eta}$, we deduce that
$(\tilde{p}_{\eta}+g_{\eta})^{2}=g_{\eta}^{2}$ yielding $\alpha=0$ or $-2$,
i.e. $M=M_{\eta}^{+}$ or $M=M_{\eta}^{-}$. ∎

##  3 Symmetrization and second proof of main results in dimension $N\geq 5$

###  3.1 A symmetrization of scalar functions

In this section, we consider a spherical average rearrangement which is
probably known to the experts. See e.g. [52, Chapter 1, Section 9] for a
similar rearrangement in the context of the Laplace operator. Let $1\leq
q<\infty$. For a function $g\in L^{q}(B^{N},\mathbb{R})$, define a radial
symmetrization $\check{g}$ of $g$ by

$\check{g}(r)=\Big{\\{}\fint_{\mathbb{S}^{N-1}}|g(r\theta)|^{q}\,d\sigma(\theta)\Big{\\}}^{1/q}\geq
0,\quad r\in(0,1).$ (3.1)

When $q=2$, we can also think of this as a rearrangement in the spherical
harmonic decomposition of $g$.

######  Theorem 11.

Let $N\geq 2$, $1\leq q<\infty$, $g\in L^{q}(B^{N},\mathbb{R})$ and
$\check{g}$ be associated to $g$ by (3.1). We have the following conclusions.

  1. (i)

The map $g\mapsto\check{g}$ is a $1$-Lipschitz continuous map from
$L^{q}(B^{N},\mathbb{R})$ into itself:

$\|\check{g}-\check{h}\|_{L^{q}(B^{N},\mathbb{R})}\leq\|g-h\|_{L^{q}(B^{N},\mathbb{R})}.$

Moreover,
$\int_{\mathbb{S}^{N-1}}|\check{g}(r\theta)|^{q}\,d\sigma(\theta)=\int_{\mathbb{S}^{N-1}}|g(r\theta)|^{q}\,d\sigma(\theta)$
for a.e. $r\in(0,1)$.

  2. (ii)

Let $G:[0,\infty)\times[0,\infty)\rightarrow[0,\infty)$ be continuous. If $G$
is convex in the second variable, then

$\displaystyle\int_{B^{N}}G(r,|\check{g}(x)|^{q})\,dx\leq\int_{B^{N}}G(r,|g(x)|^{q})\,dx.$

In particular, for any $q<p<\infty$,

$\int_{B^{N}}|\check{g}|^{p}\,dx\leq\int_{B^{N}}|g|^{p}\,dx.$

  3. (iii)

Assume in addition that $g\in W^{1,q}(B^{N},\mathbb{R})$. Then $\check{g}\in
W^{1,q}(B^{N},\mathbb{R})$ and

$\int_{B^{N}}|\nabla\check{g}|^{q}\,dx\leq\int_{B^{N}}|\nabla g|^{q}\,dx.$
(3.2)

Equality is attained if and only if $g$ is radially symmetric and
$|g|=\check{g}$ in $(0,1)$.

###### Proof.

Proof of (i): From the definition of the radial function
$\check{g}(x)=\check{g}(r)$ we have

$\int_{\mathbb{S}^{N-1}}|\check{g}(r\theta)|^{q}\,d\sigma(\theta)=\int_{\mathbb{S}^{N-1}}|g(r\theta)|^{q}d\sigma(\theta)\text{
for a.e. }r\in(0,1),$

which implies $\check{g}\in L^{q}(B^{N})$. Also, by the reverse triangle
inequality, we have for $g,h\in L^{q}(B^{N})$ that

$\displaystyle\|\check{g}-\check{h}\|_{L^{q}(B^{N})}^{q}$
$\displaystyle=|{\mathbb{S}^{N-1}}|\int_{0}^{1}|\check{g}(r)-\check{h}(r)|^{q}\,r^{N-1}\,dr$
$\displaystyle=\int_{0}^{1}\big{|}\|g(r\cdot)\|_{L^{q}({\mathbb{S}^{N-1}})}-\|h(r\cdot)\|_{L^{q}({\mathbb{S}^{N-1}})}\big{|}^{q}\,r^{N-1}\,dr$
$\displaystyle\leq\int_{0}^{1}\|g(r\cdot)-h(r\cdot)\|_{L^{q}({\mathbb{S}^{N-1}})}^{q}\,r^{N-1}\,dr=\|g-h\|_{L^{q}(B^{N})}^{q}.$

Therefore $g\mapsto\check{g}$ is a $1$-Lipschitz continuous map on
$L^{q}(B^{N})$.

Proof of (ii): By Jensen inequality,

$\fint_{\mathbb{S}^{N-1}}G(r,|\check{g}(r\theta)|^{q})\,d\sigma(\theta)=G\left(r,\fint_{\mathbb{S}^{N-1}}|\check{g}(r\theta)|^{q}\,d\sigma(\theta)\right)\\\
=G\left(r,\fint_{\mathbb{S}^{N-1}}|g(r\theta)|^{q}\,d\sigma(\theta)\right)\leq\fint_{\mathbb{S}^{N-1}}G(r,|g(r\theta)|^{q})\,d\sigma(\theta).$

Integrating in $r$ gives the second bullet point. In particular, with
$G(r,s)=s^{p/q}$ with $p>q$, we see that the $L^{p}$-norm of $\check{g}$ is no
more than that of $g$.

Proof of (iii): Consider first the case $g$ belongs to
$\mathcal{C}^{\infty}(\bar{B}^{N})$, which is a dense subset of
$W^{1,q}(B^{N})$. For technical reasons, we introduce, for $\mu>0$,

$\check{g}_{\mu}(r)=\Big{\\{}\fint_{\mathbb{S}^{N-1}}(g(r\theta)^{2}+\mu)^{q/2}\,d\sigma(\theta)\Big{\\}}^{1/q}\geq\mu^{1/2},\quad
r\in(0,1).$

Note that $\check{g}_{\mu}\rightarrow\check{g}$ in $L^{q}(B^{N})$ as
$\mu\rightarrow 0$. We have, by Hölder’s inequality,

$\displaystyle|\check{g}_{\mu}(r)|^{q-1}|\check{g}_{\mu}^{\prime}(r)|$
$\displaystyle\leq\fint_{\mathbb{S}^{n-1}}(g(r\theta)^{2}+\mu)^{(q-1)/2}|\partial_{r}g(r\theta)|\,d\sigma(\theta)$
$\displaystyle\leq|\check{g}_{\mu}(r)|^{q-1}\Big{\\{}\fint_{\mathbb{S}^{n-1}}|\partial_{r}g(r\theta)|^{q}\,d\sigma(\theta)\Big{\\}}^{1/q}.$

As $\check{g}_{\mu}\geq\mu^{1/2}>0$, this implies

$\fint_{\mathbb{S}^{N-1}}|\nabla\check{g}_{\mu}(r\theta)|^{q}\,d\sigma(\theta)=|\check{g}_{\mu}^{\prime}(r)|^{q}\leq\fint_{\mathbb{S}^{N-1}}|\partial_{r}g(r\theta)|^{q}\,d\sigma(\theta).$

Integrating over $r\in(0,1)$ gives

$\int_{B^{N}}|\nabla\check{g}_{\mu}|^{q}\,dx\leq\int_{B^{N}}|\partial_{r}g|^{q}dx.$

This implies $\check{g}_{\mu}$ is bounded in $W^{1,q}(B^{N})$ and hence
converges weakly to $\check{g}$ in $W^{1,q}(B^{N})$ as $\mu\rightarrow 0$.
Hence

$\int_{B^{N}}|\nabla\check{g}|^{q}\,dx\leq\int_{B^{N}}|\partial_{r}g|^{q}dx,$
(3.3)

which proves (3.2) for $g\in\mathcal{C}^{\infty}(\bar{B}^{N})$.

Suppose now $g\in W^{1,q}(B^{N})$. Pick
$\\{g_{(j)}\\}\subset\mathcal{C}^{\infty}(\bar{B}^{N})$ such that
$g_{(j)}\rightarrow g$ in $W^{1,q}(B^{N})$. By (i),
$\check{g}_{(j)}\rightarrow\check{g}$ in $L^{q}(B^{N})$. Also, by (3.3),

$\int_{B^{N}}|\nabla\check{g}_{(j)}|^{q}\,dx\leq\int_{B^{N}}|\partial_{r}g_{(j)}|^{q}dx.$
(3.4)

This implies that $\check{g}_{(j)}$ is bounded in $W^{1,q}(B^{N})$ and hence
converges weakly in $W^{1,q}(B^{N})$ to $\check{g}$. Sending
$j\rightarrow\infty$ we see that (3.3) remains valid for $g\in
W^{1,q}(B^{N})$, which proves (3.2). Moreover, equality holds in (3.2) if and
only if $|\nabla g|=|\partial_{r}g|$ a.e., i.e. $g$ is radially symmetric. ∎

###  3.2 A symmetrization of gradient fields and proof of Theorem 5

Recall the symmetrization $\check{v}$ for a function $v\in
H^{1}(B^{N},\mathbb{R})$ is given by the formula (1.17):

$\check{v}(r)=-\int_{r}^{1}\Big{\\{}\fint_{\mathbb{S}^{N-1}}|\nabla
v(s\theta)|^{2}d\sigma(\theta)\Big{\\}}^{1/2}\,ds\leq 0,\quad r\in(0,1).$

We will use the following density result.

######  Lemma 12.

For $N\geq 2$, the set $\mathcal{S}$ of functions in
$\mathcal{C}^{\infty}(\bar{B}^{N})$ which are constant in a neighborhood of
the origin is dense in $H^{2}(B^{N})$. Moreover, if $v\in H^{2}(B^{N})$
verifies $\int_{{\mathbb{S}^{N-1}}}v(r\theta)\theta d\sigma(\theta)=0$ for
almost every $r\in(0,1)$, then its approximation sequence in $\mathcal{S}$ may
be chosen with the same property.

###### Proof.

It is well known that $\mathcal{C}^{\infty}(\bar{B}^{N})$ is dense in
$H^{2}(B^{N})$. Thus, to show that $\mathcal{S}$ is dense in $H^{2}(B^{N})$,
we only need to show that a given $v\in\mathcal{C}^{\infty}(\bar{B}^{N})$ can
be approximated by a sequence of functions in $\mathcal{S}$. In the proof, $C$
denotes a constant that can change between lines but depends only on the
dimension $N$. Pick a cut-off function
$\varphi\in\mathcal{C}^{\infty}(\mathbb{R})$ with $\varphi\equiv 1$ in
$(-\infty,1/2]$, $\varphi\equiv 0$ in $[1,\infty)$. For $j\geq 10$ and $x\in
B^{N}$, let

$\varphi_{(j)}(x)=\begin{cases}\varphi(j|x|)&\text{ if }N\geq 3,\\\
1-\varphi(\frac{\ln\ln\frac{1}{|x|}}{2\ln\ln j})&\text{ if }N=2.\end{cases}$

Note that $\varphi_{(j)}(x)=0$ for $|x|\geq\frac{1}{j}$ and
$\varphi_{(j)}(x)=1$ when $|x|$ is small enough. Define

$v_{(j)}=v(0)\varphi_{(j)}+v(1-\varphi_{(j)})=v-(v-v_{0})\varphi_{(j)}\in\mathcal{S},\quad
j\geq 1.$

We estimate

$\displaystyle|v(x)-v(0)|\leq\|\nabla v\|_{L^{\infty}(B^{N})}|x|,$
$\displaystyle\|\varphi_{(j)}\|_{L^{2}(B^{N})}\leq Cj^{-N/2},$
$\displaystyle\|\nabla\varphi_{(j)}\|_{L^{2}(B^{N})}+\|r\nabla^{2}\varphi_{(j)}\|_{L^{2}(B^{N})}\leq
C\omega_{N}(j)\text{ with }\omega_{N}(j)=\begin{cases}Cj^{-(N-2)/2}&\text{ if
}N\geq 3,\\\ \frac{C}{(\ln j\ln\ln j)^{1/2}}&\text{ if }N=2.\end{cases}$

We thus have

$\displaystyle\|(v-v(0))\varphi_{(j)}\|_{L^{2}(B^{N})}$ $\displaystyle\leq
Cj^{-N/2}\|v\|_{L^{\infty}(B^{N})},$
$\displaystyle\|\nabla[(v-v(0))\varphi_{(j)}]\|_{L^{2}(B^{N})}$
$\displaystyle\leq Cj^{-N/2}\|\nabla
v\|_{L^{\infty}(B^{N})}+C\omega_{N}(j)\|v\|_{L^{\infty}(B^{N})},$
$\displaystyle\|\nabla^{2}[(v-v(0))\varphi_{(j)}]\|_{L^{2}(B^{N})}$
$\displaystyle\leq
Cj^{-N/2}\|\nabla^{2}v\|_{L^{\infty}(B^{N})}+C\omega_{N}(j)\|\nabla
v\|_{L^{\infty}(B^{N})}.$

Clearly, these estimates imply that $v_{(j)}\rightarrow v$ in $H^{2}(B^{N})$.
We have proved that $\mathcal{S}$ is dense in $H^{2}(B^{N})$

Now suppose $v\in H^{2}(B^{N})$ and $\int_{{\mathbb{S}^{N-1}}}v(r\theta)\theta
d\sigma(\theta)=0$. Let $v_{(j)}\in\mathcal{S}$ be such that
$v_{(j)}\rightarrow v$ in $H^{2}(B^{N})$. Define
$\tilde{v}_{(j)}(r\theta)=v_{(j)}(r\theta)-\sum_{k=1}^{N}v_{(j),k}(r)\phi_{k}(\theta)$
where
$v_{(j),k}(r)=\int_{{\mathbb{S}^{N-1}}}v_{(j)}(r\theta)\phi_{k}(\theta)d\sigma(\theta)$.
It is clear that $\int_{{\mathbb{S}^{N-1}}}\tilde{v}_{(j)}(r\theta)\theta
d\sigma(\theta)=0$. Since $v_{(j)}$ is constant near $0$, $v_{(j),k}$ is
supported away from $0$, and so $\tilde{v}_{(j)}\in\mathcal{S}$. Finally,
since the map $w\in H^{2}(B^{N})\mapsto(r\theta\mapsto
w_{k}(r)\phi_{k}(\theta))$ is continuous in $H^{2}(B^{N})$ and $v_{k}\equiv 0$
for $k=1,\ldots,N$ we have

$\lim_{j\rightarrow\infty}\|\tilde{v}_{(j)}-v\|_{H^{2}(B^{N})}\leq\lim_{j\rightarrow\infty}\|v_{(j)}-v\|_{H^{2}(B^{N})}+\lim_{j\rightarrow\infty}\Big{\|}\sum_{k=1}^{N}v_{(j),k}(r)\phi_{k}(\theta)\Big{\|}_{H^{2}(B^{N})}=0.$

The proof is complete. ∎

###### Proof of Theorem 5.

Proof of (i): By Cauchy-Schwarz’ inequality,

$\displaystyle\check{v}(r)^{2}$
$\displaystyle=\Big{\\{}\int_{r}^{1}\Big{[}\fint_{\mathbb{S}^{N-1}}|\nabla
v(s\theta)|^{2}d\sigma(\theta)\Big{]}^{1/2}\,ds\Big{\\}}^{2}$
$\displaystyle\leq\Big{\\{}\int_{r}^{1}s^{1-N}ds\Big{\\}}\Big{\\{}\int_{r}^{1}s^{N-1}\fint_{\mathbb{S}^{N-1}}|\nabla
v(s\theta)|^{2}d\sigma(\theta)\,ds\Big{\\}}.$

Hence $\check{v}(r)$ is well-defined and finite in $(0,1)$; in fact,
$|\check{v}(r)|\leq C_{N}r^{-\frac{N-2}{2}}\|\nabla v\|_{L^{2}(B^{N})}$ for
$N\geq 3$ (resp. $|\check{v}(r)|\leq C\sqrt{\log(1/r)}\|\nabla
v\|_{L^{2}(B^{2})}$ when $N=2$). In particular, $\check{v}\in L^{2}(B^{N})$.
Moreover, by the definition of $\check{v}$ we have
$\int_{\mathbb{S}^{N-1}}|\nabla\check{v}(r,\theta)|^{2}\,d\sigma(\theta)=\int_{\mathbb{S}^{N-1}}|\nabla
v(r\theta)|^{2}d\sigma(\theta)$ for a.e. $r\in(0,1)$. As $\check{v}(1)=0$,
these imply that $\check{v}\in H_{0}^{1}(B^{N})$.

As in the proof of (i) in Theorem 11, the map $\nabla v\mapsto\nabla\check{v}$
is a $1$-Lipschitz continuous map from $L^{2}(B^{N},\mathbb{R}^{N})$ into
itself. By Poincaré’s inequality, the map $v\mapsto\check{v}$ is a Lipschitz
continuous map from $H^{1}(B^{N})$ into $H_{0}^{1}(B^{N})$.

Proof of (ii): This is similar to that in the proof of Theorem 11 and is
omitted.

Proof of (iii): By density and (i), it suffices to consider
$v\in\mathcal{C}_{c}^{\infty}(B^{N})$.

Let $A(r)=\fint_{{\mathbb{S}^{N-1}}}|v(r\theta)|^{p}\,d\sigma(\theta)$. We
have, by Hölder’s inequality

$\displaystyle|A^{\prime}(r)|$ $\displaystyle\leq
p\fint_{{\mathbb{S}^{N-1}}}|v(r\theta)|^{p-1}|\partial_{r}v(r\theta)|\,d\sigma\leq
p\Big{\\{}\fint_{{\mathbb{S}^{N-1}}}|v(r\theta)|^{2(p-1)}\,d\sigma\Big{\\}}^{1/2}\check{v}^{\prime}(r)$
$\displaystyle\leq p\,A(r)^{\frac{p-1}{p}}\,\check{v}^{\prime}(r),$

where we have used $2(p-1)\leq p$ when $1\leq p\leq 2$.

Fix some $\mu>0$. Then
$|\frac{d}{dr}(\mu+A(r))^{1/p}|\leq\check{v}^{\prime}(r)$. This together with
$A(1)=0$ (since $v=0$ on $\partial B^{N})$ implies

$(\mu+A(r))^{1/p}\leq\mu^{1/p}+\int_{r}^{1}\check{v}^{\prime}(r)\,dr=\mu^{1/p}-\check{v}(r)=\mu^{1/p}+|\check{v}(r)|.$

Sending $\mu\rightarrow 0$, we get the conclusion.

Proof of (iv): Without loss of generality, we can assume that $v=0$ on
$\partial B^{N}$ (since, on $\partial B^{N}$, $\nabla v(x)=cx$ is normal to
$\partial B^{N}$). Let $(\phi_{k})_{k=0}^{\infty}$ be an orthonormal basis of
$L^{2}(\mathbb{S}^{N-1})$ consisting of eigenfunctions of the Laplace-Beltrami
operator on ${\mathbb{S}^{N-1}}$ corresponding to eigenvalues
$0=\lambda_{0}<N-1=\lambda_{1}=\ldots=\lambda_{N}<2N=\lambda_{N+1}\leq\ldots\rightarrow\infty$.
We decompose

$v(r\theta)=\sum_{k=0}^{\infty}v_{k}(r)\phi_{k}(\theta)\text{ where
}v_{k}(r)=\int_{\mathbb{S}^{N-1}}v(r\theta)\phi_{k}(\theta)\,d\sigma(\theta).$

Note that $v_{k}\in H^{2}_{\mathrm{loc}}(0,1)$, and

$\displaystyle(\check{v}^{\prime})^{2}$
$\displaystyle=\sum_{k=0}^{\infty}\Big{[}(v_{k}^{\prime})^{2}+\frac{\lambda_{k}}{r^{2}}v_{k}^{2}\Big{]},$
(3.5) $\displaystyle\int_{B^{N}}(\Delta v)^{2}\,dx$
$\displaystyle=\sum_{k=0}^{\infty}\int_{0}^{1}r^{N-1}\Big{(}v_{k}^{\prime\prime}+\frac{N-1}{r}v_{k}^{\prime}-\frac{\lambda_{k}}{r^{2}}v_{k}\Big{)}^{2}\,dr.$
(3.6)

Note also that our hypotheses give in the case $N\in\\{3,4\\}$ that
$v_{1}=\ldots=v_{N}=0$.

We first prove inequality (1.18) when $v$ belongs to the set $\mathcal{S}$
defined in Lemma 12. Then $v_{0}\in\mathcal{C}^{\infty}([0,1])$, $v_{0}$ is
constant near $0$, $v_{k}\in\mathcal{C}_{c}^{\infty}((0,1])$ for $k\geq 1$,

$v_{0}(1)=0,v_{0}^{\prime}(1)=c\text{ and }v_{k}(1)=v_{k}^{\prime}(1)=0\text{
for }k\geq 1.$

This implies $\nabla\check{v}(x)=|c|x$ on $\partial B^{N}$ (recall that, by
definition, $\check{v}^{\prime}\geq 0$ in $(0,1)$). Also,

$\displaystyle\int_{0}^{1}r^{N-2}v_{k}^{\prime\prime}v_{k}^{\prime}\,dr$
$\displaystyle=-\frac{N-2}{2}\int_{0}^{1}r^{N-3}(v_{k}^{\prime})^{2}\,dr+\begin{cases}\frac{c^{2}}{2}&\text{
if }k=0,\\\ 0&\text{ if }k\geq 1,\end{cases}$
$\displaystyle\int_{0}^{1}r^{N-4}v_{k}^{\prime}v_{k}\,dr$
$\displaystyle=-\frac{N-4}{2}\int_{0}^{1}r^{N-5}v_{k}^{2}\,dr\text{ for }k\geq
1,$ $\displaystyle\int_{0}^{1}r^{N-3}v_{k}^{\prime\prime}v_{k}\,dr$
$\displaystyle=\int_{0}^{1}\Big{[}-r^{N-3}(v_{k}^{\prime})^{2}+\frac{(N-3)(N-4)}{2}r^{N-5}v_{k}^{2}\Big{]}\,dr\text{
for }k\geq 1.$

Inserting the above identities in (3.6), we obtain

$\int_{B^{N}}(\Delta
v)^{2}\,dx=(N-1)c^{2}+\sum_{k=0}^{\infty}\int_{0}^{1}r^{N-1}\Big{[}(v_{k}^{\prime\prime})^{2}+\frac{2\lambda_{k}+N-1}{r^{2}}(v_{k}^{\prime})^{2}+\frac{\lambda_{k}(\lambda_{k}+2(N-4))}{r^{4}}v_{k}^{2}\Big{]}\,dr.$
(3.7)

Next, note that, when $v\in\mathcal{S}$, the right hand side of (3.5) is a
smooth non-negative function and so $\check{v}^{\prime}$ is Lipschitz
continuous. Applying (3.7) to $\check{v}$, we get

$\displaystyle\int_{B^{N}}(\Delta\check{v})^{2}\,dx$
$\displaystyle=(N-1)c^{2}+\int_{0}^{1}r^{N-1}\Big{[}(\check{v}^{\prime\prime})^{2}+\frac{N-1}{r^{2}}(\check{v}^{\prime})^{2}\Big{]}\,dr.$
(3.8)

To continue, we need to estimate $\check{v}^{\prime\prime}$. For technical
reasons, we consider for $\mu>0$ a regularized version of $\check{v}$:

$\check{v}_{\mu}^{\prime}=\Big{\\{}\mu+\sum_{k=0}^{\infty}\Big{[}(v_{k}^{\prime})^{2}+\frac{\lambda_{k}}{r^{2}}v_{k}^{2}\Big{]}\Big{\\}}^{1/2}\geq\mu^{1/2}.$

Clearly $\check{v}_{\mu}^{\prime}$ is smooth and
$\check{v}_{\mu}^{\prime}\rightarrow\check{v}^{\prime}$ pointwise in $(0,1)$
as $\mu\rightarrow 0$. Now, for some $t_{k}\in\mathbb{R}$ to be chosen later,
we have by (3.5) that

$\displaystyle|\check{v}_{\mu}^{\prime}||\check{v}_{\mu}^{\prime\prime}|$
$\displaystyle=\Big{|}\sum_{k=0}^{\infty}\Big{[}v_{k}^{\prime}v_{k}^{\prime\prime}+\frac{\lambda_{k}}{r^{2}}v_{k}v_{k}^{\prime}-\frac{\lambda_{k}}{r^{3}}v_{k}^{2}\Big{]}\Big{|}$
$\displaystyle\leq|v_{0}^{\prime}||v_{0}^{\prime\prime}|+\Big{|}\sum_{k=1}^{\infty}\Big{[}v_{k}^{\prime}(v_{k}^{\prime\prime}+\frac{t_{k}}{r^{2}}v_{k})+\frac{1}{r}v_{k}(\frac{\lambda_{k}-t_{k}}{r}v_{k}^{\prime}-\frac{\lambda_{k}}{r^{2}}v_{k})\Big{]}\Big{|}$
$\displaystyle\leq|v_{0}^{\prime}||v_{0}^{\prime\prime}|+\sum_{k=1}^{\infty}\Big{[}(v_{k}^{\prime})^{2}+\frac{\lambda_{k}}{r^{2}}v_{k}^{2}\Big{]}^{1/2}\Big{[}(v_{k}^{\prime\prime}+\frac{t_{k}}{r^{2}}v_{k})^{2}+\frac{1}{\lambda_{k}}(\frac{\lambda_{k}-t_{k}}{r}v_{k}^{\prime}-\frac{\lambda_{k}}{r^{2}}v_{k})^{2}\Big{]}^{1/2}$
$\displaystyle\leq|\check{v}_{\mu}^{\prime}|\Big{\\{}|v_{0}^{\prime\prime}|^{2}+\sum_{k=1}^{\infty}\Big{[}(v_{k}^{\prime\prime}+\frac{t_{k}}{r^{2}}v_{k})^{2}+\frac{1}{\lambda_{k}}(\frac{\lambda_{k}-t_{k}}{r}v_{k}^{\prime}-\frac{\lambda_{k}}{r^{2}}v_{k})^{2}\Big{]}\Big{\\}}^{1/2}.$

Since $\check{v}_{\mu}^{\prime}\geq\mu^{1/2}>0$, this implies

$|\check{v}_{\mu}^{\prime\prime}|\leq\Big{\\{}|v_{0}^{\prime\prime}|^{2}+\sum_{k=1}^{\infty}\Big{[}(v_{k}^{\prime\prime}+\frac{t_{k}}{r^{2}}v_{k})^{2}+\frac{1}{\lambda_{k}}(\frac{\lambda_{k}-t_{k}}{r}v_{k}^{\prime}-\frac{\lambda_{k}}{r^{2}}v_{k})^{2}\Big{]}\Big{\\}}^{1/2}.$

This implies that $\\{\check{v}_{\mu}^{\prime}\\}$ is bounded in
$W^{1,\infty}((0,1))$ and converges weakly* in $W^{1,\infty}((0,1))$ to
$\check{v}^{\prime}$ as $\mu\rightarrow 0$ (since
$\check{v}_{\mu}^{\prime}\rightarrow\check{v}^{\prime}$ pointwise), and

$|\check{v}^{\prime\prime}|\leq\Big{\\{}|v_{0}^{\prime\prime}|^{2}+\sum_{k=1}^{\infty}\Big{[}(v_{k}^{\prime\prime}+\frac{t_{k}}{r^{2}}v_{k})^{2}+\frac{1}{\lambda_{k}}(\frac{\lambda_{k}-t_{k}}{r}v_{k}^{\prime}-\frac{\lambda_{k}}{r^{2}}v_{k})^{2}\Big{]}\Big{\\}}^{1/2}.$

Returning to (3.8), we get

$\displaystyle\int_{B^{N}}(\Delta\check{v})^{2}\,dx\leq(N-1)c^{2}+\int_{0}^{1}r^{N-1}\Big{[}(v_{0}^{\prime\prime})^{2}+\frac{N-1}{r^{2}}(v_{0}^{\prime})^{2}\Big{]}\,dr$
$\displaystyle\quad+\sum_{k=1}^{\infty}\int_{0}^{1}r^{N-1}\Big{[}(v_{k}^{\prime\prime}+\frac{t_{k}}{r^{2}}v_{k})^{2}+\frac{1}{\lambda_{k}}(\frac{\lambda_{k}-t_{k}}{r}v_{k}^{\prime}-\frac{\lambda_{k}}{r^{2}}v_{k})^{2}+\frac{N-1}{r^{2}}(v_{k}^{\prime})^{2}+\frac{(N-1)\lambda_{k}}{r^{4}}v_{k}^{2}\Big{]}\,dr$
$\displaystyle=(N-1)c^{2}+\int_{0}^{1}r^{N-1}\Big{[}(v_{0}^{\prime\prime})^{2}+\frac{N-1}{r^{2}}(v_{0}^{\prime})^{2}\Big{]}\,dr$
$\displaystyle\quad+\sum_{k=1}^{\infty}\int_{0}^{1}r^{N-1}\Big{[}(v_{k}^{\prime\prime})^{2}+\frac{\lambda_{k}^{-1}(\lambda_{k}-t_{k})^{2}-2t_{k}+N-1}{r^{2}}(v_{k}^{\prime})^{2}$
$\displaystyle\qquad+\frac{2\lambda_{k}(N-2)+t_{k}^{2}+t_{k}(N-4)^{2}}{r^{4}}v_{k}^{2}\Big{]}\,dr.$

Recalling (3.7), we get

$\displaystyle\int_{B^{N}}(\Delta
v)^{2}\,dx-\int_{B^{N}}(\Delta\check{v})^{2}\,dx$
$\displaystyle\geq\sum_{k=1}^{\infty}\int_{0}^{1}r^{N-1}\Big{[}\frac{\lambda_{k}-\lambda_{k}^{-1}t_{k}^{2}+4t_{k}}{r^{2}}(v_{k}^{\prime})^{2}$
$\displaystyle\qquad+\frac{\lambda_{k}^{2}-4\lambda_{k}-t_{k}^{2}-t_{k}(N-4)^{2}}{r^{4}}v_{k}^{2}\Big{]}\,dr.$
(3.9)

Case 1: If $N\geq 5$, we choose $t_{k}=0$, and using the sharp Hardy
inequality
$\int_{0}^{1}r^{N-3}(v_{k}^{\prime})^{2}\,dr\geq\frac{(N-4)^{2}}{4}\int_{0}^{1}r^{N-5}v_{k}^{2}\,dr$
to obtain from (3.9) the inequality

$\int_{B^{N}}(\Delta
v)^{2}\,dx\geq\int_{B^{N}}(\Delta\check{v})^{2}\,dx+\sum_{k=1}^{\infty}\lambda_{k}s_{k}\int_{0}^{1}r^{N-5}v_{k}^{2}\,dr,$
(3.10)

where, for $k\geq 1$,

$s_{k}=\lambda_{k}+\frac{(N-4)^{2}}{4}-4>0\quad(\text{since }\lambda_{k}\geq
N-1\geq 4).$

Inequality (1.18) thus follows.

Case 2: If $N\in\\{2,3,4\\}$, recall that our hypotheses give
$v_{1}=\ldots=v_{N}=0$. We choose $t_{k}=(2-\sqrt{5})\lambda_{k}$ in (3.9) so
that the term involving $v_{k}^{\prime}$ vanishes, and arrive again at (3.10)
but with

$s_{k}=\begin{cases}0&\text{ if }1\leq k\leq N,\\\
(\sqrt{5}-2)(4\lambda_{k}+(N-4)^{2})-4&\text{ if }k\geq N+1.\end{cases}$

As $\lambda_{k}\geq 2N$ for $k\geq N+1$, we have

$\displaystyle s_{k}$ $\displaystyle\geq(\sqrt{5}-2)(N^{2}+16)-4\geq
20\sqrt{5}-44>0\text{ for }N\in\\{2,3,4\\},k\geq N+1.$

Inequality (1.18) thus follows from (3.10).

Consider now the general case $v\in H^{2}(B^{N})$. By Lemma 12 we can select
$\\{v_{(j)}\\}\subset\mathcal{S}$ such that $v_{(j)}\rightarrow v$ in
$H^{2}(B^{N})$ as $j\rightarrow\infty$. Moreover, in case $N\in\\{2,3,4\\}$,
it holds also that
$\int_{{\mathbb{S}^{N-1}}}v_{(j)}(r\theta)\theta\,d\sigma(\theta)=0$. By
Fubini’s theorem, after passing to a subsequence, we have
$(v_{(j)})_{k}\rightarrow v_{k}$ a.e. in $(0,1)$ for the spherical harmonic
coefficients of $v_{(j)}$ and $v$. Also, by (i),
$\nabla\check{v}_{(j)}\rightarrow\nabla\check{v}$ in
$L^{2}(B^{N},\mathbb{R}^{N})$. Since $v_{(j)}\in\mathcal{S}$, we have by
(3.10)

$\int_{B^{N}}(\Delta
v_{(j)})^{2}\,dx\geq\int_{B^{N}}(\Delta\check{v}_{(j)})^{2}\,dx+\sum_{k=1}^{\infty}\lambda_{k}s_{k}\int_{0}^{1}r^{N-5}(v_{(j)})_{k}^{2}\,dr.$
(3.11)

This implies that $\\{\check{v}_{(j)}\\}$ is bounded in $H^{2}(B^{N})$. As
$\nabla\check{v}_{(j)}\rightarrow\nabla\check{v}$ in
$L^{2}(B^{N},\mathbb{R}^{N})$, this implies $\Delta\check{v}_{(j)}$ converges
weakly in $L^{2}(B^{N})$ to $\Delta\check{v}$; in particular, $\check{v}\in
H^{2}(B^{N})$. Sending $j\rightarrow\infty$ in (3.11), using the convergence
of $v_{(j)}$ to $v$ in $H^{2}(B^{N})$ on the left hand side, the weak
convergence of $\Delta\check{v}_{(j)}$ to $\Delta\check{v}$ in $L^{2}(B^{N})$
and Fatou’s lemma for the infinite sum on the right hand side, we see that
(3.10) remains valid for $v\in H^{2}(B^{N})$. This proves (1.18) for $v\in
H^{2}(B^{N})$. Also, equality occurs in (1.18) if and only if $v_{k}=0$ for
all $k\geq 1$, meaning $v$ is radially symmetric and
$|v^{\prime}|=\check{v}^{\prime}$ in $(0,1)$. ∎

###  3.3 Second proof of Theorems 1, 2 and 3 in dimension $N\geq 5$

###### Second proof of Theorem 1 in dimension $N\geq 5$.

As $s\mapsto W(1-s)$ is convex, we deduce from Theorem 5 that

$E_{\epsilon}^{GL}[\nabla u]\geq E_{\epsilon}^{GL}[\nabla\check{u}]\text{ for
all }\nabla u\in\mathcal{A}^{GL},$

where equality holds if and only if $u$ is radially symmetric. In particular,
if $\nabla u\in\mathcal{A}^{GL}$ is a minimizer of $E_{\epsilon}^{GL}$ among
gradient field configurations in $\mathcal{A}^{GL}$, then so is
$\nabla\check{u}$ with $E_{\epsilon}^{GL}[\nabla
u]=E_{\epsilon}^{GL}[\nabla\check{u}]$ and hence $u$ is radially symmetric.
The conclusion then follows from [26, Theorem 2.1] on the uniqueness of
radially symmetric critical point of $E_{\epsilon}^{GL}$ in
$\mathcal{A}^{GL}$. ∎

###### Second proof of Theorem 2 in dimension $N\geq 5$.

Observe that if $(\nabla m,M_{N+1})\in\mathcal{A}^{MM}$ and if $\check{m}$
denotes the symmetrization of $m$ by (1.17) and $\check{M}_{N+1}$ denotes the
symmetrization of $M_{N+1}$ by (3.1), then
$(\nabla\check{m},\check{M}_{N+1})\in\mathcal{A}^{MM}$ because

$|\nabla\check{m}|^{2}(r)+\check{M}_{N+1}(r)^{2}=\fint_{{\mathbb{S}^{N-1}}}\big{(}|\nabla
m|^{2}(r\theta)+M_{N+1}^{2}(r\theta)\big{)}\,d\sigma(\theta)=1.$

Thus, by Theorems 5 and 11, if $(\nabla m,M_{N+1})\in\mathcal{A}^{MM}$ is a
minimizer of $E_{\eta}^{MM}$ in $\mathcal{A}^{MM}$, the
$(\nabla\check{m},\check{M}_{N+1})$ is also a minimizer of $E_{\eta}^{MM}$ in
$\mathcal{A}^{MM}$ and $(\nabla m,M_{N+1})$ is radially symmetric. The
conclusion then follows from [26, Theorem 2.6] on the classification of
radially symmetric minimizers of $E_{\eta}^{MM}$. ∎

###### Second proof of Theorem 3 in dimension $N\geq 5$.

Let $U=(\nabla v,g)\in\mathcal{A}$ be a minimizer of $E_{\epsilon,\eta}$ in
$\mathcal{A}$. Define the symmetrization $\check{v}$ and $\check{g}$ of $v$
and $g$ as in the previous two sections with $q=2$, and let
$\check{U}=(\nabla\check{v},\check{g})$. By Theorems 5 and 11, we have

$\displaystyle\int_{{\mathbb{S}^{N-1}}}|\nabla\check{v}(r\theta)|^{2}\,d\sigma(\theta)$
$\displaystyle=\int_{{\mathbb{S}^{N-1}}}|\nabla
v(r\theta)|^{2}\,d\sigma(\theta)\text{ for a.e. }r\in(0,1),$
$\displaystyle\int_{{\mathbb{S}^{N-1}}}\check{g}(r\theta)^{2}\,d\sigma(\theta)$
$\displaystyle=\int_{{\mathbb{S}^{N-1}}}g(r\theta)^{2}\,d\sigma(\theta)\text{
for a.e. }r\in(0,1),$ $\displaystyle\int_{B^{N}}\tilde{W}(\check{g}^{2})\,dx$
$\displaystyle\leq\int_{{\mathbb{S}^{N-1}}}\tilde{W}(g^{2})\,dx,$
$\displaystyle\int_{B^{N}}(\Delta\check{v})^{2}\,dx$
$\displaystyle\leq\int_{B^{N}}(\Delta v)^{2}\,dx.$

The first two identities and the convexity of $W$ give

$\fint_{{\mathbb{S}^{N-1}}}W(1-|\check{U}|^{2})(r\theta)\,d\sigma(\theta)=W\Big{(}\fint_{{\mathbb{S}^{N-1}}}(1-|\check{U}|^{2})(r\theta)\,d\sigma(\theta)\Big{)}\\\
=W\Big{(}\fint_{{\mathbb{S}^{N-1}}}(1-|U|^{2})(r\theta)\,d\sigma(\theta)\Big{)}\leq\fint_{{\mathbb{S}^{N-1}}}W(1-|U|^{2})(r\theta)\,d\sigma(\theta).$

These estimates together implies that $E_{\epsilon,\eta}[\check{U}]\leq
E_{\epsilon,\eta}[U]$ and so $\check{U}$ is also a minimizer of
$E_{\epsilon,\eta}$ in $\mathcal{A}$ with
$E_{\epsilon,\eta}[\check{U}]=E_{\epsilon,\eta}[U]$. Returning to the equality
cases in Theorems 5 and 11, we have that $v$ and $g$ are radially symmetric,
i.e. $U$ is a radially symmetric minimizer of $E_{\epsilon,\eta}$. The
conclusion follows from [26, Theorem 2.4] on the classification of radially
symmetric minimizer of $E_{\epsilon,\eta}$. ∎

###  3.4 Symmetry for solutions to a nonlinear eigenvalue problem

For $d>0$, $1\leq p<2$ and $\lambda\in\mathbb{R}$, consider the energy
functional

$J[v]=\frac{1}{2}\|\Delta
v\|_{L^{2}(B^{N})}^{2}-\frac{\lambda}{2}\|v\|_{L^{2}(B^{N})}^{2}$

on the set

$S_{p,d}=\big{\\{}v\in H^{2}_{0}(B^{N}):\|v\|_{L^{p}(B^{N})}=d\big{\\}}.$

Let $\lambda_{1}(\Delta^{2})$ denote the first eigenvalue of the bi-Laplacian
in $H_{0}^{2}(B^{N})$. When $\lambda<\lambda_{1}(\Delta^{2})$, after adjusting
by a scaling factor to remove the Lagrange multiplier, minimizers of $J$ on
$S_{p,d}$ correspond to solutions of the elliptic problem (1.20):

$\begin{cases}\Delta^{2}v=\lambda v+|v|^{p-2}v\text{ on }B^{N},\\\
v=\partial_{r}v=0\text{ on }\partial B^{N}.\end{cases}$

(For $\lambda\geq\lambda_{1}(\Delta^{2})$, the partial differential equation
is different, namely

$\Delta^{2}v=\begin{cases}\lambda_{1}(\Delta^{2})v&\text{ if
}\lambda=\lambda_{1}(\Delta^{2}),\\\ \lambda v-|v|^{p-2}v&\text{ if
}\lambda>\lambda_{1}(\Delta^{2}),\end{cases}$

and we do not consider these cases here for simplicity.)

Problem (1.20) has been studied by many authors and a summary of known results
would go beyond the scope of the present paper. We refer the reader to e.g.
[4, 7, 14, 16, 17, 36] and the references therein.

We prove:

######  Corollary 13.

Let $N\geq 5$ and $1\leq p<2$. For $\lambda<\lambda_{1}(\Delta^{2})$,
minimizers of $J$ over $S_{p,d}$ are radially symmetric, do not change sign
and are either radially non-decreasing or radially non-increasing.

###### Proof.

Note that as $\lambda<\lambda_{1}(\Delta^{2})$, $J$ is coercive on
$H_{0}^{2}(B^{N})$. By the compactness embedding theorem, $J$ has a minimizer
over $S_{p,d}$.

Let $v$ be a minimizer of $J$ over $S_{p,d}$; in particular, $J[v]\geq 0$. By
Theorem 5, we have

$\begin{cases}\|\Delta\check{v}\|_{L^{2}(B^{N})}\leq\|\Delta
v\|_{L^{2}(B^{N})},\\\
\|\check{v}\|_{L^{p}(B^{N})}\geq\|v\|_{L^{p}(B^{N})}=d,\\\
\|\check{v}\|_{L^{2}(B^{N})}\geq\|v\|_{L^{2}(B^{N})}.\end{cases}$ (3.12)

Let

$\bar{v}=\mu\check{v}\text{ where
}\mu=d\|\check{v}\|_{L^{p}(B^{N})}^{-1}\stackrel{{\scriptstyle\eqref{Eq:C6vv}}}{{\leq}}1$

so that $\bar{v}\in S_{p,d}$. We compute, keeping in mind that $\mu\leq 1$,

$J[\bar{v}]=\mu^{2}J[\check{v}]\stackrel{{\scriptstyle\eqref{Eq:C6vv}}}{{\leq}}\mu^{2}J[v]\leq
J[v],$

where for the last inequality we use the fact that $J[v]\geq 0$. It follows
that $\bar{v}$ is also a minimizer of $J$ over $S_{p,d}$, which in turn
implies $J[\check{v}]=J[v]$ and all inequalities in (3.12) are saturated.
Appealing to the equality case in Theorem 5(iv), we see that $v$ is radially
symmetric and $\check{v}^{\prime}=|v^{\prime}|$.

It remains to prove that $v$ and $\partial_{r}v$ do not change sign. Indeed,
we have

$|v(r)|=|v(r)-v(1)|=\Big{|}\int_{r}^{1}v^{\prime}(s)\,ds\Big{|}\leq\int_{r}^{1}|v^{\prime}(s)|\,ds=\int_{r}^{1}\check{v}^{\prime}(s)\,ds=|\check{v}(r)|.$

As $\|\check{v}\|_{L^{2}(B^{N})}=\|v\|_{L^{2}(B^{N})}$, it follows that
equality is attained in the above inequality, i.e. $v^{\prime}$ does not
change sign. As $v(1)=0$, it follows also that $v$ does not change sign. ∎

##  Appendix A The negativity of $F_{\epsilon}$ in dimension $N\in\\{2,3\\}$

We now give the proof of Proposition 4 on the negativity of $F_{\epsilon}$ in
dimension $N\in\\{2,3\\}$.

###### Proof of Proposition 4.

We follow ideas from e.g. the proof of [26, Lemma 2.3], [27, Proposition 4.1],
[30, Theorem 1.7]. The main task is to show that there exists
$v\in\mathcal{C}_{c}^{2}(B^{N}\setminus\\{0\\})$ such that555Note that
$\mathcal{C}_{c}^{2}(B^{N}\setminus\\{0\\})$ is not a dense subspace of
$H_{0}^{2}(B^{N})$ in dimension $N\in\\{2,3\\}$, hence the existence of such
$v$ does not follow immediately from the sharpness of the Hardy inequality
(1.16).

$F_{*}[\nabla v]:=\int_{B^{N}}\Big{[}(\Delta v)^{2}\,-\frac{N-1}{r^{2}}|\nabla
v|^{2}\Big{]}\,dx<0.$ (A.1)

Supposing for the moment that such a $v$ has been found, we proceed to show
that $F_{\epsilon}[\nabla v]<0$ for this particular $v$ and for sufficiently
small $\epsilon>0$. Indeed, using the Hardy decomposition Lemma 7 with the
decomposition $\nabla v=f_{\epsilon}\frac{\nabla v}{f_{\epsilon}}$, noting
that $\Delta
f_{\epsilon}=\frac{N-1}{r^{2}}f_{\epsilon}-\frac{1}{\epsilon^{2}}W^{\prime}(1-f_{\epsilon}^{2})f_{\epsilon}$,
we find

$F_{\epsilon}[\nabla
v]=\int_{B^{N}}f_{\epsilon}^{2}\Big{[}\Big{|}\nabla\big{(}\frac{\nabla
v}{f_{\epsilon}}\big{)}\Big{|}^{2}-\frac{N-1}{r^{2}}\frac{|\nabla
v|^{2}}{f_{\epsilon}^{2}}\Big{]}\,dx.$

Since $f_{\epsilon}\rightarrow 1$ in $\mathcal{C}^{1}_{\rm
loc}(B^{N}\setminus\\{0\\})$ and
$v\in\mathcal{C}_{c}^{2}(B^{N}\setminus\\{0\\}$, we deduce that

$\lim_{\epsilon\rightarrow 0}F_{\epsilon}[\nabla
v]=\int_{B^{N}}\Big{[}|\nabla^{2}v|^{2}-\frac{N-1}{r^{2}}|\nabla
v|^{2}\Big{]}\,dx=F_{*}[\nabla v]<0,$

which gives the conclusion.

It remains to find $v\in\mathcal{C}_{c}^{2}(B^{N}\setminus\\{0\\})$ satisfying
(A.1). The proof of Theorem 5 suggests the ansatz

$v(x)=a(r)\frac{x_{1}}{r}.$

We are thus led to searching for $a\in\mathcal{C}_{c}^{2}((0,1))$ such that

$\int_{0}^{1}r^{N-1}\Big{[}(a^{\prime\prime})^{2}+\frac{2(N-1)}{r^{2}}(a^{\prime})^{2}+\frac{2(N-1)(N-4)}{r^{4}}a^{2}\Big{]}\,dr<0.$

We decompose $a(r)=r^{-\frac{N-4}{2}}b(r)$ and compute

$\displaystyle\int_{0}^{1}r^{N-1}(a^{\prime\prime})^{2}\,dr$
$\displaystyle=\int_{0}^{1}r^{3}\Big{(}b^{\prime\prime}-\frac{N-4}{r}b^{\prime}+\frac{(N-2)(N-4)}{4r^{2}}b\Big{)}^{2}\,dr$
$\displaystyle=\int_{0}^{1}\Big{(}r^{3}(b^{\prime\prime})^{2}+\frac{(N-2)(N-4)}{2}r(b^{\prime})^{2}+\frac{(N-2)^{2}(N-4)^{2}}{16r}b^{2}\Big{)}\,dr,$
$\displaystyle\int_{0}^{1}r^{N-3}(a^{\prime})^{2}\,dr$
$\displaystyle=\int_{0}^{1}r\Big{(}b^{\prime}-\frac{N-4}{2r}b\Big{)}^{2}\,dr$
$\displaystyle=\int_{0}^{1}\Big{(}r(b^{\prime})^{2}+\frac{(N-4)^{2}}{4r}b^{2}\Big{)}\,dr$

We thus need to find $b\in\mathcal{C}_{c}^{2}((0,1))$ such that

$\int_{0}^{1}\Big{(}r^{3}(b^{\prime\prime})^{2}+\frac{N^{2}-2N+4}{2}r(b^{\prime})^{2}+\frac{(N-4)(N^{3}+12N-16)}{16r}b^{2}\Big{)}\,dr<0.$
(A.2)

To this end, we fix a cut-off function
$\varphi\in\mathcal{C}_{c}^{\infty}([0,\infty))$ with $\varphi\equiv 1$ in
$[0,1/4]$, $\varphi\equiv 0$ in $[1/2,\infty)$. For $j>20$ large to be fixed,
we let

$b(r)=\begin{cases}\varphi(r)&\text{ if }r\geq 1/8,\\\
\varphi\big{(}\frac{\ln\ln\frac{1}{r}}{4\ln\ln j}\big{)}&\text{ if
}0<r<1/8.\end{cases}$

Then, for some constant $C$ independent of $j$, we have

$\displaystyle\int_{0}^{1}\frac{1}{r}b^{2}\,dr\geq\int_{1/j}^{1/4}\frac{dr}{r}=\ln\frac{j}{4},$
$\displaystyle\int_{0}^{1}\Big{(}r^{3}(b^{\prime\prime})^{2}+r(b^{\prime})^{2}\Big{)}\,dr\leq
C.$

Therefore, as $(N-4)(N^{3}+12N-16)<0$ for $N\in\\{2,3\\}$, we can select a
sufficiently large $j$ so that (A.2) is satisfied. ∎

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