# Uniformly rotating vortices for the lake equation

Taoufik Hmidi  ,  Haroune Houamed  New York University Abu Dhabi
Abu Dhabi
United Arab Emirates<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Emeric
Roulley  SISSA International School for Advanced Studies, Via Bonomea 265,
34136
Trieste
Italy<EMAIL_ADDRESS>and  Mohamed Zerguine  LEDPA, Université de Batna 2,
département des mathématiques, 05000
BATNA
ALGERIE<EMAIL_ADDRESS>
###### Abstract.

We investigate the existence of time-periodic vortex patch solutions, in both
simply and doubly-connected cases, for the two-dimensional lake equation where
the depth function of the lake is assumed to be non-degenerate and radial. The
proofs employ bifurcation techniques, where the most challenging steps are
related to the regularity study of some nonlinear functionals and the spectral
analysis of their linearized operators around Rankine type vortices. The main
difficulties stem from the roughness and the implicit form of the Green
function connecting the fluid vorticity and its stream function. We handle in
part these issues by exploring the asymptotic structure of the solutions to
the associated elliptic problem. As to the distribution of the spectrum, it is
tackled by a fixed-point argument through a perturbative approach.

###### Key words and phrases:

Lake equation, Euler equations, V-states, vortex, elliptic regularity,
singular kernels, fundamental solution

###### Contents

  1. 1 Introduction and main results
    1. 1.1 Aspects of interest
    2. 1.2 Main results
    3. 1.3 Challenges and roadmap of the proofs
  2. 2 On an elliptic problem with variable coefficients
    1. 2.1 A priori bounds and decomposition of solutions
    2. 2.2 Kernels and integral representation of solutions
  3. 3 Simply-connected time-periodic solutions
    1. 3.1 Contour dynamics
    2. 3.2 Spectral analysis
    3. 3.3 Regularity analysis
      1. 3.3.1 On the continuity of some singular operators
      2. 3.3.2 Regularity of the functional $\mathcal{F}$
    4. 3.4 Proof of Theorem 1.1
  4. 4 Doubly-connected time-periodic solutions
    1. 4.1 The contour dynamics equation and linearization
    2. 4.2 Proof of Theorem 1.2
  5. A Crandall–Rabinowitz’s Theorem
  6. B Back to the elliptic problem
    1. B.1 Elliptic regularity
    2. B.2 Existence and uniqueness of solutions
      1. B.2.1 Functional framework and introduction to the equations
      2. B.2.2 Compatibility of the far–boundary conditions and the source term
      3. B.2.3 Existence and uniqueness of weak solutions
    3. B.3 A notion of self-adjointness
    4. B.4 Singular kernel and proof of Proposition 2.2
  7. C A key formula

##  1\. Introduction and main results

###  1.1. Aspects of interest

In this paper, we are interested in the study of time-periodic solutions of
the lake equations, given by

(1.1) $\begin{cases}\partial_{t}v+v\cdot\nabla v+\nabla p=0,\\\
\operatorname{div}(bv)=0,\end{cases}$

supplemented with an initial datum. In this set of equations, the space
variable, denoted by $x$ hereafter, takes values in the whole plane
$\mathbb{R}^{2}$. The time-independent function
$b:\mathbb{R}^{2}\rightarrow\mathbb{R}^{+}$ describes the bottom topography
(or depth function) of the lake, whereas the vector field $v$ refers to the
velocity field of the fluid. Finally, the scalar function $p$ designates the
pressure. Further assumptions on the coefficient $b$ will be specified later
on. The equations (1.1), describing balanced flow without gravity waves, are
basically recovered from the shallow-water equations in an appropriate time-
space scale limit [24], see also [3].

One of the most important features of the equations (1.1) is their equivalent
vorticity formulation, which we recall next

(1.2)
$\begin{cases}\partial_{t}\left(\frac{\omega}{b}\right)+v\cdot\nabla\left(\frac{\omega}{b}\right)=0,\\\
\operatorname{div}(bv)=0,\\\ \omega=\operatorname{curl}v.\end{cases}$

Note that the divergence free condition in (1.2) allows us to express $bv$ as

(1.3) $bv=-\nabla^{\perp}\psi_{b},\qquad\nabla^{\perp}\overset{\rm
def}{=}\begin{pmatrix}-\partial_{x_{2}}\\\ \partial_{x_{1}}\end{pmatrix},$

where the potential $\psi_{b}$ solves the elliptic problem

(1.4) $-\operatorname{div}\left(\frac{\nabla\psi_{b}}{b}\right)=\omega.$

The lake equation, formulated as (1.2), has garnered considerable attention
over the past few decades, resulting in numerous significant contributions
from both analytical and numerical perspectives. Interested readers can find
more comprehensive series of results on the well-posedness of (1.1) in [1, 4,
35, 42, 43] and the references therein. Most of these results are established
in bounded domains, with or without island, under various assumptions on the
depth function $b$. For additional insights and progress on specific types of
solutions, such as point vortices and steady states, we refer to [6, 15, 16,
34, 44, 50].

It is worth noting that the vorticity reformulation (1.2) shares structural
similarities with the classical Euler equations, governing the motion of an
ideal incompressible fluid. This similarity is observed by setting $b\equiv 1$
in the equations above. Additionally, under reasonable conditions on the depth
function $b$, a Yudovich theory for the lake equation (1.2) can be
established, akin to its known statement for the Euler equations [51]. As a
consequence of such a theory, the structure of an initial patch

$\omega|_{t=0}=b\mathds{1}_{D_{0}},$

for any bounded domain $D_{0}$ in $\mathbb{R}^{2}$, remains unchanged by the
flow. More precisely, the solution associated with the preceding initial datum
is given by

(1.5) $\omega(t,\cdot)=b\mathds{1}_{D_{t}},\qquad D_{t}=\Psi(t,D_{0}),$

for all positive times $t\geq 0$, where $\Psi(t,\cdot)$ is the flow of the
log-Lipschitz velocity field $v$, defined as the unique solution of the
integro-differential equation

$\Psi(t,x)=x+\int_{0}^{t}v\big{(}\tau,\Psi(\tau,x)\big{)}d\tau,\quad
x\in\mathbb{R}^{2}.$

In this paper, our focus extends to a specific subclass of patches where the
transported domain $D_{t}$ is exactly a rotation, by some angular velocity
$\Omega\in\mathbb{R}$, of the initial domain, referred to as V-states. In
other words, we are seeking solutions of the form (1.5), where

(1.6) $D_{t}=D_{0}e^{{\rm i}t\Omega},$

for all times $t\geq 0$. It is important to note that, in general and
conversely to the Eulerian case, the area of the transported patch may not be
conserved. However, as a by-product of the specific ansatz (1.6), the area of
the transported domain in that case remains unchanged by the flow.

Notably, in the case where the depth function is radially symmetric, one can
prescribe a huge class of stationary solutions with radial shapes. Among these
steady states, it is readily seen that the functions

$\omega(t,\cdot)=b(|\cdot|)\mathds{1}_{\\{x\in\mathbb{R}^{2}:\;|x|<a\\}},\quad
a>0,$

and

$\omega(t,\cdot)=b(|\cdot|)\mathds{1}_{\\{x\in\mathbb{R}^{2}:\;a_{2}<|x|<a_{1}\\}},\quad
0<a_{2}<a_{1},$

are solutions of (1.2) for all $t\geq 0$.

Therefore, it is natural to seek time-periodic solutions in the vicinity of
these steady states. The justification for the existence of such solutions is
the central theme of this work, and further details on this will be provided
later on.

Note that time-periodic (and time-quasi-periodic) solutions have previously
been extensively studied and established for various fluid models. The initial
discovery in this direction for the Euler equations, specifically for
ellipses, dates back to Kirchhoff [41]. Subsequently, Deem and Zabusky [14]
identified cases with higher symmetries through numerical simulations. Burbea
[5] then proposed an analytical approach to rigorously justify their
existence, based on local bifurcation theory. Specifically, for any $m$-fold
symmetry, $m\geq 1$, there exists only one branch of V-states bifurcating from
the Ranking vortices at the angular velocity $\Omega_{m}=\frac{m-1}{2m}$.

Similar structures for the doubly-connected case have been explored in [13].
In addition to that, several other properties of the V-states have been
investigated for Euler equations, including questions about the regularity of
the boundary [8, 37], the global bifurcation [31] and the existence of
multipole vortex patches [19, 20, 27, 33, 36]. Further other aspects have also
been addressed, such as the implementation of Nash–Moser scheme and KAM
techniques to construct time-quasi-periodic solutions; see [2, 29, 32].

Similar inquiries have been discussed about other active-scalar equations such
as the generalized surface quasi-geostrophic equation (SQG) [12, 25, 26, 28,
39, 46], the quasi-geostrophic shallow-water equations [18, 38, 49] or the
Euler-$\alpha$ model [48]. A general approach generalizing the previous
results in the simply-connected case has been obtained in [40]. More recently,
bifurcation theory has been implemented for more complex coupled Hamiltonian
systems, see [30] for an interconnected layers model arising in geophysical
flows and [47] for the Vlasov–Poisson equation. At last, we also refer the
interested reader to another set of results in a three-dimensional setting
[21, 23, 22].

###  1.2. Main results

Before we get into the details of the main contribution of the paper, allow us
to setup the basis of our assumptions on the bottom topography $b$ as well as
some notations to be constantly used thereafter.

#### Overall sufficient conditions on $b$

The depth function $b$ will be assumed to enjoy certain conditions which are
consistent with the vortex-patch solutions of our interest throughout this
paper, though they might not be optimal in some sense. The sufficient
conditions on $b$ are now listed below for the sake of clarity and
convenience, even though they will not be required as all of them in every
single result of the paper. More details of that will be discussed
subsequently.

  1. •

(Regularity) We assume that the function $b$ enjoys the regularity

(Hb1) $b\in C^{2}(\mathbb{R}^{2},\mathbb{R}).$

  2. •

(Positivity) We assume that $b$ is positive, i.e.,

(Hb2) $b(x)>0,\quad\text{for all}\quad x\in\mathbb{R}^{2}.$

  3. •

(Constancy at infinity) There exists $R_{\infty}>0$ and $b_{\infty}>0$ such
that

(Hb3) $b(x)=b_{\infty},\quad\text{for all}\quad x\in\mathbb{R}^{2}\setminus
B(0,R_{\infty}).$

  4. •

(Radial symmetry) There exists $\mathtt{b}:[0,\infty)\to(0,\infty)$ such that

(Hb4) $b(x)=\mathtt{b}(|x|),\quad\text{for all}\quad x\in\mathbb{R}^{2}.$

For a mere of simplicity, with an abuse of notation, we will keep using $b$
instead of $\mathtt{b}$.

At last, we introduce the sequence

(1.7) $\Lambda_{n}(\alpha,\beta)\overset{\rm
def}{=}\int_{0}^{2\pi}K_{b}\big{(}\alpha,\beta e^{{\rm
i}\eta}\big{)}\cos(n\eta)d\eta,$

for any given $\alpha,\beta>0$ and $n\in\mathbb{N}^{*},$ where $K_{b}$ refers
to the kernel associated with the elliptic equation (1.4), which will be
defined and discussed with details in Section 2.2.

The first main result of the paper concerns the existence of time-periodic
solutions of (1.2) bifurcating from a given disc, which we state next.

######  Theorem 1.1.

Let $a>0$ and $b$ be a function satisfying (Hb1), (Hb2), (Hb3) and (Hb4).
Then, there is a positive constant $M(a,b)>0$ such that, for any
$m\in\mathbb{N}^{*}$ with

$m>M(a,b),$

there exists a curve of $m$-fold symmetric simply-connected V-states for (1.2)
bifurcating from the stationary solution

$\omega=b\mathds{1}_{\\{x\in\mathbb{R}^{2}:\;|x|<a\\}},$

at the angular velocity

$\Omega_{m}=\frac{1}{a^{2}}\int_{0}^{a}\tau b(\tau)d\tau-\Lambda_{m}(a,a).$

######  Remark 1.1.

A precise expression of the constant $M(a,b)$ is given in Proposition 3.1,
below. In particular, this constant is identically zero if $b$ is a constant
function. Moreover, it holds that

$\lim_{\left\|b^{\prime\prime}\right\|_{L^{\infty}}\to 0}M(a,b)=0$

and when $b\equiv 1$, that

$M(a,b)=0\qquad\text{and}\qquad\Omega_{m}=\frac{m-1}{2m},$

for any integer $m\geq 1$. These values of the angular velocities correspond
to the same values of Burbea’s result, previously established for Euler
equations [5]. Accordingly, Theorem 1.1 above establishes “sharp” existence of
$m$-fold symmetric $V$-states for any $m\geq 1$ (resp. large symmetries $m\gg
1$) if $\left\|b^{\prime\prime}\right\|_{L^{\infty}}$ is sufficiently small
(resp. if $\left\|b^{\prime\prime}\right\|_{L^{\infty}}$ is arbitrary large).

The second main result of the paper establishes the existence of doubly-
connected time-periodic solutions of (1.2) bifurcating from an annulus, which
is the content of the next theorem.

######  Theorem 1.2.

Let $0<a_{2}<a_{1}$ and $b$ be a function satisfying (Hb1), (Hb2), (Hb3) and
(Hb4). Then, there is a positive constant $N(a_{1},a_{2},b)>0$ such that, for
any $m\in\mathbb{N}^{*}$ with

$m>N(a_{1},a_{2},b),$

there exist two curves of $m$-fold symmetric doubly-connected V-states
bifurcating from the stationary solution of (1.2)

$\omega=b\mathds{1}_{\\{x\in\mathbb{R}^{2}:\;a_{2}<|x|<a_{1}\\}},$

at the angular velocities

$\displaystyle\Omega_{m}^{\pm}$ $\displaystyle\overset{\rm
def}{=}\frac{1}{2a_{1}^{2}}\int_{a_{2}}^{a_{1}}\tau
b(\tau)d\tau+\frac{\Lambda_{m}(a_{2},a_{2})-\Lambda_{m}(a_{1},a_{1})}{2}$
$\displaystyle\quad\pm\frac{1}{2}\sqrt{\left(\frac{1}{a_{1}^{2}}\int_{a_{2}}^{a_{1}}\tau
b(\tau)d\tau-\Lambda_{m}(a_{1},a_{1})-\Lambda_{m}(a_{2},a_{2})\right)^{2}-4\Lambda_{m}^{2}(a_{1},a_{2})}.$

######  Remark 1.2.

By setting $b\equiv 1$ in Theorem 1.2 we recover the same values of the
angular velocities for the Euler equations established in [13]. Moreover, in
that case, the condition on $m$ can be shown to align with the constraints on
the same symmetry parameter derived from the results on the Euler equations
[13].

#### Comments about the assumptions on the depth function $b$

It is apparent through the statements of our theorems above that the
assumptions (Hb1)–(Hb4) are sufficient for the construction of time-periodic
solutions of (1.2). However, we do not make any claim about their necessity
and optimality. Though, let us briefly shed light on the relevance of these
conditions to the mechanism underlying our methods and proofs of Theorems 1.1
and 1.2. Further discussion, specifically focused on the elliptic problem
(1.4), will be provided in Section 2.

  * •

Condition (Hb1) is closely tied to defining the kernel associated with the
elliptic equation (1.4). While it has been shown to be useful and has been
employed in other papers on the lake equation [1, 17, 44], it is not strictly
required in many parts in the study of the elliptic problem (1.4). More
precisely, it is particularly related to the decomposition proposed in [17] of
the kernel $K_{b}$, see (2.8). Additionally, it is notably employed in the
regularity analysis in Section 3.3.

  * •

Condition (Hb2) was commonly imposed in various other papers [1, 17, 44]. Our
proofs heavily rely on it and our approach possibly breaks-down otherwise.
Allowing the depth function to degenerate, as in [4, 35], for example, would
necessitate non-trivial changes in all the arguments presented in this paper.
Therefore, it is currently unclear whether time-periodic solutions could be
constructed if $b$ vanishes on a non-empty set.

  * •

Condition (Hb3) was conjectured in [17] as sufficient to extend all the
results therein from bounded domains to the whole plane $\mathbb{R}^{2}$ (see
[17, open problem 2, page 1493]). Additionally, our motivation for imposing
this condition is justified by the objective of studying rotating solutions,
whose construction is primarily local-in-space. In other words, roughly
speaking, solutions of the form (1.5) are barely impacted by the behavior of
the depth function at infinity. Nevertheless, condition (Hb3) can possibly be
relaxed. Indeed, we believe that most of the results in Section 2 about the
elliptic problem (1.4) can potentially be established under appropriate
assumptions on the decay at infinity of derivatives of the depth function $b$,
as is for instance discussed in [44]. Even though, we emphasize that, with
condition (Hb3) in hand, several properties of solutions of (1.4) can be
proven through direct computations via constructive proofs, which will be
detailed later on in Section 2.

  * •

It will be noticed that the condition (Hb4) appears to be necessary in the
construction of the V-states. Indeed, this is used to generate Rankine
vortices from which we can bifurcate under suitable assumptions. Thus, for the
sake of simplicity, we have decided to discuss the elliptic problem (1.9)
under the radial symmetry assumption on the depth function $b$, that is (Hb4).
Nevertheless, most of the results in Section 2 could be extended to cover more
general cases of the depth function $b$.

###  1.3. Challenges and roadmap of the proofs

The proofs of Theorems 1.1 and 1.2 will be built upon the scheme laid out by
the generalized version of implicit-functions theorem, that is, Crandall-
Rabinowitz’s Theorem A.1. To that end, in the simply connected case, the
starting point consists in observing that time-periodic solutions are patches,
by definition, which obey the following contour dynamics equation

(1.8)
$\partial_{t}z(t,\theta)\cdot\mathbf{n}\big{(}t,z(t,\theta)\big{)}+\frac{1}{b\big{(}z(t,\theta)\big{)}}\partial_{\theta}\Big{(}\psi_{b}\big{(}t,z(t,\theta)\big{)}\Big{)}=0,$

for all $t\geq 0$, and $\theta\in\mathbb{T}$, where the unknown function
$z(t,\cdot)\subset\partial D_{t}$ refers to a parametrization of the boundary
of the patch at time $t\geq 0$, whereas
$\mathbf{n}\big{(}t,z(t,\theta)\big{)}$ denotes the outward normal vector to
the boundary at the point $z(t,\theta)$ and $\psi_{b}$ is the stream function
that solves the elliptic problem (1.4). In addition to that, time-periodic
solutions will be characterized by a particular form

$z(t,\theta)=R(\theta)e^{{\rm i}(\theta+t\Omega)},$

for some function $R>0$ and some angular velocity $\Omega\in\mathbb{R}$. At
last, as is aforementioned, it is natural to look for solutions close to the
stationary state

$\omega=b\mathds{1}_{\\{x\in\mathbb{R}^{2}:\;|x|<a\\}},$

for a given radius $a>0$. Therefore, this leads us to the ansatz

$z(t,\theta)=\sqrt{a^{2}+2r(\theta)}e^{{\rm i}(\theta+t\Omega)},$

where $r:\mathbb{T}\rightarrow\mathbb{R}$. Afterwards, by incorporating the
preceding ansatz in the contour dynamics equation, one sees that the time-
variable can be discarded from the equations, whereby we arrive at the
stationary problem

$\mathcal{F}(\Omega,r)=0,$

for some functional $\mathcal{F}$. The details of that are postponed and will
be discussed in Section 3.1. Likewise in the case of doubly-connected V-states
near a given annulus, we emphasize that a similar analysis leads to a two-
dimensional functional system

$\mathcal{G}(\Omega,r_{1},r_{2})\overset{\rm
def}{=}\Big{(}\mathcal{G}_{1}(\Omega,r_{1},r_{2}),\mathcal{G}_{2}(\Omega,r_{1},r_{2})\Big{)}=0,$

for some functionals $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ which will be
precised and discussed in Section 4, as well.

The strategy is then to show that the functionals $\mathcal{F}$ and
$\mathcal{G}$ satisfy all the prerequisites in the statement of Crandall-
Rabinowitz’s Theorem A.1. This is mainly related to the regularity study of
these nonlinear functionals and the spectral analysis of their linearized
operators around the stationary solutions. In contrast to [5, 7, 26, 37], the
kernels involved in these functionals are not explicit and with a quite rough
regularity, making their analysis much more delicate. In particular, our
analysis cannot be included in the general work [40]. To give an insight onto
the encountered difficulties throughout our proofs, let us now focus on the
main aspects on to the reconstruction of the stream functions related to the
elliptic problem

(1.9) $-\operatorname{div}(\tfrac{1}{b}\nabla\psi_{b})=f,$

with data $b$, $f$ and unknown $\psi_{b}.$ We prove in Proposition 2.2 a
Dekeyzer–Schaftingen-type decomposition [17] in the form

(1.10) $\psi_{b}(x)=\int_{\mathbb{R}^{2}}K_{b}(x,y)f(y)dy,\qquad
K_{b}(x,y)=-\tfrac{1}{2\pi}\log|x-y|\sqrt{b(x)b(y)}+S_{b}(x,y),$

with $S_{b}$ being symmetric and of regularity $W^{2,p}$, for all
$p\in(1,\infty)$. Note that a similar decomposition was obtained recently in
[44], using different arguments. Such a decomposition reveals to be useful to
describe the discrete spectrum of the linearized operator at the equilibrium
state, laid out in Lemma 3.4 and Proposition 4.1. More specifically, although
no explicit form of the eigenvalues associated with this linearized operator
is available, we are able to develop a suitable asymptotic expansion of the
spectrum through a contraction argument. This is very crucial and is discussed
in Proposition 3.1.

As for the regularity of the functionals $\mathcal{F}$ and $\mathcal{G}$, we
rely on a new decomposition obtained in Lemma 2.1 taking the form, in the
particular case $k=1$ from that lemma,

$\psi_{b}(x)=-\tfrac{1}{2\pi}\int_{\mathbb{R}^{2}}\log|x-y|b(y)f(y)dy+\phi_{f}(x),$

with $\nabla\phi_{f}\in W^{2,p}(\mathbb{R}^{2})$, for any $p>2$, provided that
$f$ is compactly supported and belongs to the Lebesgue space $L^{p}$. We
believe that the generic statement of Lemma 2.1 could be used to explore more
important questions on related topics.

In the sequel, we are going to use the standard notations for Lebesgue,
Sobolev and Hölder spaces: $L^{p}(\mathbb{R}^{2})$, $H^{s}(\mathbb{R}^{2})$,
$\dot{H}^{s}(\mathbb{R}^{2})$, $C^{\alpha}(\Omega)$,
$\dot{C}^{\alpha}(\Omega)$ …, where $p\in[1,\infty]$, $s\in\mathbb{R}$,
$\alpha\in(0,1)$ and $\Omega=\mathbb{R}^{2}$ or $\mathbb{T}^{2}$. The
preceding spaces are then equipped with their standard (semi-)norms. Note that
a particular attention will be paid to homogeneous Sobolev spaces
$\dot{H}^{s}(\mathbb{R}^{2})$ in Section B.2.1.

For a mere of simplicity, we employ “$\lesssim$” instead of “$\leq C$” where
the dependance on the constant $C>0$ is not crucial. Moreover, we occasionally
use the symbol “$\lesssim_{\delta}$” to emphasize the dependance of the
preceding inequality on some parameter or function $\delta$.

##  2\. On an elliptic problem with variable coefficients

This section is dedicated to the analysis of the elliptic problem

(2.1)
$\displaystyle-\operatorname{div}\left(\frac{1}{b}\nabla\psi_{b}\right)=f,$

where $f=f(x)$ and $b=b(x)$ are two given real functions defined on the entire
domain $\mathbb{R}^{2}$. Further precise assumptions on both $f$ and $b$ will
be explicitly stated, later on. Notice that (2.1) is relevant to the lake
equation (1.1) as the stream function $\psi_{b}$ is recovered from the
vorticity $\omega$ by solving (2.1) with the source term $f=\omega$.

Our primary focus here is to discuss some qualitative and quantitative
properties of $\psi_{b}$ as the solution of (2.1) with a general source term
$f$. The main novelty in this section concerns the new decomposition of
solutions given by Lemma 2.1. Additionally, many other results are collected
here, though they may not be brand-new. For the sake of completeness, we
supply this section with a self-contained appendix (Appendix B), which
provides a collection of some complementary results and proofs that we present
in a different (simple and self-contained) fashion compared to some other
references.

###  2.1. A priori bounds and decomposition of solutions

To begin with, let us assume that, under reasonable conditions on the source
term $f$ to be specified subsequently, the elliptic problem (2.1) admits at
least a solution $\psi_{b}$. This solution can in fact be shown to be unique
in appropriate functional spaces. We defer the discussion about the existence
and uniqueness of solutions of (2.1) to Appendix B.2. Moreover, the solution
of (2.1) enjoys the classical elliptic regularities as will be stated in the
next proposition. For the sake of completeness, a full justification of this
proposition will be detailed in Appendix B.1.

######  Proposition 2.1.

Let $b$ be a $C^{1}$ function satisfying assumptions (Hb2), (Hb3) and (Hb4).
Further consider a source term $f$ with the properties that

$f\in L^{1}\big{(}(1+|x|)dx;\mathbb{R}^{2}\big{)}\cap L^{p}(\mathbb{R}^{2}),$

for some $p\in[2,\infty)$. Then, any solution of (2.1) enjoys the bound

(2.2)
$\left\|\nabla^{2}\psi_{b}\right\|_{L^{m}(\mathbb{R}^{2})}\lesssim_{b}\left\|f\right\|_{L^{m}(\mathbb{R}^{2})},$

for all $m\in(1,p]$.

######  Remark 2.1.

By Sobolev embedding, it is readily seen that the preceding proposition
entails the control

$\nabla\psi_{b}\in L^{n}(\mathbb{R}^{2}),$

for all $n\in(2,\infty)$. The endpoint case $n=2$ is not attained in general.
However, the case $2=\infty$ is reachable in the case $p>2$. This is crucial
and, once again, for a mere sake of completeness, it will be further discussed
in Appendix B.1.

######  Remark 2.2.

Note that the results of the preceding proposition can be extended to cover
similar bounds on derivatives of the solution as soon as the source term
enjoys extra Sobolev regularities. More precisely, it is straightforward to
show that, if the source term $f$ is compactly supported111Here, the
assumption on the compactness of the support of the source term is made for
the sake of simplicity. This can be relaxed under adequate assumption on the
growth of the function $b$. and belongs to $W^{k,p}(\mathbb{R}^{2})$, for some
$k\in\mathbb{N}^{*}$, then it holds that

$\left\|\nabla^{2}\psi_{b}\right\|_{\dot{W}^{j,m}(\mathbb{R}^{2})}\lesssim\left\|f\right\|_{W^{j,m}(\mathbb{R}^{2})},$

for all $m\in(1,p]$ and $j\in\llbracket 1,k\rrbracket$. Indeed, to see that
for $j=1$, we just need to expand (2.1) after applying a directional
derivative $\partial_{i}$, for $i\in\\{1,2\\}$, to write that

$\displaystyle-\operatorname{div}\left(\frac{1}{b(x)}\nabla(\partial_{i}\psi_{b})(x)\right)=\partial_{i}f(x)-\Delta\psi_{b}\frac{\partial_{i}b}{b^{2}}-\nabla\left(\frac{\partial_{i}b}{b^{2}}\right)\cdot\nabla\psi_{b}\overset{\rm
def}{=}F.$

This means that $\partial_{i}\psi_{b}$ satisfies the elliptic equation (2.1)
with the new source term $F$. The justification of our claim above for $j=1$
amount then to showing that the new source term $F$ belongs to
$L^{p}(\mathbb{R}^{2})$. This is done by virtue of a direct computation using
the results of Proposition 2.1 and the assumptions on $b$ and $f$. Moreover,
we emphasize that this procedure can straightforwardly be iterated to show the
desired control for any $j\in\llbracket 1,k\rrbracket$.

In the next simple, but essential, lemma, we establish a new decomposition of
the solution of (2.1). In particular, this new decomposition allows us to
regard (2.1) as a regular perturbation of the classical Poisson’s problem with
a suitable source term where the size of its $L^{p}$-norm is comparable to the
same norm of the original source in (2.1).

A crucial observation here, which will come in handy later on, is that the
remainder in the new expansion of $\psi_{b}$ below can be set to be regular as
much as we need, provided that $b$ is sufficiently regular. We believe that
the features of the new decomposition given in the next lemma go beyond the
scope of the present paper as it can serve in other different situations.

######  Lemma 2.1.

Let $b$ be a function that satisfies assumptions (Hb2), (Hb3) and (Hb4) and
belongs to $C^{k+1}(\mathbb{R}^{2})$, for some integer $k\geq 1$. Further
consider a compactly supported function $f$ in $L^{p}(\mathbb{R}^{2})$, for
some $p\in[2,\infty)$. Then, there are two functions $\varphi_{k}\in
L^{p}(\mathbb{R}^{2})$ and $\phi_{k}\in\dot{W}^{k+2,m}(\mathbb{R}^{2})$, for
all $m\in(1,p]$, such that any solution of (2.1) with source term $f$ can be
decomposed as

$\psi_{b}=-\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\log|\cdot-y|\varphi_{k}(y)dy+\phi_{k}.$

Furthermore, $\varphi_{k}$ is compactly supported and satisfies the bound

$\left\|\varphi_{k}\right\|_{L^{q}(\mathbb{R}^{2})}\lesssim_{b}\left\|f\right\|_{L^{q}(\mathbb{R}^{2})},$

for all $q\in[1,p]$, and $\phi_{k}$ enjoys the control

$\left\|\phi_{k}\right\|_{\dot{W}^{i+2,m}(\mathbb{R}^{2})}\lesssim_{b}\left\|f\right\|_{L^{m}(\mathbb{R}^{2})},$

for all $m\in(1,p]$ and any $i\in\llbracket 0,k\rrbracket$, as well as

$\left\|\nabla\phi_{k}\right\|_{L^{n}(\mathbb{R}^{2})}\lesssim_{b}\left\|f\right\|_{L^{n}(\mathbb{R}^{2})},$

for every $n\in(2,p]$.

######  Remark 2.3.

We emphasize that the assumption on the support of the source term can be
traded with weaker assumptions on its growth in Lebesgue spaces along with
suitable assumptions on the growth of the function $b$ in the absence of the
stronger condition (Hb3). For the sake of simplicity, we chose to restrict our
selves to the case where the condition (Hb3) and the assumption one the
compactness of the source are in action, which are reasonable hypothesis in
the context of vortex-patch solutions of (1.2).

######  Remark 2.4.

Later on, we will only need to apply the preceding lemma with $k=1$. This in
particular allows us to write that

$\psi_{b}(x)=-\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\log|x-y|b(y)f(y)dy+\phi_{f}(x),$

for all $x\in\mathbb{R}^{2}$, where the remainder term solves the equation

(2.3) $-\operatorname{div}\left(\frac{1}{b}\nabla\phi_{f}\right)=\frac{\nabla
b}{b^{2}}\cdot\nabla(\Delta)^{-1}(bf)$

and enjoys the bound

(2.4) $\nabla\phi_{f}\in W^{2,p}(\mathbb{R}^{2}).$

Notably, this remainder enjoys a better regularity compared to the
decomposition of the solution proposed in [17].

###### Proof.

Observe first that the elliptic equation (2.1) can be recast as

$-\Delta\psi_{b}=bf-\frac{\nabla b}{b}\cdot\nabla\psi_{b}.$

This inspires introducing the following expansion of $\psi_{b}$

$\psi_{b}=\sum_{j=1}^{k}\widetilde{\phi}_{j}+\phi_{k},$

where $\widetilde{\phi}_{j}$ is defined by the induction scheme

$-\Delta\widetilde{\phi}_{1}\overset{\rm
def}{=}bf\qquad\text{and}\qquad-\Delta\widetilde{\phi}_{j}\overset{\rm
def}{=}-\frac{\nabla b}{b}\cdot\nabla\widetilde{\phi}_{j-1},\quad\text{for
all}\quad j\in\llbracket 2,k\rrbracket,$

whereas the remainder term $\phi_{k}$ in the expansion above is governed by
the equation

(2.5) $-\operatorname{div}\left(\frac{1}{b}\nabla\phi_{k}\right)=-\frac{\nabla
b}{b^{2}}\cdot\nabla\widetilde{\phi}_{k}.$

Notice that the assumptions on the supports of $f$ and $b$ imply that the
right-hand sides in the previous elliptic equations are compactly supported.
Therefore, classical elliptic regularity for Poisson’s problem (or simply
consider Proposition 2.1 with $b\equiv 1$) ensure the following bounds

(2.6)
$\left\|\nabla^{2}\widetilde{\phi}_{j}\right\|_{L^{m}(\mathbb{R}^{2})}\lesssim_{b}\left\|f\right\|_{L^{m}(\mathbb{R}^{2})},$

and

(2.7)
$\left\|\nabla\widetilde{\phi}_{j}\right\|_{L^{n}(\mathbb{R}^{2})}\lesssim_{b}\left\|f\right\|_{L^{n}(\mathbb{R}^{2})},$

for all $m\in(1,p]$, $n\in(2,p]$ and $j\in\llbracket 1,k\rrbracket$. Likewise
for the remainder term, Proposition 2.1 yields the same bounds

$\left\|\nabla^{2}\phi_{k}\right\|_{L^{m}(\mathbb{R}^{2})}\lesssim_{b}\left\|f\right\|_{L^{m}(\mathbb{R}^{2})},$

and

$\left\|\nabla\phi_{k}\right\|_{L^{n}(\mathbb{R}^{2})}\lesssim_{b}\left\|f\right\|_{L^{n}(\mathbb{R}^{2})}.$

We are now ready to justify the claims of the lemma. To that end, we introduce
the functions

$\varrho_{1}\overset{\rm
def}{=}bf\qquad\text{and}\qquad\varrho_{j}\overset{\rm def}{=}-\frac{\nabla
b}{b}\cdot\nabla\widetilde{\phi}_{j-1},\quad\text{for all}\quad j\in\llbracket
2,k\rrbracket.$

Accordingly, we define $\varphi_{k}$ as

$\varphi_{k}\overset{\rm def}{=}\sum_{j=1}^{k}\varrho_{j},$

whereby obtaining that

$\sum_{j=1}^{k}\widetilde{\phi}_{j}=(-\Delta)^{-1}\varphi_{k}=-\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\log|\cdot-y|\varphi_{k}(y)dy.$

Moreover, due to the bounds established in the first part of the proof
together with the support assumption on $\nabla b$ deduced from (Hb3), it is
readily seen that

$\left\|\varphi_{k}\right\|_{L^{q}(\mathbb{R}^{2})}\lesssim_{b}\left\|f\right\|_{L^{q}(\mathbb{R}^{2})},$

for all $q\in[1,p]$. Let us now be more precise about the regularity of the
summands $\widetilde{\phi}_{j}$, which will allow us to show that the
remainder $\phi_{k}$ enjoys a better regularity. To see that, we proceed by
induction to show that

$\widetilde{\phi}_{j}\in\dot{W}^{j+1,n}(\mathbb{R}^{2}),$

for any $j\in\llbracket 1,k\rrbracket$ and $n\in(1,p]$, with

$\left\|\nabla^{j+1}\widetilde{\phi}_{j}\right\|_{L^{n}(\mathbb{R}^{2})}\lesssim_{b}\left\|f\right\|_{L^{n}(\mathbb{R}^{2})}.$

Indeed, the previous claim is already proven for $j=1$, whereas its proof for
$j\in\llbracket 2,k\rrbracket$ only relies on the observation that any
solution of Poisson’s equation with a source term belonging to
$\dot{W}^{n,q}(\mathbb{R}^{2})$, for some $n\in\mathbb{N}$ and
$q\in(1,\infty)$, belongs itself to $\dot{W}^{2+n,q}(\mathbb{R}^{2})$. Thus,
repeating this procedure and utilizing the $C^{1+k}(\mathbb{R}^{2})$
regularity of $b$, altogether with assumption (Hb3), leads to the desired
regularity for $\widetilde{\phi}_{j}$. At the end, we particularly have shown
that

$\nabla\widetilde{\phi}_{k}\in\dot{W}^{k,p}(\mathbb{R}^{2}),$

with

$\left\|\nabla\widetilde{\phi}_{k}\right\|_{\dot{W}^{k,p}(\mathbb{R}^{2})}\lesssim_{b}\left\|f\right\|_{L^{p}(\mathbb{R}^{2})}.$

Therefore, it follows that the source term in (2.5) is compactly supported
(due to assumption (Hb3)) and belongs to $W^{k,p}(\mathbb{R}^{2})$ (by virtue
of the preceding bound altogether with (2.7)). Thus, in view of Remark 2.2, we
deduce that

$\left\|\phi_{k}\right\|_{\dot{W}^{k+2,m}(\mathbb{R}^{2})}\lesssim_{b}\left\|f\right\|_{L^{m}(\mathbb{R}^{2})},$

for all $m\in(1,p]$. At last, combining the previous bound with (2.6), we
deduce by a direct interpolation inequality, that

$\left\|\phi_{k}\right\|_{\dot{W}^{i+2,m}(\mathbb{R}^{2})}\lesssim_{b}\left\|f\right\|_{L^{m}(\mathbb{R}^{2})},$

for all $m\in(1,p]$ and $i\in\llbracket 0,k\rrbracket$, thereby completing the
proof of the lemma. ∎

###  2.2. Kernels and integral representation of solutions

Here, we show the existence of a kernel associated with the elliptic operator

$-\operatorname{div}(b^{-1}\nabla\cdot).$

Several properties of this kernel will also be established in the next
proposition which will serve in the upcoming sections of this article. The
solution of the following elliptic equation

(2.8)
$\displaystyle-\operatorname{div}_{x}\left(\frac{1}{b(x)}\nabla_{x}S_{b}(x,y)\right)=\frac{1}{2\pi}\log|x-y|\sqrt{b(y)}\Delta_{x}\left(\frac{1}{\sqrt{b(x)}}\right),$

for all $x\in\mathbb{R}^{2}$ and any given $y\in\mathbb{R}^{2}$, is also
relevant in a sense that will be made precise, now.

######  Proposition 2.2.

Let $b$ be a function satisfying assumptions (Hb1), (Hb2), (Hb3) and (Hb4).
Then, for any given $y\in\mathbb{R}^{2}$, there is a unique solution222Here,
the uniqueness of the solution is understood in the sense of Theorem B.1.
$S_{b}(\cdot,y)$ of (2.8) enjoying the bounds

$\nabla^{2}_{x}S_{b}(x,y)\in
L^{\infty}_{\mathrm{loc}}\big{(}(dy;\mathbb{R}^{2});L^{m}(dx;\mathbb{R}^{2})\big{)},$

for all $m\in(1,p]$, and

$\nabla_{x}S_{b}(x,y)\in
L^{\infty}_{\mathrm{loc}}\big{(}(dy;\mathbb{R}^{2});L^{n}(dx;\mathbb{R}^{2})\big{)},$

for all $n\in(2,p]$, as well as

$S_{b}\in L^{\infty}_{\mathrm{loc}}(\mathbb{R}^{2}\times\mathbb{R}^{2}).$

Moreover, for any compactly supported function $f$ in $L^{p}(\mathbb{R}^{2})$,
for some $p\in[2,\infty)$, the unique solution of (2.1) can be represented as

(2.9) $\psi_{b}(x)=\int_{\mathbb{R}^{2}}K_{b}(x,y)f(y)dy,$

for any $x\in\mathbb{R}^{2}$, where we set

(2.10) $K_{b}(x,y)\overset{\rm
def}{=}-\frac{1}{2\pi}\log|x-y|\sqrt{b(x)b(y)}+S_{b}(x,y),$

for any $x,y\in\mathbb{R}^{2}$ with $x\neq y$.

Furthermore, $K_{b}(x,y)$ is symmetric for any $x,y\in\mathbb{R}^{2}$ with
$x\neq y$, i.e.

(2.11) $K_{b}(x,y)=K_{b}(y,x),$

and it is stable by rotations and involution transformation in the sense that

(2.12) $K_{b}(\mathcal{R}_{\theta}\cdot x,\mathcal{R}_{\theta}\cdot
y)=K_{b}(x,y),\qquad K_{b}(\mathcal{S}x,\mathcal{S}y)=K_{b}(x,y),$

for any $\theta\in[0,2\pi]$, where $\mathcal{R}_{\theta}$ and $\mathcal{S}$
denote the rotation matrix with angle $\theta$ and the involution operator,
respectively, i.e.

$\mathcal{R}_{\theta}\overset{\rm
def}{=}\begin{pmatrix}\cos(\theta)&-\sin(\theta)\\\
\sin(\theta)&\cos(\theta)\end{pmatrix},$

and

$\mathcal{S}x\overset{\rm def}{=}|x|e^{-i\eta},$

for any given $x=|x|e^{{\rm i}\eta}\in\mathbb{R}^{2}$, for some
$\eta\in[0,2\pi]$

######  Remark 2.5.

The solution of the elliptic problem (2.1) defined through the integral
representation (2.9) satisfies all the bounds on the solution of (2.1),
previously established in Proposition 2.1. This will come in handy, later on.

For a mere of a clear presentation, we defer the proof of this proposition to
Appendix B.4. Let us emphasize in passing that the definition of $K_{b}$ in
terms of the solution $S_{b}$ of (2.8) was established at a first time in
bounded domains in [17]. Very recently, similar results in the case of the
whole plane were claimed in [44]. On the other hand, for the sake of
completeness, the full justification of Proposition 4.1 will be discussed with
details in Appendix B.4, using different arguments compared to [44]. Our proof
tends to be more constructive under suitable assumptions on the function $b$.

We conclude this section by stating and proving a computational lemma which
will be useful the next sections.

######  Lemma 2.2.

Let $\alpha,\beta\in(0,\infty)$ and $K_{b}$ be the kernel associated with the
elliptic problem (2.1) as introduced in the preceding proposition. Then, it
holds that

$\frac{1}{\alpha}\int_{0}^{2\pi}\int_{0}^{\beta}\partial_{x_{1}}K_{b}\big{(}\alpha,\rho
e^{{\rm i}\eta}\big{)}b(\rho)\rho d\rho
d\eta=-\frac{b(\alpha)}{\alpha^{2}}\int_{0}^{\min\\{\alpha,\beta\\}}\tau
b(\tau)d\tau.$

###### Proof.

The proof follows from a direct computation, employing the explicit expression
of solutions of the elliptic equation (2.1). To see that, we first write

$\displaystyle\int_{0}^{2\pi}\int_{0}^{\beta}\partial_{x_{1}}K_{b}\big{(}\alpha,\rho
e^{{\rm i}\eta}\big{)}b(\rho)\rho d\rho d\eta$
$\displaystyle=\left(\int_{0}^{2\pi}\int_{0}^{\beta}\partial_{x_{1}}K_{b}\big{(}x,\rho
e^{{\rm i}\eta}\big{)}b(\rho)\rho d\rho d\eta\right)\Big{|}_{x=(\alpha,0)}$
$\displaystyle=\left(\partial_{x_{1}}\int_{0}^{2\pi}\int_{0}^{\beta}K_{b}\big{(}x,\rho
e^{{\rm i}\eta}\big{)}b(\rho)\rho d\rho d\eta\right)\Big{|}_{x=(\alpha,0)}$
$\displaystyle=\left(\partial_{x_{1}}\int_{\mathbb{R}^{2}}K_{b}(x,y)b(|y|)\mathds{1}_{B_{\beta}}dy\right)\Big{|}_{x=(\alpha,0)},$

where $B_{\beta}$ denotes the ball centered at the origin of radius $\beta$.
Therefore, by virtue of Proposition 2.2, we see that the function

$x\mapsto\psi(x)\overset{\rm
def}{=}\int_{\mathbb{R}^{2}}K_{b}(x,y)b(|y|)\mathds{1}_{B_{\beta}}dy$

solves the equation

$-\operatorname{div}\left(\frac{1}{b}\nabla\psi\right)=b\mathds{1}_{B_{\beta}},$

whence, since $b\mathds{1}_{B_{\beta}}$ is a radially symmetric function, we
obtain the explicit formula

$\psi(x)=-\int_{0}^{|x|}\frac{b(s)}{s}\int_{0}^{\min\\{s,\beta\\}}\tau
b(\tau)d\tau ds+C,$

for some constant $C\in\mathbb{R}$ and all $x\in\mathbb{R}^{2}$. Deriving the
preceding expression of $\psi$ in the direction $x_{1}$, then, evaluating the
resulting formula at the point $x=(\alpha,0)$ and incorporating it into the
computations at the beginning of the proof yields the desired identity and
concludes the proof of the lemma. ∎

##  3\. Simply-connected time-periodic solutions

This section is devoted to the proof of Theorem 1.1. We shall split it into
several subsection for better readability. Some arguments from this section
will also be useful in the doubly-connected case, which will be the subject of
Section 4, later on.

#### Notations

Before we get into the detailed analysis of the contour dynamics equation,
allow us to set up a few notations that will be employed throughout the sequel
of this paper.

In the sequel, we will also agree the identification
$\mathbb{C}\approx\mathbb{R}^{2}$ and we recall that the scalar product of two
complex numbers $z_{1}=x_{1}+{\rm i}y_{1}$ and $z_{2}=x_{2}+{\rm i}y_{2}$ is
given by

$z_{1}\cdot z_{2}=\text{Re}(z_{1}\overline{z_{2}})=x_{1}x_{2}+y_{1}y_{2}.$

We will often use the notation

$\int_{\mathbb{T}}=\frac{1}{2\pi}\int_{0}^{2\pi}$

for a mere simplification of our proofs, where $\mathbb{T}\overset{\rm
def}{=}\mathbb{R}/2\pi\mathbb{Z}$.

At last, for consistency of notation in Section 4, we now introduce the
function

(3.1) $Q(\alpha,\beta)\overset{\rm
def}{=}\frac{1}{\alpha^{2}}\int_{\min\\{\alpha,\beta\\}}^{\max\\{\alpha,\beta\\}}\tau
b(\tau)d\tau=\int_{\min\\{1,\frac{\beta}{\alpha}\\}}^{\max\\{1,\frac{\beta}{\alpha}\\}}\tau
b(\alpha\tau)d\tau,$

for any given $\alpha>0$ and $\beta\geq 0,$ which will serve in this section
and the next one, too.

###  3.1. Contour dynamics

Here, we establish an equivalent reformulation of (1.2) in the case of
periodic uniformly rotating solutions. To that end, we begin with considering
a parametrization of the boundary of the initial domain $D_{0}$ which is close
to a disc of radius $a>0$, namely

(3.2) $\displaystyle z:\mathbb{T}$ $\displaystyle\mapsto\partial D_{0}$
$\displaystyle\theta$ $\displaystyle\mapsto R(\theta)e^{{\rm
i}\theta}\overset{\rm def}{=}\sqrt{a^{2}+2r(\theta)}e^{{\rm i}\theta}.$

######  Lemma 3.1 (Contour equation).

Let $\Omega\in\mathbb{R}.$ The radial deformation defined through (3.2) gives
rise to a uniformly rotating, at angular velocity $\Omega$, vortex patch
solution to the lake equation (1.2) if and only if it satisfies the nonlinear
contour equation

$\mathcal{F}(\Omega,r)=0,$

where, for any $\theta\in\mathbb{T}$, we denote

(3.3) $\mathcal{F}(\Omega,r)(\theta)\overset{\rm def}{=}\Omega
r^{\prime}(\theta)+\frac{1}{b\big{(}R(\theta)\big{)}}\partial_{\theta}\left(\int_{0}^{2\pi}\int_{0}^{R(\eta)}K_{b}\big{(}R(\theta)e^{{\rm
i}\theta},\rho e^{{\rm i}\eta}\big{)}b(\rho)\rho d\rho d\eta\right),$

and $K_{b}$ is the kernel associated with the elliptic equation (2.1).

Moreover, the trivial deformation, corresponding to $r\equiv 0$, is a solution
to the preceding contour equation for all $\Omega\in\mathbb{R}$.

###### Proof.

We first proceed to obtain the contour dynamics equation for a general
parametrization of the patch; then we particularly look after the case of a
rotating domain.

Let us consider, for any time $t>0,$ a parametrization
$z(t,\cdot):\mathbb{T}\rightarrow\partial D_{t}$ of the boundary. Then,
according to [37, page 174], we have, for any $\theta\in\mathbb{T}$ and $t\geq
0$, that

$\partial_{t}z(t,\theta)\cdot\textbf{n}\big{(}t,z(t,\theta)\big{)}=v\big{(}t,z(t,\theta)\big{)}\cdot\textbf{n}\big{(}t,z(t,\theta)\big{)},$

where we denoted $\textbf{n}\big{(}t,z(t,\theta)\big{)}$ the inward normal
vector to the boundary $\partial D_{t}$ of the patch at the point
$z(t,\theta)$, i.e., we agree that

$\textbf{n}\big{(}t,z(t,\theta)\big{)}\overset{\rm def}{=}{\rm
i}\partial_{\theta}z(t,\theta).$

Now, recalling that $\psi_{b}$ is the stream function corresponding to the
divergence-free vector field $-bv$, defined through (1.3), and using the
notation

$\nabla=2\partial_{\overline{z}},\quad\text{and}\quad\nabla^{\perp}=2{\rm
i}\partial_{\overline{z}},$

we then compute that

$\displaystyle\partial_{\theta}\Big{(}\psi_{b}\big{(}t,z(t,\theta)\big{)}\Big{)}$
$\displaystyle=\nabla\psi_{b}\big{(}t,z(t,\theta)\big{)}\cdot\partial_{\theta}z(t,\theta)$
$\displaystyle=\textnormal{Re}\left(2\partial_{\overline{z}}\psi_{b}\big{(}t,z(t,\theta)\big{)}\overline{\partial_{\theta}z(t,\theta)}\right)$
$\displaystyle=\textnormal{Im}\left(2{\rm
i}\partial_{\overline{z}}\psi_{b}\big{(}t,z(t,\theta)\big{)}\overline{\partial_{\theta}z(t,\theta)}\right)$
$\displaystyle=-b\big{(}z(t,\theta)\big{)}\textnormal{Im}\left(v\big{(}t,z(t,\theta)\big{)}\overline{\partial_{\theta}z(t,\theta)}\right)$
$\displaystyle=-b\big{(}z(t,\theta)\big{)}v\big{(}t,z(t,\theta)\big{)}\cdot\textbf{n}\big{(}t,z(t,\theta)\big{)}.$

Accordingly, we obtain the general equation of motion

(3.4)
$\partial_{t}z(t,\theta)\cdot\textbf{n}\big{(}t,z(t,\theta)\big{)}+\frac{1}{b\big{(}z(t,\theta)\big{)}}\partial_{\theta}\Big{(}\psi_{b}\big{(}t,z(t,\theta)\big{)}\Big{)}=0.$

Now, we particularize by looking for periodic solutions where $D_{t}$ is a
rotation of the initial domain $D_{0}$, namely

(3.5) $D_{t}=e^{{\rm i}\Omega t}D_{0}.$

This leads to consider the following parametrization for the boundary
$\partial D_{t}$

$z(t,\theta)\overset{\rm def}{=}e^{{\rm i}\Omega t}z(\theta)=R(\theta)e^{{\rm
i}(\Omega t+\theta)},$

where $z$ is the parametrization of $\partial D_{0}$ introduced in (3.2). On
the one hand, it is readily seen that

$\partial_{t}z(t,\theta)={\rm i}\Omega e^{{\rm i}\Omega t}R(\theta)e^{{\rm
i}\theta},$

and

$\partial_{\theta}z(t,\theta)=e^{{\rm i}\Omega t}\left({\rm
i}R(\theta)+\tfrac{r^{\prime}(\theta)}{R(\theta)}\right)e^{{\rm i}\theta},$

whereby, it follows that

$\displaystyle\partial_{t}z(t,\theta)\cdot\textbf{n}\big{(}t,z(t,\theta)\big{)}$
$\displaystyle=\text{Im}\left(\partial_{t}z(t,\theta)\overline{\partial_{\theta}z(t,\theta)}\right)$
(3.6) $\displaystyle=\Omega r^{\prime}(\theta).$

On the other hand, the radial symmetry property (Hb4) of the bottom topography
$b$ implies that

(3.7) $b\big{(}z(t,\theta)\big{)}=b\big{(}R(\theta)\big{)}.$

In addition, in view of the representation (2.9) , we write that

$\psi_{b}\big{(}t,z(t,\theta)\big{)}=\int_{D_{t}}K_{b}\big{(}z(t,\theta),y\big{)}b(y)dy.$

Therefore, we deduce by employing (2.12), (3.5), making the change of
variables $y\rightarrow ye^{{\rm i}t\Omega}$ and utilizing (Hb4) that

$\displaystyle\psi_{b}\big{(}t,z(t,\theta)\big{)}$
$\displaystyle=\int_{D_{0}}K_{b}\big{(}R(\theta)e^{{\rm i}(\Omega
t+\theta)},ye^{{\rm i}\Omega t}\big{)}b\big{(}ye^{{\rm i}\Omega t}\big{)}dy$
(3.8) $\displaystyle=\int_{D_{0}}K_{b}\big{(}R(\theta)e^{{\rm
i}\theta},y\big{)}b(y)dy.$

Hence, inserting (3.6), (3.7) and (3.8) into (3.4) infers that

$\Omega
r^{\prime}(\theta)+\frac{1}{b\big{(}R(\theta)\big{)}}\partial_{\theta}\left(\int_{D_{0}}K_{b}\big{(}z(\theta),y\big{)}b(y)dy\right)=0,$

which implies in turn, via polar change of variables, that

$\Omega
r^{\prime}(\theta)+\frac{1}{b\big{(}R(\theta)\big{)}}\partial_{\theta}\left(\int_{0}^{2\pi}\int_{0}^{R(\eta)}K_{b}\big{(}z(\theta),\rho
e^{{\rm i}\eta}\big{)}b(\rho)\rho d\rho d\eta\right)=0.$

This derives the contour dynamics as it is claimed in Lemma 3.1, above.

In addition to that, by employing the identity (2.12), followed by a change of
variables, we see, for all $\theta\in[0,2\pi]$, that

$\int_{0}^{2\pi}\int_{0}^{a}K_{b}\big{(}ae^{{\rm i}\theta},\rho e^{{\rm
i}\eta}\big{)}b(\rho)\rho d\rho
d\eta=\int_{0}^{2\pi}\int_{0}^{a}K_{b}\big{(}a,\rho e^{{\rm
i}\eta}\big{)}b(\rho)\rho d\rho d\eta,$

thereby yielding that

(3.9) $\partial_{\theta}\left(\int_{0}^{2\pi}\int_{0}^{a}K_{b}\big{(}ae^{{\rm
i}\theta},\rho e^{{\rm i}\eta}\big{)}b(\rho)\rho d\rho d\eta\right)=0.$

Subsequently, it follows, for all $\Omega\in\mathbb{R}$, that $(\Omega,0)$ is
a zero of the functional $\mathcal{F}$. The proof of the lemma is now
completed. ∎

Observe, from the proof above, that the functional $\mathcal{F}(\Omega,r)$ can
be recast, for any $\theta\in\mathbb{T}$, as

(3.10) $\mathcal{F}(\Omega,r)(\theta)=\Omega
r^{\prime}(\theta)+\frac{1}{b\big{(}R(\theta)\big{)}}\partial_{\theta}\left(\psi_{b}\big{(}R(\theta)e^{{\rm
i}\theta}\big{)}\right),$

where $\psi_{b}\overset{\rm def}{=}\psi_{b}(0,\cdot)$ is the unique solution
of the elliptic problem (2.1) with source term $f=b\mathds{1}_{D_{0}}$. The
preceding representation will come in handy in the regularity study of the
contour equation, later on.

We conclude this section by a simple lemma where we establish important
symmetry properties of the functional $\mathcal{F}$.

######  Lemma 3.2 (Symmetry properties of the functional $\mathcal{F}$).

Let $\Omega\in\mathbb{R}$, $r\in C^{1+\alpha}(\mathbb{T}),$ for some
$\alpha\in(0,1)$, and $\mathcal{F}(\Omega,r)$ be given by (3.3). If moreover
$r$ is an even function, i.e.,

$r(-\theta)=r(\theta),\quad\text{for all}\quad\theta\in[0,2\pi],$

then, $\mathcal{F}(\Omega,r)$ is an odd function, i.e.

$\mathcal{F}(\Omega,r)(-\theta)=-\mathcal{F}(\Omega,r)(\theta),\quad\text{for
all}\quad\theta\in[0,2\pi].$

Furthermore, if $r$ is a $m-$fold symmetric function, for some
$m\in\mathbb{N}^{*}$, i.e.

$r(\theta+\tfrac{2\pi}{m})=r(\theta),\quad\text{for
all}\quad\theta\in[0,2\pi],$

then, $\mathcal{F}(\Omega,r)$ is also a $m$-fold symmetric function.

###### Proof.

It is readily seen that, if $r$ is an even function, then $r^{\prime}$ is odd.
Likewise for the $m-$fold symmetry, i.e., if $r$ is a $m-$fold symmetric
function, then so is $r^{\prime}$. Thus, regarding the expression of
$\mathcal{F}$ in (3.3), we can restrict our focus on showing that the function

$\theta\mapsto\int_{0}^{2\pi}\int_{0}^{R(\eta)}K_{b}\big{(}R(\theta)e^{{\rm
i}\theta},\rho e^{{\rm i}\eta}\big{)}b(\rho)\rho d\rho d\eta$

is even and $m$-fold symmetric, as soon as $r$ is an even $m$-fold symmetric
function. The parity part follows by applying the second symmetry property of
the kernel $K_{b}$ given by (2.12), followed by the change of variables
$\eta\to-\eta$. As for the $m$-fold symmetry property, it is shown by
employing the first symmetry property of the kernel $K_{b}$ given by (2.12),
followed by the change of variables $\eta\to\eta+\frac{2\pi}{m}$. The proof of
Lemma 3.2 is now completed. ∎

###  3.2. Spectral analysis

In this section, we study the linearized operator associated with the
functional $\mathcal{F}$ at the equilibrium $r=0$ and we show that it acts as
a Fourier multiplier. We emphasize, however, that all the computations in this
section will be completed with a detailed regularity analysis in Section 3.3,
below.

To begin, let us now compute the differential of the functional $\mathcal{F}$
at any point $r$.

######  Lemma 3.3.

Let $\mathcal{F}$ be given by (3.3). Let $m\in\mathbb{N}^{*}$,
$\alpha\in(0,1)$ and $\varepsilon>0$ be a given small parameter in such a way
that the function $\mathcal{F}$ is differentiable according to Proposition
3.2. Then, the differential of $\mathcal{F}$ at
$r\in\mathcal{B}_{m,\varepsilon}^{\alpha}$ in the direction $h\in
X_{m}^{\alpha}$ is given by

$d_{r}\mathcal{F}(\Omega,r)[h]=\Omega h^{\prime}+\frac{1}{b\circ
R}\partial_{\theta}\Big{(}Q(r)h+P(r)[h]\Big{)}+L(r)h,$

where we set

$\displaystyle P(r)[h](\theta)$ $\displaystyle\overset{\rm
def}{=}\int_{0}^{2\pi}h(\eta)K_{b}\big{(}R(\theta)e^{{\rm
i}\theta},R(\eta)e^{{\rm i}\eta}\big{)}b\big{(}R(\eta)\big{)}d\eta,$
$\displaystyle Q(r)(\theta)$ $\displaystyle\overset{\rm def}{=}\frac{e^{{\rm
i}\theta}}{R(\theta)}\cdot\int_{0}^{2\pi}\int_{0}^{R(\eta)}\nabla_{x}K_{b}\big{(}R(\theta)e^{{\rm
i}\theta},\rho e^{{\rm i}\eta}\big{)}b(\rho)\rho d\rho d\eta,$ $\displaystyle
L(r)(\theta)$ $\displaystyle\overset{\rm
def}{=}\frac{-b^{\prime}\big{(}R(\theta)\big{)}}{R(\theta)b^{2}\big{(}R(\theta)\big{)}}\partial_{\theta}\left(\int_{0}^{2\pi}\int_{0}^{R(\eta)}K_{b}\big{(}R(\theta)e^{{\rm
i}\theta},\rho e^{{\rm i}\eta}\big{)}b(\rho)\rho d\rho d\eta\right),$

for all $\theta\in\mathbb{T}$.

###### Proof.

Recalling that $\mathcal{F}$ is given by (3.3), we only focus on
differentiating

$r\mapsto\frac{1}{b\big{(}R(\theta)\big{)}}\qquad\text{and}\qquad
r\mapsto\int_{0}^{R(\eta)}K_{b}\big{(}R(\theta)e^{{\rm i}\theta},\rho e^{{\rm
i}\eta}\big{)}b(\rho)\rho d\rho.$

For the first function above, it is readily seen that

$d_{r}\left(\frac{1}{b\circ R}\right)[h]=-\frac{b^{\prime}\circ
R}{R\times(b^{2}\circ R)}h.$

As for the second one, by a direct computation, we find that

$\displaystyle d_{r}\bigg{(}\int_{0}^{R(\eta)}$ $\displaystyle
K_{b}\big{(}R(\theta)e^{{\rm i}\theta},\rho e^{{\rm i}\eta}\big{)}b(\rho)\rho
d\rho\bigg{)}[h]$ $\displaystyle=K_{b}\big{(}R(\theta)e^{{\rm
i}\theta},R(\eta)e^{{\rm i}\eta}\big{)}b\big{(}R(\eta)\big{)}h(\eta)$
$\displaystyle\quad\quad\quad+\left(\frac{e^{{\rm
i}\theta}}{R(\theta)}\cdot\int_{0}^{R(\eta)}\nabla_{x}K_{b}\big{(}R(\theta)e^{{\rm
i}\theta},\rho e^{{\rm i}\eta}\big{)}b(\rho)\rho d\rho\right)h(\theta).$

At last, gathering the foregoing two identities completes the proof of the
lemma. ∎

Now, we show that the linearized operator at the equilibrium $r=0$ acts as a
Fourier multiplier whose coefficients are precise in the next lemma.

######  Lemma 3.4.

Let $m\in\mathbb{N}^{*}$, $a>0$, $\Omega\in\mathbb{R}$ and $\alpha\in(0,1).$
Then, for any $h\in X_{m}^{\alpha}$ taking the form

$h(\theta)=\sum_{n=1}^{\infty}h_{n}\cos(nm\theta),$

for all $\theta\in\mathbb{T}$ and some $h_{n}\in\mathbb{R}$, it holds that

$d_{r}\mathcal{F}(\Omega,0)[h](\theta)=-\sum_{n=1}^{\infty}nm\big{(}\Omega-Q(a,0)+\Lambda_{nm}(a,a)\big{)}h_{n}\sin(nm\theta),$

where $Q(a,0)$ and $\big{(}\Lambda_{n}(a,a)\big{)}_{n\in\mathbb{N}^{*}}$ are
defined by (3.1) and (1.7), respectively.

###### Proof.

First, observing that

$P(0)[h](\theta)=b(a)\sum_{n=1}^{\infty}h_{n}\int_{0}^{2\pi}K_{b}\big{(}ae^{{\rm
i}\theta},ae^{{\rm i}\eta}\big{)}\cos(nm\eta)d\eta,$

yields, in view of (2.12), that

$P(0)[h](\theta)=b(a)\sum_{n=1}^{\infty}h_{n}\int_{0}^{2\pi}K_{b}\big{(}a,ae^{{\rm
i}(\eta-\theta)}\big{)}\cos(nm\eta)d\eta.$

Therefore, by a change of variables, we infer that

$\displaystyle P(0)[h](\theta)$
$\displaystyle=b(a)\sum_{n=1}^{\infty}h_{n}\int_{0}^{2\pi}K_{b}\big{(}a,ae^{{\rm
i}\eta}\big{)}\cos\big{(}nm(\eta+\theta)\big{)}d\eta$
$\displaystyle=b(a)\cos(nm\theta)\sum_{n=1}^{\infty}h_{n}\int_{0}^{2\pi}K_{b}\big{(}a,ae^{{\rm
i}\eta}\big{)}\cos(nm\eta)d\eta$
$\displaystyle\quad-b(a)\sin(nm\theta)\sum_{n=1}^{\infty}h_{n}\int_{0}^{2\pi}K_{b}\big{(}a,ae^{{\rm
i}\eta}\big{)}\sin(nm\eta)d\eta.$

Now, the second identity in (2.12) implies that the following function is even

$\eta\mapsto K_{b}(a,ae^{{\rm i}\eta}).$

As a consequence, we obtain the oddness of the function

$\eta\mapsto K_{b}(a,ae^{{\rm i}\eta})\sin(nm\eta)$

and by $2\pi$-periodicity, we deduce that

$\int_{0}^{2\pi}K_{b}\big{(}a,ae^{{\rm i}\eta}\big{)}\sin(nm\eta)d\eta=0,$

thereby establishing that

$P(0)[h](\theta)=b(a)\sum_{n=1}^{\infty}h_{n}\cos(nm\theta)\int_{0}^{2\pi}K_{b}\big{(}a,ae^{{\rm
i}\eta}\big{)}\cos(nm\eta)d\eta.$

Consequently, it follows that

$\frac{1}{b\circ
R(r)}\partial_{\theta}\Big{(}P(r)[h]\Big{)}\big{|}_{r=0}=-\sum_{n=1}^{\infty}h_{n}\sin(nm\theta)\Lambda_{nm}(a,a).$

Next, we take care of $Q(0)$. To that end, observe first that, for any
$\theta\in\mathbb{T}$,

$\displaystyle Q(0)(\theta)$ $\displaystyle=\frac{e^{{\rm
i}\theta}}{a}\cdot\int_{0}^{2\pi}\int_{0}^{a}\nabla_{x}K_{b}\big{(}ae^{{\rm
i}\theta},\rho e^{{\rm i}\eta}\big{)}b(\rho)\rho d\rho d\eta$
$\displaystyle=\frac{1}{a}\int_{0}^{2\pi}\int_{0}^{a}\partial_{a}\Big{(}K_{b}\big{(}ae^{{\rm
i}\theta},\rho e^{{\rm i}\eta}\big{)}\Big{)}b(\rho)\rho d\rho d\eta$
$\displaystyle=\frac{1}{a}\int_{0}^{2\pi}\int_{0}^{a}\partial_{a}\Big{(}K_{b}\big{(}a,\rho
e^{{\rm i}(\eta-\theta)}\big{)}\Big{)}b(\rho)\rho d\rho d\eta.$

Thus, by performing a change of variables, again, we find that

$Q(0)(\theta)=\frac{1}{a}\int_{0}^{2\pi}\int_{0}^{a}\partial_{x_{1}}K_{b}\big{(}a,\rho
e^{{\rm i}\eta}\big{)}b(\rho)\rho d\rho d\eta.$

Thus, by the explicit formula for the right-hand side given in Lemma 2.2, we
deduce that

$Q(0)(\theta)=-b(a)Q(a,0),$

which in turn yields

$\frac{1}{b\circ
R(r)}\partial_{\theta}\Big{(}Q(0)[h]\Big{)}\big{|}_{r=0}=-Q(a,0)\sum_{n=1}^{\infty}h_{n}\sin(nm\theta).$

In addition, we deduce from (3.9) that

(3.11) $L(0)\equiv 0.$

All in all, gathering the preceding identities concludes the proof of the
lemma. ∎

In the next lemma, we perform a detailed asymptotic analysis of the specific
Fourier coefficients $\Lambda_{n}$ that appear in the expression of the
linearized operator $d_{r}\mathcal{F}$ from the preceding lemma. The results
of the next lemma will be also crucial in the spectral analysis of the doubly-
connected V-states, which will be discussed in Section 4, later on.

######  Proposition 3.1.

Let $\alpha,\beta>0$ and the sequence
$\big{(}\Lambda_{n}(\alpha,\beta)\big{)}_{n\in\mathbb{N}^{*}}$ be as defined
in (1.7). Then, the following holds:

  1. (1)

The terms of the sequence
$\big{(}\Lambda_{n}(\alpha,\beta)\big{)}_{n\in\mathbb{N}^{*}}$ expand as

(3.12)
$\Lambda_{n}(\alpha,\beta)=\Lambda_{n}(\beta,\alpha)=\frac{\sqrt{b(\alpha)b(\beta)}}{2n}\mathtt{m}^{n}(\alpha,\beta)+f_{n}(\alpha,\beta),$

where

$\mathtt{m}(\alpha,\beta)\overset{\rm
def}{=}\min\left\\{\frac{\alpha}{\beta},\frac{\beta}{\alpha}\right\\}$

and

(3.13) $f_{n}(\alpha,\beta)\overset{\rm
def}{=}\int_{0}^{2\pi}S_{b}\big{(}\alpha,\beta e^{{\rm
i}\eta}\big{)}\cos(n\eta)d\eta,$

with $S_{b}$ being defined as the solution of (2.8).

  2. (2)

Defining, on $[0,\infty)$, the function

$\displaystyle r\mapsto\Theta(r)$ $\displaystyle\overset{\rm
def}{=}\frac{d}{dr}\left(r\frac{d}{dr}\left(\frac{1}{\sqrt{b(r)}}\right)\right)$

and setting

$A\overset{\rm def}{=}\max\\{\alpha,\beta,R_{\infty}\\},$

where $R_{\infty}>0$ being the radius of the ball defined in (Hb3), it holds
that

(3.14) $\displaystyle|f_{n}(\alpha,\beta)|\leq
2A\sqrt{{b(\beta)}}\frac{\mathtt{m}^{n}(\alpha,\beta)}{n^{2}}\left(\delta_{\alpha,\beta}\left(\frac{1}{n}-1\right)+1\right)\|\Theta\|_{L^{\infty}(\mathbb{R}^{+})},$

for any $n\in\mathbb{N}^{*}$ satisfying

$n\geq\frac{A}{2}\left\|\Theta\right\|_{L^{\infty}(\mathbb{R}^{+})},$

where $\delta_{\alpha,\beta}$ denotes the Kronecker-delta-function of entries
$\alpha$ and $\beta$.

  3. (3)

The sequence $\big{(}\Lambda_{n}(\alpha,\beta)\big{)}_{n>M(b)}$ is positive,
where

(3.15) $M(b)\overset{\rm
def}{=}16A\left\|\Theta\right\|_{L^{\infty}(\mathbb{R}^{+})}\max\left\\{1,\frac{1}{\sqrt{b(\alpha)}}\right\\}.$

  4. (4)

Moreover, in the case $\alpha=\beta$, the sequence
$\big{(}\Lambda_{n}(\alpha,\alpha)\big{)}_{n>M(b)}$ is decreasing.

###### Proof.

(1) Fix $n\in\mathbb{N}^{*}$ and $\alpha,\beta>0$ and observe that the
symmetry property in (3.12) is obtained as a consequence of (2.12) by
implementing the change of variable $\eta\to-\eta.$ This leaves us with the
proof of the second identity of (3.12). To that end, we first write, by using
the decomposition (2.10) of the kernel $K_{b}$, that

(3.16)
$\Lambda_{n}(\alpha,\beta)=-\frac{\sqrt{b(\alpha)b(\beta)}}{2\pi}\int_{0}^{2\pi}\log|\alpha-\beta
e^{\mathrm{i}\eta}|\cos(n\eta)d\eta+\int_{0}^{2\pi}S_{b}\big{(}\alpha,\beta
e^{{\rm i}\eta}\big{)}\cos(n\eta)d\eta.$

Now, by virtue of (C.1), we find that

$\displaystyle-\frac{\sqrt{b(\alpha)b(\beta)}}{2\pi}\int_{0}^{2\pi}\log|\alpha-\beta
e^{\mathrm{i}\eta}|\cos(n\eta)d\eta=\frac{\sqrt{b(\alpha)b(\beta)}}{2n}\mathtt{m}^{n}(\alpha,\beta).$

Hence, identifying the second integral in (3.16) with the definition of
$f_{n}$ given by (3.13) concludes the proof of (3.12).

(2) Since $S_{b}(\cdot,y)$ solves the elliptic problem (2.8), then it can be
represented, by applying Proposition 2.2, as

$S_{b}(x,y)=-\frac{\sqrt{b(y)}}{2\pi}\int_{\mathbb{R}^{2}}K_{b}(x,z)\log|y-z|\Delta\left(\frac{1}{\sqrt{b(z)}}\right)dz.$

Therefore, using the decomposition (2.10) in the preceding identity leads to
the following splitting

$\displaystyle S_{b}(x,y)$
$\displaystyle=\frac{\sqrt{b(y)b(x)}}{2\pi}\int_{\mathbb{R}^{2}}\log|x-z|\log|y-z|\Delta\left(\frac{1}{\sqrt{b(z)}}\right)\sqrt{b(z)}dz$
$\displaystyle\quad+\frac{\sqrt{b(y)}}{2\pi}\int_{\mathbb{R}^{2}}S_{b}(x,z)\log|y-z|\Delta\left(\frac{1}{\sqrt{b(z)}}\right)dz$
$\displaystyle\overset{\rm
def}{=}\widetilde{S}_{b,1}(x,y)+\widetilde{S}_{b,2}(x,y).$

We now take care of each term separately. Before we get into the details of
that, note first that using the radial symmetry of the bottom topography $b$
and writing the Laplace operator in polar coordinates yields

$\Delta\left(\frac{1}{\sqrt{b(|z|)}}\right)=\frac{1}{r}\frac{d}{dr}\left(r\frac{d}{dr}\left(\frac{1}{\sqrt{b(r)}}\right)\right)=\frac{1}{r}\Theta(r),$

for all $z\in\mathbb{R}^{2}$ with $r=|z|$. Thus, for $\widetilde{S}_{b,1}$, we
proceed by utilizing a polar change of variables and twice the identity C.1 to
find that

$\displaystyle\int_{0}^{2\pi}\widetilde{S}_{b,1}$
$\displaystyle\big{(}\alpha,\beta e^{\mathrm{i}\eta}\big{)}\cos(n\eta)d\eta$
$\displaystyle=\frac{\sqrt{b(\beta)}}{4\pi^{2}}\int_{0}^{\infty}\Theta(r)\int_{0}^{2\pi}\log|\alpha-
re^{\mathrm{i}\theta}|\int_{0}^{2\pi}\log|\beta e^{\mathrm{i}\eta}-re^{{\rm
i}\theta}|\cos(n\eta)d\eta d\theta dr$
$\displaystyle=\frac{\sqrt{b(\beta)}}{2\pi}\int_{0}^{\infty}\Theta(r)\int_{0}^{2\pi}\log|\alpha-
re^{\mathrm{i}\theta}|\left(-\frac{\mathtt{m}^{n}(r,\beta)}{2n}\right)\cos(n\theta)d\theta
dr$
$\displaystyle=\frac{\sqrt{b(\beta)}}{4n^{2}}\int_{0}^{\infty}\Theta(r)\mathtt{m}^{n}(r,\alpha)\mathtt{m}^{n}(r,\beta)dr,$

whereby, we have shown that

$\int_{0}^{2\pi}\widetilde{S}_{b,1}\big{(}\alpha,\beta
e^{\mathrm{i}\eta}\big{)}\cos(n\eta)d\eta=\frac{\sqrt{b(\beta)}}{4n^{2}}U_{n}(\alpha,\beta),$

where we set

$U_{n}(\alpha,\beta)\overset{\rm
def}{=}\int_{0}^{\infty}\Theta(r)\mathtt{m}^{n}(r,\alpha)\mathtt{m}^{n}(r,\beta)dr.$

Then, we control $U_{n}$ by employing the elementary computation

(3.17)
$\int_{0}^{\infty}\mathtt{m}^{n}(r,\alpha)\mathtt{m}^{n}(r,\beta)dr=\mathtt{m}^{n}(\alpha,\beta)\left(|\alpha-\beta|+\frac{\min(\alpha,\beta)}{2n+1}+\frac{\max(\alpha,\beta)}{2n-1}\right)$

to obtain that

$\displaystyle|U_{n}(\alpha,\beta)|$
$\displaystyle\leqslant\|\Theta\|_{L^{\infty}(\mathbb{R}^{+})}\int_{0}^{\infty}\mathtt{m}^{n}(r,\alpha)\mathtt{m}^{n}(r,\beta)dr$
$\displaystyle=\|\Theta\|_{L^{\infty}(\mathbb{R}^{+})}\mathtt{m}^{n}(\alpha,\beta)\left(|\alpha-\beta|+\frac{\min(\alpha,\beta)}{2n+1}+\frac{\max(\alpha,\beta)}{2n-1}\right),$

thereby arriving at the bound

(3.18) $|U_{n}(\alpha,\beta)|\leq C_{n}\,\mathtt{m}^{n}(\alpha,\beta),$

where $C_{n}$ is defined as

(3.19) $C_{n}\overset{\rm
def}{=}\begin{cases}4A\|\Theta\|_{L^{\infty}(\mathbb{R}^{+})},&\text{if}\quad\alpha\neq\beta,\vspace{0.2cm}\\\
\displaystyle\frac{2A}{n}\|\Theta\|_{L^{\infty}(\mathbb{R}^{+})},&\text{if}\quad\alpha=\beta,\end{cases}$

where $A$ is defined in the statement of the proposition. Now, we expand the
term involving $\widetilde{S}_{b,2}$ by proceeding in a similar way as before
to write

$\displaystyle\int_{0}^{2\pi}\widetilde{S}_{b,2}$ $\displaystyle(\alpha,\beta
e^{{\rm i}\eta})\cos(n\eta)d\eta$
$\displaystyle=\frac{\sqrt{b(\beta)}}{2\pi}\int_{0}^{\infty}\Theta(r)\int_{0}^{2\pi}S_{b}\big{(}\alpha,re^{{\rm
i}\theta}\big{)}\left(\int_{0}^{2\pi}\log|\beta e^{{\rm i}\eta}-re^{{\rm
i}\theta}|\cos(n\eta)d\eta\right)d\theta dr$
$\displaystyle=-\frac{\sqrt{b(\beta)}}{2n}\int_{0}^{\infty}\Theta(r)\mathtt{m}^{n}(r,\beta)\left(\int_{0}^{2\pi}S_{b}\big{(}\alpha,re^{{\rm
i}\theta}\big{)}\cos(n\theta)d\theta\right)dr$
$\displaystyle=-\frac{\sqrt{b(\beta)}}{2n}\int_{0}^{\infty}\Theta(r)\mathtt{m}^{n}(r,\beta)f_{n}(\alpha,r)dr.$

Therefore, noticing that the hypothesis (Hb3) implies that $\Theta$ is
compactly supported in the ball $B_{R_{\infty}}\subset B_{A}$, it follows from
the previous computations that

(3.20)
$f_{n}(\alpha,\beta)=\frac{\sqrt{b(\beta)}}{4n^{2}}U_{n}(\alpha,\beta)-\frac{\sqrt{b(\beta)}}{2n}\int_{0}^{A}\Theta(r)\mathtt{m}^{n}(r,\beta)f_{n}(\alpha,r)dr.$

Now, note that the sequence $f_{n}$ is uniformly bounded due to its definition
(3.13) and the bounds on function $S_{b}$ from Proposition 2.2 which imply
that

$\sup_{n\in\mathbb{N}^{*}}\left\|f_{n}\right\|_{L^{\infty}\left([0,A]^{2}\right)}\leq
2\pi\left\|S_{b}\right\|_{L^{\infty}\left([0,A]^{2}\right)}<\infty.$

Thus, for any $\varepsilon\in(0,1)$, defining

$g_{n,\varepsilon}(\alpha)\overset{\rm
def}{=}\sup_{\beta\in[0,A]}\left|\frac{f_{n}(\alpha,\beta)}{\sqrt{b(\beta)}(\mathtt{m}^{n}(\alpha,\beta)+\varepsilon)}\right|$

it follows

$\displaystyle\frac{f_{n}(\alpha,\beta)}{\sqrt{b(\beta)}(\mathtt{m}^{n}(\alpha,\beta)+\varepsilon)}$
$\displaystyle\leq\frac{C_{n}}{4n^{2}}+g_{n,\varepsilon}(\alpha)\frac{\left\|\Theta\right\|_{L^{\infty}(\mathbb{R}^{+})}}{2n}\int_{0}^{A}\mathtt{m}^{n}(r,\beta)\Big{(}\mathtt{m}^{n}(r,\alpha)+\varepsilon\Big{)}dr$
$\displaystyle\leq\frac{C_{n}}{4n^{2}}+g_{n,\varepsilon}(\alpha)\frac{A\left\|\Theta\right\|_{L^{\infty}(\mathbb{R}^{+})}}{n},$

for the fact that

$\sup_{c_{1},c_{2}\in(0,\infty)}\mathtt{m}^{n}(c_{1},c_{2})\leq 1.$

Now, choosing $n\in\mathbb{N}^{*}$ in such a way that

$n\geq\frac{A\left\|\Theta\right\|_{L^{\infty}(\mathbb{R}_{+})}}{2},$

we find

$g_{n,\varepsilon}(\alpha)\leq\frac{C_{n}}{2n^{2}},$

which, upon taking the limit $\varepsilon\to 0$, yields

(3.21)
$|f_{n}(\alpha,\beta)|\leq\frac{C_{n}\sqrt{b(\beta)}}{2n^{2}}\mathtt{m}^{n}(\alpha,\beta),$

which completes the justification of the bound on $f_{n}$ after noticing that

$C_{n}\leq
4A\left(\delta_{\alpha,\beta}\left(\frac{1}{n}-1\right)+1\right)\|\Theta\|_{L^{\infty}(\mathbb{R}^{+})}.$

(3) We establish now the positivity of $\Lambda_{n}$ for $n$ sufficiently
large. To that end, we write, in view of (3.19) and (3.21), that

$\displaystyle\Lambda_{n}(\alpha,\beta)$
$\displaystyle\geq\frac{\sqrt{b(\alpha)b(\beta)}}{2n}\mathtt{m}^{n}(\alpha,\beta)-|f_{n}(\alpha,\beta)|$
$\displaystyle\geq\frac{\mathtt{m}^{n}(\alpha,\beta)}{2n^{2}}\left(n\sqrt{b(\alpha)b(\beta)}-4A\sqrt{b(\beta)}\|\Theta\|_{L^{\infty}\left(\mathbb{R}^{+}\right)}\right).$

Therefore, it follows that

$\Lambda_{n}(\alpha,\beta)>0,$

for all $n\in\mathbb{N}^{*}$ with

$n>\frac{4A}{\sqrt{b(\alpha)}}\left\|\Theta\right\|_{L^{\infty}(\mathbb{R}^{+})},$

which yields the non-negativity of $\Lambda_{n}(\alpha,\beta)$ under the
stronger condition stated in the proposition

$n>M(b),$

where $M(b)$ is introduced in (3.15).

(4) Now, let us assume that $\alpha=\beta$ and we compute, for any $n\geq 1$,
that

$\displaystyle\Lambda_{n}(\alpha,\alpha)-\Lambda_{n+1}(\alpha,\alpha)$
$\displaystyle=\frac{b(\alpha)}{2n(n+1)}+f_{n}(\alpha,\alpha)-f_{n+1}(\alpha,\alpha)$
$\displaystyle\geq\frac{b(\alpha)}{2n(n+1)}-|f_{n}(\alpha,\alpha)|-|f_{n+1}(\alpha,\alpha)|.$

Hence, noticing that the bound (3.19) and (3.21) imply

$|f_{n}(\alpha,\alpha)|\leq\frac{2A\left\|\Theta\right\|_{L^{\infty}(\mathbb{R}^{+})}\sqrt{b(\alpha)}}{n^{3}},$

for all $n\geq N(\alpha,\alpha,b),$ it then follows that

$\Lambda_{n}(\alpha,\alpha)-\Lambda_{n+1}(\alpha,\alpha)\geq\frac{b(\alpha)}{4n^{2}}-\frac{4A\|\Theta\|_{L^{\infty}\left(\mathbb{R}^{+}\right)}\sqrt{b(\alpha)}}{n^{3}}\cdot$

Consequently, for a given $n_{0}\in\mathbb{N}^{*}$, we deduce that the
sequence $\big{(}\Lambda_{n}(\alpha,\alpha)\big{)}_{n\geq n_{0}}$ is
decreasing if

$n_{0}>A\left\|\Theta\right\|_{L^{\infty}(\mathbb{R}^{+})}\max\left\\{\frac{1}{2},\frac{16}{\sqrt{b(\alpha)}}\right\\},$

which is clearly implied by the simpler condition that $n_{0}>M(b)$ as claimed
in the statement of the proposition and concludes its proof. ∎

###  3.3. Regularity analysis

The aim of this section is to provide a detailed analysis of the regularity
properties of the functional $\mathcal{F}$. This analysis allows us in
particular to justify the computations laid out in Section 3.2, above, which
will also serve in the doubly connected case, later on.

In the sequel of the paper, for a fixed integer $m\in\mathbb{N}^{*}$, and real
parameter $\alpha\in(0,1)$, the spaces $X_{m}^{\alpha}$ and $Y_{m}^{\alpha}$
are defined by

$X_{m}^{\alpha}\overset{\rm def}{=}\left\\{f\in
C^{1+\alpha}(\mathbb{T}):f(\theta)=\sum_{n=1}^{\infty}f_{n}\cos(nm\theta),\
f_{n}\in\mathbb{R},\ \theta\in\mathbb{T}\right\\}$

and

$Y_{m}^{\alpha}\overset{\rm def}{=}\left\\{g\in
C^{\alpha}(\mathbb{T}):g(\theta)=\sum_{n=1}^{\infty}g_{n}\sin(nm\theta),\
g_{n}\in\mathbb{R},\ \theta\in\mathbb{T}\right\\}$

and equipped with the usual $C^{1+\alpha}$ and $C^{\alpha}$ norms,
respectively. Moreover, for a given $\varepsilon>0$, we set

$\mathcal{B}^{\alpha}_{m,\varepsilon}\overset{\rm def}{=}\big{\\{}f\in
X_{m}^{\alpha}:\left\|f\right\|_{C^{1+\alpha}}<\varepsilon\big{\\}}.$

In addition to that, allow us to clarify some vectorial notations that will be
exclusively used in this section. For any given two vectors
$u=(u_{1},u_{2})\in\mathbb{R}^{2}$ and $v=(v_{1},v_{2})\in\mathbb{R}^{2}$, the
bilinear application

$u\otimes v:\mathbb{R}^{2}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$

is defined, for any $x,y\in\mathbb{R}^{2}$, by

$(u\otimes v)(x,y)\overset{\rm def}{=}(u\cdot x)(v\cdot y).$

Applying it to the canonical basis $(e_{1},e_{2})$ of $\mathbb{R}^{2}$, we
obtain, for any $(i,j)\in\\{1,2\\}^{2}$, that

$(u\otimes v)(e_{i},e_{j})=(u\cdot e_{i})(v\cdot e_{j})=u_{i}v_{j},$

which gives the following representation of the tensor-product

$u\otimes v\overset{\rm def}{=}\begin{pmatrix}u_{1}v_{1}&u_{1}v_{2}\\\
u_{2}v_{1}&u_{2}v_{2}\end{pmatrix}\in M_{2}(\mathbb{R}).$

Now, for given two matrices $A=(A_{ij})_{1\leq i,j\leq 2}$ and
$B=(B_{ij})_{1\leq i,j\leq 2}$ in $M_{2}(\mathbb{R}),$ we define the double
contraction of $A$ and $B$ by

$A:B\overset{\rm def}{=}\sum_{1\leq i,j\leq 2}A_{ij}B_{ij}.$

In particular, one also sees that, for given vectors $u=(u_{1},u_{2})$,
$u^{\prime}=(u_{1}^{\prime},u_{2}^{\prime})$, $v=(v_{1},v_{2})$ and
$v^{\prime}=(v_{1}^{\prime},v_{2}^{\prime})$ in $\mathbb{R}^{2}$, we have

$\big{(}u\otimes v\big{)}:\big{(}u^{\prime}\otimes
v^{\prime}\big{)}=\sum_{1\leq i,j\leq
2}u_{i}v_{j}u_{i}^{\prime}v_{j}^{\prime}=\left(\sum_{i=1}^{2}u_{i}u_{i}^{\prime}\right)\left(\sum_{j=1}^{2}v_{j}v_{j}^{\prime}\right)=(u\cdot
u^{\prime})(v\cdot v^{\prime}).$

####  3.3.1. On the continuity of some singular operators

For a later use, we now discuss some fundamental results which will allow us
to show the continuity of operators with singular kernels. We first recall an
important lemma whose proof can be found in [39, Lemma 2.6] and [13, Lemma 1].

######  Lemma 3.5.

Let $\alpha\in[0,1)$ and consider a measurable complex-valued function $K$,
defined on $\mathbb{T}\times\mathbb{T}\setminus\\{(\theta,\theta),\
\theta\in\mathbb{T}\\}$. Assume that there exists a constant $C>0$ such that

$|K(\theta,\eta)|\leq\frac{C}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|^{\alpha}},\quad\text{for
all}\quad\theta\neq\eta\in\mathbb{T},$

and that $\theta\mapsto K(\theta,\eta)$ is differentiable with

$|\partial_{\theta}K(\theta,\eta)|\leq\frac{C}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|^{1+\alpha}},\quad\text{for
all}\quad\theta\neq\eta\in\mathbb{T}.$

Then, the complex-valued function defined on $\mathbb{T}$, for any $f\in
L^{\infty}(\mathbb{T})$, by

$\theta\mapsto\mathcal{T}f(\theta)\overset{\rm
def}{=}\int_{0}^{2\pi}K(\theta,\eta)f(\eta)d\eta,$

belongs to $C^{\alpha}(\mathbb{T})$ and satisfies

$\left\|\mathcal{T}f\right\|_{C^{\alpha}}\lesssim_{\alpha}C\left\|f\right\|_{L^{\infty}}.$

Now, we establish three essential consequences of Lemma 3.5 which will come in
handy in the regularity analysis of contour dynamics, later on.

######  Corollary 3.1.

Let $a>0$ and $(\alpha,s)\in(0,1)\times[0,1]$. Consider a real-valued
functions $r\in C^{1+\alpha}(\mathbb{T})$ with

(3.22) $\left\|r\right\|_{C^{1+\alpha}}\leq\varepsilon_{0},$

for some small parameter $\varepsilon_{0}>0$. Accordingly, we define the
function

(3.23) $\theta\mapsto R(\theta)\overset{\rm def}{=}\sqrt{a^{2}+2r(\theta)}.$

Then, the function

$\theta\in\mathbb{T}\mapsto\mathcal{T}_{s}f(\theta)\overset{\rm
def}{=}\int_{0}^{2\pi}\log\left|R(\theta)e^{{\rm i}\theta}-sR(\eta)e^{{\rm
i}\eta}\right|f(\eta)d\eta,$

defined for any $f\in L^{\infty}(\mathbb{T})$, belongs to
$C^{\alpha}(\mathbb{T})$ and enjoys the bound

$\left\|\mathcal{T}_{s}f\right\|_{C^{\alpha}}\leq
C_{\alpha}\left\|f\right\|_{L^{\infty}},$

for some constant $C_{\alpha}>0$ which is independent of $s$.

###### Proof.

The proof directly follows by applying Lemma 3.5 provided that we show that
the kernel defined by

$(\theta,\eta)\mapsto K_{s}(\theta,\eta)\overset{\rm
def}{=}\log\left|R(\theta)e^{{\rm i}\theta}-sR(\eta)e^{{\rm i}\eta}\right|$

satisfies the assumptions of Lemma 3.5 for a constant $C$ which is independent
of $s$. To that end, observe first, by a continuity argument together with the
smallness assumption (3.22), that

(3.24)
$\inf_{\theta\in\mathbb{T}}R(\theta)\geq\frac{a}{2}\quad\text{and}\quad\sup_{\theta\in\mathbb{T}}R(\theta)\leq
2a.$

Besides, writing, for any $s\in[0,1]$ and any $\theta,\eta\in\mathbb{T}$, that

(3.25) $\displaystyle\left|R(\theta)e^{{\rm i}\theta}-sR(\eta)e^{{\rm
i}\eta}\right|^{2}$
$\displaystyle=R^{2}(\theta)+s^{2}R^{2}(\eta)-2sR(\theta)R(\eta)\cos(\eta-\theta)$
$\displaystyle=\big{(}R(\theta)-sR(\eta)\big{)}^{2}+4sR(\theta)R(\eta)\sin^{2}\left(\tfrac{\theta-\eta}{2}\right),$

it then follows, by continuity and using (3.25) and (3.24), that there exists
a small parameter $s_{0}\in(0,1)$ in such a way that

$\left|R(\theta)e^{{\rm i}\theta}-sR(\eta)e^{{\rm
i}\eta}\right|\geq\frac{R(\theta)}{2}\geq\frac{a}{4}\left|\sin\left(\tfrac{\theta-\eta}{2}\right)\right|,\quad\text{for
all}\quad s\in[0,s_{0}].$

On the other hand, by utilizing (3.25) and (3.24) again, we obtain that

$\left|R(\theta)e^{{\rm i}\theta}-sR(\eta)e^{{\rm i}\eta}\right|\geq
2\sqrt{s_{0}R(\theta)R(\eta)}\left|\sin\left(\tfrac{\theta-\eta}{2}\right)\right|\geq
a\sqrt{s_{0}}\left|\sin\left(\tfrac{\theta-\eta}{2}\right)\right|,\quad\text{for
all}\quad s\in[s_{0},1].$

Putting together the last two estimates leads to the lower bound

(3.26) $\inf_{s\in[0,1]}\left|R(\theta)e^{{\rm i}\theta}-sR(\eta)e^{{\rm
i}\eta}\right|\geq\frac{a\sqrt{s_{0}}}{4}\left|\sin\left(\tfrac{\theta-\eta}{2}\right)\right|,\quad\text{for
all}\quad\theta,\eta\in\mathbb{T}.$

Moreover, using the fact that

(3.27) $x\mapsto x^{\gamma}\big{|}\log x\big{|}\in
L^{\infty}_{\mathrm{loc}}(\mathbb{R}^{+}),$

for all $\gamma>0$, we then deduce that

$|K_{s}(\theta,\eta)|\leq\frac{C}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|^{\gamma}},$

for all $\eta\neq\theta\in\mathbb{T}$ and some constant $C>0$ which is
independent of $s$.

Likewise, computing that

$\partial_{\theta}K_{s}(\theta,\eta)=\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\cdot\frac{R(\theta)e^{{\rm i}\theta}-sR(\eta)e^{{\rm
i}\eta}}{\left|R(\theta)e^{{\rm i}\theta}-sR(\eta)e^{{\rm
i}\eta}\right|^{2}},$

and utilizing the bounds (3.22), (3.24) and (3.26), we infer that

$|\partial_{\theta}K_{s}(\theta,\eta)|\leq\frac{C}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|}\leq\frac{C}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|^{1+\alpha}},$

for all $\eta\neq\theta\in\mathbb{T}$, up to suitably change the constant $C$,
above. At last, Lemma 3.5 applies to conclude the proof of Corollary 3.1. ∎

######  Corollary 3.2.

Let $\alpha,s$, $R_{1}$ and $R_{2}$ be respectively defined in terms of
$r_{1}$ and $r_{2}$ as in Corollary 3.1 by also assuming that $r_{1}$ and
$r_{2}$ satisfy the smallness assumption (3.22). Further define the following
operator acting on $L^{\infty}(\mathbb{T})$ by

$f\mapsto\mathcal{\widetilde{T}}_{s}f\overset{\rm
def}{=}\int_{0}^{2\pi}\widetilde{K}_{s}(\cdot,\eta)f(\eta)d\eta,$

where we set, for any $\theta\neq\eta\in\mathbb{T}$,

$\widetilde{K}_{s}(\theta,\eta)\overset{\rm
def}{=}\log\left|R_{1}(\theta)e^{{\rm i}\theta}-sR_{1}(\eta)e^{{\rm
i}\eta}\right|-\log\left|R_{2}(\theta)e^{{\rm i}\theta}-sR_{2}(\eta)e^{{\rm
i}\eta}\right|.$

Then, $\mathcal{\widetilde{T}}_{s}f\in C^{\alpha}(\mathbb{T})$ and it holds,
for any $\beta\in[0,\alpha)$, that

$\left\|\mathcal{\widetilde{T}}_{s}f\right\|_{C^{\alpha}}\lesssim_{\alpha}\left\|r_{1}-r_{2}\right\|_{C^{1}}^{\beta}\left\|f\right\|_{L^{\infty}}.$

###### Proof.

First remark that the mean value theorem implies, for all $x,y\in\mathbb{R}$,
that

$\big{|}\log x-\log y\big{|}\leq\frac{|x-y|}{\min\\{x,y\\}}\cdot$

Thus, combining this estimate with (3.26) and employing the triangular
inequality, yields

$\displaystyle\big{|}\widetilde{K}_{s}(\theta,\eta)\big{|}$
$\displaystyle\lesssim\frac{\Big{|}\left|R_{1}(\theta)e^{{\rm
i}\theta}-sR_{1}(\eta)e^{{\rm i}\eta}\right|-\left|R_{2}(\theta)e^{{\rm
i}\theta}-sR_{2}(\eta)e^{{\rm
i}\eta}\right|\Big{|}}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|}$
$\displaystyle\lesssim\frac{\left\|R_{1}-R_{2}\right\|_{L^{\infty}}}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|}\cdot$

Now, observing that

(3.28) $R_{1}-R_{2}=2\frac{r_{1}-r_{2}}{R_{1}+R_{2}},$

and combining this identity with (3.24), it is readily seen then that

$\left\|R_{1}-R_{2}\right\|_{L^{\infty}}\lesssim\left\|r_{1}-r_{2}\right\|_{L^{\infty}},$

which implies in turn that

$\big{|}\widetilde{K}_{s}(\theta,\eta)\big{|}\lesssim\frac{\left\|r_{1}-r_{2}\right\|_{L^{\infty}}}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|}\cdot$

On the other hand, we deduce from (3.27) that

$|\widetilde{K}_{s}(\theta,\eta)|\leq\frac{C}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|^{\gamma}},$

for any $\gamma>0$. Hence, in view of the preceding two bounds, we write, for
any $\theta\neq\eta\in\mathbb{T}$ and $\beta\in(0,1)$ to be chosen in a
moment, that

$\displaystyle\big{|}\widetilde{K}_{s}(\theta,\eta)\big{|}$
$\displaystyle=\big{|}\widetilde{K}(\theta,\eta)\big{|}^{\beta}\big{|}\widetilde{K}(\theta,\eta)\big{|}^{1-\beta}$
$\displaystyle\lesssim\frac{\left\|r_{1}-r_{2}\right\|_{L^{\infty}}^{\beta}}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|^{\beta+\gamma(1-\beta)}}\cdot$

Therefore, choosing $\beta>0$ small enough by setting, for instance, that

$\gamma=\frac{\alpha-\beta}{1-\beta}\quad\text{and}\quad\beta\in(0,\alpha),$

we end up with the desired control

(3.29)
$|\widetilde{K}_{s}(\theta,\eta)|\lesssim\frac{\left\|r_{1}-r_{2}\right\|_{L^{\infty}}^{\beta}}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|^{\alpha}}\cdot$

Likewise, we first observe that

$\displaystyle|\partial_{\theta}\widetilde{K}_{s}(\theta,\eta)|$
$\displaystyle\leq\frac{\left|\partial_{\theta}\left(R_{1}(\theta)e^{{\rm
i}\theta}\right)-\partial_{\theta}\left(R_{2}(\theta)e^{{\rm
i}\theta}\right)\right|}{\left|R_{2}(\theta)e^{{\rm
i}\theta}-sR_{2}(\eta)e^{{\rm i}\eta}\right|}$
$\displaystyle\quad+\left|\partial_{\theta}\left(R_{1}(\theta)e^{{\rm
i}\theta}\right)\right|\left|\frac{R_{1}(\theta)e^{{\rm
i}\theta}-sR_{1}(\eta)e^{{\rm i}\eta}}{\left|R_{1}(\theta)e^{{\rm
i}\theta}-sR_{2}(\eta)e^{{\rm i}\eta}\right|^{2}}-\frac{R_{2}(\theta)e^{{\rm
i}\theta}-sR_{2}(\eta)e^{{\rm i}\eta}}{\left|R_{2}(\theta)e^{{\rm
i}\theta}-sR_{2}(\eta)e^{{\rm i}\eta}\right|^{2}}\right|.$

Then writing, in view of the elementary bounds

$\left|\frac{X_{1}}{|X_{1}|^{2}}-\frac{X_{2}}{|X_{2}|^{2}}\right|\leq
3\frac{|X_{1}-X_{2}|}{\min\\{|X_{1}|,|X_{2}|\\}^{2}},$

and

$\left|\frac{X_{1}}{|X_{1}|^{2}}-\frac{X_{2}}{|X_{2}|^{2}}\right|\leq\frac{2}{\min\\{|X_{1}|,|X_{2}|\\}},$

with the values

$X_{k}=X_{k}(\theta,\eta)=R_{k}(\theta)e^{{\rm i}\theta}-sR_{k}(\eta)e^{{\rm
i}\eta},\qquad k\in\\{1,2\\},$

that

$\displaystyle\left|\frac{X_{1}}{|X_{1}|^{2}}-\frac{X_{2}}{|X_{2}|^{2}}\right|$
$\displaystyle=\left|\frac{X_{1}}{|X_{1}|^{2}}-\frac{X_{2}}{|X_{2}|^{2}}\right|^{\alpha}\left|\frac{X_{1}}{|X_{1}|^{2}}-\frac{X_{2}}{|X_{2}|^{2}}\right|^{1-\alpha}$
$\displaystyle\lesssim_{\alpha}\frac{|X_{1}-X_{2}|^{\alpha}}{\min\\{|X_{1}|,|X_{2}|\\}^{1+\alpha}},$

it follows, by further utilizing the bound (3.26), that

$\displaystyle|\partial_{\theta}\widetilde{K}_{s}(\theta,\eta)|$
$\displaystyle\lesssim_{\alpha}\frac{\left\|R_{1}-R_{2}\right\|_{C^{1}}}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|}+\frac{\left\|R_{1}-R_{2}\right\|_{L^{\infty}}^{\alpha}}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|^{1+\alpha}}\cdot$

Now, notice that deriving (3.28) and estimating the resulting identity in
$L^{\infty}$, yields

$\left\|R_{1}-R_{2}\right\|_{C^{1}}\lesssim\left\|r_{1}-r_{2}\right\|_{C^{1}}.$

Hence, since $\beta<\alpha$ and $r_{1},r_{2}$ satisfy the smallness condition
(3.22), we arrive at the control

$\displaystyle|\partial_{\theta}\widetilde{K}_{s}(\theta,\eta)|$
$\displaystyle\lesssim\frac{\left\|r_{1}-r_{2}\right\|_{C^{1}}^{\alpha}}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|^{1+\alpha}}$
$\displaystyle\lesssim\frac{\left\|r_{1}-r_{2}\right\|_{C^{1}}^{\beta}}{\left|\sin\left(\frac{\theta-\eta}{2}\right)\right|^{1+\alpha}}\cdot$

At last, combining the preceding control with (3.29) allows us to apply Lemma
3.5, thereby completing the proof of Corollary 3.2. ∎

We conclude this section by stating the following corollary whose proof can be
done along the same lines as shown for Corollaries 3.1 and 3.2.

######  Corollary 3.3.

Under the assumptions of Corollaries 3.1 and 3.2, further consider a complex
valued function $h\in C^{1+\alpha}(\mathbb{T})$ and define the operators
$\mathcal{T}_{h,R}$ and $\mathcal{\widetilde{T}}_{h}$ by

$f\mapsto\mathcal{T}_{h,R}f\overset{\rm
def}{=}\int_{0}^{2\pi}K_{h,R}(\cdot,\eta)f(\eta)d\eta,$

and

$f\mapsto\mathcal{\widetilde{T}}_{h}f\overset{\rm
def}{=}\int_{0}^{2\pi}\widetilde{K}_{h}(\cdot,\eta)f(\eta)d\eta,$

where, we set, for any $\theta\neq\eta\in\mathbb{T}$, that

$K_{h,R}(\theta,\eta)\overset{\rm def}{=}\frac{R(\theta)e^{{\rm
i}\theta}-R(\eta)e^{{\rm i}\eta}}{|R(\theta)e^{{\rm i}\theta}-R(\eta)e^{{\rm
i}\eta}|^{2}}\cdot\left(\frac{h(\theta)e^{{\rm
i}\theta}}{R(\theta)}-\frac{h(\eta)e^{{\rm i}\eta}}{R(\eta)}\right),$

and

$\widetilde{K}_{h}(\theta,\eta)\overset{\rm
def}{=}K_{h,R_{1}}(\theta,\eta)-K_{h,R_{2}}(\theta,\eta).$

Then, $\mathcal{T}_{h,R}f$ and $\mathcal{\widetilde{T}}_{h}f$ belong to
$C^{\alpha}(\mathbb{T})$ with

$\left\|\mathcal{T}_{h,R}f\right\|_{C^{\alpha}}\lesssim_{\alpha}\left\|h\right\|_{C^{1+\alpha}}\left\|f\right\|_{L^{\infty}},$

and

$\left\|\mathcal{\widetilde{T}}_{h}f\right\|_{C^{\alpha}}\lesssim_{\alpha}\left\|r_{1}-r_{2}\right\|_{C^{1}}^{\beta}\left\|h\right\|_{C^{1+\alpha}}\left\|f\right\|_{L^{\infty}},$

for all $\beta\in(0,\alpha)$.

The next two lemmas can be seen as variations of Corollaries 3.1 and 3.2. Even
though their statements are similar, the proofs of these lemmas require a
careful attention and they do not straitforwardly follow from the statements
of the previous results of this section.

######  Lemma 3.6.

Let $f$ be an element of $L^{\infty}(\mathbb{T})$ and $R$ be as in Corollary
3.1. Then, the function defined, for any
$(\rho,\theta)\in\mathbb{R}^{+}\times\mathbb{T}$, by

$A(\rho,\theta)\overset{\rm def}{=}\int_{0}^{2\pi}\log\left|\rho
e^{i\theta}-R(\eta)e^{i\eta}\right|f(\eta)d\eta$

enjoys the bound

$\left\|\partial_{\rho}A\right\|_{L^{p}\big{(}(0,\delta)\times\mathbb{T}\big{)}}\lesssim\left\|f\right\|_{L^{\infty}(\mathbb{T})},$

for any finite $\delta>0$ and any $p\in[1,\infty)$.

######  Remark 3.1.

A direct consequence of the preceding lemma is that the function

$x\mapsto\int_{0}^{2\pi}\log\left|x-R(\eta)e^{i\eta}\right|f(\eta)d\eta$

belongs to $W^{1,p}_{\mathrm{loc}}(\mathbb{R}^{2})$ and, up to a
multiplicative constant, obeys the same bound from the lemma in terms of the
given function $f$.

###### Proof.

To get started, noticing

$\partial_{\rho}A(\rho,\theta)=\int_{0}^{2\pi}\frac{\rho-R(\eta)+2R(\eta)\sin^{2}(\frac{\eta-\theta}{2})}{\big{(}\rho-R(\eta)\big{)}^{2}+4\rho
R(\eta)\sin^{2}(\frac{\eta-\theta}{2})}f(\eta)d\eta,$

and recalling the bounds from (3.24)

(3.30) $\frac{a}{2}\leq R(\eta)\leq 2a,\quad\text{for
all}\quad\eta\in\mathbb{T},$

yields that

$\left(\frac{\rho-R(\eta)+2R(\eta)\sin^{2}(\frac{\eta-\theta}{2})}{\big{(}\rho-R(\eta)\big{)}^{2}+4\rho
R(\eta)\sin^{2}(\frac{\eta-\theta}{2})}\right)\mathds{1}_{\\{0<\rho<\frac{a}{4}\\}\cup\\{4a<\rho<\delta\\}}=O(1),$

for all $\theta,\eta\in\mathbb{T}$. Therefore, we deduce that

(3.31)
$\left\|\partial_{\rho}A\right\|_{L^{p}\big{(}(0,\frac{a}{4})\cup(4a,\delta)\times\mathbb{T}\big{)}}\lesssim\left\|f\right\|_{L^{\infty}(\mathbb{T})}.$

It remains now to establish the same control on the compact domain
$(\frac{a}{4},4a)$, where the variable $\rho$ is away from the origin but
within the singular set of the function

$D(\rho,\theta,\eta)\overset{\rm
def}{=}\frac{\rho-R(\eta)+2R(\eta)\sin^{2}(\frac{\eta-\theta}{2})}{\big{(}\rho-R(\eta)\big{)}^{2}+4\rho
R(\eta)\sin^{2}(\frac{\eta-\theta}{2})}\mathds{1}_{\\{\frac{a}{4}<\rho<4a\\}}.$

To that end, letting $\varepsilon\in(0,1)$ to be chosen in a little while, we
first notice the bound

(3.32) $\displaystyle|D(\rho,\theta,\eta)|$
$\displaystyle\leq\left(\frac{1}{\big{(}\rho
R(\eta)\big{)}^{\varepsilon}}\frac{1}{\big{|}\rho-R(\eta)\big{|}^{1-2\varepsilon}\sin^{2\varepsilon}\left(\frac{\theta-\eta}{2}\right)}+\frac{1}{\rho}\right)\mathds{1}_{\\{\frac{a}{4}<\rho<4a\\}}$
$\displaystyle\lesssim\left(\frac{1}{\big{|}\rho-R(\eta)\big{|}^{1-2\varepsilon}\sin^{2\varepsilon}\left(\frac{\theta-\eta}{2}\right)}+1\right)\mathds{1}_{\\{\frac{a}{4}<\rho<4a\\}},$

which is obtained by decomposing the nominator in the definition of the
function $D$ along with the elementary interpolation inequality

(3.33) $c_{1}^{2}+c_{2}^{2}\gtrsim
c_{1}^{2\varepsilon}c_{2}^{2(1-\varepsilon)},$

which is valid for any positive numbers $c_{1}$ and $c_{2}$ and any parameter
$\varepsilon\in[0,1]$. Thus, for any given $p\in(1,\infty)$, we obtain that

$\left\|D(\cdot,\theta,\eta)\right\|_{L^{p}(\mathbb{R}^{+})}\lesssim
1+\frac{1}{\sin^{2\varepsilon}\left(\frac{\theta-\eta}{2}\right)}\left(\int_{\frac{a}{4}}^{4a}\frac{d\rho}{\big{|}\rho-R(\eta)\big{|}^{p(1-2\varepsilon)}}\right)^{\frac{1}{p}},$

whence, by making an adequate change of variables and choosing $\varepsilon$
sufficiently close to $\frac{1}{2}$ in such a way that

(3.34) $p(1-2\varepsilon)<1\quad\text{and}\quad 0<\varepsilon<\frac{1}{2},$

we infer that

$\displaystyle\left\|D(\cdot,\theta,\eta)\right\|_{L^{p}(\mathbb{R}^{+})}$
$\displaystyle\lesssim
1+\frac{1}{\sin^{2\varepsilon}\left(\frac{\theta-\eta}{2}\right)}\left(\int_{0}^{4a}\frac{d\sigma}{\sigma^{p(1-2\varepsilon)}}\right)^{\frac{1}{p}}$
$\displaystyle\lesssim
1+\frac{1}{\sin^{2\varepsilon}\left(\frac{\theta-\eta}{2}\right)},$

for any $\theta,\eta\in\mathbb{T}$. Hence, by virtue of the choice
$\varepsilon<\frac{1}{2}$, it follows that

$\int_{0}^{2\pi}\left\|D(\cdot,\theta,\eta)\right\|_{L^{p}(\mathbb{R}^{+})}|f(\eta)|d\eta\lesssim\left\|f\right\|_{L^{\infty}(\mathbb{T})},$

where we have utilized the fact that

$\eta\mapsto\frac{1}{\sin^{2\varepsilon}\left(\frac{\theta-\eta}{2}\right)}\in
L^{1}(\mathbb{T}),$

uniformly in $\theta\in\mathbb{T}$ as soon as $\varepsilon<\frac{1}{2}$.
Consequently, we deduce that

$\left\|\partial_{\rho}A\right\|_{L^{p}\big{(}(\frac{a}{4},4a)\times\mathbb{T}\big{)}}\lesssim\left\|f\right\|_{L^{\infty}(\mathbb{T})}.$

All in all, combining this bound with (3.31) concludes the proof of the lemma.
∎

The next lemma in this section establishes a continuity aspect with respect to
the functional $R$ in the preceding lemma.

######  Lemma 3.7.

Let $R_{1}$ and $R_{2}$ be respectively defined in terms of $r_{1}$ and
$r_{2}$ as in Corollary 3.1. Furtehr define, for $i\in\\{1,2\\}$ and any given
function $f\in L^{\infty}(\mathbb{T})$,

$A_{i}(\rho,\theta)\overset{\rm def}{=}\int_{0}^{2\pi}\log\left|\rho
e^{i\theta}-R_{i}(\eta)e^{i\eta}\right|f(\eta)d\eta,$

for any $\theta,\eta\in\mathbb{T}$. Then, it holds that

$\left\|\partial_{\rho}A_{1}-\partial_{\rho}A_{2}\right\|_{L^{p}\big{(}(0,\delta)\times\mathbb{T}\big{)}}\lesssim\left\|R_{1}-R_{2}\right\|_{L^{\infty}(\mathbb{T})}^{\beta}\left\|f\right\|_{L^{\infty}(\mathbb{T})},$

for any $\beta\in[0,\frac{1}{p})$ and $\delta>0$.

###### Proof.

We begin with noticing that

$\displaystyle\left|\frac{\rho-
R_{1}(\eta)+2R_{1}(\eta)\sin^{2}(\frac{\eta-\theta}{2})}{\big{(}\rho-
R_{1}(\eta)\big{)}^{2}+4\rho
R_{1}(\eta)\sin^{2}(\frac{\eta-\theta}{2})}-\frac{\rho-
R_{2}(\eta)+2R_{2}(\eta)\sin^{2}(\frac{\eta-\theta}{2})}{\big{(}\rho-
R_{2}(\eta)\big{)}^{2}+4\rho
R_{2}(\eta)\sin^{2}(\frac{\eta-\theta}{2})}\right|\mathds{1}_{\\{0<\rho<\frac{a}{4}\\}\cup\\{4a<\rho<\delta\\}}$
$\displaystyle\qquad=O\left(|R_{1}(\eta)-R_{2}(\eta)|\right),$

for all $\theta,\eta\in\mathbb{T}$, whose justification solely relies on the
bound (3.30) and the fact that the variable $\rho$ is localized away from the
potential values of $R(\eta)$. Thus, it follows that

(3.35)
$\left\|\partial_{\rho}A_{1}-\partial_{\rho}A_{2}\right\|_{L^{p}\big{(}(0,\frac{a}{4})\cup(4a,\delta)\times\mathbb{T}\big{)}}\lesssim\left\|R_{1}-R_{2}\right\|_{L^{\infty}(\mathbb{T})}\left\|f\right\|_{L^{\infty}(\mathbb{T})}.$

Now, we focus our attention on obtaining a similar bound within the domain
$(\frac{a}{4},4a)\times\mathbb{T}$ where the function

$\displaystyle D_{1}(\rho,\theta,\eta)-D_{2}(\rho,\theta,\eta)$
$\displaystyle\quad\overset{\rm def}{=}\left(\frac{\rho-
R_{1}(\eta)+2R_{1}(\eta)\sin^{2}(\frac{\eta-\theta}{2})}{\big{(}\rho-
R_{1}(\eta)\big{)}^{2}+4\rho
R_{1}(\eta)\sin^{2}(\frac{\eta-\theta}{2})}-\frac{\rho-
R_{2}(\eta)+2R_{2}(\eta)\sin^{2}(\frac{\eta-\theta}{2})}{\big{(}\rho-
R_{2}(\eta)\big{)}^{2}+4\rho
R_{2}(\eta)\sin^{2}(\frac{\eta-\theta}{2})}\right)\mathds{1}_{\\{\frac{a}{4}<\rho<4a\\}}$

is singular. Here, we need to perform additional splittings. We proceed by
introducing the notation

$\alpha=\alpha(\eta)\overset{\rm def}{=}|R_{1}(\eta)-R_{2}(\eta)|\leq 1$

and, accordingly, writing

$\displaystyle|D_{1}(\rho,\theta,\eta)-D_{2}(\rho,\theta,\eta)|$
$\displaystyle\leq\Big{(}|D_{1}(\rho,\theta,\eta)|+|D_{2}(\rho,\theta,\eta)|\Big{)}\mathds{1}_{\\{|\rho-
R_{1}(\eta)|\leq 2\alpha\\}\cap\\{|\rho-R_{2}(\eta)|\leq 2\alpha\\}}$
$\displaystyle\quad+\big{|}D_{1}(\rho,\theta,\eta)-D_{2}(\rho,\theta,\eta)\big{|}\mathds{1}_{\\{|\rho-
R_{1}(\eta)|>2\alpha\\}\cap\\{|\rho-R_{2}(\eta)|\geq\alpha\\}}$
$\displaystyle\quad+\big{|}D_{1}(\rho,\theta,\eta)-D_{2}(\rho,\theta,\eta)\big{|}\mathds{1}_{\\{|\rho-
R_{2}(\eta)|>2\alpha\\}\cap\\{|\rho-R_{1}(\eta)|\geq\alpha\\}}$
$\displaystyle\leq\Big{(}|D_{1}(\rho,\theta,\eta)|+|D_{2}(\rho,\theta,\eta)|\Big{)}\mathds{1}_{\\{|\rho-
R_{1}(\eta)|\leq 2\alpha\\}\cap\\{|\rho-R_{2}(\eta)|\leq 2\alpha\\}}$
$\displaystyle\quad+2\big{|}D_{1}(\rho,\theta,\eta)-D_{2}(\rho,\theta,\eta)\big{|}\mathds{1}_{\\{|\rho-
R_{1}(\eta)|>\alpha\\}\cap\\{|\rho-R_{2}(\eta)|\geq\alpha\\}}$
$\displaystyle\overset{\rm
def}{=}\mathcal{D}_{1}(\rho,\theta,\eta)+\mathcal{D}_{2}(\rho,\theta,\eta).$

Notice that the validity of this decomposition is due to the partition

$\mathbb{R}^{+}\times\mathbb{T}=\left(\\{|\rho-R_{1}(\eta)|\leq
2\alpha\\}\cap\\{|\rho-R_{2}(\eta)|\leq 2\alpha\\}\right)\cup\\{|\rho-
R_{1}(\eta)|>2\alpha\\}\cup\\{|\rho-R_{2}(\eta)|>2\alpha\\},$

and the triangular inequality

$|\rho-R_{1}(\eta)|\geq|\rho-R_{2}(\eta)|-\alpha\geq\alpha,$

in the case $|\rho-R_{2}(\eta)|>2\alpha$ and vice versa. Next, we take care of
$\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ separately. To that end, by means of
the same arguments laid out in the proof of Lemma 3.6 to estimate the function
$D$ therein (see in particular (3.32)), we infer that

$\displaystyle\left\|\mathcal{D}_{1}(\cdot,\theta,\eta)\right\|_{L^{p}(\mathbb{R}^{+})}$
$\displaystyle\lesssim\sum_{i=1}^{2}\left(\int_{\frac{a}{4}}^{4a}\mathds{1}_{\\{|\rho-
R_{i}(\eta)|\leq 2\alpha\\}}d\rho\right)^{\frac{1}{p}}$
$\displaystyle\quad+\frac{1}{\sin^{2\varepsilon}\left(\frac{\theta-\eta}{2}\right)}\sum_{i=1}^{2}\left(\int_{\frac{a}{4}}^{4a}\frac{\mathds{1}_{\\{|\rho-
R_{i}(\eta)|\leq 2\alpha\\}}}{\big{|}\rho-
R_{i}(\eta)\big{|}^{p(1-2\varepsilon)}}d\rho\right)^{\frac{1}{p}}$
$\displaystyle\lesssim\alpha^{\frac{1}{p}}+\frac{1}{\sin^{2\varepsilon}\left(\frac{\theta-\eta}{2}\right)}\left(\int_{0}^{2\alpha}\frac{d\sigma}{\sigma^{p(1-2\varepsilon)}}\right)^{\frac{1}{p}}$
$\displaystyle\lesssim\alpha^{\frac{1}{p}}+\frac{1}{\sin^{2\varepsilon}\left(\frac{\theta-\eta}{2}\right)}\alpha^{\frac{1}{p}-(1-2\varepsilon)},$

for all $\theta,\eta\in\mathbb{T}$, where the smallness condition (3.34) on
the choice of $\varepsilon\in(0,\frac{1}{2})$ was employed in the last line.
Hence, it follows that

(3.36)
$\int_{0}^{2\pi}\left\|\mathcal{D}_{1}(\cdot,\theta,\eta)\right\|_{L^{p}(\mathbb{R}^{+})}|f(\eta)|d\eta\lesssim\left\|R_{1}-R_{2}\right\|_{L^{\infty}(\mathbb{T})}^{\beta}\left\|f\right\|_{L^{\infty}(\mathbb{T})},$

uniformly in $\theta\in\mathbb{T}$, where we set

(3.37) $\beta\overset{\rm def}{=}\frac{1}{p}-(1-2\varepsilon)>0.$

We turn our attention now to the estimate of $\mathcal{D}_{2}$. To that end,
we write that

$\displaystyle(D_{1}$ $\displaystyle-
D_{2})(\rho,\theta,\eta)=\frac{\big{(}R_{2}(\eta)-R_{1}(\eta)\big{)}\big{(}1-2\sin^{2}(\tfrac{\eta-\theta}{2})\big{)}}{\big{(}\rho-
R_{2}(\eta)\big{)}^{2}+4\rho R_{2}(\eta)\sin^{2}(\frac{\eta-\theta}{2})}$
$\displaystyle\qquad\qquad\qquad+\Big{(}\rho-
R_{1}(\eta)+2R_{1}(\eta)\sin^{2}(\tfrac{\eta-\theta}{2})\Big{)}$
$\displaystyle\quad\times\left(\frac{1}{\big{(}\rho-
R_{1}(\eta)\big{)}^{2}+4\rho
R_{1}(\eta)\sin^{2}(\frac{\eta-\theta}{2})}-\frac{1}{\big{(}\rho-
R_{2}(\eta)\big{)}^{2}+4\rho
R_{2}(\eta)\sin^{2}(\frac{\eta-\theta}{2})}\right)$
$\displaystyle=\frac{\big{(}R_{2}(\eta)-R_{1}(\eta)\big{)}\big{(}1-2\sin^{2}(\frac{\eta-\theta}{2})\big{)}}{\big{(}\rho-
R_{2}(\eta)\big{)}^{2}+4\rho R_{2}(\eta)\sin^{2}(\frac{\eta-\theta}{2})}$
$\displaystyle\qquad\qquad\qquad+\Big{(}\rho-
R_{1}(\eta)+2R_{1}(\eta)\sin^{2}\left(\tfrac{\eta-\theta}{2}\right)\Big{)}\big{(}R_{2}(\eta)-R_{1}(\eta)\big{)}$
$\displaystyle\quad\times\left(\frac{(\rho-R_{1}(\eta))+(\rho-
R_{2}(\eta))+2\rho\sin^{2}(\frac{\eta-\theta}{2})}{\left(\big{(}\rho-
R_{1}(\eta)\big{)}^{2}+4\rho
R_{1}(\eta)\sin^{2}(\frac{\eta-\theta}{2})\right)\left(\big{(}\rho-
R_{2}(\eta)\big{)}^{2}+4\rho
R_{2}(\eta)\sin^{2}(\frac{\eta-\theta}{2})\right)}\right).$

Therefore, employing (3.30) once again together with the interpolation
inequality (3.33) for suitable values of parameters leads to the control

$\displaystyle\big{|}(D_{1}$ $\displaystyle-D_{2})(\rho,\theta,\eta)\big{|}$
$\displaystyle\lesssim\big{|}R_{1}(\eta)-R_{2}(\eta)\big{|}\left(1+\frac{1}{|\rho-
R_{1}(\eta)|^{2(1-\varepsilon)}\sin^{2\varepsilon}(\frac{\theta-\eta}{2})}+\frac{1}{|\rho-
R_{2}(\eta)|^{2(1-\varepsilon)}\sin^{2\varepsilon}(\frac{\theta-\eta}{2})}\right),$

for all $\rho\in(\frac{a}{4},4a)$ and any $\varepsilon\in(0,\frac{1}{2})$.
Thus, we find that

$\displaystyle\left\|\mathcal{D}_{2}(\cdot,\theta,\eta)\right\|_{L^{p}(\mathbb{R}^{+})}$
$\displaystyle\lesssim\alpha\left(\left(\int_{\frac{a}{4}}^{4a}d\rho\right)^{\frac{1}{p}}+\frac{1}{\sin^{2\varepsilon}(\frac{\theta-\eta}{2})}\sum_{i=1}^{2}\left(\int_{\frac{a}{4}}^{4a}\frac{\mathds{1}_{\\{|\rho-
R_{i}(\eta)|\geq\alpha\\}}}{|\rho-
R_{i}(\eta)|^{2p(1-\varepsilon)}}d\rho\right)^{\frac{1}{p}}\right)$
$\displaystyle\lesssim\alpha\left(1+\frac{1}{\sin^{2\varepsilon}(\frac{\theta-\eta}{2})}\left(\int_{\alpha}^{6a}\frac{d\sigma}{\sigma^{2p(1-\varepsilon)}}\right)^{\frac{1}{p}}\right)$
$\displaystyle\lesssim\alpha\left(1+\frac{1}{\sin^{2\varepsilon}(\frac{\theta-\eta}{2})}\alpha^{\frac{1}{p}-2(1-\varepsilon)}\right)$
$\displaystyle\lesssim\alpha^{\beta}\left(1+\frac{1}{\sin^{2\varepsilon}(\frac{\theta-\eta}{2})}\right),$

where, $\beta>0$ is introduced in (3.37). Thus, it follows that

$\int_{0}^{2\pi}\left\|\mathcal{D}_{2}(\cdot,\theta,\eta)\right\|_{L^{p}(\mathbb{R}^{+})}|f(\eta)|d\eta\lesssim\left\|R_{1}-R_{2}\right\|_{L^{\infty}(\mathbb{T})}^{\beta}\left\|f\right\|_{L^{\infty}(\mathbb{T})},$

uniformly in $\theta\in\mathbb{T}$, whence, combined with (3.36), we deduce
that

$\left\|\partial_{\rho}A_{1}-\partial_{\rho}A_{2}\right\|_{L^{p}\big{(}(\frac{a}{4},4a)\times\mathbb{T}\big{)}}\lesssim\left\|R_{1}-R_{2}\right\|^{\beta}_{L^{\infty}(\mathbb{T})}\left\|f\right\|_{L^{\infty}(\mathbb{T})}.$

At last, putting this bound and (3.35) together concludes the proof of the
lemma. ∎

We conclude this section with the following simple lemma that establishes a
continuity property of composed Hölder-functions. This will serve to simply
the conclusion of the proof of Proposition 3.2.

######  Lemma 3.8.

Let $m_{1}$ and $m_{2}$ be two Lipschitz-continuous functions on $\mathbb{T}$.
Further consider a function $\Phi\in
C^{\kappa}_{\mathrm{loc}}(\mathbb{R}^{2})$, for some $\kappa\in(0,1)$. Then,
it holds that

$\left\|\Phi(m_{1})-\Phi(m_{2})\right\|_{C^{\alpha}(\mathbb{T})}\leq
C_{\Phi}\left\|m_{1}-m_{2}\right\|_{C^{1}(\mathbb{T})}^{\kappa-\alpha},$

for any $\alpha\in[0,\kappa]$ and for some constant $C_{\Phi}>0$ depending on
the $C^{\kappa}$-norm of $\Phi$.

###### Proof.

We only focus our attention on the estimate of the difference in
$\dot{C}^{\kappa}$ as its control in $L^{\infty}$ easily follows from the fact
that the supremum of continuous functions is attained on compact sets. Now, we
introduce the parameter

$\delta\overset{\rm def}{=}\left\|m_{1}-m_{2}\right\|_{C^{1+\alpha}}$

and we write that

$\displaystyle\sup_{|\theta_{1}-\theta_{2}|\leq 1}$
$\displaystyle\frac{\left|\Big{(}\Phi\big{(}m_{1}(\theta_{1})\big{)}-\Phi\big{(}m_{2}(\theta_{1})\big{)}\Big{)}-\Big{(}\Phi\big{(}m_{1}(\theta_{2})\big{)}-\Phi\big{(}m_{2}(\theta_{2})\big{)}\Big{)}\right|}{|\theta_{1}-\theta_{2}|^{\alpha}}$
$\displaystyle\quad\leq\sup_{|\theta_{1}-\theta_{2}|\leq\delta}\left(\frac{\left|\Phi\big{(}m_{1}(\theta_{1})\big{)}-\Phi\big{(}m_{1}(\theta_{2})\big{)}\right|}{|\theta_{1}-\theta_{2}|^{\alpha}}+\frac{\left|\Phi\big{(}m_{2}(\theta_{1})\big{)}-\Phi\big{(}m_{2}(\theta_{2})\big{)}\right|}{|\theta_{1}-\theta_{2}|^{\alpha}}\right)$
$\displaystyle\qquad\qquad+\sup_{\delta\leq|\theta_{1}-\theta_{2}|\leq
1}\left(\frac{\left|\Phi\big{(}m_{1}(\theta_{1})\big{)}-\Phi\big{(}m_{2}(\theta_{1})\big{)}\right|}{|\theta_{1}-\theta_{2}|^{\alpha}}+\frac{\left|\Phi\big{(}m_{1}(\theta_{2})\big{)}-\Phi\big{(}m_{2}(\theta_{2})\big{)}\right|}{|\theta_{1}-\theta_{2}|^{\alpha}}\right).$

Therefore, employing the assumption that $\Phi$ belongs to
$C^{\kappa}(B_{\bar{m}})$, with

$\bar{m}\overset{\rm
def}{=}\max\\{\left\|m_{1}\right\|_{L^{\infty}(\mathbb{T})},\left\|m_{2}\right\|_{L^{\infty}(\mathbb{T})}\\},$

it follows that

$\displaystyle\sup_{|\theta_{1}-\theta_{2}|\leq 1}$
$\displaystyle\frac{\left|\Phi\big{(}m_{1}(\theta_{1})\big{)}-\Phi\big{(}m_{2}(\theta_{1})\big{)}-\Phi\big{(}m_{1}(\theta_{2})\big{)}-\Phi\big{(}m_{2}(\theta_{2})\big{)}\right|}{|\theta_{1}-\theta_{2}|^{\alpha}}$
$\displaystyle\quad\leq\left\|\Phi\right\|_{C^{\kappa}(B_{\bar{m}})}\sup_{|\theta_{1}-\theta_{2}|\leq\delta}\left(\frac{\left|m_{1}(\theta_{1})-m_{1}(\theta_{2})\right|^{\kappa}}{|\theta_{1}-\theta_{2}|^{\alpha}}+\frac{\left|m_{2}(\theta_{1})-m_{2}(\theta_{2})\right|^{\kappa}}{|\theta_{1}-\theta_{2}|^{\alpha}}\right)$
$\displaystyle\qquad\qquad+\left\|\Phi\right\|_{C^{\kappa}(B_{\bar{m}})}\sup_{\delta\leq|\theta_{1}-\theta_{2}|\leq
1}\left(\frac{\left\|m_{1}-m_{2}\right\|_{L^{\infty}(\mathbb{T})}^{\kappa}}{|\theta_{1}-\theta_{2}|^{\alpha}}\right).$

Hence, by employing the Lipschitz-regularity assumption on the functions
$m_{1}$ and $m_{2}$, we arrive at the bound

$\displaystyle\sup_{|\theta_{1}-\theta_{2}|\leq 1}$
$\displaystyle\frac{\left|\Big{(}\Phi\big{(}r_{1}(\theta_{1})\big{)}-\Phi\big{(}r_{2}(\theta_{1})\big{)}\Big{)}-\Big{(}\Phi\big{(}r_{1}(\theta_{2})\big{)}-\Phi\big{(}r_{2}(\theta_{2})\big{)}\Big{)}\right|}{|\theta_{1}-\theta_{2}|^{\alpha}}\leq
2\left\|\Phi\right\|_{C^{\kappa}(B_{\bar{m}})}\delta^{\kappa-\alpha},$

thereby completing the proof of the lemma. ∎

####  3.3.2. Regularity of the functional $\mathcal{F}$

In Proposition 3.2 below, we rigorously show that the functional $\mathcal{F}$
is differentiable in a well-chosen functional setting. This will allow us to
compute the linearized operator and study its spectral properties, later on.

######  Proposition 3.2.

Let $\alpha\in(0,1)$ and $m\in\mathbb{N}^{*}.$ There exists $\varepsilon>0$
such that the functional

$\mathcal{F}:\mathbb{R}\times\mathcal{B}^{\alpha}_{m,\varepsilon}\rightarrow
Y_{m}^{\alpha}$

is well-defined and is of class $C^{1}.$ Moreover, the partial derivative
$\partial_{\Omega}d_{r}\mathcal{F}$ exists in the sense that

$\partial_{\Omega}d_{r}\mathcal{F}:\mathbb{R}\times\mathcal{B}^{\alpha}_{m,\varepsilon}\rightarrow\mathcal{L}(X_{m}^{\alpha},Y_{m}^{\alpha})$

and is continuous.

###### Proof.

The symmetry properties of the functional $\mathcal{F}$ are already
established in Lemma 3.2, above. Hence, we only focus here on justifying the
regularity and continuity of $\mathcal{F}$ as well as its derivatives.
Recalling first the contour representation (3.10), it is then readily seen
that the claims of Proposition 3.2 above are reduced to showing that the
mapping

$r\mapsto\frac{1}{b(R\big{(}\theta)\big{)}}\partial_{\theta}\left(\psi_{b}\big{(}R(\theta)e^{{\rm
i}\theta}\big{)}\right):\mathcal{B}^{\alpha}_{m,\varepsilon}\rightarrow
C^{\alpha}(\mathbb{T})$

is well-defined and is of class $C^{1}$, for some $\varepsilon>0$ and all
$(m,\alpha)\in\mathbb{N}^{*}\times(0,1)$. Moreover, note, due to the
assumptions on $b$ and $r$, that we can further restrict the justification of
the preceding claims to the functional

$r\mapsto\partial_{\theta}\left(\psi_{b}\big{(}R(\theta)e^{{\rm
i}\theta}\big{)}\right):\mathcal{B}^{\alpha}_{m,\varepsilon}\rightarrow
C^{\alpha}(\mathbb{T}).$

To that end, we first employ the decomposition from Lemma 2.1 to write that

$\partial_{\theta}\left(\psi_{b}\big{(}R(\theta)e^{{\rm
i}\theta}\big{)}\right)=-\frac{1}{2\pi}\underbrace{\partial_{\theta}\left(\int_{D_{0}}\log|R(\theta)e^{{\rm
i}\theta}-y|b^{2}(|y|)dy\right)}_{\overset{\rm
def}{=}\mathcal{H}(r)(\theta)}+\underbrace{\partial_{\theta}\left(\phi_{R}\big{(}R(\theta)e^{{\rm
i}\theta}\big{)}\right)}_{\overset{\rm def}{=}\mathcal{K}(r)(\theta)},$

where the remainder $\phi_{R}\in W^{3,p}_{\mathrm{loc}}(\mathbb{R}^{2})$ is
determined by (2.3) and can be recast in terms of the kernel $K_{b}$ as

(3.38) $\phi_{R}(x)=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}K_{b}(x,z)\frac{\nabla
b}{b^{2}}(z)\cdot\nabla_{z}\int_{D_{0}}\log|z-y|b(y)dydz,$

for all $x\in\mathbb{R}^{2}.$ Next, we will take care of each term separately.

#### Regularity of $\mathcal{H}$

By a straightforward computation, we find that

$\displaystyle\mathcal{H}(r)(\theta)$
$\displaystyle=-\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\cdot\int_{D_{0}}\nabla_{y}\Big{(}\log|R(\theta)e^{{\rm
i}\theta}-y|\Big{)}b^{2}(|y|)dy$
$\displaystyle=\int_{0}^{2\pi}\log|R(\theta)e^{{\rm i}\theta}-R(\eta)e^{{\rm
i}\eta}|b^{2}\big{(}R(\eta)\big{)}\partial_{\eta}\left({\rm i}R(\eta)e^{{\rm
i}\eta}\right)\cdot\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)d\eta$
$\displaystyle\quad+2\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\cdot\int_{0}^{2\pi}\int_{0}^{R(\eta)}\log|R(\theta)e^{{\rm
i}\theta}-\rho e^{{\rm i}\eta}|(bb^{\prime})(\rho)e^{{\rm i}\eta}\rho d\rho
d\eta,$

where we have utilized Gauss–Green theorem in the second identity, above.
Therefore, by further employing the identity

(3.39) $\partial_{\eta}\left({\rm i}f(\eta)e^{{\rm
i}\eta}\right)\cdot\partial_{\theta}\left(g(\theta)e^{{\rm
i}\theta}\right)=\partial_{\theta\eta}^{2}\big{(}f(\theta)g(\eta)\sin(\theta-\eta)\big{)},$

we arrive at the expansion

$\mathcal{H}(r)(\theta)=\mathcal{H}_{1}(r)(\theta)+\mathcal{H}_{2}(r)(\theta),$

where

$\displaystyle\mathcal{H}_{1}(r)(\theta)$ $\displaystyle\overset{\rm
def}{=}\int_{0}^{2\pi}\log|R(\theta)e^{{\rm i}\theta}-R(\eta)e^{{\rm
i}\eta}|b^{2}\big{(}R(\eta)\big{)}L(r)(\theta,\eta)d\eta,$ $\displaystyle
L(r)(\theta,\eta)$ $\displaystyle\overset{\rm
def}{=}\partial_{\theta\eta}^{2}\big{(}R(\theta)R(\eta)\sin(\theta-\eta)\big{)}$

and

$\mathcal{H}_{2}(r)(\theta)\overset{\rm
def}{=}2\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\cdot\int_{0}^{2\pi}\int_{0}^{R(\eta)}\log|R(\theta)e^{{\rm
i}\theta}-\rho e^{{\rm i}\eta}|(bb^{\prime})(\rho)e^{{\rm i}\eta}\rho d\rho
d\eta.$

We now deal with each term in the preceding decompositions separately, and we
will only prove their $C^{1}$ regularities, for the fact that the functionals
above are well defined can be easily justified by making use of the same
arguments below. First, we compute that

$\displaystyle d_{r}\big{(}\mathcal{H}_{1}(r)\big{)}[h](\theta)$
$\displaystyle=\int_{0}^{2\pi}\nabla_{x}\log|R(\theta)e^{{\rm
i}\theta}-R(\eta)e^{{\rm i}\eta}|\cdot\left(\frac{h(\theta)e^{{\rm
i}\theta}}{R(\theta)}-\frac{h(\eta)e^{{\rm
i}\eta}}{R(\eta)}\right)L(r)(\theta,\eta)b^{2}\big{(}R(\eta)\big{)}d\eta$
$\displaystyle\quad+\int_{0}^{2\pi}\log|R(\theta)e^{{\rm
i}\theta}-R(\eta)e^{{\rm
i}\eta}|\partial_{\theta\eta}^{2}\left(\frac{h(\eta)}{R(\eta)}R(\theta)\sin(\theta-\eta)\right)b^{2}\big{(}R(\eta)\big{)}d\eta$
$\displaystyle\quad+\partial_{\theta}\left(\frac{h(\theta)}{R(\theta)}e^{{\rm
i}\theta}\right)\cdot\int_{0}^{2\pi}\log|R(\theta)e^{{\rm
i}\theta}-R(\eta)e^{{\rm i}\eta}|\partial_{\eta}\left({\rm i}R(\eta)e^{{\rm
i}\eta}\right)b^{2}\big{(}R(\eta)\big{)}d\eta$
$\displaystyle\quad+2\int_{0}^{2\pi}\log|R(\theta)e^{{\rm
i}\theta}-R(\eta)e^{{\rm
i}\eta}|L(r)(\theta,\eta)(bb^{\prime})\big{(}R(\eta)\big{)}\frac{h(\eta)}{R(\eta)}d\eta,$

where we have used the identities

$d_{r}\big{(}L(r)\big{)}[h](\theta,\eta)=\partial_{\theta\eta}^{2}\left(\frac{h(\eta)}{R(\eta)}R(\theta)\sin(\theta-\eta)\right)+\partial_{\theta\eta}^{2}\left(\frac{h(\theta)}{R(\theta)}R(\eta)\sin(\theta-\eta)\right)$

and, in view of (3.39), that

$\displaystyle\partial_{\theta\eta}^{2}\left(\frac{h(\theta)}{R(\theta)}R(\eta)\sin(\theta-\eta)\right)$
$\displaystyle=\partial_{\eta}\left({\rm i}R(\eta)e^{{\rm
i}\eta}\right)\cdot\partial_{\theta}\left(\frac{h(\theta)}{R(\theta)}e^{{\rm
i}\theta}\right).$

Now, it is more convenient to utilize (3.39), again, to write that

$L(r)(\theta,\eta)=\partial_{\eta}\left({\rm i}R(\eta)e^{{\rm
i}\eta}\right)\cdot\partial_{\theta}\left(R(\theta)e^{{\rm i}\theta}\right)$

and then that

$d_{r}\big{(}\mathcal{H}_{1}(r)\big{)}[h](\theta)=g_{1,1}(\theta)\cdot\int_{0}^{2\pi}T_{1}(\theta,\eta)f_{1,1}(\eta)d\eta+\sum_{i=2}^{4}g_{1,i}(\theta)\cdot\int_{0}^{2\pi}T_{0}(\theta,\eta)f_{1,i}(\eta)d\eta,$

where, we set

$\displaystyle T_{1}(\theta,\eta)$ $\displaystyle\overset{\rm
def}{=}\nabla_{x}\log|R(\theta)e^{{\rm i}\theta}-R(\eta)e^{{\rm
i}\eta}|\cdot\left(\frac{h(\theta)e^{{\rm
i}\theta}}{R(\theta)}-\frac{h(\eta)e^{{\rm i}\eta}}{R(\eta)}\right),$
$\displaystyle T_{0}(\theta,\eta)$ $\displaystyle\overset{\rm
def}{=}\log|R(\theta)e^{{\rm i}\theta}-R(\eta)e^{{\rm i}\eta}|$

and

$\begin{array}[]{ll}f_{1,1}(\eta)\overset{\rm def}{=}\partial_{\eta}\left({\rm
i}R(\eta)e^{{\rm
i}\eta}\right)b^{2}\big{(}R(\eta)\big{)},&g_{1,1}(\theta)\overset{\rm
def}{=}\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right),\vspace{0.2cm}\\\ f_{1,2}(\eta)\overset{\rm
def}{=}\partial_{\eta}\left({\rm i}\frac{h(\eta)}{R(\eta)}e^{{\rm
i}\eta}\right)b^{2}\big{(}R(\eta)\big{)},&g_{1,2}(\theta)\overset{\rm
def}{=}\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right),\vspace{0.2cm}\\\ f_{1,3}(\eta)\overset{\rm
def}{=}\partial_{\eta}\left({\rm i}R(\eta)e^{{\rm
i}\eta}\right)b^{2}\big{(}R(\eta)\big{)},&g_{1,3}(\theta)\overset{\rm
def}{=}\partial_{\theta}\left(\frac{h(\theta)}{R(\theta)}e^{{\rm
i}\theta}\right),\vspace{0.2cm}\\\ f_{1,4}(\eta)\overset{\rm
def}{=}2\frac{h(\eta)}{R(\eta)}\partial_{\eta}\left({\rm i}R(\eta)e^{{\rm
i}\eta}\right)(bb^{\prime})\big{(}R(\eta)\big{)},&g_{1,4}(\theta)\overset{\rm
def}{=}\partial_{\theta}\left(R(\theta)e^{{\rm i}\theta}\right).\end{array}$

Likewise, we compute that

$\displaystyle d_{r}\big{(}\mathcal{H}_{2}(r)\big{)}[h](\theta)$
$\displaystyle=\partial_{\theta}\left(\frac{h(\theta)}{R(\theta)}e^{{\rm
i}\theta}\right)\cdot\int_{D_{0}}\log|R(\theta)e^{{\rm
i}\theta}-y|\nabla_{y}\left(b^{2}(|y|)\right)dy$
$\displaystyle\quad+\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\cdot\int_{D_{0}}\left(\frac{h(\theta)e^{{\rm
i}\theta}}{R(\theta)}\cdot\nabla_{x}\Big{[}\log|R(\theta)e^{{\rm
i}\theta}-y|\Big{]}\right)\nabla_{y}\left(b^{2}(|y|)\right)dy$
$\displaystyle\quad+2\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\cdot\int_{0}^{2\pi}\log|R(\theta)e^{{\rm
i}\theta}-R(\eta)e^{{\rm
i}\eta}|(bb^{\prime})\big{(}R(\eta)\big{)}h(\eta)e^{{\rm i}\eta}d\eta.$

Therefore, using that
$\nabla_{x}\big{(}\log|x-y|\big{)}=-\nabla_{y}\big{(}\log|x-y|\big{)},$ we get

$\displaystyle\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\cdot\int_{D_{0}}$ $\displaystyle\left(\frac{h(\theta)e^{{\rm
i}\theta}}{R(\theta)}\cdot\nabla_{x}\Big{(}\log|R(\theta)e^{{\rm
i}\theta}-y|\Big{)}\right)\nabla_{y}\left(b^{2}(|y|)\right)dy$
$\displaystyle=-\left(\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\otimes\frac{h(\theta)e^{{\rm
i}\theta}}{R(\theta)}\right):\int_{D_{0}}\nabla_{y}\left(b^{2}(|y|)\right)\otimes\nabla_{y}\Big{(}\log|R(\theta)e^{{\rm
i}\theta}-y|\Big{)}dy.$

By further employing Green–Gauss theorem to write that

$\displaystyle\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\cdot\int_{D_{0}}\left(\frac{h(\theta)e^{{\rm
i}\theta}}{R(\theta)}\cdot\nabla_{x}\Big{(}\log|R(\theta)e^{{\rm
i}\theta}-y|\Big{)}\right)\nabla_{y}\left(b^{2}(|y|)\right)dy$
$\displaystyle\quad=2\left(\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\otimes\frac{h(\theta)e^{{\rm
i}\theta}}{R(\theta)}\right):\int_{0}^{2\pi}\log|R(\theta)e^{{\rm
i}\theta}-R(\eta)e^{{\rm i}\eta}|(bb^{\prime})\big{(}R(\eta)\big{)}e^{{\rm
i}\eta}\otimes\partial_{\eta}\left({\rm i}R(\eta)e^{{\rm i}\eta}\right)d\eta$
$\displaystyle\quad+\left(\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\otimes\frac{h(\theta)e^{{\rm
i}\theta}}{R(\theta)}\right):\int_{D_{0}}\log|R(\theta)e^{{\rm
i}\theta}-y|\nabla_{y}^{2}\left(b^{2}(|y|)\right)dy,$

it then follows that

$d_{r}\big{(}\mathcal{H}_{2}(r)\big{)}[h](\theta)=\sum_{i=1}^{2}g_{2,i}(\theta)\cdot\int_{0}^{2\pi}T_{0}(\theta,\eta)f_{2,i}(\eta)d\eta+\Upsilon_{1}(\theta)+\Upsilon_{2}(\theta),$

where, we set

$\begin{array}[]{ll}f_{2,1}(\eta)\overset{\rm
def}{=}2(bb^{\prime})\big{(}R(\eta)\big{)}h(\eta)e^{{\rm
i}\eta},&g_{2,1}(\theta)\overset{\rm
def}{=}\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right),\vspace{0.2cm}\\\ f_{2,2}(\eta)\overset{\rm
def}{=}2bb^{\prime}\big{(}R(\eta)\big{)}e^{{\rm
i}\eta}\otimes\partial_{\eta}\left({\rm i}R(\eta)e^{{\rm
i}\eta}\right),&g_{2,2}(\theta)\overset{\rm
def}{=}\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\otimes\frac{h(\theta)e^{{\rm
i}\theta}}{R(\theta)}\end{array}$

and

$\displaystyle\Upsilon_{1}(\theta)$ $\displaystyle\overset{\rm
def}{=}\partial_{\theta}\left(\frac{h(\theta)}{R(\theta)}e^{{\rm
i}\theta}\right)\cdot\int_{D_{0}}\log|R(\theta)e^{{\rm
i}\theta}-y|\nabla_{y}\left(b^{2}(|y|)\right)dy,$
$\displaystyle\Upsilon_{2}(\theta)$ $\displaystyle\overset{\rm
def}{=}\left(\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\otimes\frac{h(\theta)e^{{\rm
i}\theta}}{R(\theta)}\right):\int_{D_{0}}\log|R(\theta)e^{{\rm
i}\theta}-y|\nabla_{y}^{2}\left(b^{2}(|y|)\right)dy.$

Now, assuming for a moment that

$\Upsilon_{1},\Upsilon_{2}\in C^{\alpha}(\mathbb{T})$

and noticing that $g_{j,k}$ and $f_{j,k}$ belong to $C^{\alpha}(\mathbb{T})$
and $L^{\infty}(\mathbb{T})$, respectively, for all possible values of the
integer parameters $(j,k)$ above, we conclude, by virtue of Corollaries 3.1,
3.2 and 3.3, that

$d_{r}\mathcal{H}\in\mathcal{L}(X_{m}^{\alpha},Y_{m}^{\alpha}).$

Let us now show that $\Upsilon_{2}$ belongs to $C^{\alpha}(\mathbb{T})$. To
that end, we first notice that

$\sup_{\theta\in\mathbb{T}}\left|\int_{D_{0}}\log|R(\theta)e^{{\rm
i}\theta}-y|\nabla_{y}^{2}\left(b^{2}(|y|)\right)dy\right|\lesssim\sup_{x\in
B_{\sigma}}\int_{D_{0}}\left|\log|x-y|\right|dy\left\|\nabla^{2}\left(b^{2}\right)\right\|_{L^{\infty}(D_{0})},$

for any ball $B_{\sigma}$ whose radius satisfies

$\rho\geq\left\|R\right\|_{L^{\infty}(\mathbb{T})}.$

Moreover, writing that

$\int_{D_{0}}\log|R(\theta)e^{{\rm
i}\theta}-y|\nabla_{y}^{2}\left(b^{2}(|y|)\right)dy=w\left(R(\theta)e^{{\rm
i}\theta}\right),$

where it is seen that $w$ solves Poisson’s equation

$\Delta w=\nabla^{2}(b^{2})\mathds{1}_{D_{0}},$

it then follows that

$\left\|w\left(R(\cdot)e^{{\rm
i}\cdot}\right)\right\|_{\dot{C}^{\alpha}(\mathbb{T})}\lesssim\left\|R\right\|_{C^{1}(\mathbb{T})}\left\|w\right\|_{\dot{C}^{\alpha}(\mathbb{R}^{2})}\lesssim\left\|R\right\|_{C^{1}(\mathbb{T})}\left\|w\right\|_{\dot{W}^{1,p_{\alpha}}(\mathbb{R}^{2})},$

where we set

(3.40) $p_{\alpha}\overset{\rm def}{=}\frac{2}{1-\alpha}\in(2,\infty).$

Thus, by virtue of Proposition 2.1 and (Hb1), we deduce that

$\left\|w\left(R(\cdot)e^{{\rm
i}\cdot}\right)\right\|_{\dot{C}^{\alpha}(\mathbb{T})}\lesssim\left\|R\right\|_{C^{1}(\mathbb{T})}\left\|\nabla^{2}(b^{2})\right\|_{L^{p_{\alpha}}(D_{0})}<\infty.$

All in all, combining these bounds with the assumption that $r\in
C^{1+\alpha}(\mathbb{T})$ yields $\Upsilon_{2}\in C^{\alpha}(\mathbb{T})$. In
addition to that, we emphasize that the same preceding arguments apply to
estimate $\Upsilon_{1}$, entailing the same conclusion that $\Upsilon_{1}\in
C^{\alpha}(\mathbb{T})$, too.

Now, we are left with justifying the continuity of functional

$r\mapsto d_{r}\mathcal{H}(r)$

from $\mathcal{B}_{m,\varepsilon}^{\alpha}$ into
$\mathcal{L}(X_{m}^{\alpha},Y_{m}^{\alpha})$, i.e., we claim that

$\sup_{\left\|h\right\|_{C^{1+\alpha}}<1}\left\|d_{r}\mathcal{H}(r_{1})[h]-d_{r}\mathcal{H}(r_{2})[h]\right\|_{C^{\alpha}}\to
0,$

as

$\left\|r_{1}-r_{2}\right\|_{C^{1+\alpha}}\to 0.$

For the sake of clarity, we keep the same notations for all the functionals
introduced above and they will be further indexed, hereafter, by $r_{1}$ or
$r_{2}$ to precise the dependence on the functions $r_{1}$ or $r_{2}$,
respectively.

Below, we will only focus on the analysis of non-classical terms
$\Upsilon_{1}$ and $\Upsilon_{2}$, given by the expansion of
$d_{r}\mathcal{H}_{2}$, above, for the treatment of the remaining terms
follows by a direct application of Corollaries 3.1, 3.2 and 3.3. Note that we
can further restrict our proof below to $\Upsilon_{2}$, for the same analysis
applies to $\Upsilon_{1}$, as well. To that end, we first write that

$\displaystyle\Upsilon_{2,r_{1}}(\theta)-\Upsilon_{2,r_{2}}(\theta)$
$\displaystyle=\big{(}\Pi_{r_{1}}(\theta)-\Pi_{r_{2}}(\theta)\big{)}:J_{r_{1}}(\theta)+\Pi_{r_{2}}(\theta):\big{(}J_{r_{1}}(\theta)-J_{r_{2}}(\theta)\big{)},$

where, we denote, for any $\theta\in\mathbb{T}$, that

$\Pi_{r}(\theta)\overset{\rm
def}{=}\left(\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\otimes\frac{h(\theta)e^{{\rm i}\theta}}{R(\theta)}\right)$

and

$J_{r}(\theta)\overset{\rm def}{=}\int_{D_{0,r}}\log|R(\theta)e^{{\rm
i}\theta}-y|\nabla_{y}^{2}\left(b^{2}(|y|)\right)dy.$

Therefore, it is readily seen that $J_{r_{1}}\in C^{\alpha}(\mathbb{T})$ and
that

$\left\|\Pi_{r_{1}}-\Pi_{r_{2}}\right\|_{C^{\alpha}}\lesssim\left\|h\right\|_{C^{\alpha}}\left\|r_{1}-r_{2}\right\|_{C^{1+\alpha}}.$

Now, we write, for any $\theta\in\mathbb{T}$, that

$\displaystyle\big{(}J_{r_{1}}-J_{r_{2}}\big{)}(\theta)$
$\displaystyle=\int_{D_{0,r_{1}}}\left(\log|R_{1}(\theta)e^{{\rm
i}\theta}-y|-\log|R_{2}(\theta)e^{{\rm
i}\theta}-y|\right)\nabla_{y}^{2}\left(b^{2}(|y|)\right)dy$
$\displaystyle\quad+\left(\int_{D_{0,r_{1}}}-\int_{D_{0,r_{2}}}\right)\log|R_{2}(\theta)e^{{\rm
i}\theta}-y|\nabla_{y}^{2}\left(b^{2}(|y|)\right)dy$
$\displaystyle\overset{\rm def}{=}\Delta_{1}(\theta)+\Delta_{2}(\theta).$

Thus, by denoting

$x_{s}(\theta)\overset{\rm def}{=}sR_{1}(\theta)+(1-s)R_{2}(\theta),$

and employing Taylor’s expansion, we obtain, for any $\theta\in\mathbb{T}$,
that

$\displaystyle\Delta_{1}(\theta)$
$\displaystyle=\big{(}R_{1}(\theta)-R_{2}(\theta)\big{)}e^{{\rm
i}\theta}\cdot\int_{0}^{1}\int_{D_{0,r_{1}}}\nabla_{x}\log|x_{s}(\theta)e^{{\rm
i}\theta}-y|\nabla_{y}^{2}\left(b^{2}(|y|)\right)dyds$
$\displaystyle=-2\pi\big{(}R_{1}(\theta)-R_{2}(\theta)\big{)}e^{{\rm
i}\theta}\cdot\int_{0}^{1}\Big{(}\nabla\Delta^{-1}\big{(}\mathds{1}_{D_{r_{1}}}\nabla^{2}(b^{2})\big{)}\Big{)}\big{(}x_{s}(\theta)e^{{\rm
i}\theta}\big{)}ds.$

In the sequel of the proof, we will be using the bound

$\sup_{\theta\in\mathbb{T}}R_{k}(\theta)\leq 2a,\qquad k\in\\{1,2\\},$

which has been shown in (3.24). Thus, it follows, by the composition law

$\displaystyle\left\|\Big{(}\nabla\Delta^{-1}\big{(}\mathds{1}_{D_{r_{1}}}\nabla^{2}(b^{2})\big{)}\Big{)}\big{(}x_{s}(\cdot)e^{{\rm
i}\cdot}\big{)}\right\|_{C^{\alpha}(\mathbb{T})}$
$\displaystyle\qquad\qquad\qquad\lesssim\left(1+\left\|R_{1}\right\|_{C^{1}(\mathbb{T})}+\left\|R_{2}\right\|_{C^{1}(\mathbb{T})}\right)\left\|\nabla\Delta^{-1}(\mathds{1}_{D_{r_{1}}}\nabla^{2}(b^{2}))\right\|_{C^{\alpha}(B_{2a})},$

that

$\displaystyle\left\|\Delta_{1}\right\|_{C^{\alpha}(\mathbb{T})}$
$\displaystyle\lesssim\left\|r_{1}-r_{2}\right\|_{C^{\alpha}(\mathbb{T})}\left(1+\left\|R_{1}\right\|_{C^{1}(\mathbb{T})}+\left\|R_{2}\right\|_{C^{1}(\mathbb{T})}\right)\left\|\nabla\Delta^{-1}(\mathds{1}_{D_{r_{1}}}\nabla^{2}(b^{2}))\right\|_{C^{\alpha}(B_{2a})}.$

Now, as the function

$v\overset{\rm def}{=}\Delta^{-1}(\mathds{1}_{D_{r_{1}}}\nabla^{2}(b^{2}))$

is a solution of Poisson’s equation

$\Delta v=\mathds{1}_{D_{r_{1}}}\nabla^{2}(b^{2}),$

the bounds

$\left\|\nabla
v\right\|_{L^{\infty}(\mathbb{R}^{2})}\lesssim\left\|\nabla^{2}(b^{2})\right\|_{L^{p_{\alpha}}(D_{r_{1}})}$

and

$\left\|\nabla
v\right\|_{\dot{C}^{\alpha}(\mathbb{R}^{2})}\lesssim\left\|v\right\|_{\dot{W}^{2,p_{\alpha}}(\mathbb{R}^{2})}\lesssim\left\|\nabla^{2}(b^{2})\right\|_{L^{p_{\alpha}}(D_{r_{1}})}$

follow by a direct application of the Lemma B.1 and Proposition 2.1, thanks to
the assumptions (Hb1) and (Hb3) on the depth function $b$, where $p_{\alpha}$
is defined in (3.40). Hence, we deduce that

$\displaystyle\left\|\Delta_{1}\right\|_{C^{\alpha}(\mathbb{T})}$
$\displaystyle\lesssim\left\|r_{1}-r_{2}\right\|_{C^{\alpha}(\mathbb{T})}\left\|\nabla^{2}(b^{2})\right\|_{L^{p_{\alpha}}(D_{r_{1}})},$

which takes care of the estimate on $\Delta_{1}$. As for $\Delta_{2}$, we
first perform a change of variables in polar coordinates to rewrite it, for
any $\theta\in\mathbb{T}$, as

$\displaystyle\Delta_{2}(\theta)$
$\displaystyle=\int_{0}^{2\pi}\int_{0}^{1}\log|R_{2}(\theta)e^{{\rm
i}\theta}-sR_{1}(\eta)e^{{\rm i}\eta}|u_{1}(s,\eta)dsd\eta$
$\displaystyle\quad-\int_{0}^{2\pi}\int_{0}^{1}\log|R_{2}(\theta)e^{{\rm
i}\theta}-sR_{2}(\eta)e^{{\rm i}\eta}|u_{2}(s,\eta)dsd\eta,$

where we denote, for $k\in\\{1,2\\}$, that

$u_{r_{k}}(s,\eta)\overset{\rm
def}{=}sR^{2}_{k}(\eta)\left(\nabla_{y}^{2}\Big{(}b^{2}\big{(}|y|\big{)}\Big{)}\right)\Big{|}_{y=sR_{k}(\eta)}.$

Therefore, it follows that

$\displaystyle\Delta_{2}(\theta)$
$\displaystyle=\int_{0}^{2\pi}\int_{0}^{1}\left(\log|R_{2}(\theta)e^{{\rm
i}\theta}-sR_{1}(\eta)e^{{\rm i}\eta}|-\log|R_{2}(\theta)e^{{\rm
i}\theta}-sR_{2}(\eta)e^{{\rm i}\eta}|\right)u_{r_{2}}(s,\eta)dsd\eta$
$\displaystyle\quad+\int_{0}^{2\pi}\int_{0}^{1}\log|R_{1}(\theta)e^{{\rm
i}\theta}-sR_{1}(\eta)e^{{\rm
i}\eta}|\left(u_{r_{1}}-u_{r_{2}}\right)(s,\eta)dsd\eta.$

Hence, by virtue of Corollaries 3.1 and 3.2, we infer, for all
$\beta\in(0,\alpha)$, that

$\left\|\Delta_{2}\right\|_{C^{\alpha}}\lesssim\left\|r_{1}-r_{2}\right\|_{C^{1}}^{\beta}\left\|u_{r_{2}}\right\|_{L^{\infty}([0,1]\times\mathbb{R}^{2})}+\left\|u_{r_{1}}-u_{r_{2}}\right\|_{L^{\infty}([0,1]\times\bar{B}_{2a})}.$

Thus, as the function $u_{r_{1}}-u_{r_{2}}$ is continuous, whence its maximum
on the closed set $[0,1]\times\bar{B}_{2a}$ is attained, we deduce that

$\left\|\Delta_{2}\right\|_{C^{\alpha}}\lesssim\left\|r_{1}-r_{2}\right\|_{C^{1}}^{\beta}\left\|u_{r_{2}}\right\|_{L^{\infty}(\mathbb{R}^{2})}+|(u_{r_{1}}-u_{r_{2}})(s_{0},\eta_{0})|,$

for some $s_{0},\eta_{0}\in[0,1]\times\bar{B}_{2a}$. Hence, due to the
preceding point-wise bound and assumption (Hb1), it is readily seen that

$\lim_{\left\|r_{1}-r_{2}\right\|_{C^{1+\alpha}}\to
0}\left\|\Delta_{2}\right\|_{C^{\alpha}}=0.$

All in all, combining all the preceding bounds, we finally deduce that

$\lim_{\left\|r_{1}-r_{2}\right\|_{C^{1+\alpha}}\to
0}\left(\sup_{\left\|h\right\|_{C^{1+\alpha}}<1}\left\|\Upsilon_{2,r_{1}}-\Upsilon_{2,r_{2}}\right\|_{C^{\alpha}}\right)=0,$

which concludes the proof of the desired regularity properties for
$\mathcal{H}$.

#### Regularity of $\mathcal{K}$

The desired regularity of that term stems from the high-regularity of the
perturbation $\phi$, which is the crux of the new decomposition in Lemma 2.1.
This is a generic property and can be observed in other settings. We proceed
first by writing, for any $\theta\in\mathbb{T}$, that

$\mathcal{K}(r)(\theta)=\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\cdot\nabla\phi_{R}\left(R(\theta)e^{{\rm i}\theta}\right).$

Therefore, by virtue of the fact that $\phi_{R}\in
W^{3,p}_{\mathrm{loc}}(\mathbb{R}^{2})$, for all $p\in(1,\infty)$, it is then
readily seen that

$\theta\mapsto\mathcal{K}(r)(\theta)\in C^{\alpha}(\mathbb{T}),$

as soon as $r\in C^{1+\alpha}(\mathbb{T})$. Moreover, we compute, for any
$h\in C^{1+\alpha}(\mathbb{T})$, that

$\displaystyle d_{r}\big{(}\mathcal{K}(r)\big{)}[h](\theta)$
$\displaystyle=\partial_{\theta}\left(\frac{h(\theta)}{R(\theta)}e^{{\rm
i}\theta}\right)\cdot\nabla\phi_{R}\left(R(\theta)e^{{\rm i}\theta}\right)$
$\displaystyle\quad+\left(\partial_{\theta}\left(R(\theta)e^{{\rm
i}\theta}\right)\otimes\frac{h(\theta)}{R(\theta)}e^{{\rm
i}\theta}\right):\nabla^{2}\phi_{R}\left(R(\theta)e^{{\rm i}\theta}\right)$
$\displaystyle\quad+\partial_{\theta}\left(\frac{h(\theta)}{R(\theta)}e^{{\rm
i}\theta}\right)\cdot\nabla\mathcal{E}_{R}\left(R(\theta)e^{{\rm
i}\theta}\right),$

where the last term above comes from differentiating the expression (3.38)
with respect to $R$ (inside the integral) and is given by

(3.41) $\mathcal{E}_{R}(x)\overset{\rm
def}{=}\int_{\mathbb{T}}\int_{0}^{\infty}K_{b}(x,\rho e^{{\rm
i}\theta})\frac{b^{\prime}}{b^{2}}(\rho)\partial_{\rho}\left(\int_{0}^{2\pi}\log|\rho
e^{{\rm i}\theta}-R(\eta)e^{{\rm
i}\eta}|b\big{(}R(\eta)\big{)}d\eta\right)\rho d\rho d\theta,$

for all $x\in\mathbb{R}^{2}$. In particular, by virtue of Lemma 3.6 and the
assumption (Hb3) on the function $b$, one sees that

$(r,\theta)\mapsto\frac{b^{\prime}}{b^{2}}(\rho)\partial_{\rho}\left(\int_{0}^{2\pi}\log|\rho
e^{{\rm i}\theta}-R(\eta)e^{{\rm i}\eta}|b\big{(}R(\eta)\big{)}d\eta\right)\in
L^{p}(\mathbb{R}^{+}\times\mathbb{T}),$

for all $p\in[1,\infty)$. Thus, by means of Proposition 2.1 (see also Remark
2.5), we deduce that

$\nabla\mathcal{E}_{R}\in
W^{1,p_{\alpha}}_{\mathrm{loc}}(\mathbb{R}^{2})\hookrightarrow
C_{\mathrm{loc}}^{\alpha}(\mathbb{R}^{2})$

with $p_{\alpha}$ being defined in (3.40).

Hence, by a similar argument, relying on the
$W^{2,p_{\alpha}}_{\mathrm{loc}}$-regularity of $\nabla\phi_{R}$, which is
again a consequence of Remark 2.4, together with the symmetry properties
established in Lemma 3.2, we deduce that

$d_{r}\mathcal{K}\in\mathcal{L}(X_{m}^{\alpha},Y_{m}^{\alpha}).$

We are now left with proof of continuity of $d_{r}\mathcal{K}$ with respect to
the $r$-variable. To that end, we first write by means of a direct computation
that

$\displaystyle\left\|d_{r}\mathcal{K}(r_{1})[h]-d_{r}\mathcal{K}(r_{2})[h]\right\|_{C^{\alpha}}$
$\displaystyle\leq\left\|R_{1}\right\|_{C^{1}}\left\|(\nabla\phi_{R_{1}},\nabla\mathcal{E}_{R_{1}})\right\|_{C^{\alpha}(B_{2a})}\left\|\partial_{\theta}\left(\frac{h}{R_{1}}e^{{\rm
i}\cdot}\right)-\partial_{\theta}\left(\frac{h}{R_{2}}e^{{\rm
i}\cdot}\right)\right\|_{C^{\alpha}}$
$\displaystyle+\left\|\nabla\phi_{R_{1}}\big{(}R_{1}e^{{\rm
i}\cdot}\big{)}-\nabla\phi_{R_{2}}\big{(}R_{2}e^{{\rm
i}\cdot}\big{)}\right\|_{C^{\alpha}}\left\|\partial_{\theta}\left(\frac{h}{R_{2}}e^{{\rm
i}\cdot}\right)\right\|_{C^{\alpha}}$
$\displaystyle+\left\|\partial_{\theta}\left(R_{1}e^{{\rm
i}\cdot}\right)\otimes\frac{h}{R_{1}}-\partial_{\theta}\left(R_{2}e^{{\rm
i}\cdot}\right)\otimes\frac{h}{R_{2}}\right\|_{C^{\alpha}}\left\|R_{1}\right\|_{C^{1}}\left\|\nabla^{2}\phi_{R_{1}}\right\|_{C^{\alpha}(B_{2a})}$
$\displaystyle+\left\|\nabla\mathcal{E}_{R_{1}}\big{(}R_{1}e^{{\rm
i}\cdot}\big{)}-\nabla\mathcal{E}_{R_{2}}\big{(}R_{2}e^{{\rm
i}\cdot}\big{)}\right\|_{C^{\alpha}}\left\|\partial_{\theta}\left(\frac{h}{R_{2}}e^{{\rm
i}\cdot}\right)\right\|_{C^{\alpha}}$
$\displaystyle+\left\|\partial_{\theta}\left(R_{2}e^{{\rm
i}\cdot}\right)\otimes\frac{h}{R_{2}}\right\|_{C^{\alpha}}\left\|\nabla^{2}\phi_{R_{1}}\left(R_{1}e^{{\rm
i}\cdot}\right)-\nabla^{2}\phi_{R_{2}}\left(R_{2}e^{{\rm
i}\cdot}\right)\right\|_{C^{\alpha}}.$

The most challenging terms in this collection of bounds are

(3.42) $\displaystyle\left\|\nabla\mathcal{E}_{R_{1}}\big{(}R_{1}e^{{\rm
i}\cdot}\big{)}-\nabla\mathcal{E}_{R_{2}}\big{(}R_{2}e^{{\rm
i}\cdot}\big{)}\right\|_{C^{\alpha}}$
$\displaystyle\lesssim\left\|R_{1}\right\|_{C^{1}}\left\|\nabla\mathcal{E}_{R_{1}}-\nabla\mathcal{E}_{R_{2}}\right\|_{C^{\alpha}(B_{2a})}$
$\displaystyle\quad+\left\|\nabla\mathcal{E}_{R_{2}}\big{(}R_{1}e^{{\rm
i}\cdot}\big{)}-\nabla\mathcal{E}_{R_{2}}\big{(}R_{2}e^{{\rm
i}\cdot}\big{)}\right\|_{C^{\alpha}}$

and

(3.43) $\displaystyle\left\|\nabla^{2}\phi_{R_{1}}\left(R_{1}e^{{\rm
i}\cdot}\right)-\nabla^{2}\phi_{R_{2}}\left(R_{2}e^{{\rm
i}\cdot}\right)\right\|_{C^{\alpha}}$
$\displaystyle\lesssim\left\|R_{1}\right\|_{C^{1}}\left\|\nabla^{2}\phi_{R_{1}}-\nabla^{2}\phi_{R_{2}}\right\|_{C^{\alpha}(B_{2a})}$
$\displaystyle\quad+\left\|\nabla^{2}\phi_{R_{2}}\left(R_{1}e^{{\rm
i}\cdot}\right)-\nabla^{2}\phi_{R_{2}}\left(R_{2}e^{{\rm
i}\cdot}\right)\right\|_{C^{\alpha}},$

whereas the remaining ones can be estimated in a similar, or even easier, way.
Notably, we already have prepared the ground to estimate them due to the
analysis laid out in the previous lemmas. In particular, noticing that

$\nabla\mathcal{E}_{R_{2}},\nabla^{2}\phi_{R_{2}}\in
W^{1,p_{\kappa}}_{\mathrm{loc}}(\mathbb{R}^{2})\hookrightarrow
C^{\kappa}_{\mathrm{loc}}(\mathbb{R}^{2}),$

for $\kappa\in(0,1)$ being defined as

$\kappa\overset{\rm
def}{=}1-\frac{2}{q_{\kappa}}>1-\frac{2}{p_{\alpha}}=\alpha,$

it then follows by a direct application of Lemma 3.8 that

$\left\|\nabla^{2}\phi_{R_{2}}\left(R_{1}e^{{\rm
i}\cdot}\right)-\nabla^{2}\phi_{R_{2}}\left(R_{2}e^{{\rm
i}\cdot}\right)\right\|_{C^{\alpha}}\lesssim\left\|R_{1}-R_{2}\right\|_{C^{1}}^{\kappa-\alpha}$

and

$\left\|\nabla\mathcal{E}_{R_{2}}\left(R_{1}e^{{\rm
i}\cdot}\right)-\nabla\mathcal{E}_{R_{2}}\left(R_{2}e^{{\rm
i}\cdot}\right)\right\|_{C^{\alpha}}\lesssim\left\|R_{1}-R_{2}\right\|_{C^{1}}^{\kappa-\alpha}.$

This takes care of the last terms in the right-hand sides of (3.42) and
(3.43). On the other hand, the first terms in the right-hand sides of these
inequalities will be controlled by using a different argument. More precisely,
owing to the definition (3.41), employing Sobolev embedding and the elliptic-
gain of regularity (see Proposition 2.1 and Remark 2.5), we infer that

$\displaystyle\left\|\nabla\mathcal{E}_{R_{1}}-\nabla\mathcal{E}_{R_{1}}\right\|_{C^{\alpha}(B_{2a})}$
$\displaystyle\lesssim\left\|\mathcal{E}_{R_{1}}-\mathcal{E}_{R_{1}}\right\|_{\dot{W}^{2,p_{\alpha}}(\mathbb{R}^{2})}$
$\displaystyle\lesssim\left\|\frac{b^{\prime}}{b^{2}}(\rho)\partial_{\rho}(\mathcal{A}_{1}-\mathcal{A}_{2})(\rho,\theta)\right\|_{L^{p}(\rho
d\rho d\theta;(0,R_{\infty})\times\mathbb{T})},$

where $R_{\infty}$ is the constant introduced in assumption (Hb3) and with

$\mathcal{A}_{i}(\rho,\theta)\overset{\rm def}{=}\int_{\mathbb{T}}\log|\rho
e^{{\rm i}\theta}-R_{i}(\eta)e^{{\rm i}\eta}|b\big{(}R(\eta)\big{)}d\eta.$

Thus, by a direct application of Lemma 3.7, it is readily seen that

$\left\|\nabla\mathcal{E}_{R_{1}}-\nabla\mathcal{E}_{R_{1}}\right\|_{C^{\alpha}(B_{2a})}\lesssim\left\|R_{1}-R_{2}\right\|_{L^{\infty}}^{\beta},$

for any $\beta\in[0,\frac{1}{p_{\alpha}})$.

At last, it only remains to estimate the first term in (3.43). To that end, we
notice that the representation (3.38) entails that

$\phi_{R_{1}}-\phi_{R_{2}}=\phi_{\widetilde{R}}(x),$

where $\phi_{\widetilde{R}}$ is given, for any $x\in\mathbb{R}^{2}$, by

$\phi_{\widetilde{R}}(x)\overset{\rm
def}{=}\frac{1}{2\pi}\int_{\mathbb{R}^{2}}K_{b}(x,z)\frac{\nabla