text:
\section{Introduction and main results}

The Dirichlet-Neumann operator plays an important role in fluid mechanics, for example in the Craig-Sulem-Zhakarov \cite{CS,Zak1}
formulation of the water waves equations (cfr. Section \ref{sec:stokes}),
and in several other branches of analysis, as in
the theory of inverse problems. Roughly speaking it  is defined as the linear
operator  which maps
the Dirichlet datum of a harmonic function in a
given domain into the normal derivative at its boundary
 (Neumann datum).
The Dirichlet-Neumann operator is nonlinear with respect to the boundary of the domain.
In view of many applications it is important
to determine
its regularity in different function spaces.

Several results about the analyticity of the Dirichlet-Neumann operator
acting between Sobolev spaces,  with respect to the variation of the boundary,
have been proved, starting with the pioneering works of Coifmann-Meyer \cite{CM}, Craig-Schanz-Sulem
\cite{CSS}, Craig-Nicholls \cite{CN} and Lannes \cite{LannesLivre} where we
refer for an extended bibliography.
We also mention the deep analysis of the Dirichlet-Neumann operator in \cite{AIM,AD,ABZ2,Wu1}, on which we will comment later.

\smallskip
The major
aim of this paper is to prove a further analyticity result for the Dirichlet-Neumann operator
$ G(\eta ) $ defined in \eqref{DN1}
on the unbounded domain $ \mathbb{T}^d \times \{ y \leq \eta (x) \} $, where
$ \mathbb{T}^d := (\mathbb{R} / 2 \pi \mathbb{Z})^d $ is the standard  $ d$-dimensional flat
torus, in any space dimension $ d \geq 1  $.
Assuming that $\eta (x) $ is analytic, we prove in
Theorem \ref{thm:DN}  the analyticity of the
map  $ \eta \mapsto G(\eta) $ acting between suitable spaces of analytic
periodic functions. The delicate point of this result
is that $ \eta $ and $ \psi $ are assumed to
have the {\it same} regularity (if
$ \eta $ is more regular than $ \psi $ the result is simpler).
Following  Lannes \cite{Lannes,LannesLivre} and Alazard-Burq-Zuily \cite{ABZ}
we make use of a regularizing diffeomorphism
to flatten the domain to the half cylinder,  in which the transformed
harmonic function solves  a perturbed elliptic equation. Then
the proof relies on a perturbative approach to invert the transformed Laplacian
over suitable spaces of functions $ u(x,y) $ which are analytic in $ x $, with Sobolev
regularity in $ y $ and
decay to zero as $ y \to -\infty $, cfr. \eqref{spaziok}.
The key step is obtain  linear elliptic regularity estimates for the
Poisson equation in these spaces, see Lemma \ref{lem:ell5}.
Then the elliptic estimates for the modified problem are obtained
by a perturbative argument
 differently from  \cite{ABZ}.

\smallskip

As a consequence of  Theorem \ref{thm:DN},
we derive an analytic bifurcation result of analytic Stokes waves --i.e. space periodic traveling solutions, which look stationary in a  moving frame with constant speed-- of  the pure gravity  water waves equations in infinite depth, see Theorem \ref{LeviCivita}.
Existence of traveling waves which are constant in one space dimension, i.e.
are $ 1$-dimensional waves,
dates back to  classical works of
Levi-Civita \cite{LC}, Nekrasov \cite{Nek} and Struik \cite{Struik},
 in the twenties of the last century.  Then Lewy \cite{Lewy} proved
that a traveling wave  which is at least $ C^1 $
is actually analytic. Theorem \ref{LeviCivita}
proves in addition that small amplitude  Stokes waves depend
analytically on the amplitude taking values in a space of analytic functions.
In finite depth and with  surface tension, a result of this kind
is proved in Nicholls-Reitich \cite{NR} by a power series expansion approach.

In this paper we deduce Theorem \ref{LeviCivita}  by the analytic
Crandall-Rabinowitz
bifurcation theorem from a simple eigenvalue, as presented in
the book of Ambrosetti-Prodi \cite{AP}, thanks to the analytic estimates
of the Dirichlet-Neumann operator obtained in Theorem
\ref{thm:DN}.

\smallskip

In addition to their interests per se --traveling waves have
 fundamental importance in fluid mechanics--,
 these results have  been used in the study of the
Benjiamin-Feir instability of the  Stokes waves in \cite{BMV}.

\smallskip

We now state precisely our results. Along this paper we use the following notation.
We denote the spatial variables by
$ (x,y)  \in \mathbb{T}^d \times \mathbb{R} $, $ d \geq 1 $,  where $ \mathbb{T}^d := (\mathbb{R}/ 2 \pi \mathbb{Z})^d $
is  the standard flat torus.
 The symbol $\nabla$ denotes the gradient
$$
\nabla := (\partial_{x_j})_{j=1, \ldots, d} \qquad \text{and}
\qquad \Delta := \sum_{j=1}^d \partial_{x_j}^2 \, ,
\quad \Delta_{x,y} := \Delta + \partial_y^2 \, .
$$
 A dot will denote the standard scalar product in $\mathbb{R}^d$.
Moreover
$ \mathbb{N} := \{1, 2, \ldots \} $ and $ \mathbb{N}_0 = \{0\} \cup \mathbb{N}  $.

\subsection{Dirichlet-Neumann operator}

We consider the cylindrical domain
\begin{equation}\label{Deta}
{\cal D}_\eta := \big\{  (x,y) \in \mathbb{T}^d \times \mathbb{R} \ : \ y < \eta (x) \big\} \, , \quad d \geq 1  \, ,
\end{equation}
delimited by the graph $ \partial {\cal D}_\eta =  \{ y = \eta (x) \} $ of a periodic function $ \eta (x) $, and, given
a periodic Dirichlet datum
$ \psi (x) $,  we consider the unique harmonic function $\Phi(x,y)$
solving the system
\begin{equation}\label{Phi}
\begin{cases}
\Delta_{x,y} \Phi = 0  & \mbox{ in } {	\cal D}_\eta \\
\Phi(x, y ) = \psi(x) & \mbox{ at } y = \eta(x) \\
\partial_y \Phi(x,y) \to 0 & \mbox{ as } y \to - \infty \, .
\end{cases}
\end{equation}
The Dirichlet-Neumann operator $G(\eta) $ is then defined as the linear operator
\begin{equation}
\label{DN1}
\begin{aligned}
[G(\eta) \psi](x) & := \sqrt{1+|\nabla \eta|^2} \,  \partial_n \Phi_{|y = \eta (x) } \\
& = (\partial_y \Phi) (x, \eta(x)) - \nabla \eta(x) \cdot (\nabla \Phi)(x, \eta(x))
\end{aligned}
\end{equation}
where $ n $ denotes the exterior normal
$$
n :=  \frac{1}{\sqrt{1 + |\nabla \eta |^2}} \vet{- \nabla \eta}{1}  \, ,
\quad
\partial_n := \frac{1}{\sqrt{1 + |\nabla \eta |^2}}(\partial_y - \nabla \eta \cdot \nabla ) \, .
$$
The reason of the name
``Dirichlet-Neumann"  is that the operator $G(\eta) $  maps the Dirichlet datum $ \psi (x) $ of
the harmonic function  $ \Phi (x,y) $  into the (normalized) normal derivative
$ \partial_n \Phi $
 at the
boundary $ \partial {\cal D}_\eta = \{ y = \eta (x) \} $ (Neumann datum).

\begin{rmk}
In \eqref{Phi}  it is equivalent to require the boundary condition
$ \nabla \Phi (x,y) \to 0 $
as $ y \to - \infty $, see Remark \ref{remdecay}. Actually
$ \nabla \Phi (x,y)  $ decays to zero exponentially fast as $ y \to - \infty $.
\end{rmk}

Simple algebraic properties of the Dirichlet-Neumann operator are recalled in Appendix \ref{app:DN}.

Since Calderon it is known that
the Dirichlet-Neumann operator $ G(\eta) $ is, if $ \eta $ is a $ C^\infty $ function,
a classical pseudo-differential operator, elliptic of order $ 1 $, with an asymptotic expansion
in classical decreasing symbols.
For the flat surface $ \eta (x) = 0 $, the Dirichlet-Neumann operator is the Fourier multiplier
$$
G(0) = |D| = (-\Delta)^{\frac12}
$$
as follows by the elementary calculus \eqref{ell3}.
In space dimension $ d = 1 $
the Dirichlet-Neumann operator is equal to $|D|$ up to infinitely many
times regularizing operators, see e.g.  \cite{BM,BBHM}.
If $ \eta (x) $ has a finite smoothness, Lannes \cite{Lannes,LannesLivre}
proved  an analogous expansion in symbols  with finite smothness.

The Dirichlet-Neumann operator is a nonlinear map with respect to the boundary
of the domain
$ \partial {\cal D}_\eta $.
The analytic dependence with respect to $ \eta $
of the Dirichlet-Neumann operator $ \eta \mapsto G(\eta )$
has been first established in the two dimensional setting by Coifman-Meyer
\cite{CM}, and in the three dimensional setting by
Craig, Schanz and Sulem \cite{CSS}, showing
that, if  $ \eta \in C^{k+1} $, $ \psi \in H^{k+1}$, $ k \in \mathbb{N} $, then
$ G(\eta)[\psi] \in H^k $ is analytic in  $C^{k+1} \cap \{ \| \eta \|_{C^1}< r \} $ for $r$  sufficiently small.

In view of of application to water waves
Craig-Nicholls \cite{CN}, Wu
\cite{Wu1,Wu2},  and Lannes \cite{Lannes,LannesLivre} proved,
 with different approaches, that if
 $ \eta, \psi $ have the same Sobolev regularity $H^s$
then $  G(\eta) [\psi ] \in H^{s-1} $.
In particular Lannes proved tame estimates using {\it regularizing} diffeomorphisms
to  straighten the domain.

The paralinearization of $ G(\eta) \psi $, which enables to prove optimal
estimates for the action of the Dirichlet-Neumann operator, has been obtained in
Alazard-Metivier \cite{AIM},
Alazard-Delort \cite{AD},
and Alazard-Burq-Zuily \cite{ABZ1,ABZ2}
in rough domains, using a variational analysis to construct the solution
and applying elliptic regularity theory.
The paralinearization of  the Dirichlet-Neumann operator in $ d = 1 $
with a multilinear expansion in $ \eta $ is proved in Berti-Delort \cite{BD}, by using a
paradifferential parametrix \`a la Boutet de Monvel.

Finally we mention the work of Alazard-Burq-Zuily \cite{ABZ}
for the study of the Dirichlet-Neumann operator acting in  analytic function spaces,
making use of a regularizing diffeomorphism as in \cite{Lannes}, variational methods and
 elliptic regularity analysis.

\smallskip

In this paper we prove an analyticity result  (Theorem \ref{thm:DN})
for the Dirichlet-Neumann map $ \eta \mapsto  G(\eta) \psi $ in
the cylindrical domain $ {\cal D}_\eta $ defined in \eqref{Deta},
acting between spaces of periodic analytic functions defined in \eqref{def:Hsigmas} below.
We suppose that the functions  $\eta$ and $\psi$
belong to the spaces of   periodic
functions
\begin{equation} \label{def:Hsigmas}
H^{\sigma, s} := H^{\sigma, s}(\mathbb{T}^d):=
\Big\{ u (x) = \sum_{k \in \mathbb{Z}^d}  u_k e^{\mathrm{i}\, k \cdot x} \, : \ \
\| u \|_{H^{\sigma, s}}^2 := \sum_{k \in \mathbb{Z}^d}  e^{2\sigma |k|_1}  \langle k \rangle^{2s}
\, |  u_k|^2 < \infty  \Big\}
\end{equation}
where,  for any $ k = (k_1, \ldots, k_d) \in \mathbb{Z}^d $, we set
$$
|k|_1 := |k_1|+\dots+|k_d| \, , \quad
\langle k \rangle := \max(1, |k|) \, , \quad
| k| := \Big( \sum_{j=1}^d k_j^2 \Big)^{1/2} \, .
$$
Clearly, if the dimension $ d = 1 $ then $ |k| = |k |_1 $.

If $ \sigma = 0 $ the space $ H^{0, s}  $ is the usual Sobolev space $ H^s $.
If  $ \sigma > 0 $, a periodic function $ u (x) $ belongs to
$H^{\sigma, s}(\mathbb{T}^d)$, if and only if
it admits an analytic extension in the strip $ |y|_{\infty}
:= \max\{ |y_1|, \ldots, |y_d| \} < \sigma $ and
the traces at the boundaries $ u( \cdot  + \mathrm{i}\, y ) $,  $ |y|_\infty = \sigma $,
belong to   the Sobolev space $H^s := H^s(\mathbb{T}^d)$.
In  Appendix
\ref{App1A} we prove this characterization, together with the property that
 the spaces $ H^{\sigma,s} $
form, for $ s > d/ 2 $, an algebra with respect to the product of functions
and satisfy tame estimates.
\smallskip

The main result of this section is the following theorem.

Let $ B^{\sigma, s}(r) $ denote the open ball in
$ H^{\sigma,s} $ of center $ 0 $ and radius $ r > 0  $.
\begin{teo}\label{thm:DN}
{\bf (Dirichlet-Neumann operator)}
Let $\sigma\geq 0$ and $s $, $ s_0$
such that $s+\frac12,\, s_0 \in \mathbb{N} $,
and  $s-\frac32\geq s_0 > \frac{d+1}{2} $. Then there exists $\epsilon_0 :=\epsilon_0(s)>0$ such that
the Dirichlet-Neumann operator map\footnote{$H^{\sigma, s}\cap B^{\sigma, s_0}(\epsilon_0) $ is an open set in the $H^{\sigma,s}$ topology. }
$$
\eta \mapsto G(\eta) \, , \quad
H^{\sigma, s}\cap B^{\sigma, s_0+\frac32}(\epsilon_0)  \to \mathcal{L}(H^{\sigma, s}, H^{\sigma, s-1}) \, ,
$$
is analytic and fulfills the tame estimate
\begin{equation}\label{tameGeta}
\| G(\eta)\psi \|_{H^{\sigma,s-1}} \leq C(s)\big( \|\psi\|_{H^{\sigma,s}} +\|\eta \|_{H^{\sigma,s}}\|\psi\|_{H^{\sigma,s_0+\frac32}}  \big) \, .
\end{equation}
\end{teo}
We remark that, in Theorem \ref{thm:DN}, the functions
$ \eta, \psi $ have the same analytic regularity.
The proof of such result, given in Section \ref{sec:DN},
relies on a regularizing flattening method (following \cite{Lannes, ABZ}) together with a perturbative argument in
suitable functional spaces.

\subsection{Stokes waves}\label{sec:stokes}

As an application of Theorem \ref{thm:DN} we prove that
$ 1$-dimensional Stokes waves solutions of the pure gravity
water waves equations in deep water
are analytic functions belonging to the spaces $ H^{\sigma,s} $,
and moreover depend  analytically with respect to
 the amplitude  parameter.
  Clearly  $ 1 $-dimensional traveling waves are also $ 2$d-traveling waves
which are constant in one space direction, so it extends to higher dimensional Stokes waves.
We first present the
water waves equations.
\\[1mm]{\bf The pure gravity water waves equations.}
We consider the Euler equations for a bi-dimensional incompressible,
inviscid, irrotational fluid under the action of  gravity,
filling the
region ${ \cal D}_\eta $ defined in  \eqref{Deta} with $ d = 1 $,
\begin{equation}\label{water}
\begin{cases}
\partial_t \Phi + \frac12
\big( ( \partial_x \Phi )^2 + (\partial_y \Phi)^2 \big) + g \eta = 0  \ \
\qquad     \text{ at} \ y = \eta (x)  \cr
\partial_t \eta = \partial_y \Phi - (\partial_x \eta) \,
(\partial_x \Phi) \qquad \qquad \qquad \quad \ \,  \text{at} \  y = \eta (x)  \cr
\Delta_{x,y} \Phi =0 \qquad  \qquad \qquad \qquad \      \qquad  \qquad  \ \quad \text{in} \ {\cal D}_{\eta}  \cr
\partial_y \Phi \to 0  \qquad  \qquad \ \ \,  \qquad \qquad \qquad \qquad \quad \   \text{ as} \
y \to  - \infty  \, ,
\end{cases}
\end{equation}
where $ g >  0 $ is the acceleration of gravity.
The irrotational velocity field is the gradient
of the harmonic scalar potential $\Phi=\Phi(t,x,y) $,
determined by its trace $ \psi(t,x)=\Phi(t,x,\eta(t,x)) $ at the free surface
$ y = \eta (t, x ) $.
Actually $\Phi (t, \cdot ) $ is
the unique solution of the elliptic equation \eqref{Phi}.
The time evolution of the fluid is determined by the first
two boundary conditions in \eqref{water}  at the free surface.
The first states  that the pressure of the fluid
is equal, at the free surface, to the constant atmospheric pressure  (dynamic boundary condition) and the second one
that the fluid particles  remain, along the evolution, on the free surface   (kinematic
boundary condition).

As shown by Zakharov \cite{Zak1} and Craig-Sulem \cite{CS},
the evolutionary system \eqref{water} amounts to
the following equations for the unknowns $ (\eta (t,x), \psi (t,x)) $,
\begin{equation}\label{WWeq}
 \eta_t  = G(\eta)\psi \, , \quad
  \psi_t  =
- g \eta - \dfrac{\psi_x^2}{2} + \dfrac{1}{2(1+\eta_x^2)} \big( G(\eta) \psi + \eta_x \psi_x \big)^2 \, ,
\end{equation}
where  $G(\eta)$ is
the Dirichlet-Neumann operator in \eqref{DN1}.
In addition the equations \eqref{WWeq} are the Hamiltonian system
\begin{equation}\label{HSWW}
\partial_t \eta = \nabla_\psi \mathcal{H} \, , \quad \partial_t \psi = - \nabla_\eta \mathcal{H} \, ,
\end{equation}
 where $ \nabla_{\eta}, \nabla_{\psi} $
 denote the $ L^2$-gradients of the Hamiltonian
$$
\mathcal{H}(\eta,\psi) :=
 \frac12 \int_{\mathbb{T}} \left( \psi \,G(\eta)\psi +g \eta^2 \right) \mathrm{d} x \, ,
$$
which  is the sum of the kinetic  energy (cfr. \eqref{DNSP})
 and potential gravitational energy  of the fluid.
 Actually, as proved in \cite{CS,Zak1}, the $ L^2$-gradient with respect to
 $ \eta  $ of the kinetic energy
\begin{equation} \label{kine}
K(\eta, \psi) := \frac12 (\psi, G(\eta) \psi)_{L^2}  \stackrel{\eqref{DNSP}}
= \frac12 \int_{{\cal D}_\eta}
|\nabla \Phi|^2 \, \mathrm{d} x \, ,
\end{equation}
is equal to
\begin{equation}\label{Kform}
\nabla_\eta K(\eta, \psi) =  -  \frac12 \psi_x^2  +  \dfrac{1}{2(1+\eta_x^2)} \big( G(\eta) \psi + \eta_x \psi_x \big)^2 \, ,
\end{equation}
yielding the equivalence between \eqref{HSWW} and \eqref{WWeq}.

 We also remark that the water waves equations \eqref{WWeq}
are  invariant under space translations namely, by
\eqref{DN-inv},
$$
{\mathcal H} \circ \tau_\theta = {\mathcal H } \, , \quad \forall \theta \in \mathbb{R}^d \, .
$$
In addition, the water waves equations
are reversible with respect to the involution
$$
\rho\vet{\eta(x)}{\psi(x)} := \vet{\eta(-x)}{-\psi(-x)}, \quad \text{i.e. }
\mathcal{H} \circ \rho = \mathcal{H} \, ,
$$
as a consequence of \eqref{DN-rev}.
\\[1mm]
{\bf The Stokes waves.}
Noteworthy solutions of \eqref{WWeq} are the so-called  Stokes waves, namely traveling
solutions of the form
\begin{equation}\label{profiles}
\eta(t,x)=\breve \eta(x-ct) \, , \quad
\psi(t,x)=\breve \psi(x-ct) \, ,
\end{equation}
for some real  $c $ (the speed)  and  $2\pi$-periodic functions  $(\breve \eta (x), \breve \psi (x)) $ (the profiles).
In a reference frame in translational motion with constant speed $c$,
 the water waves equations \eqref{WWeq} then become,
 by using the translation invariance property
\eqref{DN-inv},
\begin{equation}\label{travelingWW}
\eta_t  = c\eta_x+G(\eta)\psi \, , \quad
 \psi_t  = c\psi_x - g \eta - \dfrac{\psi_x^2}{2} + \dfrac{1}{2(1+\eta_x^2)} \big( G(\eta) \psi + \eta_x \psi_x \big)^2  \, .
\end{equation}
The Stokes waves profiles $(\breve \eta, \breve \psi)$ in \eqref{profiles}
are then equilibrium steady solutions
of \eqref{travelingWW}, namely solve the system
\begin{equation}\label{travelingWWstokes}
c\eta_x+G(\eta)\psi = 0 \, , \quad
c\psi_x - g \eta - \dfrac{\psi_x^2}{2} +
\dfrac{1}{2(1+\eta_x^2)} \big( G(\eta) \psi + \eta_x \psi_x \big)^2
= 0 \, .
\end{equation}
The next theorem is the main bifurcation result of small amplitude Stokes waves
proved in this paper.
We denote by $B(r):= \{ x \in \mathbb{R} \colon \  |x| < r\}$ the real ball with center 0 and radius $r$.
 \begin{teo}\label{LeviCivita}
{\bf (Stokes waves)} For any  $ \sigma \geq 0 $,
$ s > 5/ 2 $ and $k\in\mathbb{N}$, there exists  $ \epsilon_0 := \epsilon_0 (\sigma,s,k) > 0 $
and a unique family  of
 solutions
 $$
 (\eta_\epsilon(x), \psi_\epsilon(x), c_\epsilon) \in H^{\sigma,s} (\mathbb{T})\times H^{\sigma,s}(\mathbb{T})\times \mathbb{R}
 $$
  of the system  \eqref{travelingWWstokes},
  parameterized by
  $|\epsilon| \leq \epsilon_0$,
  such that
  \begin{enumerate}
  \item
 the map
 $ \epsilon \mapsto (\eta_\epsilon, \psi_\epsilon, c_\epsilon)  $,
$  B(\epsilon_0) \to H^{\sigma,s} (\mathbb{T})\times H^{\sigma,s}(\mathbb{T})\times \mathbb{R} $
 is analytic;
 \item
 $\eta_\epsilon (x) $ is even,  $ \eta_\epsilon (x) $ has zero average,
 $\psi_\epsilon (x) $ is odd;
\item the solutions $ (\eta_\epsilon(x), \psi_\epsilon(x), c_\epsilon) $ have the expansion
\begin{equation}\label{HH}
 (\eta_\epsilon(x), \psi_\epsilon(x)) = \epsilon (\sqrt{k} \cos (kx), \sqrt{g} \sin (kx)) + O(\epsilon^2) \, , \quad c_\epsilon \to
 \sqrt{\frac{g}{k}}
 \ \ \text{as} \ \ \epsilon \to 0 \, .
 \end{equation}
 \end{enumerate}
\end{teo}

 Theorem \ref{LeviCivita} is proved in Section \ref{sec:Sto}.
 Let us make some comments on the result.
 \\[1mm]
1. As already mentioned in the introduction, the first rigorous bifurcation
proof of  small amplitude  Stokes waves for pure
  gravity water waves  goes back to  Levi-Civita \cite{LC} and
 Nekrasov \cite{Nek} in deep water, and Struik \cite{Struik} in finite depth.
 We refer to the monographs of Ambrosetti-Prodi \cite{AP}
 and Buffoni-Toland \cite{BuT} for a complete presentation.
Concerning regularity, it is known  since  Lewy \cite{Lewy} that a Stokes wave which is at least $ C^1 $
is actually analytic.  Theorem \ref{LeviCivita}
proves in addition that the Stokes waves $  (\eta_\epsilon(x), \psi_\epsilon(x)) $  depend
analytically on the amplitude $ \epsilon $ taking values in a space of analytic functions $ H^{\sigma,s} 	\times H^{\sigma,s}   $.
In finite depth and in presence of surface tension, an analyticity result of this kind
 is proved in Nicholls-Reitich \cite{NR}, by a power series expansion.
We also mention Plotnikov-Toland \cite{PlotTol} for  related results about analytic continuation of Stokes waves.

 Existence of traveling water waves has been also proved by
 Zeidler \cite{Zei} under the effect of capillary forces and
 Martin  \cite{Martin}, Walh\'en \cite{Wh1} also for constant vorticity flows.
 We expect that, thanks to  Theorem \ref{thm:DN},
 an analyticity result for the Stokes waves, analogous to  Theorem \ref{LeviCivita},
holds also in these cases.
\\[1mm]
2. Higher order Taylor expansions of the Stokes waves in $ \epsilon $ are known, see e.g.
\cite{Fenton}, \cite{NS}, \cite{NR}.
 We remark that Theorem \ref{LeviCivita} proves the convergence of the Taylor
 series of the Stokes waves in $ \epsilon $, taking values in spaces of analytic periodic functions.
\\[2mm]
3.  {\it Quasi-periodic traveling waves.}
More general $ 1$d  time quasi-periodic traveling Stokes waves
  have been recently obtained in Berti-Franzoi-Maspero \cite{BFM,BFM2}, with or without
  surface tension,   and Feola-Giuliani \cite{FG}, by means
  of a Nash-Moser implicit function iterative scheme.
We remark that these solutions  are not steady in any moving frame. This implies
a small divisor problem.
\\[1mm]
4. {\it Higher space dimension: existence.}
{For three dimensional fluids,   in addition to Stokes waves,
also  traveling wave solutions which are
nontrivially periodic  in both spatial directions  are known},
for example forming hexagonal patterns.
 Their existence was first  proved in  Craig-Nicholls
\cite{CN} for  gravity-capillary  water waves, by applying
variational bifurcation arguments \`a la Weinstein-Moser, exploiting the Hamiltonian
nature \eqref{HSWW} of the water waves equations.
The surface tension allows to apply, in the bifurcation analysis,
the standard implicit function theorem.
On the other hand the existence of
$ 2$d pure gravity doubly-periodic traveling wave solutions is a small divisor problem.
In this case, solutions with Sobolev regularity were constructed   by
Iooss-Plotinkov
\cite{IP-Mem-2009,IP2} by means of a Nash-Moser implicit function theorem,
requiring suitable Diophantine conditions on the speed vector.
\\[1mm]
5. {\it
 Regularity.}
In higher space dimensions a regularity result \`a la Lewy \cite{Lewy}, i.e.
a traveling wave surface which is at least $ C^1 $
is actually analytic,
has been  proved for gravity-capillary water waves
by Craig-Matei \cite{CM}.
For
pure gravity waves, a   result of this kind is false,
 because the system is no more elliptic. This feature is the counterpart of the small divisor problem arising in the existence proof of Iooss-Plotinkov \cite{IP-Mem-2009,IP2}.
Assuming Diophantine conditions on the speed vector,
Alazard-Metivier  \cite{AIM} proved that the periodic traveling waves
constructed in  \cite{IP-Mem-2009,IP2}, which have Sobolev regularity,
are  indeed $ C^\infty $.
\\[1mm]
6. We finally note that, for larger values of the amplitude
$ \epsilon $,  the regularity of the traveling wave
solutions
may break down. Indeed it is well known that
large traveling waves have cusps, as proved in the celebrated works about the Stokes
conjecture of Amick, Fraenkel, Toland \cite{AFT} and Plotinkov \cite{Plot}.

\section{Analyticity of the Dirichlet-Neumann operator}\label{sec:DN}

In this section we prove Theorem \ref{thm:DN} concerning the analyticity of the Dirichlet-Neumann operator.
The first step is to straighten  the free surface.
\\[1mm]
{\bf Regularizing diffeomorphism.}
Following \cite{Lannes,ABZ} we apply the regularizing
change of variables
\begin{equation}
\label{smooth.chanhe}
 x = x' \, , \qquad y = \rho(x', y') \, ,  \quad \rho(x', y') := y' + e^{y'|D|}\eta(x') \, ,
\end{equation}
where $e^{y|D|} $ is the Fourier multiplier
$$
\left(e^{y|D|} g\right)(x):=  \sum_{k \in \mathbb{Z}^d}  g_k \, e^{y|k|} \, e^{\mathrm{i}\, k \cdot x} \, ,
\quad \forall \  g(x) = \sum_{k \in \mathbb{Z}^d}  g_k  \, e^{\mathrm{i}\, k \cdot x} \, .
$$
Note that
$$
\rho(x', 0) = \eta(x') \, ,   \quad \lim_{y' \to - \infty} \rho(x',y')-y' = \eta_0 \, ,
$$
and, since
$$
\partial_{y'} \rho(x',y') = 1 + e^{y|'D|} |D| \eta \, ,
$$
 if
 $ \sup_{y'<0} \| e^{y'|D|} |D| \eta \|_{L^\infty (\mathbb{T}^d)} < 1 $
the change of coordinates \eqref{smooth.chanhe}  is a diffeomorphism between  the domain $ {\cal D}_\eta = \{(x,y) \, \colon \,  y \leq \eta(x)\}$ and the flat half-cylinder  $\{ (x',y')
\, \colon \,  y' \leq 0 \} = \mathbb{T}^d \times \mathbb{R}_{\leq 0}$ where $ \mathbb{R}_{\leq 0} := (-\infty, 0] $.
By the change of variables \eqref{smooth.chanhe}  the derivatives $\partial_y$ and $\nabla_x$ become respectively
$$
\Lambda_1 = \frac{1}{\partial_{y'} \rho} \partial_{y'} \, , \qquad
\Lambda_2 = \nabla_{x'} - \frac{\nabla_{x'} \rho}{\partial_{y'}\rho} \partial_{y'} \, ,
$$
and the transformed harmonic function
$$
\varphi(x',y') := \Phi(x', y' + \rho(x',y'))
$$
solves the elliptic problem
\begin{equation}
\label{ell2.0}
\begin{cases}
(\Lambda_1^2 + \Lambda_2^2)\varphi = 0 \\
\varphi(x,0) = \psi(x)    \\
\partial_y \varphi(x,y) \to 0 & \mbox{ as } y \to - \infty \, .
\end{cases}
\end{equation}
By means of chain rule, system \eqref{ell2.0}  is rewritten (cfr.  \cite{ABZ})
as the perturbed elliptic problem    (we rename the variables $x',y' $ as $ x,y$)
\begin{equation}
\label{ell2}
\begin{cases}
\Delta_{x,y} \varphi = F(\eta)[\varphi]
  \\
\varphi(x,0) = \psi(x)    \\
\partial_y \varphi(x,y) \to 0 \quad \mbox{ as } y \to - \infty \, ,
\end{cases}
\end{equation}
where
\begin{equation}\label{def:f}
F(\eta)[\varphi]:=    \left( \alpha(\eta) \partial_y^2  + \beta(\eta) \Delta + \gamma(\eta) \cdot \nabla\partial_y + \delta(\eta ) \partial_y \right) \varphi
\end{equation}
with, since $ \nabla \rho(x,y) = e^{y|D|} \nabla \eta $ and
$ \partial_{y} \rho(x,y) = 1 + e^{y|D|} |D| \eta $,
 \begin{equation}
 \label{f}
 \begin{aligned}
&\alpha(\eta) :=  1- \frac{1+|\nabla \rho|^2}{\partial_y \rho} =
\frac{e^{y|D|} |D| \eta- | e^{y|D|} \nabla \eta|^2}{1+e^{y|D|} |D| \eta } \, ,
\\
&\beta(\eta) := 1-\partial_y \rho = - e^{y|D|} |D| \eta \, , \\
&\gamma(\eta) := 2 \nabla \rho = 2 e^{y|D|}\nabla \eta \, , \\
&\delta(\eta) :=\frac{1}{\partial_y \rho } \Big( -2 \nabla\rho \cdot \nabla \partial_y \rho + \partial_y \rho \Delta \rho
+ \frac{1+|\nabla \rho|^2}{\partial_y \rho} \partial_y^2 \rho \Big)\, .
 \end{aligned}
 \end{equation}
In the new variables \eqref{smooth.chanhe},  the Dirichlet-Neumann operator defined in \eqref{DN1} becomes
\begin{equation}
\label{DN2}
\begin{aligned}
[G(\eta)\psi] (\cdot )& =

- \nabla\eta\cdot \nabla\varphi(\cdot, 0) +
\frac{1+ |\nabla\eta|^2(\cdot)}{1+(|D|\eta)(\cdot)} \,  (\partial_y \varphi)(\cdot, 0) \, .
\end{aligned}
\end{equation}

\paragraph{Function spaces.}
In order to state our main existence result for the solutions  of \eqref{ell2},
we introduce some function spaces.
Given $s \in \mathbb{N}_0 $, $ \sigma,  a  \geq 0 $, we define
\begin{equation}
\label{spaziok}
\mathcal{H}^{\sigma, s, a}:=
\Big\{
u(x,y) = \sum_{k \in \mathbb{Z}^d}   u_k (y) e^{\mathrm{i}\, k \cdot x}
\, \colon \, \mathbb{T}^d \times (-\infty, 0] \to \mathbb{C} \ \ \mbox{:} \ \    \| u \|_{\sigma, s, a} < \infty
\Big\}  \,
\end{equation}
endowed with  the norm
\begin{align}
\| u \|_{\sigma, s, a}^2 & :=
\sum_{j = 0}^s
\| \partial_y^j u \|_{L^{2,a}( \mathbb{R}_{\leq 0},H^{\sigma,s-j})}^2   \label{migliore} \\
& =
\sum_{j = 0}^s
\int_{-\infty}^0 \| \partial_y^j u(\cdot, y)\|_{H^{\sigma,s-j}}^2 \, e^{-2 a y} \mathrm{d} y  \notag \\
& =
\sum_{j = 0}^s \int_{-\infty}^0 \sum_{k \in \mathbb{Z}^d}
 \, e^{2 \sigma |k|_1 } \, \langle k \rangle^{2(s-j)} \,
| \partial_y^j  u_k  (y)|^2 e^{2 a |y|} \mathrm{d} y  \notag \\
& =
\sum_{j = 0}^s \sum_{k \in \mathbb{Z}^d}
e^{2 \sigma |k|_1 } \, \langle k \rangle^{2(s-j)} \,  \| \partial_y^j u_k \|_{L^{2,a}}^2
\label{migliore4}
\end{align}
where,  given a Hilbert space $ X $, we have used the notation
\begin{equation}\label{defuL2a}
\| u \|_{L^{2,a}( \mathbb{R}_{\leq 0},X)}^2  := \int_{-\infty}^{0} \| u(y) \|_X^2 e^{-2 a y} \,  \mathrm{d} y
= \int_{-\infty}^{0} \| u(y) \|_X^2 e^{2 a |y|} \, \mathrm{d} y  \, .
\end{equation}

\begin{rmk}
For $ \sigma = a = 0$,  the space $ H^{0,s,0} $ coincides with the
Sobolev space $ H^s ( \mathbb{T}^d \times \mathbb{R}_{\leq 0 }) $ of
$ L^2 $ functions $ u : \mathbb{T}^d \times \mathbb{R}_{\leq 0 } \to \mathbb{C} $
possessing weak derivatives $ \partial^\alpha u $ in $ L^2 $, for any multiindex $ \alpha \in \mathbb{N}^{d+1}$ with modulus $ | \alpha| \leq s $,
with equivalent norm
$ \| u \|_s^2 = \sum_{\alpha \in \mathbb{N}^{d+1}, |\alpha| \leq s} \| \partial^\alpha u \|_{L^2}^2 $.
\end{rmk}

We point out that, for any $ s \in \mathbb{N} $,
$$
\|u\|_{\sigma,s,a}^2 = \|u\|^2_{L^{2,a}(\mathbb{R}_{\leq 0}, H^{\sigma,s})} + \|\partial_y u\|_{\sigma,s-1,a}^2 \, ,
$$
and, by  \eqref{migliore} and $  \| \partial_{x_i} v \|_{H^{\sigma,s-1}}
 \leq  \| v  \|_{H^{\sigma,s}}  $,   we directly get the following simple lemma.
\begin{lem}\label{lem:prop}
Let $ s \in \mathbb{N} $, $ \sigma \geq 0 $, $ a \geq 0 $.
The linear maps
$$
\partial_{x_i} : \mathcal{H}^{\sigma, s, a} \mapsto \mathcal{H}^{\sigma, s-1, a} \, , \   \forall i =1,\dots,d,
\quad \partial_y : \mathcal{H}^{\sigma, s, a} \mapsto \mathcal{H}^{\sigma, s-1, a} \, ,
$$
are continuous.
\end{lem}

We also denote
\begin{equation}\label{coplusH}
\mathbb{C} \oplus \mathcal{H}^{\sigma, s, a}
:= \big\{ c + u(x,y) \, , \ c \in \mathbb{C} \, , \ u \in \mathcal{H}^{\sigma, s, a} \big\} \, ,\quad \Pi : \mathbb{C} \oplus \mathcal{H}^{\sigma, s, a} \to \mathcal{H}^{\sigma, s, a}, \ \Pi [c + u] = u \, ,
\end{equation}
and, with a small abuse of notation, given a function
$g  \in \mathbb{C} \oplus \mathcal{H}^{\sigma, s, a}$, we denote  its norm by
$\| g\|_{\sigma, s, a}:=  \| \Pi g \|_{\sigma, s, a} + |g - \Pi g | $.
The function spaces $\mathcal{H}^{\sigma, s, a}  $ and $\mathbb{C} \oplus \mathcal{H}^{\sigma, s, a} $ are
modeled to mimic the decay
of the harmonic function
$ \underline{\varphi} $ in
\eqref{ell3} as $ y \to - \infty $, cfr.  Lemma \ref{lem:und.varphi}.

\smallskip

We now list a series of properties
of the  spaces $\mathcal{H}^{\sigma, s, a}$ used in the sequel; we defer their proofs in
  Appendix \ref{sec:A2}.

\begin{lem}[{\bf Trace}]\label{lem:trace}
Let  $ \sigma \geq 0 $, $ s  \in \mathbb{R} $. Then one has
\begin{equation}\label{uinfty}
\|u \|_{C^0(\mathbb{R}_{\leq 0},H^{\sigma,s})}\leq \|u\|_{L^{2}(\mathbb{R}_{\leq 0},H^{\sigma,s+\frac12})} + \|\partial_y u\|_{L^{2}(\mathbb{R}_{\leq 0},H^{\sigma,s-\frac12})}.
\end{equation}
In particular,
the  trace operator
\begin{equation} \label{optraccia}
\Gamma (u):= u(\cdot, 0) := u\vert_{y= 0}
\end{equation}
is, for any  $s\in \mathbb{N}_0$, $a \geq 0$,   a linear bounded
 map between $ \mathcal{H}^{\sigma, s+1, a} \to H^{\sigma, s+\frac12}$,
satisfying
\begin{equation}
\label{est:trace}
\| \Gamma (u) \|_{H^{\sigma, s+ \frac12}} \leq   \| u \|_{\sigma, s+1, 0} \leq  \| u \|_{\sigma, s+1, a} \, .
\end{equation}
\end{lem}

If  $s   > \frac{d+1}{2}$, the space $  \mathcal{H}^{\sigma, s, a} $  is an algebra with respect to the product of functions and the following tame estimates hold.

\begin{prop}[{\bf Tame}]\label{lem:algebray}
 Let $\sigma, a \geq 0$ and $s \geq s_0  > \frac{d+1}{2}$, $s,s_0 \in \mathbb{N}$.
 Then there exist positive constants $C_s \geq 1 $ (non-decreasing in $ s $)  such that, for any
$u  \in \mathcal{H}^{\sigma, s, 0}$ and  $ v \in \mathcal{H}^{\sigma, s, a}$,
\begin{equation}
\label{algebrassa}
\| u v \|_{\sigma, s, a} \leq C_s \big(
\| u \|_{\sigma, s, 0} \, \| v \|_{\sigma, s_0, a } +
  \| u \|_{\sigma, s_0, 0} \, \| v \|_{\sigma, s, a } \big) \, .
\end{equation}
In particular one has
\begin{equation}\label{potenze}
\| u^j \|_{\sigma,s,a}\leq \big(2 C_s \|u\|_{\sigma,s_0,a}\big)^{j-1} \|u\|_{\sigma,s,a}\, , \quad \forall j \geq 1 \, .
\end{equation}
\end{prop}

The next lemma proves the continuity
of the harmonic function  $e^{y|D|}g $, which solves the Dirichlet-Neumann
elliptic problem  \eqref{ell3},
with respect to  the Dirichlet datum $ g $ at  $ y = 0 $.

\begin{lem}[{\bf Harmonic propagator}]\label{lem:und.varphi}
Let $\sigma\geq 0$ and $s + \frac12 \in \mathbb{N}$.
Then, for any $g \in H^{\sigma, s}$,  the function
$$
( e^{y|D|}g)(x) := \sum_{k\in \mathbb{Z}^d} g_k \, e^{y|k|} \, e^{\mathrm{i}\, k\cdot x}
$$
belongs to
$ \mathbb{C} \oplus \mathcal{H}^{\sigma, s+\frac12 , a} $, $a \in (0,1)$,
and the linear map
$$
H^{\sigma, s} \to
\mathcal{H}^{\sigma, s+\frac12, a} \, ,
\quad g \mapsto   \Pi [ e^{y|D|}g ] = e^{y|D|}g -  g_0 \, ,
$$
is continuous.
\end{lem}

We now come back to  Theorem \ref{thm:DN}.
The  key result of its proof is the following proposition regarding the solution of the  elliptic problem \eqref{ell2}.

The parameter $a \in (0,1)$
plays a technical
role  in studying the decay as $y \to -\infty$ of the solution of the elliptic problem
\eqref{ell2.0} (see in particular Lemma \ref{lem:T}).
{\em  In the sequel  we fix $a = \frac12$.}

\begin{prop}\label{prop:varphi}
Let $\sigma\geq 0$  and $s,\, s_0$ such that
$s+\frac12 $, $ s_0 \in \mathbb{N} $ and
$s-\frac32\geq s_0 > \frac{d+1}{2} $. Then there exist $\epsilon_0 :=\epsilon_0(s)>0$ and,
 for any $ \eta \in H^{\sigma,s}\cap B^{\sigma,s_0+\frac32}(\epsilon_0)$ and $\psi \in H^{\sigma, s}$,
 a unique solution $\varphi \in \mathbb{C} \oplus \mathcal{H}^{\sigma, s+\frac12, a}$
  of the elliptic problem \eqref{ell2}, satisfying
\begin{equation}\label{tamevarphi}
\| \Pi \varphi \|_{\sigma,s+\frac12,a} \leq C(s)\big( \|\psi\|_{H^{\sigma,s}} +\|\eta \|_{H^{\sigma,s}}\|\psi\|_{H^{\sigma,s_0+\frac32}}  \big) \, .
\end{equation}
Moreover  $\varphi = \Psi(\eta)[\psi]$, where
$\Psi$ is an analytic map  $H^{\sigma,s}\cap B^{\sigma,s_0+\frac32}(\epsilon_0) \to$ $
  \mathcal{L}(H^{\sigma,s}, \, \mathbb{C} \oplus \mathcal{H}^{\sigma, s+\frac12, a} )$, and
  $\Psi(0) \psi= e^{y|D|}\psi$.
\end{prop}
Postponing the proof of this proposition, we first use it to deduce Theorem  \ref{thm:DN}.

\smallskip

\noindent{\bf Proof of Theorem  \ref{thm:DN}.}
By Proposition \ref{prop:varphi}, for any
$ \eta \in H^{\sigma, s}\cap B^{\sigma,s_0+\frac32}(\epsilon_0)$ and $ \psi \in  H^{\sigma, s}$,
there exists a unique solution $ \varphi \in \mathbb{C} \oplus \mathcal{H}^{\sigma, s+ \frac12, a} $
of
\eqref{ell2}. The Dirichlet-Neumann operator is computed in \eqref{DN2}.
Since $ \varphi (x,0) = \psi (x) $, using the  trace operator $  \Gamma(u)  = u(\cdot, 0) $
in \eqref{optraccia}, and recalling the definition of $ \Pi $ in \eqref{coplusH},
we rewrite   \eqref{DN2} as
\begin{align}
G(\eta)\psi   &= - \nabla\eta\cdot \nabla \psi+
 \frac{1+ |\nabla\eta|^2}{1+(|D|\eta)} \,  \Gamma[ \partial_y \varphi ] \notag
 \\ \label{DN3bis}
 &=\underbrace{ - \nabla\eta\cdot \nabla \psi}_{=: G_1(\eta)\psi}
 +
 \underbrace{ \Gamma[ \partial_y \Pi \varphi ]}_{=: G_2(\eta)\psi}
 +
\underbrace{ \frac{|\nabla\eta|^2-(|D|\eta)}{1+(|D|\eta)} \,  \Gamma[ \partial_y \Pi \varphi ]}_{=:  G_3(\eta)\psi} \, .
\end{align}
We prove that each map
\begin{equation}\label{Giana}
G_i :
H^{\sigma, s}\cap B^{\sigma, s_0+\frac32}(\epsilon_0)  \to \mathcal{L}(H^{\sigma, s}, H^{\sigma, s-1}) \, ,
\ \  i =1, 2, 3 \, , \quad \text{is analytic} \, ,
\end{equation}
and fulfills the tame estimate
\eqref{tameGeta}.
Regarding $G_1(\eta)\psi$, it suffices to note that it is  linear in $\eta$ and  by \eqref{tameHs},
\begin{equation} \label{stima1}
\|\nabla\eta\cdot \nabla \psi\|_{H^{\sigma,s-1}} \lesssim_s \|\eta \|_{H^{\sigma,s_0+\frac12}} \|  \psi\|_{H^{\sigma,s}} + \|\eta \|_{H^{\sigma,s}} \|  \psi\|_{H^{\sigma,s_0+\frac12}} \, .
 \end{equation}
Next we consider $G_2(\eta)\psi =  \Gamma[\partial_y \Pi \Psi(\eta)\psi] $.
By Lemma \ref{lem:prop} and  \ref{lem:trace}, the map $\varphi \mapsto  \Gamma[\partial_y \Pi \varphi] \in \mathcal{L}(\mathbb{C} \oplus \mathcal{H}^{\sigma, s+\frac12, a}, H^{\sigma, s-1})$ which, together with the
analyticity  of $\eta \mapsto \Psi(\eta)$ stated in Proposition \ref{prop:varphi}, implies  the analyticity of $\eta \mapsto G_2(\eta)$ as in \eqref{Giana}.
Moreover by
\eqref{est:trace}, Lemma \ref{lem:prop} and \eqref{tamevarphi}, we have
\begin{equation}\label{stima2}
\| \Gamma[\partial_y  \Pi \varphi]\|_{H^{\sigma,s-1}} \leq \|\partial_y \Pi \varphi \|_{\sigma,s-\frac12,0} \lesssim_s
 \|\psi\|_{H^{\sigma,s}} +\|\eta \|_{H^{\sigma,s}}\|\psi\|_{H^{\sigma,s_0+\frac32}}  \, .
\end{equation}
Finally consider $G_3(\eta)\psi = f(\eta) G_2(\eta)\psi$, where
$f(\eta)$ is the multiplication operator by the function
\begin{equation}\label{domsvif}
 f(\eta) = \frac{|\nabla\eta|^2-(|D|\eta)}{1+(|D|\eta)}
 =  \big( |\nabla\eta|^2-(|D|\eta) \big) \sum_{j=0}^\infty (-|D|\eta)^j \, .
 \end{equation}
By Lemma \ref{lem:surfacingsubalgebra} we have that
$ \| (|D| \eta)^j \|_{H^{\sigma, s-1}}
\leq ( C(s) \| \eta \|_{H^{\sigma, s_0 + \frac32}})^j
   \|\eta \|_{H^{\sigma,s}} $ for any  $  j \in \mathbb{N} $,
and therefore  $ f(\eta)$ in \eqref{domsvif}
is bounded,  on the domain  $H^{\sigma, s}\cap B^{\sigma, s_0+\frac32}(\epsilon_0)  $, by
\begin{equation}\label{boufeta}
\| f(\eta) \|_{H^{\sigma, s-1}}
\lesssim_s
   \| \eta \|_{H^{\sigma,s}} \ .
\end{equation}
Moreover  $f(\eta)$ in \eqref{domsvif}
  is a series of analytic functions
 uniformly convergent on the sets $B^{\sigma,s}(R) \cap B^{\sigma, s_0+\frac32}(\epsilon_0) $, $\forall R >0$.
Thus, by Weierstrass theorem,
$\eta \mapsto f(\eta)$ is analytic on
$B^{\sigma,s}(R) \cap B^{\sigma, s_0+\frac32}(\epsilon_0)$, and, by the arbitrariness of $R$, on the whole open set
$H^{\sigma, s}\cap B^{\sigma, s_0+\frac32}(\epsilon_0)  \to H^{\sigma, s-1} $.
We conclude that also $G_3  (\eta) $  is analytic
as stated  in \eqref{Giana}.
Finally,  \eqref{boufeta} and \eqref{stima2} imply that $ G_3 (\eta) $ satisfies
the tame estimate \eqref{tameGeta}.
\qed

\begin{rmk}
It  follows from the proof that $G(0)\psi = G_2(0)\psi = \Gamma[\partial_y \Pi \Psi(0)\psi]$, which, together with  $\Psi(0) \psi= e^{y|D|}\psi$,  recovers the identity $G(0)\psi = |D| \psi$.
\end{rmk}
\vspace{1em}

The final paragraph is devoted to the  proof of Proposition \ref{prop:varphi}.

\paragraph{Proof of Proposition \ref{prop:varphi}: the perturbative argument.}
We look for a solution $\varphi$ of \eqref{ell2} of the form
\begin{equation}
\label{eq:varphi}
 \varphi(x, y) =  \underline{\varphi}(x, y) + u(x, y)
\end{equation}
where $ \underline{\varphi}$ is the harmonic solution of
\begin{equation}
\label{ell3}
\begin{cases}
\Delta_{x,y} \underline{\varphi} =0  \\
\underline{\varphi}(x,0) = \psi(x) &  \\
\partial_y \underline{\varphi}(x,y) \to 0 & \mbox{ as } y \to - \infty  \, ,
\end{cases}
\qquad
\mbox{ i.e. }  \quad
\underline{\varphi}(x,y) := e^{y|D|}\psi(x) \, ,
\end{equation}
whereas
$u $ solves the elliptic problem
\begin{equation}\label{ell4}
\begin{cases}
\Delta_{x,y} u = F(\eta)[\phi + u]  \, , \qquad   \\
u(x,0) =0 &  \\
\partial_y u(x,y) \to 0 \qquad \mbox{ as } y \to - \infty \, ,
\end{cases}
\end{equation}
with $\phi :=\underline{\varphi}$.
The harmonic function $ \underline{\varphi} = e^{y|D|} \psi  $
ia estimated by Lemma \ref{lem:und.varphi}.
\begin{rmk}\label{remdecay}
Also the derivative $ \partial_x  \underline{\varphi} (x,y) \to 0 $ as $ y \to - \infty $.
Actually any solution of \eqref{Phi} satisfies $ \nabla \Phi (x,y) \to 0 $ as
$ y \to - \infty $. Indeed let $ a $ such that $ \mathbb{T} \times \{ y = - a \} \subset {\cal D}_\eta $.
Since the harmonic function
$ \Phi (x, y )$ is analytic then $ \vartheta (x) := \Phi (x, -a)$   is analytic as well. Thus
(cfr. \eqref{ell3}) we can represent
$ \Phi (x,y) = \sum_{k \in \mathbb{Z}^d}   \vartheta_k \, e^{|k| (y+a)} \, e^{\mathrm{i}\, k\cdot x} $, which proves that
$$
\nabla \Phi (x,y) = \sum_{k \in \mathbb{Z}^d \setminus \{0\}}
\mathrm{i}\, k \,  \vartheta_k \, e^{|k| (y+a)} \, e^{\mathrm{i}\, k\cdot x}  \to 0 \quad \text{as}
\quad y \to - \infty \, ,
$$
actually  exponentially fast.
\end{rmk}
The solution of system \eqref{ell4} is given  by  the following lemma.
\begin{lem}\label{lem:lasol}
Let $\sigma\geq 0$ and $s,\, s_0$ such that
$s+\frac12,\, s_0 \in \mathbb{N} $ and $s-\frac32\geq s_0 > \frac{d+1}{2} $. Then there exist $\epsilon_0 :=\epsilon_0(s)>0$ and a unique analytic map
\begin{align*}
\eta \mapsto U(\eta) \, , \quad
U: H^{\sigma,s}\cap B^{\sigma, s_0+\frac32}(\epsilon_0) \longrightarrow \mathcal{L} \big( \mathbb{C} \oplus \mathcal{H}^{\sigma, s+\frac12,a}\big)\, ,
\end{align*}
such that $u = U(\eta)[\phi]=U(\eta)[\Pi \phi]$, with $\Pi $ in \eqref{coplusH},
solves  \eqref{ell4}, satisfying
\begin{equation}\label{tameu}
\|\Pi U(\eta)[\phi]\|_{\sigma,s+\frac12,a} \leq C(s )\big( \|\eta \|_{H^{\sigma,s_0+\frac32}}\|\Pi \phi\|_{\sigma,s+\frac12,a} +\|\eta \|_{H^{\sigma,s}}\|\Pi \phi\|_{\sigma,s_0+2,a}  \big) \, .
\end{equation}
\end{lem}
The proof of  Lemma \ref{lem:lasol} relies on
Lemmata \ref{lem:ell5} and \ref{lem:non} below.

Given a function $ g(x, y) $ defined in $ \mathbb{T}^d \times (- \infty, 0 )$, we  first
 consider the linear elliptic problem
\begin{equation}\label{ell5}
\begin{cases}
\Delta_{x,y} u = g  \\
u(x,0) =0 &  \\
\partial_y u(x,y) \to 0 & \mbox{ as } y \to - \infty  \, .
\end{cases}
\end{equation}
The following key lemma is proved in Appendix \ref{sec:sol.ell}.
\begin{lem}[{\bf Elliptic regularity}]\label{lem:ell5}
Fix $\sigma \geq 0$, $s \in \mathbb{N}_0 $  and $a \in (0,1)$.
For any  $g \in \mathcal{H}^{ \sigma, s,  a}$, the  elliptic problem
\eqref{ell5}
has a unique solution
$u := L (g)  \in \mathbb{C} \oplus  \mathcal{H}^{\sigma, s+2, a} $.
The linear map
$$
L  :  \mathcal{H}^{ \sigma, s,  a}\to \mathbb{C} \oplus  \mathcal{H}^{ \sigma, s+2,  a} \, , \quad
 g \mapsto L(g) \, ,
$$
is continuous, i.e. there exists $ C_{a} > 0 $ such that
$ \| Lg \|_{ \sigma, s+2, a} \leq C_{a} \| g\|_{\sigma, s,a} $.
\end{lem}

Thanks to Lemma \ref{lem:ell5}, we recast the nonlinear elliptic problem \eqref{ell4} into the equation
\begin{equation}
\label{ell10}
\big(\mathrm{Id} - L \circ F(\eta)\big) [u] = L\circ F(\eta)[\phi] \, .
\end{equation}
Note that the linear operator $\mathrm{Id} - L \circ F(\eta) $ depends non-linearly on $\eta$
and that, recalling \eqref{def:f},
$ F(\eta)[\phi] =
F(\eta) [\Pi \phi ] $
depends only on the component $ \Pi \phi \in  \mathcal{H}^{\sigma,s,a}$ of $ \phi $
defined in \eqref{coplusH},
for the presence of the derivatives $ \partial_{y},  \partial_{yy}, \nabla\partial_{y} $.
In the next lemma we study the regularity of the nonlinear map $\eta \mapsto F(\eta)$.
\begin{lem}
\label{lem:non}
Let $\sigma \geq 0$,  $s + \frac12, s_0 \in \mathbb{N}$  with
$ s-\frac32  \geq s_0 > \frac{d+1}{2} $.
There exists $\epsilon_0 :=\epsilon_0(s) >0$ such that the nonlinear map
\begin{alignat*}{2}
F : H^{\sigma,s}&\cap B^{\sigma, s_0+\frac32}(\epsilon_0) &&\to \mathcal{L}\big(\mathbb{C} \oplus \mathcal{H}^{\sigma, s+\frac12, a}, \mathcal{H}^{\sigma, s-\frac32,a} \big) \, , \\
&\; \eta &&\mapsto \qquad
\big\{\, \phi \mapsto F(\eta)[\phi]\, \big\}  \, ,
\end{alignat*}
defined in \eqref{def:f} is analytic and satisfies the tame estimate
\begin{equation}\label{stimalineare}
\|F(\eta)[\phi]\|_{\sigma,s-\frac32,a} \leq C(s) \big( \|\eta\|_{H^{\sigma,s_0+\frac32}} \|\Pi\phi\|_{\sigma,s+\frac12,a} + \|\eta\|_{H^{\sigma,s}} \|\Pi\phi\|_{\sigma,s_0+2,a}  \big) \, .
\end{equation}
\end{lem}
\begin{proof}
We write $F(\eta)[\phi]$ in \eqref{def:f} as
$$
F(\eta)[\phi] =
 \mathcal{F}_1[\alpha(\eta),  \phi]
 +
  \mathcal{F}_2[\beta(\eta),\phi]
  +
   \sum_{j=1}^d \mathcal{F}_{3j}[\gamma_j(\eta), \phi]
   +
    \mathcal{F}_4[\delta(\eta),\phi]
$$
with   bilinear maps
\begin{align*}
\mathcal{F}_1[g, \phi]:= g \partial_y^2 \phi \, , \quad
\mathcal{F}_2[g, \phi]:= g \Delta \phi \, , \quad
\mathcal{F}_{3j}[g, \phi]:=  g \partial_{x_j} \partial_y \phi \, , \quad
\mathcal{F}_4[g,\phi]:= g \partial_y \phi  \, .
\end{align*}
  In view of \eqref{algebrassa}, Lemma \ref{lem:prop} and \eqref{coplusH}, each of these maps is   bounded $\mathcal{H}^{\sigma, s-\frac32, a} \times (\mathbb{C} \oplus \mathcal{H}^{\sigma, s+\frac12,a}) \to \mathcal{H}^{\sigma, s-\frac32,a}$ and any $ \mathcal{F} \in \{\mathcal{F}_1, \mathcal{F}_2, \mathcal{F}_{3j}, \mathcal{F}_4 \} $
  fulfills the tame  estimates
\begin{equation}
\label{Fi}
\| \mathcal{F} [g, \phi] \|_{\sigma, s-\frac32,a} \lesssim_s  \| g \|_{\sigma, s-\frac32,a}
\|\Pi \phi \|_{\sigma, s_0+2,a} + \| g \|_{\sigma, s_0,a}
\|\Pi \phi \|_{\sigma, s+\frac12,a}   \, .
\end{equation}
We claim that  the maps
\begin{equation}\label{mappeanaliti}
\begin{aligned}
& H^{\sigma, s}  \cap B^{\sigma,s_0+\frac12}(\epsilon_0) \to \mathcal{H}^{\sigma, s-\frac12, a} \, ,
\quad \eta \mapsto \alpha(\eta) \, , \beta(\eta) \, , \gamma_j(\eta) \, , \quad j = 1, \ldots, d \, , \\
& H^{\sigma, s} \cap B^{\sigma,s_0+\frac32}(\epsilon_0)  \to \mathcal{H}^{\sigma, s-\frac32, a} \, , \quad \eta \mapsto \delta(\eta) \, ,
\end{aligned}
\end{equation}
are analytic and, for any $s \geq s_0+\frac32 $, $ j=1,\dots, d $,
\begin{equation}\label{midestimate}
\|\alpha(\eta) \|_{\sigma,s-\frac12,a},\, \|\beta(\eta) \|_{\sigma,s-\frac12,a},\, \|\gamma_j(\eta) \|_{\sigma,s-\frac12,a},\, \|\delta(\eta) \|_{\sigma,s-\frac32,a} \leq C(s) \|\eta\|_{H^{\sigma,s}} \, .
\end{equation}
It is clear that these properties, together with
\eqref{Fi},
 imply the Lemma.

Let us consider first $\alpha(\eta)$, defined in \eqref{f}, which we rewrite as
$$
\alpha(\eta) = \Big(1- \frac{1}{\partial_y \rho(\eta)}\Big)+ \Big(1- \frac{1}{\partial_y \rho(\eta)}\Big)|\nabla \rho(\eta)|^2 -  |\nabla \rho(\eta)|^2 \, .
$$
We first prove that   $\eta \mapsto 1- \frac{1}{\partial_y \rho(\eta)}$ is analytic
as a  map
$H^{\sigma, s} \cap B^{\sigma,s_0}(\epsilon_0)  \to \mathcal{H}^{\sigma, s-\frac12, a}$.
We first note that  Lemma \ref{lem:und.varphi} implies
\begin{equation}\label{boundpay}
\|  \partial_y \rho(\eta)  - 1\|_{\sigma, s-\frac12 , a}
=  \| e^{y|D|} |D| \eta \|_{\sigma, s-\frac12, a} \lesssim_s \| \eta \|_{H^{\sigma,s}} \, .
\end{equation}
Then by \eqref{potenze} and \eqref{boundpay} the series
\begin{equation}
\label{neu.rho}
1- \frac{1}{\partial_y \rho(\eta)}  =
-  \sum_{j \geq 1}   (  1- \partial_y \rho(\eta) )^j
\end{equation}
is bounded by
\begin{align}
\Big\|  \frac{1}{\partial_y \rho(\eta)}-1 \Big\|_{\sigma, s-\frac12, a}
&\leq \|\partial_y \rho(\eta)-1  \|_{\sigma,s-\frac12,a} \sum_{j \geq 1}  \big(2C_s \|\partial_y \rho(\eta)-1\|_{\sigma,s_0,a}\big)^{j-1} \leq C(s) \|\eta  \|_{H^{\sigma,s} }  \label{fracest}
\end{align}
provided
$  \| \eta \|_{H^{\sigma,s_0+ \frac12}} < \epsilon_0(s)$ is small enough.
The  series \eqref{neu.rho} of analytic functions in uniformly
convergent in $\mathcal{H}^{\sigma, s-\frac12,a}$ on the domain $\eta \in
B^{\sigma,s}(R)  \cap B^{\sigma,s_0}(\epsilon_0)$, $ \forall R > 0 $,
thus it defines
an analytic  map
on $ H^{\sigma, s}  \cap B^{\sigma,s_0}(\epsilon_0) $.
Moreover the linear map  $ \eta \mapsto \nabla \rho(\eta) =
e^{y|D|}  \nabla \eta  $ is, by Lemma \ref{lem:und.varphi},
bounded between $ H^{\sigma, s} \to \mathcal{H}^{\sigma, s-\frac12,a} $.
Therefore  $\alpha(\eta)$ is the product of analytic functions  $H^{\sigma, s} \cap B^{\sigma,s_0+\frac12}(\epsilon_0) \to \mathcal{H}^{\sigma, s-\frac12,a}$, and using the tame estimate
\eqref{algebrassa} we get  \eqref{midestimate}.

The analyticity  and the estimates of the functions
$\eta \mapsto \beta(\eta), \gamma_j (\eta)$, $ j = 1, \ldots, d$
stated in \eqref{mappeanaliti} follow similarly.
Finally consider  $\delta(\eta)$ in \eqref{f}.
The  biggest loss of derivatives follows from  the linear  maps
$\eta \mapsto \Delta \rho(\eta) $, $ \partial_y^2\rho(\eta) $, $ \nabla \partial_y\rho(\eta) $
which, by Lemmata \ref{lem:prop} and \ref{lem:und.varphi},
are bounded between $ H^{\sigma, s} \to \mathcal{H}^{\sigma, s-\frac32,a} $.
Moreover $\delta(\eta)$ satisfies the estimate
$\| \delta(\eta) \|_{H^{\sigma, s-\frac32,a}} \leq C(s, \| \eta\|_{H^{\sigma, s_0+\frac32}} ) \, \| \eta\|_{H^{\sigma, s}} $
for any $ s -\frac32 \geq s_0$.
\end{proof}

\begin{proof}[Proof of Lemma \ref{lem:lasol}] For any $s\geq s_0+\frac32$ such that $s+\frac12\in\mathbb{N}$,
by Lemmata \ref{lem:ell5} and \ref{lem:non}, the map
$$
\eta \mapsto P(\eta):= L\circ  F(\eta) \, , \quad
H^{\sigma,s}\cap B^{\sigma, s_0+\frac12}(\epsilon_0) \to \mathcal{L}(\mathbb{C} \oplus \mathcal{H}^{\sigma, s+\frac12, a}) \, ,
$$
is analytic and, for positive constants $ C(s) \geq C'(s_0) > 0 $,
 in view of \eqref{coplusH},
\begin{equation}\label{indbase}
\begin{aligned}
&
\| P(\eta)[\phi] \|_{\sigma, s+\frac12, a} \leq C(s) \, \big(  \| \eta \|_{H^{\sigma, s_0+\frac32}}\|\Pi\phi \|_{\sigma, s+\frac12, a}+\| \eta \|_{H^{\sigma, s}}\|\Pi\phi \|_{\sigma, s_0+2, a} \big) \\
& \| P(\eta)[\phi] \|_{\sigma, s_0+2, a} \leq C'(s_0) \,
\| \eta \|_{H^{\sigma, s_0+\frac32}}  \|\Pi\phi \|_{\sigma, s_0+2, a}  \, .
\end{aligned}
\end{equation}
We claim that, for any $\eta \in H^{\sigma,s}\cap B^{\sigma, s_0+\frac32}(\epsilon_0) $
and $  \epsilon_0 (s) > 0 $ small enough, the operator $\mathrm{Id}-P(\eta) $ is invertible in $\mathcal{L}(\mathbb{C} \oplus \mathcal{H}^{\sigma, s+\frac12, a})$ and the inverse map
 \begin{equation}\label{Neumannseries}
 \eta \mapsto (\mathrm{Id} - P(\eta))^{-1} = \sum_{j=0}^\infty P(\eta)^j[\phi] \, , \quad H^{\sigma,s}\cap B^{\sigma, s_0+\frac32}(\epsilon_0) \to \mathcal{L}(\mathbb{C} \oplus \mathcal{H}^{\sigma, s+\frac12, a})  \, ,
 \end{equation}
 is analytic.
 As each $\eta \mapsto P(\eta)^j$ is analytic $ H^{\sigma,s}\cap B^{\sigma, s_0+\frac12}(\epsilon_0) \to \mathcal{L}(\mathbb{C} \oplus \mathcal{H}^{\sigma, s+\frac12, a})$,
the claim follows by proving that the series \eqref{Neumannseries} converges uniformly  in
$\mathcal{L}(\mathbb{C} \oplus \mathcal{H}^{\sigma, s+\frac12, a})$ for
$\eta \in B^{\sigma,s}(R)\cap B^{\sigma, s_0+\frac32}(\epsilon_0) $ for any $R >0$.
By \eqref{indbase}  we have, for any $ j \in \mathbb{N} $,
\begin{equation}\label{indbassa}
\| P(\eta)^j[\phi] \|_{\sigma, s_0+2, a} \leq \big(C'(s_0) \| \eta \|_{H^{\sigma, s_0+\frac32}}\big)^j \|\Pi\phi \|_{\sigma, s_0+2, a} \, ,
\end{equation}
and, by induction, we prove that
\begin{equation}\label{indalta}
\| P(\eta)^j[\phi] \|_{\sigma, s+\frac12, a} \leq C(s)^j \|\eta \|_{H^{\sigma, s_0+\frac32}}^{j-1}  \big(  \| \eta \|_{H^{\sigma, s_0+\frac32}}\|\Pi\phi \|_{\sigma, s+\frac12, a}\!+\!j\| \eta \|_{H^{\sigma, s}}\|\Pi\phi \|_{\sigma, s_0+2, a} \big) \, .
\end{equation}
Indeed, for $j=1$ this is \eqref{indbase}.
Then assuming that  \eqref{indalta} holds for $j$, we get
\begin{equation*}
\| P(\eta)^{j+1}[\phi] \|_{\sigma, s+\frac12, a} \stackrel{\eqref{indbase}}{\leq} C(s) \, \big(\underbrace{\|\eta \|_{H^{\sigma, s_0+\frac32}}\|P(\eta)^j[\phi] \|_{\sigma, s+\frac12, a}}_{=:A}+\underbrace{\| \eta \|_{H^{\sigma, s}}\| P(\eta)^j[\phi] \|_{\sigma, s_0+2, a}}_{=:B} \big).
\end{equation*}
By the inductive hypothesis the first term is bounded by
\begin{equation*}
A \leq C(s)^{j} \|\eta \|_{H^{\sigma, s_0+\frac32}}^{j}  \, \big(  \| \eta \|_{H^{\sigma, s_0+\frac32}}\|\Pi\phi \|_{\sigma, s+\frac12, a}+j\| \eta \|_{H^{\sigma, s}}\|\Pi\phi \|_{\sigma, s_0+2, a} \big) \, ,
\end{equation*}
whereas, by \eqref{indbassa},
\begin{equation*}
B\leq (C'(s_0) \| \eta \|_{H^{\sigma, s_0+\frac32}})^{j} \| \eta \|_{H^{\sigma, s}} \|\Pi\phi \|_{\sigma, s_0+2, a} \, ,
\end{equation*}
and we deduce, as $ C'(s_0) \leq C(s) $,  that
\begin{equation*}
\| P(\eta)^{j+1}[\phi] \|_{\sigma, s+\frac12, a} \leq C(s)^{j+1} \|\eta \|_{H^{\sigma, s_0+\frac32}}^{j}  \, \big(  \| \eta \|_{H^{\sigma, s_0+\frac32}}\|\Pi\phi \|_{\sigma, s+\frac12, a}+(j+1)\| \eta \|_{H^{\sigma, s}}\|\Pi\phi \|_{\sigma, s_0+2, a} \big) \, ,
\end{equation*}
which proves \eqref{indalta} at the step $ j+1 $.

By \eqref{indalta},
the series in \eqref{Neumannseries} is bounded by
\begin{align}
\|(\mathrm{Id}-P(\eta))^{-1}[\phi]\|_{\sigma,s+\frac12,a} &\leq \sum_{j \geq 0} \| P(\eta)^j[\phi] \|_{\sigma,s+\frac12,a}  \notag \\ \notag
&\leq \|\Pi\phi \|_{\sigma, s+\frac12, a} +
\sum_{j \geq 1} C(s)^j \|\eta \|_{H^{\sigma, s_0+\frac32}}^{j-1}\   \| \eta \|_{H^{\sigma, s_0+\frac32}}\|\Pi\phi \|_{\sigma, s+\frac12, a}  \\
&\qquad \qquad \qquad \, +\sum_{j \geq 1} C(s)^j \|\eta \|_{H^{\sigma, s_0+\frac32}}^{j-1}\,j\,\| \eta \|_{H^{\sigma, s}}\|\Pi\phi \|_{\sigma, s_0+2, a} \notag \\
&\leq 2\|\Pi\phi\|_{\sigma,s+\frac12,a}+
C \|\eta\|_{H^{\sigma,s}} \|\Pi\phi\|_{\sigma,s_0+2,a}
\label{finalest}
\end{align}
provided
$ \| \eta \|_{H^{\sigma, s_0+\frac32}} < \epsilon_0(s)$ is sufficiently small.
In particular this shows the claim on the uniform convergence of the series on $B^{\sigma,s}(R)\cap B^{\sigma, s_0+\frac32}(\epsilon_0) $ for any $R >0$.

The analytic map
\begin{align*}
U: H^{\sigma,s}\cap B^{\sigma, s_0+\frac32}(\epsilon_0) \longrightarrow \mathcal{L} \big( \mathbb{C} \oplus \mathcal{H}^{\sigma, s+\frac12,a}\big) \, , \quad
 U(\eta)[ \phi]:= (\mathrm{Id} - L \circ F(\eta) )^{-1}\big[L\circ F(\eta)[\phi]\big] \, ,
\end{align*}
defines the unique solution $u = U(\eta)[\phi]$
of  \eqref{ell10} and, consequently, of system \eqref{ell4}.
By \eqref{finalest} and \eqref{indbase} we deduce
\eqref{tameu}. This  proves Lemma \ref{lem:lasol}.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:varphi}]
It follows with $ \varphi = \Psi(\eta)[\psi]= e^{y|D|}\psi + U (\eta)[e^{y|D|}\psi] $, see
\eqref{eq:varphi}, \eqref{ell3} and Lemma \ref{lem:lasol}.
\end{proof}

\section{Analyticity of the Stokes wave}\label{sec:Sto}

In this section we prove Theorem \ref{LeviCivita}.
With the aid of the analyticity result of Theorem \ref{thm:DN}, the bifurcation
proof is classical.
We report it for completeness.
It is based on the application of the analytic Crandall-Rabinowitz  Theorem \ref{thm:CrandallRabinowitz} below.  For the proof
we refer e.g to
\cite{BuT}, and  Theorem 4.1 in Chap. 5 of \cite{AP} for its smooth version.
\begin{teo}[Crandall-Rabinowitz bifurcation Theorem]\label{thm:CrandallRabinowitz}
Let $X,Y$ be Banach spaces and $  U \subset X$ be an open neighbourhood of 0. Let  $F:U\times \mathbb{R} \to Y$, $ F(u, c ) $, be an analytic map satisfying
 $F(0,c)=0$ for any $c \in \mathbb{R}$. Let $c^*$ be such  that
 $ L := \mathrm{d}_u F(0,c^*) \in {\mathcal L} (X,Y)$ is not invertible and
\begin{enumerate}
\item
$ \text{Ker} ( L )   = \textup{span} \langle u^* \rangle $, $ u^* \in X $, is  $ 1 $-dimensional;
\item the range  $ R := \text{Rng} (L ) $  is closed and $\textup{codim } R = 1 $;
\item {(\it {transversality})}
$ \; \partial_c \mathrm{d}_u F(0,c^*)[u^*] \notin R   \, . $
\end{enumerate}
Then there exist $\epsilon_*  >0$ and an analytic function
$$
(-\epsilon_*,\epsilon_*) \to   U\times \mathbb{R} , \ \ \epsilon \mapsto (u_\epsilon, c_\epsilon) \, , \   \quad \quad
 u_\epsilon = \epsilon u^* + O(\epsilon^2) \, , \ \  c_\epsilon = c^* + O(\epsilon) \, ,
  $$
such that
  $F(u_\epsilon, c_\epsilon) = 0$ for any $| \epsilon | < \epsilon_* $.
\end{teo}

Theorem \ref{LeviCivita}  is proved by applying Theorem \ref{thm:CrandallRabinowitz} to
the nonlinear operator
\begin{equation}\label{def:F}
\begin{aligned}
& F:\; \big( H^{\sigma,s}_{\mathtt{ev}_0} \cap B^{\sigma,s_0}(\epsilon_0) \big)
\times  H^{\sigma,s}_{\mathtt{odd}}
\times \mathbb{R} \longrightarrow   H^{\sigma,s-1}_{\mathtt{odd}}
\times  H^{\sigma,s-1}_{\mathtt{ev}_0}\, , \ \sigma \geq 0 \, , \  s > 5/ 2  \, , \\
& \qquad  F(\eta,\psi,c):= \Big(c\eta_x+G(\eta)\psi \ , \
c\psi_x - g \eta - \dfrac{\psi_x^2}{2} +
\dfrac{1}{2(1+\eta_x^2)} \big( G(\eta) \psi + \eta_x \psi_x \big)^2   \Big)
\end{aligned}
\end{equation}
where
$ H^{\sigma,s}_{\mathtt{ev}_0} $, respectively $ H^{\sigma,s}_{\mathtt{odd}} $, denote the  space of even, respectively odd, and average-free real valued functions in $ H^{\sigma,s}$ defined in \eqref{def:Hsigmas},
and $ \epsilon_0 := \epsilon_0(\sigma,s,s_0) > 0 $ is provided by Theorem \ref{thm:DN}. Note that a
real  function $ (\eta, \psi) \in H^{\sigma,s}_{\mathtt{ev}_0} \times H^{\sigma,s}_{\mathtt{odd}}  $ admits a Fourier series expansion
\begin{equation}\label{svifou}
\vet{\eta(x)}{\psi(x)} = \sum_{k \geq 1} \vet{\eta_k \cos(kx)}{\psi_k \sin(kx)} \quad \text{with norm} \quad
\| (\eta,\psi) \|_{H^{\sigma,s}}^2 \simeq \sum_{k \geq 1}
 e^{2 \sigma |k|} \langle k \rangle^{2s} (\eta_k^2 + \psi_k^2)  \, .
\end{equation}
The fact that the nonlinear operator $ F  $ in \eqref{def:F} maps
a pair of functions $ (\eta, \psi )$ which are odd/even in $ x $
 into a pair of functions which are even/odd in $ x $ is verified thanks to the reversibility property \eqref{DN-rev}.   Moreover, the second component of $ F $

 has zero average thanks to the following  lemma.
\begin{lem}\label{lem:averagefree}
 Let $G ( \eta) $ be the Dirichlet-Neumann operator defined in \eqref{DN1}. Then
\begin{equation}\label{DNinf}
\int_\mathbb{T} -  \frac12 \psi_x^2  +  \dfrac{1}{2(1+\eta_x^2)} \big( G(\eta) \psi + \eta_x \psi_x \big)^2
\, \mathrm{d} x  = 0 \, .
\end{equation}
\end{lem}
\begin{proof}
By \eqref{DN-inf}, the kinetic energy
$ K(\eta, \psi) = \frac12 (\psi, G(\eta) \psi)_{L^2}  $ in \eqref{kine} satisfies
$ K(\eta+m,\psi) = K(\eta,\psi) $ for any $ m \in \mathbb{R} $.
Thus
$$
0 = \frac{\mathrm{d}}{\mathrm{d} m} K(\eta+m,\psi)
	= \mathrm{d}_{\eta} K(\eta,\psi)[1] = (\nabla_\eta K(\eta, \psi), 1)_{L^2}
	= \int_{\mathbb{T}} \nabla_\eta K(\eta, \psi) \, \mathrm{d} x \, .
$$
In view of  \eqref{Kform}, the identity  \eqref{DNinf} is proved.
\end{proof}

\smallskip
We now start verifying the assumptions of the Crandall-Rabinowitz Theorem \ref{thm:CrandallRabinowitz}.
First, by Theorem \ref{thm:DN}, the nonlinear operator
 $ F $ defined in \eqref{def:F} is {\it analytic}. Moreover, by inspection,
$$
F(0,0,c) = 0 \, , \quad \forall c \in \mathbb{R} \, .
$$
The possible bifurcation values of non-trivial solutions of $ F(\eta, \psi, c ) = 0 $
are those speeds $ c $ such that
the  linearized operator
\begin{equation}\label{linF}
\begin{aligned}
\mathrm{d}_{(\eta,\psi)} F(0,0,c):\; H^{\sigma,s}_{\mathtt{ev}_0} \times H^{\sigma,s}_{\mathtt{odd}} \to& H^{\sigma,s-1}_{\mathtt{odd}} \times  H^{\sigma,s-1}_{\mathtt{ev}_0} \, , \quad
 \vet{\hat\eta}{\hat\psi} \mapsto& \begin{bmatrix} c\partial_x & |D| \\ -g & c \partial_x \end{bmatrix}\vet{\hat\eta}{\hat\psi} \, ,
\end{aligned}
\end{equation}
has a nontrivial kernel. In the next lemma we characterize such values.
\begin{lem}\label{lem:ker}
{\bf (Bifurcation speeds)}
The kernel of $\mathrm{d}_{(\eta,\psi)} F(0,0,c)$ in \eqref{linF} is nontrivial if and only if
\begin{equation}\label{ilc}
c= \pm \sqrt{\frac{g}{k}} \qquad \text{for  some} \  k \in \mathbb{N} \, .
\end{equation}
For any  $ k \in \mathbb{N} $, the Kernel of $ L := \mathrm{d}_{(\eta,\psi)} F(0,0, c^*_k) $,
where we set $ c^*_k := \sqrt{\frac{g}{k}}$,
is one dimensional and
\begin{equation}\label{kernelL}
\text{Ker}(L)= \langle u^* \rangle  \quad \text{with} \quad
u^* := \vet{\sqrt{k}\cos(kx)}{\sqrt{g}\sin(kx)} \, .
\end{equation}
\end{lem}

\begin{proof}
By the Fourier expansion
  \eqref{svifou}, it results that the kernel of $\mathrm{d}_{(\eta,\psi)} F(0,0,c) $ is nontrivial if
and only if at least one of the matrices
$ \begin{bmatrix} - ck   & k \\ -g &  c k \end{bmatrix}$, $k \in \mathbb{N}$, has zero determinant. This is  verified provided  $c^2 k = g $ for some  $k \in \mathbb{N} $, i.e. \eqref{ilc} holds.
In addition, a vector
$ \vet{\eta(x)}{\psi(x)} = \sum_{j \geq 1} \vet{\eta_j \cos(jx)}{\psi_j \sin(jx)}$
belongs to the Kernel of
$  \mathrm{d}_{(\eta,\psi)} F(0,0,c_k^* ) $ if and only if
\begin{equation}
\label{CR1.sys}
\begin{bmatrix}
- c^*_k j & j \\
-g & c^*_k j
\end{bmatrix}\vet{\eta_j}{\psi_j} =  0 \, , \quad \forall j \geq 1 \, .
\end{equation}
If  $j \neq k$ then
\begin{equation}
\label{det.ck}
\det \begin{bmatrix}
- c^*_k j & j \\
-g & c^*_k j
\end{bmatrix}  = - (c^*_k)^2 j^2 + g j
= j^2 \big( (c^*_j)^2 - (c^*_k)^2\big) \neq 0 \, ,
\end{equation}
since  the map $k \mapsto (c^*_k)^2 = g / k $ is  injective on $\mathbb{N}$.
Hence $\eta_j = \psi_j = 0$ for any $j \neq k$. On the other hand, if  $j = k$
then \eqref{CR1.sys} is solved provided  $\sqrt{g} \eta_k =\sqrt{k}\psi_k$, proving
\eqref{kernelL}.
\end{proof}

We apply Theorem  \ref{thm:CrandallRabinowitz} with
$ c^*_k := \sqrt{\frac{g}{k}}$. By Lemma \ref{lem:ker}
 assumption 1 holds.
The next lemma verifies the assumptions 2)-3).
\begin{lem}\label{lem:CrandallRabinowitzHP}
The range  $ R:=\text{Rng} L$,  $ L = \mathrm{d}_{(\eta,\psi)} F(0,0, c^*_k) $,  is
\begin{equation}\label{formaR}
R= \left\lbrace
 \vet{f}{g} \in H^{\sigma,s-1}_{\mathtt{odd}}(\mathbb{T})\times  H^{\sigma,s-1}_{\mathtt{ev}_0}(\mathbb{T})
  \, \colon
  \ \
  \vet{f(x)}{g(x)} =  \vet{f_k \sin(kx)}{ c_k^* f_k \cos( kx)}  +
  \sum_{j\geq 1, j \neq k} \vet{f_j \sin(jx)}{g_j \cos(jx)}
\right\rbrace \, .
\end{equation}
 In particular $R$ is closed and $\mathrm{codim}\, R =1 $.

The vector $(\partial_c \mathrm{d}_{(\eta,\psi)} F)(0,0, c^*_k)\vet{\sqrt{k}\cos(kx)}{\sqrt{g}\sin(kx)}$ does not belong to $R$.
\end{lem}

\begin{proof}
A vector  $\vet{f}{g}\in H^{\sigma,s-1}_{\mathtt{odd}}(\mathbb{T})\times  H^{\sigma,s-1}_{\mathtt{ev}_0}(\mathbb{T}) $ belongs to $ R $ if and only if
there is $\vet{\eta}{\psi} \in H^{\sigma,s}_{\mathtt{ev}_0} \times  H^{\sigma,s}_{\mathtt{odd}}$
such that, recalling  \eqref{linF} and \eqref{svifou},
\begin{equation}
\label{CR2.sys}
\begin{bmatrix}
- c^*_k j & j \\
-g & c^*_k j
\end{bmatrix}\vet{\eta_j}{\psi_j} =
\vet{f_j}{g_j} \quad \forall j \geq 1
\quad \text{where} \quad
\vet{f(x)}{g(x)} = \sum_{j\geq 1} \vet{f_j \sin(jx)}{g_j \cos(jx)}  \, .
\end{equation}
For any $j \neq k$, by \eqref{det.ck}, system \eqref{CR2.sys} has the unique solution
\begin{equation}
\label{sol.1}
\eta_j = \frac1g \frac{\sqrt{k}}{k-j}(\sqrt{g}f_j-\sqrt{k}g_j) \, , \quad \psi_j = \frac1{j\sqrt{g}}\frac{\sqrt{k}}{k-j}(\sqrt{kg}f_j-jg_j) \, .
\end{equation}
 If  $j = k$, the system \eqref{CR2.sys}  is solvable if and only if
\begin{equation}
\label{fkgk}
\sqrt{g} f_k = \sqrt{k} g_k
\end{equation}
and  a solution is  $\eta_k = - \frac{1}{\sqrt{kg}}f_k$, $\psi_k = 0$.
By  \eqref{sol.1} we deduce that
$|\eta_j|,  |\psi_j| \leq \frac{C_k}{j} (|f_j| + |g_j|)$, for any $ j \in \mathbb{N} \setminus \{k \}$, implying
that $ (\eta, \psi) \in H^{\sigma,s} $, actually
$$
\| \eta \|_{H^{\sigma,s}},\; \| \psi \|_{H^{\sigma,s}}  \leq C_k (\|f\|_{H^{\sigma,s-1}}+\|g\|_{H^{\sigma,s-1}}) \, .
$$
In conclusion,
the range $ R $ of $ L $ has the form \eqref{formaR},
by \eqref{fkgk} and $ c^*_k = \sqrt{g/k} $.

Finally differentiating \eqref{linF} one computes
 $$
 (\partial_c \mathrm{d}_{(\eta,\psi)} F)(0,0,c^*_k)\vet{\sqrt{k}\cos(kx)}{\sqrt{g}\sin(kx)}= {\scriptsize \begin{bmatrix} \partial_x \ \ 0 \\ 0 \ \ \partial_x \end{bmatrix}}\vet{\sqrt{k}\cos(kx)}{\sqrt{g}\sin(kx)} =\vet{-k^{\frac32}\sin(kx)}{k\sqrt{g}\cos(kx)} $$
 which  does not belong to the range $R$ in \eqref{formaR}.
\end{proof}

All the  assumptions of the Crandall-Rabinowitz Theorem are verified,  proving Theorem \ref{LeviCivita}.

redpajama_set_name: RedPajamaArXiv


Bungie's next big video game release since the Halo series is upon us. This September 9, 2014 is when gamers can pick up Destiny and to celebrate the release, Bungie is inviting you, yes you, to their studio.
Announced through the Bungie blog, community manager, Deej, has stated that those who are available on September 8, 2014 can come down to the studio and expect some free goodies. Now there is a limit but if you're one of the lucky first 300, you get the top tier package.
Mind you, this is all free. From Bungie, the announcement was made that the first 300 attendees will get a chance to tour the studio and enjoy some multiplayer action of the upcoming Destiny video game.
If you don't make the first 300 then no problem. The first 1,000 in line will not be leaving with nothing more than memories. Bungie has noted that these first 1,000 will be laving with a few surprises, what those surprises are may be unknown but this is certainly an event you will want to join in on if you live in the area.


Treasury Secretary Jacob Lew speaks about the U.S. economy at the Council on Foreign Relations in Washington, D.C., Jan. 16, 214.
The U.S. has accumulated $17.3 trillion in debt over the years and the figure grows by the day. The government suspended limits on the debt ceiling in October, but is reinstating it on February 7. At that point, Treasury officials can use what they call "extraordinary measures" to keep paying the country's bills even as it edges closer to running out of cash.


Planning a picnic? This would be a perfect addition.
Meanwhile, in a medium skillet, heat the olive oil over medium heat. Add the garlic and sauté until fragrant, about 2 minutes. Remove from heat and add the anchovies, capers, and vinegar; stir to combine.
Drain the pasta, reserving 1/2 cup of the cooking liquid. Transfer the pasta to a large mixing bowl, and then toss with the garlic oil mixture until well coated. Allow the pasta to cool to room temperature.


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Home > News > Rolls-Royce just proved that the rich don't give a damn about the pandemic
Rolls-Royce just proved that the rich don't give a damn about the pandemic
CY Foong · Jun 6, 2021 12:00 PM
USD 28 million (RM 115.6 million) is plenty of cash for us regular people. For the ultra-rich, it might as well be pocket change. Rolls-Royce has just recently announced the launch of the coach-built Rolls-Royce Boat-Tail and it is the most expensive brand-new car on sale ever.
You wouldn't be surprised that even in this time of crisis with the world slowly recovering from the pandemic (not sure about our side though), someone would commission an exclusive topless Rolls with a price tag of around 1,605 Perodua Ativa AVs.
A proper fridge for your bubbly beverage, sir
I could simply wax poetic about the exquisite features of the rebodied Rolls-Royce Dawn including a double refrigerator used to store the unnamed owner's favourite vintage champagne or just copy straight from the Spirit of Ecstasy's press release.
An automatic parasol to hide from the anti-establishment...I mean the sun, sir
Well, I'm not and this is not some tattling publication or a report on the nouveau riche. If you would like to take a look and admire the decadence of the ultra-wealthy, you can look elsewhere.
The rich are getting richer…
It is funny how this pandemic has unearthed some unfairness in the world, especially if you're from a working-class background or anyone else that doesn't belong in the 1%. This even applies to automakers.
Last year, most of the world saw a downturn in sales throughout the year but only one segment thrived and ironically, posted its best profits ever – the luxury segment.
Also read: What are some of the best-selling cars globally?
Rolls-Royce posted its best quarter in its 116-year history with 1,380 units sold in Q1 2021. That is a 62% rise compared to the same quarter in 2020. As billionaires reaped whatever they earned during the pandemic, they flaunt it like it's Black Friday, filling their garages with Cullinans, Ghosts, and Phantoms.
Living with one for a day might change my attitude
The Spirit of Ecstasy wasn't the only brand suffering winning from success. Former partners and now rivals, Bentley saw its best sales ever in 2020 with 11,206 vehicles sold. This was even despite a 7-week closure of its Crewe factory.
But not all that are rich are treated equally
Regular carmakers could only wish for those kinds of profits in a year that saw fewer people buying new cars as a way to adapt to the new normal. The ball clearly only favours the rich, but some luxury brands still saw losses in the year.
Lamborghini saw 7,430 units sold in 2020, down by just only 9% compared to 2019. Sales surged in the second half of the year once Italy allowed its factories to reopen. Aston Martin also saw sales slipping in 2020 but slowly rebounded in Q1 2021 following a rise in sales of its DBX SUV.
Ferrari also saw a slump in sales in 2020 but reported a gain in profits. Q1 2021 also saw the Prancing Horse recording a rise in sales with 2,771 units sold according to its latest investor's report.
Though we could see a rise in sales for most car manufacturers worldwide the rest of the year, the reality is that we mere mortals are still lying in the dark.
One of the Bentley models commissioned by the Sultan of Brunei
Besides, these luxury brands would have one-percenters buyers from all over the world who are willing to spill whatever chump change they have to commission something special. Bentley wouldn't exist today if some Southeast Asian royalty didn't ask to build some unique one-offs.
Flaunting is the new trigger warning?
Personally, I'm not turned off by these exquisite cars and I'm sure most of you aren't either. Remember when the world was surrounding the hype of the Bugatti Veyron as the world's fastest car? Double standard was just a sarcastic way to mock biased football referees.
It is still fascinating to see some exquisite cars being launched even if we don't have the means to own one. Just looking at it can spark a sensation coined by a spaniel-haired scribe known as "the fizz".
The Aston Martin Victor, a cool one-off built during these trying times
There's no denying that cars like the Aston Martin Victor or the Rolls-Royce Boat-Tail can inspire awe to us regular folks who wanted to dream big. In this time of crisis though, flaunting something so exquisite might be a bit of a turn-off.
For the lucky three owners (including Beyoncé and Jay-Z) of the Boat-Tail, their ride is certainly exquisite and will be a valuable asset. To us mere folks though, the Boat-Tail might as well be an unfair reminder that it's simply unattainable.
CY Foong
Traded advertising for a career that fits his passion for cars. Enjoys spotting cars during his free time and has a soft spot for Japanese Kei cars but drives a thirsty manual sedan.
Body TypeConvertible
Capacity- L
Transmission-
Horsepower- PS
4 other Rolls Royce models
What's Genesis? Meet the Korean Lexus and a recent addition to the Sultan of Johor's collection
Mercedes-Benz: Challenges with EV charging just as difficult in Germany as it is in Malaysia
Mercedes-Benz Malaysia to partner with local companies to rollout EV charging facilities
Batu Tiga was once Malaysia's motor racing playground, what the heck happened?
Priced from just RM 174k in China, here's a closer look at the XC90 twin's, the Lynk & Co 09
Jerrica
Land Rover Defender: Not just a British icon, but the 4x4 that built Malaysia
Mamak Talk
Found these two ads online, credits to the owner who kept them saved in digital form. Imagine seeing a Mercedes and Rolls Royce ad back in the day, it can't get anymore fancy that this.
Real feeling from a Toyota Vios 2014 owner. How much did I pay for it: 700 per month How long have I bought it: 6 years Do I still like it: Yes What's the best part of my car: Fuel efficiency, a comfy car for this class, less repair What's the worst thing of my car: Absorbers quite low and easily to scratch while go through speed bumper if loaded with 5 person. What's the next car I want to buy: CX30, CX5, X70, X50 or Corolla Cross
Hanson Ang
2016 Rolls Royce Dawn Dawn
RM 717,800TBC
RM 1,087,053TBC
Rolls Royce Dawn Images
Rolls Royce Dawn Reviews
Rolls Royce Dawn Specs
Rolls Royce Dawn Fuel Consumption
Rolls Royce Dawn Colors
Rolls Royce Dawn Price

redpajama_set_name: RedPajamaCommonCrawl


Should Google make an effort to reach China, or should they leave that market alone?
"Apple's one of those big international bruisers, so this China Mobile deal shows that the government has figured out how Apple can be part of the market without dominating it. That'll be fine for Apple; with its huge profit margin and China's huge population, the company only needs to skim off the top of the Chinese market to succeed. The real loser here, of course, is Google.
Bring some of that money back into the United States!


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Smoke alert US there are 5 Promo Codes, coupons, discounts for you to choose from including 5 promo codes in February 2019. We also install these in homes in our service area. You could get a particular amount of deal or discount on every Promo Code you have. When you purchase, youll see a box where you should input the code to obtain special discounts. Code - Was Expired, have expired, but occasionally are re-published. Insert coupons in different envelopes for each store. You may also contact us to feedback us on these smoke alert US coupon codes. We offer dryer vent cleaning, a free home fire safety inspections, and free family life safety classes.


I'm damned if I can 'get' this game . Though I WILL persevere.
I remembered that I was pretty damn pants at this when I originally played it but I do remember that I got a little way into the main game before something else shiny got my attention.
There must be some knack, some sweet route to take for maximum grinding on this intro level.
Any hints to get me through the intro?
I know how you feel. I never beat the tutorial. How in the hell do you nail 30 tricks in a row? Are there secret moves I don't know about?
I'm going to continue with the main game and maybe come back to the intro later.
So the green rails can be linked in a loop Does that utilise a wall grind from the advertising board to get you over to the second set of rails? I guess I'm going to have to work on my wall grinds as I have only been able to pull one off by accident up to now.
Don't that just look oh so tasty?
Squeeeeeeezed through the second level. It took me 3 atempts and I have to admit I only just got it on the last attempt.
To begin with it was a matter of learning the level, then to figure out what the best tags would be to do before the riot police turned up.
On the final and successfull attempt I thought I had it complete but found that I still had one more tag to go. It was only because I was standing right next to it in the playground area that I manged to finish with only a few seconds left on the clock.
Have given up for the night after my swearing at Garam sent me hoarse "stupid bleedin golden eyed FREAK!"
Last edited by frood42 on Tue Sep 23, 2008 5:21 pm, edited 1 time in total.
Holy cow -- that was quick!
frood42 wrote: Squeeeeeeezed through the second level. It took me 3 atempts and I have to admit I only just got it on the last attempt.
I'm still not a master myself.
So don't let the lack of combo-grinding get you down.
I just got Mew to join and cleared the No. 540 level. I'm cheating slightly in that I did the first few levels a few months ago and I'm just picking up where I left off.
Thanks for kicking me into revisiting the game though.
btw, has anyone cleared the tutorial? what route did you take to get the 50 tricks?
I tried the tutorial on my first time with the game. The tutorial is NUTS hard, it's ridiculous. I managed 30 tricks without doing loops, but I think its impossible to do that without linking rails - 20 is "easy" as you can "just" link the rail (well, 2 half-rails that are very close and on the same line) that finishes near the upward stairs and go up, around and down the stairs on the other side without doing air tricks, just because the rail + stairs is long enough for that.
For 30 I moved this course one rail back, so I needed to do one rail-to-rail jump, I also think I needed to go to the opposite side going down the stairs and grind a bit on the truck but maybe I already had 30 by then (it was sort of simultaneous).
50 is ludicrous though, I think you need to go on the same rail twice somehow or reverse on the stairs? Any tips on getting long jumps, as I haven't quite understood how to control when he does the long reach (trick) jumps. The dude also tends to go to the side when doing a trick (the one where he does the splits in the air), is this my controller fault or is this "normal"?
BTW I'd like someone to explain to me how to consistently do turn arounds. I have managed to do it some times, but not at all consistent, and to do the jump twist that would be helpful. The tutorial said I needed to "push forward then back when dashing", but I can't really push forward when dashing as to dash you need to keep the stick forward? I think I managed by doing forward-back-forward when dashing, but about 90% of the times the guy just brakes from the dash... I tried so many times I think I got motion sick from it. Turn around can't be this hard so I must be getting something wrong.
Did you capture it with a Pokeball?


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Plados sinks
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The new ?Plados - Kitchen Experience? catalogue for the best way to experience everyday life in the kitchen
We have produced the new catalogue Plados - Kitchen Experience 2022-2023 the graphic design with style and freshness, trying to bring out the full range of the quality and the sophistication of our products.
In fact, what is contained within the catalogue is the result of 30 years of experience and awareness of the importance of persone, land, research and development, and sustainability.
Our commitment to creating increasingly functional, environmentally friendly and durable products is reflected in a wide range of new sinks, accessories, mixers and home appliances.
Kitchen experience is our philosophy of inhabiting domestic spaces, where our products find their placeone to be experienced in complete freedom and comfort while respecting our planet.
Elegant design and beauty are perfectly symbiotic with the innovative technologies which cover all our product lines.
Kitchen experience sums up our commitment to concretely improving the lifestyle of persone, so that time spent in the kitchen and every gesture are experiences of good living, well-being, health, care and attentionone, for us and nature.
kitchen experience is our contribution to a better futurefor everyone.
Download the Plados 2022 catalogue or visit the Downloads
Plados Telma launches Rinnova, the all-green sink
Words create worlds and the world we want to help create is a greener world in which sustainability, both environmental and economic, improves the quality of life of persone. For this reason, the word we have chosen is a verb, renovate, whose meaning refers to the business of making or returning to new, renovating, making...
Rinnova Plados Telma sinks
Care and maintenanceone of the kitchen sink
To ensure that the kitchen is always a safe and hygienic environment, the sink must be properly cared for and maintainedone. In addition to eliminating dangerous sources of bacteria, regular cleaning will also allow you to extend the life cycle of the mixer and the sink itself. In this article we offer you...
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All kitchen trends for 2023 converge on sustainability, both from the point of view of materials and finishes and from the point of view of technology applied to appliances in order to reduce waste and save resources. As far as materials are concerned, in addition to wood, stone and metal, new materials are entering the trend...
green kitchens 2023
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Storms; trees felling down, crops getting hurt etc.
this would be influenced by weather too.
old men in villages beeing able to tell you more about the weather than your weatherloreskill, like early warnings from storms.
Yes! It always feels weird when the character walks through torrential rain for hours, even when ill, yet seems hardly affected.
On consideration: more realistic/complex weather effects might make changing clothes more relevant. Not sure if that would end up interesting - or just a hassle...?


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Top 10 Things to See When Visiting Sydney with Kids in 2023
by admin May 23, 2022, 12:13 pm
Harbor City, Sydney isn't a typical metropolis that is packed only with skyscrapers, residential, business districts, traffic and malls. The city is one of the liveliest places with a friendly community. Visiting Sydney, especially with kids has a lot of activities, and places to visit which will provide them with a great time and experience of a lifetime. Children and their parents both will love this amazing place. It is a city where parents can breathe a sigh of relief as the kids won't be bored even for a day. While there are plenty of parks, and beachside reserves with lots of space, lots of activities require no money. Children can kick balls in the park, climb a tree, swim in the ocean pool, or indulge in whale watching.
Among the fun things to do in Sydney this weekend one can take their kids for a stroll through walkable pathways. As there are attractions spread throughout the city, there is never a dull weekend or summer holiday for little darlings.
Here are 10 Things to See When Visiting Sydney with Kids.
1. Relish A Family Picnic At The Royal Botanical Garden
Sydney prides itself on its cosmopolitan landscape which is infused with scenic gardens which will make anyone admire the greenery. Among the popular kid's attractions in Sydney that locals love is having a family picnic at the Royal Botanical garden. The place has 30 hectares of lush greenery and is a great place for enjoying a picnic with kids. All that is needed is a mat and a basket full of favourite food and one can be ready for a picnic with the scenic beauty of the harbour.
The Royal Botanical Garden offers a kid-friendly space for kids to play around and enjoy soothing weather. Among the fun activities at the Garden, one can stroll through the park admiring the flora and fauna or take a ride in Choo Choo Express. Kids can catch a view of wildlife and join kiddle activities during the school holidays.
Timing: 7:00 am to 5 p.m. or 8 p.m. based on the season.
Cost: Cho Cho express ride costs between AUD 10 for AUD 5 for children (3-17) years. Park entry is free.
2. Make Friends With Sea Creatures At Sydney Sea Life Aquarium
Marine life has always been a place of interest for kids. Especially when they find it in front of them, they are always overjoyed and it expands their knowledge about life on earth. Visiting Sydney Sealife Aquarium, kids will have a chance to experience a plethora of marine sea life. The trivia and facts shared about sea life animals are informative and will improve the knowledge of the kids. It is an excellent audio-visual experience that kids will enjoy the most.
Kids can indulge in feeding activities and get more interactive experiences with the sea creatures like dugongs, sharks, and penguins. There is a penguin boat expedition that will make the kids go face-to-face with penguin families. To visit this fun and exciting Sydney Attraction for families, one can take a small walk from Town Hall or Wynyard Station. Kids must not miss the activity of a touch pool where they can touch real starfish, coral and shark eggs.
Timing: The park is open for the entire week from 10:00 am to 6:00 pm.
Cost: AUD 28 to AUD 85
3. Entertain The Kids With Experiments At Powerhouse Museum
The museum tempts the kids to learn through the demonstration and exhibits at the museum. The venue fulfils both criteria of 'do' and 'see' in Sydney, especially for the kids. It is among the kid's attractions in Sydney, which will keep the kids engaged in the quest of learning something new. The building is a 4-storey learning and activity centre located at Darling Harbor where kids can learn through play activities. Visit the Powerhouse Museum with kids to have a fun-filled productive day with experiments and knowledge exploration. Kids will learn things about robots, electricity, magnetism, sound waves, space visit, transport and many more.
The museum focuses on science and technology providing kids with a world of introduction in the areas. Kids visiting with parents can learn about the modern-day scientific challenges which involve issues like climate change. It is meant to raise awareness among the kids so that they can contribute to sustaining the environment. There are around 1600 matchbox cars, vintage trains and out-of-commission aeroplanes. For comic geeks, there is a separate room for Star Wars and Harry Potter activities for the kids to take part in and live their fantasies and characters.
Timing: 10:00 am to 5:00 pm open daily
Cost: No charges for visitors aged between 16 and below. The admission ticket above the age costs AUD 11.
4. Hobnob With The Koalas At Taronga Zoo
Visiting Australia, and missing out on Koalas is not a good idea. The Taronga Zoo for Sydney Fun activities is exciting for both parents and kids. The Zoo is located on the right bank of Sydney Harbour. The Taronga Zoo is home to around 250 species of animals. Once your kids are done having fun with the cute Koalas, they can join in watching animal shows and activities like feeding giraffes from their hands. While taking a walk through the zoo, kids will appreciate various life on earth and learn their names. Parents have to be ready with the trivia to answer the questions which the kids will throw at them.
If parents are lucky, they can come across a friendly and patient guide from the zoo who will fulfil the queries of the kids. Taronga Zoo has some exotic animal collections like Asiatic elephants and koalas. The Zoo is located precisely at Bradleys Head Road in Mosman and it can be accessed best by taking a ferry from the Circular Quay or Darling Harbor. Families then must take a fun ride through the cable car to reach the Taronga Zoo.
Timing: Taronga Zoo is open daily from 9:30 to 5:00 pm.
Cost: Tickets to the zoo cost AUD 24.3 (4-15 years ago) and AUD 42.3 (16 years and above). Family tickets will provide a discount of 20 percent.
5. Traversing Cockatoo Island
If you are thinking about what to do in Sydney for kids after visiting all the above places, the answer is to visit Cockatoo Island. It is the largest island in Sydney Harbour which was named after the sulphur-crested cockatoo which is common on the island. Enjoy the ferry ride with the kids, appreciating the view of the city and sea and taking a short ferry ride to Cockatoo Island with the kids. Consider Cockatoo Island as a historical, cultural experience.
The visitors will be treated with prison buildings, shipping docks, and a reformatory which is declared a World Heritage Site by UNESCO. The parents must ensure that the kids can take the 45-minute tour of convict history to get a better understanding of the place. Additionally, there is an interesting Convict Clues Challenge which will channel their little inner detective skills. Camping is another activity that families can take to have a spooky night out on this historical island. The kids will love it and the parents too.
Timing: 10 am to 3 pm on a daily basis
Teaching children about the natural world should be seen as one of the most important events in their lives. ~ Thomas Berry
6. 'Surf And Turf' At Bondi Beach With A Surfing Lesson
Visit Bondi beach to help the kids experience the adrenaline rush of adventure sports through surfing. It is one of the most popular beaches in Sydney where kids get the chance to learn surfing. It is a local recommendation especially when families have kids who have just reached their teens. The private lessons given at Bondi Beach will help the kids learn how to ride the waves. The professional trainers are well-trained and know how to engage the kids in the surfing activity.
Timing: Tuesday, Wednesday, Thursday, and Friday (9 am to 5 pm), Saturday (9 am – 3 pm)
7. Discover Indigenous Sydney
The kids are often curious and have an eagerness to learn. One can take this opportunity to dive deep into discovering the Indigenous culture hidden in the history of Australia. The Royal Botanical Garden can be a go-to venue for taking the Aboriginal Heritage Tour. The tour is conducted on Wednesday, Friday, and Saturday. A kid-friendly guide will give an insight into the culture, and practice of the native indigenous people, and the use of their plants as food and medicine.
Kids will learn to appreciate culture and embrace diversity. This interactive tour is a great place, to begin with. If the kids find interest and want to dig deeper, take the indigenous story tour at the Australian Museum and the Museum of Sydney which comes with live exhibits from the native people of Australia.
Timing: Aboriginal Heritage tours are run every Wednesday, Friday and Saturday from 10 am.
Price: AUD 13.05 for kids above 7 years. Free for kids who are below the age of seven years.
8. Jet Set GO!
Want to pack in some thrilling experience of Speed for the kids? Sydney Harbor offers a thrilling jet boat ride to kick up the thrill of speed on the water. The ride takes a high-speed lap of Sydney Harbor which mixes the thrill of speed with a sightseeing experience. The ride is perfect for the kids to give them the thrill of high-speed throttle which will make them scream to the top of their voices. The ride is safe and the riders are given all safety precautions and safety jackets. The riders are trained professionals who can provide the best experience for the riders. The Jet boat ride is located at the Eastern Pontoon of the Circular Quay.
Timing: Daily rides are available from 9 am to 5 pm.
Price: AUD 89 for adults, AUD 49 for children under 16 years. AUD 229 for a family of 2 adults and 2 kids.
9. "Housing" Experience At Susannah Place
If the parents are looking for some exclusive experience for the kids in Sydney, visiting Susannah Place will provide the perfect experience. Susannah Place is located at the Rocks which provides a glimpse of life in history. The venue has a couple of old terraces which have been conserved to provide a glance at bygone era life. The kids will be taken back into history where they can have fun with their parents, and experience living in small dwellings. A friendly guide will share stories about people of the past, their daily chores, and how they had fun when there were no electronic gadgets or modern amenities.
There is a corner store that sells authentic goods venues in case the kids want to buy a souvenir as a mark of the visit. It is the best place to visit during bad weather in Sydney as all activities are indoors. Susannah Place Museum and Corner shop are included in the tour which will take about an hour. Susannah Place is located at 58-64 Gloucester Street, The Rocks.
Timing: Tours are conducted on Thursdays and Fridays which starts at 10 am, 11 am, 1:30 pm, 2:30 pm and 4 pm.
Price: AUD 12 for adults, AUD 10 for children above the age of 5, and AUD 32 per family with 2 adults and 2 children, there is no fee for children under the age of 5.
10. Unadulterated Fun At Luna Park
If the children are longing for a full day of amusing activities, Luna Park is the perfect destination. Despite all the activities of learning, and touring, it is natural for the kids to have a day where they will have pure fun and excitement. Luna Park is a heritage-listed transport exchange that was built in 1935. For kids and adults who like fun and thrill, Luna Park is second to none. A thrilling Wild Mouse roller coaster is a kid's favourite ride where the legendary Rotor sticks to the wall and the floor vanishes.
The park is filled with loads of slides, and games where the kids will never feel bored. Kids and families can enjoy a great scenic sunset from the giant Ferris Wheel. Other rides can provide a great overview of the iconic Sydney Opera House and Sydney Harbor Bridge. The Luna Park is located at 1 Olympic Drive, Milson's Point, New South Wales.
Timing: Monday 10 am – 3 pm. Friday and Saturday 11 am – 8 pm. Sunday 11 am – 6 pm. The park remains closed on Tuesday and Thursday except for NSW school holidays.
Price: The pricing at Luna Park varies from AUD 44 (adults) and AUD 34 (children) for unlimited rides for a day.
End Words
The Harbor City of Sydney can be hardly matched by any other in terms of offering activities for kids. Sydney offers many other locations and activities for the kids and their interests. Visiting the activities and venues will not make the kids feel left out as they can become friends with local kids who are friendly as their parents. What elevates the experience of visiting Sydney with kids is its friendly community. The sheer variety of kids' activities and fun places to visit offered by Sydney is diverse, and there is something for kids with various types of interests. The weekends are fun packed and so are the weekdays as some places have special activities which are available on specific days and keep the kids engaged.
Which Are The Places Where Kids Can Have Fun In Sydney?
Luna Park Sydney, House Museum, Blackwattle Bay Park, Cockatoo Island, Royal National Park, The Australian Botanic Garden Mount Annan, The Rocks, Sydney Motorsport Park Eastern Creek, and Australian Museum are among the top places which are kid friendly.
Is Sydney Kids Friendly?
Sydney is a perfect city for kids and teenagers too. The city is packed with beaches, bridges and bucket-listable iconic venues. Especially when families visit with kids or teens, Sydney will provide a variety of experiences and fun-filled activities. It will be a memory that will be cherished throughout life.
What Are The Free Events For Kids In Sydney Australia?
There are plenty of free events at the weekends where you can put your kid. Some of the fun-filled free-of-cost activity lists are given below.
3 days of free Brazilian Jiu-jitsu and self-defence classes for kids and teens.
Chanukah in Lane Cove Plaza
Christmas Workshop at Centre of Fun
Dooleys Lidcombe Kids Bingo
Australian crafting workshop for kids above the age of 5
Crafty Card Making Workshop for kids and playgroups in NSW.
What Are Sunday Activities For Kids In Sydney?
There are Sunday activities in Sydney that the kids will love and so will their parents. Some of the activities will please the parents and kids equally. Here are some of the exciting activities available in Sydney on Sunday.
Kids can dive underwater in the Sydney Aquarium. It is a must-see experience for the kids where they can see and touch Sea creatures.
Kids' creative activity at the Museum of Contemporary art will help them enhance their creativity.
Get in an educational walk for the kids which will take them through topics like the Indigenous culture of Australia.
Ice cream and street art adventure is an exciting Sunday activity that the kids can enjoy.
Engage with the local kids at Sydney Park where they can have a fun time with football or any other sports. The local kids are friendly and they are quite open to inviting other kids to play with them during the weekend. Sydney park offers a play area where they can indulge in outdoor sports.
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