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\AtBeginDocument{{\noindent\small
2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela.\newline
{\em Electronic Journal of Differential Equations},
Conference 13, 2005, pp. 95--99.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{95}

\begin{document}

\title[\hfilneg EJDE/Conf/13 \hfil Solution of a spectral problem]
{Solution of a spectral problem for the  curl operator
on a cylinder}

\author[R. Saks \& C. J. Vanegas \hfil EJDE/Conf/13 \hfilneg]
{Romen Saks, Carmen Judith Vanegas}  % in alphabetical order

\address{Romen Saks \hfill\break
Bashkir State University\\
32, Frunze street\\
450074 Ufa, Russia }
\email{saks@ic.bashedu.ru}

\address{Carmen Judith Vanegas \hfill\break
Department of Mathematics \\
Universidad Sim\'{o}n Bol\'{\i}var \\
Sartenejas-Edo. Miranda \\
P. O. Box 89000, Venezuela}
\email{cvanegas@usb.ve}

\date{}
\thanks{Published May 30, 2005.}
\subjclass[2000]{35P05, 35J25, 35J55}
\keywords{Spectral theory; partial differential operators; \hfill\break\indent
boundary value problems; elliptic equations}

\begin{abstract}
 In this work, we give method to construct an explicit solutions
 to one spectral problem
 with  the $\mathop{\rm curl} $ operator on a bounded cylinder.
 The eigen-values of this operator are square roots of
 eigen-values of Laplace operator (with Dirichlet boundary
 condition) and zero.
 The eigen-functions related to this problem are found using  some
 results from complex analysis.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

The eigenvalue problem for the  $\mathop{\rm curl}$ operator has
important applications in plasma physics, where the
eigen-functions of the $\mathop{\rm curl}$ operator are called
free-decay fields. In \cite{chaken} the free-decay fields have
been found as the sum of a poloidal and a toroidal vector fields
using a quite particular method. In the theory of fusion plasma, a
free-decay field is called Taylor state which is considered the
final state that makes the energy a minimum in order to leave the
plasma in equilibrium \cite{tay}. The free-decay fields play also
an important roll to study turbulence in plasma \cite{mtv}. From a
mathematical point of view, we have the studies in \cite{yogi} and
\cite{sak2}.  In \cite{yogi} spectral properties of $\mathop{\rm
curl}$ operator in various function spaces is considered, while in
\cite{sak2} the eigenvalue problem for the $\mathop{\rm curl}$
operator on periodic vector functions is studied. From this same
point of view, we study the spectrum of the $\mathop{\rm curl} $
operator on a bounded cylinder
\[
G:= \{(x_1, x_2, x_3)\in \mathbb{R}^3 : x_1^2+x_2^2 < R^2,\;0\leq x_3 \leq l\}.
\]
We consider the eigenvalue problem
\begin{gather}\label{eqp 1}
\mathop{\rm curl} \textbf{u} = \lambda \textbf{u} \quad \mbox{in } G, \\ 
\label{eqp 2} u_3 \big |_{\partial G}  =  0, \quad u_2
\big|_{\gamma} = 0, \quad u_1 \big |_p = 0\,.
\end{gather}
Here $\textbf{u} = (u_1(\textbf{x}), u_2(\textbf{x}),
u_3(\textbf{x}))$ is a vector value function of a class
$C^{1}(G)\cap C(\overline{G})$, $\partial G$ is the boundary of
$G$, $\gamma$ is the circle of radius $R$ in the plane $x_3 = 0$,
and $p$ is an arbitrary fixed point taken on $\gamma$. In this
work $C^\alpha (G) $ denotes the space of $\alpha$-H\"older
continuous functions defined on $G$ and $C^{k,\alpha}(G)$ denotes
the space of functions defined on $G$ and possessing there
($\alpha$-H\"older) continuous derivatives up to order $k$. The
subspace of $C^{k,\alpha}(G)$ consisting of all the functions of
compact support in $ G\setminus\partial G$ will be denoted by $
C^{k,\alpha}_o(G)$.

\section{The eigenvalues of the {\rm curl} operator}

If $\lambda = 0$,  $\mathop{\rm curl} \textbf{u} = 0$ on $G$, and
for $w_k \in C_0^{2}(G)$, $k= 1, 2, \dots $, then
\begin{gather*}
\textbf{u}_k =  \nabla w_k(\textbf{x}), \\ 
\textbf{ u}_k \big|_{\partial G} = 0\,;
\end{gather*}
i.e.,  we have infinitely many solutions $(0, \nabla w_k)$, $k= 1,
2, \dots $ of the eigenvalue problem (\ref{eqp 1}), (\ref{eqp 2}).

For the case  $\lambda \neq 0$ , we apply the  divergence and curl
operators to (\ref{eqp 1}), and obtain
 $$ \mathop{\rm div}
\textbf{u} = 0 , \quad \mbox{and} \quad -\Delta u_j = \lambda^2
u_j, \quad \mbox{for }  j=1, 2, 3. $$

 For $j = 3 $, let us
consider  the eigenvalue problem
\begin{gather}\label{u3}
-\Delta u_3  =  \lambda^2 u_3 \quad \mbox{in }  G, \\
 \label{u3 f}
u_3  =  0 \quad \mbox{on } \partial G.
\end{gather}
The solutions of this problem are well known. We shift to
cylindrical coordinates $(r,\theta, z)$ obtaining, after a
straightforward calculation, the  eigenvalues $\lambda_{\kappa}^2$
with
\begin{equation}\label{autova}
\lambda_{\kappa}= \sqrt{\frac{\rho^2_{k, j}}{R^2} + \frac{m^2
\pi^2}{l^2}}, \quad k=0, 1,\dots ; \; j= 1, 2, \dots ;\;  m= 1,2,
\dots ,
\end{equation}
where $ \kappa=(k,j,m) $ is multi-index and $\rho_{k, j}$ are the
positive roots of the Bessel function $J_k(z) $.

 The corresponding
eigen-functions are
\begin{equation}\label{autofu}
u^\kappa_3 = \frac{\sqrt{2}}{\sqrt{l\pi}\,R\,|J'_k(\rho_{kj})|}
\,J_k(\rho_{kj}\frac{r}{R})\exp(ik\theta)\sin(\frac{m\pi}{l}z).
\end{equation}
As  $\lambda_{\kappa}^2>0$ the real and imaginary parts of
$u^\kappa_3$ are eigen-functions also and they form the real
orthonormal basis in the space $L_2(G)$.

 We have the following result.


\begin{theorem} \label{thm1}
The eigenvalue $\lambda = 0$ of  problem \eqref{eqp 1}-\eqref{eqp
2} has an infinite multiplicity. In the case $\lambda\ne 0$, we
find the eigenvalues $\pm \lambda_{\kappa}$ for 
\eqref{eqp 1}-\eqref{eqp 2}   through the eigenvalues $\lambda_{\kappa}^2$ of
the problem \eqref{u3}-\eqref{u3 f}); and $\lambda_{\kappa}$  is
given by \eqref{autova}; the multiplicities of $\lambda_{\kappa}$
and $-\lambda_{\kappa}$ are equal and finite. The spectrum for
problem \eqref{eqp 1}-\eqref{eqp 2} is a point spectrum and does
not have finite limits point.
\end{theorem}

\section{Determining eigen-functions via complex analysis}

In this section, we obtain the eigen-functions of  problem
(\ref{eqp 1})-(\ref{eqp 2}) using some results from complex
analysis. We  write the eigen-functions as $u^\kappa=(u_1^\kappa,
u_2^\kappa, \underline{u}_3^\kappa)$, where
$\underline{u}_3^\kappa$ has been already found in the former
section and it is the real (or imaginary) part of $u^\kappa_3$
represented by the expression (\ref{autofu}). To find $u_1^\kappa$
and $u_2^\kappa$ we consider the complex function
 $\omega= u^\kappa_2 + i\,u^\kappa_1$. After this definition and
 substituting $\lambda $
by $ \lambda_\kappa $ and $ u_3 $ by $\underline{ u}_3^\kappa $,
we obtain the following complex form of  (\ref{eqp 1}):
\begin{gather}\label{eqmega 1}
\partial_3 \omega - i\lambda_\kappa\omega = 2 i \partial_z \underline{u}^\kappa_3  \\
\label{eqmega 2} 2\mathop{\rm Re}\partial_{\bar z}\omega -
\lambda_\kappa \underline{u}^\kappa_3  = 0\,,
\end{gather}
where   $z = x_1 + i\,x_2$ and $$\frac{\partial}{\partial z}=
\frac{1}{2}(\frac{\partial}{\partial x_1} -
i\,\frac{\partial}{\partial x_2})$$.

The general solution of the differential equation  (\ref{eqmega
1}) is
\begin{equation}\label{eqfc 1}
\omega(\textbf{x})= \omega^\kappa_0(\textbf{x})
+ \omega_1(\textbf{x'})\exp(i\lambda_\kappa x_3),\quad
\textbf{x'}=(x_1,x_2)\,,
\end{equation}
where $$ \omega^\kappa_0(\textbf{x})= 2i \int^{x_3}_0
\exp(i\lambda_\kappa (x_3-t))\partial_z
\underline{u}^\kappa_3(\textbf{x'},t)dt $$ and
$\omega_1(\textbf{x'}) $ is a function in $C^{2,\alpha}$ which
will be specified later on.

 We observe that
$\omega^\kappa_0(\textbf{x})\in C^{2,\alpha}(G)$ if $u^\kappa_3
\in C^{3,\alpha}(G) $.

On the left side of (\ref{eqmega 2}) we replace $\omega$ by the particular
solution $\omega^\kappa_0$ of (\ref{eqmega 1}) and this side will be called
$V_0$.
We will need the following lemma.

\begin{lemma} \label{lemma1}
If $\underline{u}^\kappa_3\in C^{3,\alpha}(G)$ is a solution of
\eqref{u3} (with $\lambda^2= \lambda_\kappa^2 $) on $G$ then $V_0$
satisfies, on $G$, the equation
\begin{equation}\label{lem 1}
\frac{\partial^2 V_0}{\partial x_3^2} + \lambda_\kappa^2 V_0 = 0.
\end{equation}
Moreover $V_0$ can be represented in the form
\begin{equation}
V_0(\textbf{x})=\mathop{\rm Re}(v_0(\textbf{x'})
\exp(i\lambda_\kappa x_3)), \label{lem 2}
\end{equation}
where
\[
v_0(\textbf{x'})=(V_0(\textbf{x}) - \frac{i}{\lambda_\kappa}
\frac{\partial V_0(\textbf{x})}{\partial x_3})\big |_{
x_3=0}=i(\partial_3\underline{u}^\kappa_3)\big |_{x_3=0}.
%exp(-i\lambda_\kappa )for
\]
\end{lemma}

\begin{proof}
When we apply the matrix differential operator
$\begin{pmatrix}
\partial_2 & -\partial_1\\
\partial_1 & \partial_2
\end{pmatrix}$, where $\partial_i = \frac{\partial}{\partial\, x_i}$
for $i=1,2$, to the first two equations of system $\mathop{\rm
curl} \underline{u} = \lambda \underline{u} $,
 where $ \lambda=\lambda_\kappa $, $\underline{u} = (\underline{u}^\kappa_1, \underline{u}^\kappa_2,
 \underline{u}_3^ {\kappa})$
and $\omega^\kappa_0 = \underline{u}^\kappa_2 + i
\underline{u}^\kappa_1$, and taking into account (\ref{u3}), we
obtain the equations
\begin{gather*}
-\partial_3 (\lambda \mathop{\rm div} \underline{u} ) + \lambda^2
V_0 = 0
\\
\partial_3V_0 + \lambda \mathop{\rm div} \underline{u} = 0\,.
\end{gather*}
Because $V_0$ and $\partial_3V_0 $ belong to the space $C^{1,\alpha}(G)$,
we obtain the desired result.
\end{proof}

Substituting the $\omega$ given by (\ref{eqfc 1}) in the left side of
(\ref{eqmega 2}),  we obtain
$$
2\mathop{\rm Re}\partial_{\bar z} (\omega_1(\textbf{x'})
\exp(i\lambda_\kappa x_3)) + V_0 = 0.
$$
Using  representation (\ref{lem 2}),  this last equation can also be
written as
$$
\mathop{\rm Re}((2\partial_{\bar z}\omega_1(\textbf{x'})
+ v_0(\textbf{x'}))\exp(i\lambda_\kappa x_3))=0,
$$
from which it follows
\begin{equation}\label{eqla}
2\partial_{\bar z}\omega_1(\textbf{x'}) + v_0(\textbf{x'})=0.
\end{equation}
Therefore, the solvability condition for the last equation of system
(\ref{eqp 1}) is
\begin{equation}\label{solvabi}
\partial_{\bar z}\omega_1(\textbf{x'}) = - \frac{1}{2} v_0(\textbf{x'})
\end{equation}
It  is known \cite{BCT} that the general solution of the inhomogeneous
Cauchy-Riemann system (\ref{solvabi}) is
\begin{equation}\label{solcr}
\omega_1(\textbf{x'})= \Phi(z) - \frac{1}{2\pi} \int_\mathcal{D}
\frac{v_0(\xi, \eta)}{z-\zeta} d\xi d\eta, \quad \zeta=\xi+i\eta,
\end{equation}
where $\mathcal{D}$ is the disc  $ |\textbf{x}|< R , x_3 = 0 $ and
$\Phi(z)$ is a function in $ C^{2, \alpha} (\bar {\mathcal{D}}) $,
holomorphic in $\mathcal{D}$ which will be determined soon.

Taking (\ref{eqfc 1}) and (\ref{solcr}) into account  we have
\begin{equation}\label{soluw}
\omega(\textbf{x})= \omega^\kappa_2(\textbf{x})
+ \Phi(z)\exp(i\lambda_\kappa x_3)
\end{equation}
where $$ \omega^\kappa_2(\textbf{x})= \omega^\kappa_0(\textbf{x})
+ \Big(\frac{1}{2\pi} \int_\mathcal{D} \frac{v_0(\xi, \eta)}{\zeta
- z} d\xi d\eta \Big )\exp(i\lambda_\kappa x_3) $$ which belongs
to  $C^{2,\alpha}(G)\cap C^{\alpha}(\bar G)$ \cite{vla}. If
$x_3=0$ the function  $\omega^\kappa_0(\textbf{x})=0$ by
definition. Since $u^\kappa_2 + iu^\kappa_1 = \omega(\textbf{x})$,
according to the solution (\ref{soluw}), we obtain
\begin{gather}\label{rephi}
\mathop{\rm Re}\Phi(z)  =  - \mathop{\rm
Re}\omega^\kappa_2({x',0}) \quad \mbox{on } \gamma \\
\label{imphi} \mathop{\rm Im}\Phi(z)\Big|_p  =  - \mathop{\rm
Im}\omega^\kappa_2({p}), \quad \mbox{for }  p=(p_1, p_2, 0)\in
\gamma.
\end{gather}
Now we use  (\ref{rephi}) and (\ref{imphi}), and the Schwartz
Formula \cite{musk} to specify the function $\Phi(z)$: $$ \Phi(z)
=-\frac{1}{2\pi i}\int_{\gamma} \mathop{\rm Re}
\omega^\kappa_2({t,0})\frac{t+z}{t-z}\frac{dt}{t} + i\,C_1. $$
where $$ C_1 = - \mbox{Im}\Big[\omega^\kappa_2({p',0})+
\frac{1}{2\pi i}\int_{\gamma}\mathop{\rm Re}\omega^\kappa_2({t,0})
\frac{t+z}{t-z}\frac{dt}{t} \Big], \quad \mbox{for} \quad z=
p_1+ip_2. $$

Therefore, we have
\begin{gather}\label{solu1}
u^\kappa_1 (\textbf{x})  =  \mathop{\rm Im}\big[\omega^\kappa_2(\textbf{x})
+ \Phi(z)\exp(i \lambda_\kappa x_3) \big]\\
\label{solu2}
u^\kappa_2 (\textbf{x})  =  \mathop{\rm Re}\big[\omega^\kappa_2(\textbf{x})
+ \Phi(z)\exp(i \lambda_\kappa x_3) \big]\,.
\end{gather}

Then we arrive to the following result.

\begin{theorem} \label{thm2}
The components of the vector value eigen-function
$\textbf{u}^\kappa$ of problem \eqref{eqp 1}-\eqref{eqp 2}
associated to a positive eigen-value $\lambda_\kappa$ expressed by
\eqref{autova} are given by \eqref{solu1}, \eqref{solu2} and real
(or imaginary) part of \eqref{autofu} respectively. Moreover, if
we replace $\lambda_\kappa$ by $-\lambda_\kappa$ in \eqref{eqfc
1},..., \eqref{solu1}, \eqref{solu2}, we obtain the eigen-function
$\textbf u_-^\kappa$  of problem \eqref{eqp 1}-\eqref{eqp 2}
associated to a negative eigen-value $-\lambda_\kappa$.
\end{theorem}
%The same assertion is true for
% eigen-value $-\lambda_$.
Thus, we have shown: on one side for any solution
$(\lambda_\kappa,\textbf{u}^\kappa)$  of the problem \eqref{eqp
1}-\eqref{eqp 2} a pair $(\nu_\kappa,v^\kappa)$ is a solution of
the problem \eqref{u3}~-\eqref{u3 f}, where
$\nu_\kappa=\lambda^2_\kappa$ and $v^\kappa=(\textbf{u}^\kappa,
\textbf{e}_3)= u_3^\kappa$   is a projection  of the
vector-function $\textbf{u}^\kappa$ on the ax  of cylinder
$\textbf{e}_3$. Evidently,  if the pair
$(-\lambda_\kappa,\textbf{u}^\kappa_-)$  is another solution  of
the problem \eqref{eqp 1}-\eqref{eqp 2} a pair
$(\nu_\kappa,v^\kappa_-)$    is also a solution of the problem
\eqref{u3}- \eqref{u3 f} (with same $\nu_\kappa=\lambda^2_\kappa$
and $v^\kappa_-=(\textbf{u}^\kappa_-, \textbf{e}_3)\neq v^\kappa$
in general case).

On other side, for any solution  $(\nu_\kappa,v^\kappa)$ of the
problem \eqref{u3}- \eqref{u3 f} (with a real function
$v^\kappa$), we have constructed two
 solutions $(\lambda_\kappa,\textbf{u}^\kappa)$
   and $(-\lambda_\kappa,\textbf{u}^\kappa_-)$ of the problem \eqref{eqp
1}-\eqref{eqp 2} such that $ \lambda_\kappa=\sqrt {\nu_\kappa}$
and $(u^\kappa,\textbf{e}_3)=(\textbf{u}^\kappa_-, \textbf{e}_3)=
v^\kappa $.

Now Theorem 2.1 follows from these relations and properties of
eigen-values of the problem \eqref{u3}-\eqref{u3 f} (see
\cite{vla}, f.e.).


\noindent\textbf{Remark.} Later (in 2004) we have calculated the
components of eigen-functions $\textbf{u}^\kappa$ and
$\textbf{u}^\kappa_-$ directly using  the series representation of
\eqref{autofu}, which in variables $(x_1,x_2,x_3)$  has the form
\begin{equation}\label{serautofu}
u^\kappa_3 = a_\kappa
(x_1+ix_2)^k\sin(\frac{m\pi}{l}x_3)\sum_{p=0}^\infty
\frac{(-1)^p\rho_{k,j}^{2p}(x_1^2+x_2^2)^p}{(2R)^{2p}p!(p+k)!}.
\end{equation}
These results will be published.

 A short review of the physical background for the eigenvalue
problem of the $\mathop{\rm curl}$ operator in a 3-dimensional
bounded domain, with another boundary condition, can be found in
\cite{yogi}.


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\bibitem{mtv} Montgomery D., Turnel L., Vahala G.;
  \emph{Three-dimensional magnetohydrodynamic
turbulence in cylindrical geometry}, Phys. Fluids 21 (1978), 757-764 .

\bibitem{musk} Muskhelishvili N. I.;
 \emph{Singular integral equations}, Noordhoff, Groningen - Holland (1953).

\bibitem{sak2} Saks R. S.;
 \emph{On the spectrum of the operator $\mathop{\rm curl}$}, Functional-Analytic
and Complex Methods, their Interactions, and Applications to Partial
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Kong, 58-63 (2001).

\bibitem{tay} Taylor J. B.;
 \emph{Relaxation of toroidal and generation of reverse magnetic fields},
Phys. Rev. Lett. 33 (1974), No. 19, 1139-1141.

\bibitem{BCT}  Tutschke W.;
  \emph{Complex methods in the theory of initial value problems},
Complex Methods for Partial Differential Equations,
Kluwer Academic Publishers, Dordrecht-Boston-London (1999).

\bibitem{vla} Vladimirov V. S.,
 \emph{Equations of mathematical Physics}, Marcel Dekker, INC., New York (1971).

\bibitem{yogi} Yoshida Z., Giga Y.;
 \emph{Remarks on spectra of operator rot}, Math. Z. 204 (1990),
235-245 .

\end{thebibliography}

\end{document}