\documentclass[reqno]{amsart} \usepackage{hyperref} \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 $$\label{autova} \lambda_{\kappa}= \sqrt{\frac{\rho^2_{k, j}}{R^2} + \frac{m^2 \pi^2}{l^2}}, \quad k=0, 1,\dots ; \; j= 1, 2, \dots ;\; m= 1,2, \dots ,$$ 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 $$\label{autofu} u^\kappa_3 = \frac{\sqrt{2}}{\sqrt{l\pi}\,R\,|J'_k(\rho_{kj})|} \,J_k(\rho_{kj}\frac{r}{R})\exp(ik\theta)\sin(\frac{m\pi}{l}z).$$ 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 $$\label{eqfc 1} \omega(\textbf{x})= \omega^\kappa_0(\textbf{x}) + \omega_1(\textbf{x'})\exp(i\lambda_\kappa x_3),\quad \textbf{x'}=(x_1,x_2)\,,$$ 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 $$\label{lem 1} \frac{\partial^2 V_0}{\partial x_3^2} + \lambda_\kappa^2 V_0 = 0.$$ Moreover $V_0$ can be represented in the form $$V_0(\textbf{x})=\mathop{\rm Re}(v_0(\textbf{x'}) \exp(i\lambda_\kappa x_3)), \label{lem 2}$$ 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 $$\label{eqla} 2\partial_{\bar z}\omega_1(\textbf{x'}) + v_0(\textbf{x'})=0.$$ Therefore, the solvability condition for the last equation of system (\ref{eqp 1}) is $$\label{solvabi} \partial_{\bar z}\omega_1(\textbf{x'}) = - \frac{1}{2} v_0(\textbf{x'})$$ It is known \cite{BCT} that the general solution of the inhomogeneous Cauchy-Riemann system (\ref{solvabi}) is $$\label{solcr} \omega_1(\textbf{x'})= \Phi(z) - \frac{1}{2\pi} \int_\mathcal{D} \frac{v_0(\xi, \eta)}{z-\zeta} d\xi d\eta, \quad \zeta=\xi+i\eta,$$ 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 $$\label{soluw} \omega(\textbf{x})= \omega^\kappa_2(\textbf{x}) + \Phi(z)\exp(i\lambda_\kappa x_3)$$ 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 $$\label{serautofu} u^\kappa_3 = a_\kappa (x_1+ix_2)^k\sin(\frac{m\pi}{l}x_3)\sum_{p=0}^\infty \frac{(-1)^p\rho_{k,j}^{2p}(x_1^2+x_2^2)^p}{(2R)^{2p}p!(p+k)!}.$$ 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}. \begin{thebibliography}{0} \bibitem{chaken} Chandrasekhar S., Kendall P. C.; \emph{On force-free magnetic fields}, Astrophys J. 126 (1957) p. 457-460. \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 Differential Equations, World Scientific, New Jersey-London-Singapore-Hong 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}