2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela. Electronic Journal of Differential Equations, Conference 13, 2005, pp. 95–99. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

SOLUTION OF A SPECTRAL PROBLEM FOR THE CURL OPERATOR ON A CYLINDER

ROMEN SAKS, CARMEN JUDITH VANEGAS

Abstract. In this work, we give method to construct an explicit solutions to one spectral problem with the curl operator on a bounded cylinder. The eigenvalues 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.

1. Introduction

The eigenvalue problem for the curl operator has important applications in plasma physics, where the eigen-functions of the curl operator are called free-decay fields. In [1] 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 [5]. The free-decay fields play also an important roll to study turbulence in plasma [2]. From a mathematical point of view, we have the studies in [8] and [4]. In [8] spectral properties of curl operator in various function spaces is considered, while in [4] the eigenvalue problem for the curl operator on periodic vector functions is studied. From this same point of view, we study the spectrum of the curl operator on a bounded cylinder

G := {(x_{1}, x_{2}, x_{3}) ∈ ℝ^{3} : x_{1}^{2} + x_{2}^{2} < R^{2}, 0 ≤ x_{3} ≤ l}.

We consider the eigenvalue problem

curl u = λu in G,  (1.1)
u_{3}|_{∂G} = 0, u_{2}|_{γ} = 0, u_{1}|_{p} = 0 .  (1.2)

Here u = (u_{1}(x), u_{2}(x), u_{3}(x)) is a vector value function of a class C^{1}(G) ∩ C(\overline{G}), ∂G is the boundary of G, γ is the circle of radius R in the plane x_{3} = 0, and p is an arbitrary fixed point taken on γ. In this work C^{α}(G) denotes the space of α-Hölder continuous functions defined on G and C^{k,α}(G) denotes the space of functions defined on G and possessing there (α-Hölder) continuous derivatives up to order k. The subspace of C^{k,α}(G) consisting of all the functions of compact support in G \ ∂G will be denoted by C_{o}^{k,α}(G).

2000 Mathematics Subject Classification. 35P05, 35J25, 35J55.

Key words and phrases. Spectral theory; partial differential operators;

boundary value problems; elliptic equations.

\begin{matrix}c \\ ◯ \end{matrix}2005 Texas State University - San Marcos.

Published May 30, 2005.

2. The eigenvalues of the curl operator

If λ = 0, curl u = 0 on G, and for w_{k} ∈ C_{0}^{2}(G), k = 1, 2, …, then

u_{k} = ∇w_{k}(x),
u_{k}|_{∂G} = 0 ;

i.e., we have infinitely many solutions (0, ∇w_{k}), k = 1, 2, … of the eigenvalue problem (1.1), (1.2).

For the case λ ≠ 0 , we apply the divergence and curl operators to (1.1), and obtain

div u = 0, and − Δu_{j} = λ^{2}u_{j}, for j = 1, 2, 3.

For j = 3, let us consider the eigenvalue problem

−Δu_{3} = λ^{2}u_{3} in G,  (2.1)
u_{3} = 0 on ∂G.  (2.2)

The solutions of this problem are well known. We shift to cylindrical coordinates (r, θ, z) obtaining, after a straightforward calculation, the eigenvalues λ_{κ}^{2} with

λ_{κ} = \sqrt{\frac{ρ_{k,j}^{2}}{R^{2}} + \frac{m^{2}π^{2}}{l^{2}}} , k = 0, 1, …; j = 1, 2, …; m = 1, 2, …,  (2.3)

where κ = (k, j, m) is multi-index and ρ_{k,j} are the positive roots of the Bessel function J_{k}(z).

The corresponding eigen-functions are

u_{3}^{κ} = \frac{\sqrt{2}}{\sqrt{lπ} R |J′_{k}(ρ_{kj})|} J_{k}(ρ_{kj} \frac{r}{R}) exp(ikθ) sin(\frac{mπ}{l} z).  (2.4)

As λ_{κ}^{2} > 0 the real and imaginary parts of u_{3}^{κ} are eigen-functions also and they form the real orthonormal basis in the space L_{2}(G).

We have the following result.

Theorem 2.1. The eigenvalue λ = 0 of problem (1.1)-(1.2) has an infinite multiplicity. In the case λ ≠ 0, we find the eigenvalues ±λ_{κ} for (1.1)-(1.2) through the eigenvalues λ_{κ}^{2} of the problem (2.1)-(2.2)); and λ_{κ} is given by (2.3); the multiplicities of λ_{κ} and −λ_{κ} are equal and finite. The spectrum for problem (1.1)-(1.2) is a point spectrum and does not have finite limits point.

3. Determining eigen-functions via complex analysis

In this section, we obtain the eigen-functions of problem (1.1)-(1.2) using some results from complex analysis. We write the eigen-functions as u^{κ} = (u_{1}^{κ}, u_{2}^{κ}, \underline{u}_{3}^{κ}), where \underline{u}_{3}^{κ} has been already found in the former section and it is the real (or imaginary) part of u_{3}^{κ} represented by the expression (2.4). To find u_{1}^{κ} and u_{2}^{κ} we consider the complex function ω = u_{2}^{κ} + i u_{1}^{κ}. After this definition and substituting λ by λ_{κ} and u_{3} by \underline{u}_{3}^{κ}, we obtain the following complex form of (1.1):

∂_{3}ω − iλ_{κ}ω = 2i∂_{z}\underline{u}_{3}^{κ}  (3.1)
2 Re ∂z̄ω − λ_{κ}\underline{u}_{3}^{κ} = 0 ,  (3.2)

where z = x_{1} + i x_{2} and

\frac{∂}{∂z} = \frac{1}{2}(\frac{∂}{∂x_{1}} − i \frac{∂}{∂x_{2}} )

.

The general solution of the differential equation (3.1) is

ω(x) = ω_{0}^{κ}(x) + ω_{1}(x’) exp(iλ_{κ}x_{3}), x’ = (x_{1}, x_{2}) ,  (3.3)

where

ω_{0}^{κ}(x) = 2i ∫_{0}^{x_{3}} exp(iλ_{κ}(x_{3} − t))∂_{z}\underline{u}_{3}^{κ}(x’, t)dt

and ω_{1}(x’) is a function in C^{2,α} which will be specified later on.

We observe that ω_{0}^{κ}(x) ∈ C^{2,α}(G) if u_{3}^{κ} ∈ C^{3,α}(G).

On the left side of (3.2) we replace ω by the particular solution ω_{0}^{κ} of (3.1) and this side will be called V_{0}. We will need the following lemma.

Lemma 3.1. If \underline{u}_{3}^{κ} ∈ C^{3,α}(G) is a solution of (2.1) (with λ^{2} = λ_{κ}^{2}) on G then V_{0} satisfies, on G, the equation

\frac{∂^{2}V_{0}}{∂x_{3}^{2}} + λ_{κ}^{2}V_{0} = 0.  (3.4)

Moreover V_{0} can be represented in the form

V_{0}(x) = Re(v_{0}(x’) exp(iλ_{κ}x_{3})),  (3.5)

where

v_{0}(x’) = (V_{0}(x) − \frac{i}{λ_{κ}} \frac{∂V_{0}(x)}{∂x_{3}})|_{x_{3}=0} = i(∂_{3}\underline{u}_{3}^{κ})|_{x_{3}=0}.

Proof. When we apply the matrix differential operator (\begin{matrix}∂_{2} & −∂_{1} \\ ∂_{1} & ∂_{2} \end{matrix}), where ∂_{i} = \frac{∂}{∂ x_{i}}

for i = 1, 2, to the first two equations of system curl \underline{u} = λ\underline{u}, where λ = λ_{κ}, \underline{u} = (\underline{u}_{1}^{κ}, \underline{u}_{2}^{κ}, \underline{u}_{3}^{κ}) and ω_{0}^{κ} = \underline{u}_{2}^{κ} + i\underline{u}_{1}^{κ}, and taking into account (2.1), we obtain the equations

−∂_{3}(λ div \underline{u}) + λ^{2}V_{0} = 0
∂_{3}V_{0} + λ div \underline{u} = 0 .

Because V_{0} and ∂_{3}V_{0} belong to the space C^{1,α}(G), we obtain the desired result. □

Substituting the ω given by (3.3) in the left side of (3.2), we obtain

2 Re ∂z̄(ω_{1}(x’) exp(iλ_{κ}x_{3})) + V_{0} = 0.

Using representation (3.5), this last equation can also be written as

Re((2∂z̄ω_{1}(x’) + v_{0}(x’)) exp(iλ_{κ}x_{3})) = 0,

from which it follows

2∂z̄ω_{1}(x’) + v_{0}(x’) = 0.  (3.6)

Therefore, the solvability condition for the last equation of system (1.1) is

∂z̄ω_{1}(x’) = −\frac{1}{2}v_{0}(x’)  (3.7)

It is known [6] that the general solution of the inhomogeneous Cauchy-Riemann system (3.7) is

ω_{1}(x’) = Φ(z) − \frac{1}{2π} ∫_{D} \frac{v_{0}(ξ, η)}{z − ζ}dξdη, ζ = ξ + iη,  (3.8)

where D is the disc |x| < R, x_{3} = 0 and Φ(z) is a function in C^{2,α}(D̄), holomorphic in D which will be determined soon.

Taking (3.3) and (3.8) into account we have

ω(x) = ω_{2}^{κ}(x) + Φ(z) exp(iλ_{κ}x_{3})  (3.9)

where

ω_{2}^{κ}(x) = ω_{0}^{κ}(x) + (\frac{1}{2π} ∫_{D} \frac{v_{0}(ξ, η)}{ζ − z}dξdη) exp(iλ_{κ}x_{3})

which belongs to C^{2,α}(G) ∩ C^{α}(Ḡ) [7]. If x_{3} = 0 the function ω_{0}^{κ}(x) = 0 by definition. Since u_{2}^{κ} + iu_{1}^{κ} = ω(x), according to the solution (3.9), we obtain

Re Φ(z) = − Re ω_{2}^{κ}(x′, 0) on γ  (3.10)
Im Φ(z)|_{p} = − Im ω_{2}^{κ}(p), for p = (p_{1}, p_{2}, 0) ∈ γ.  (3.11)

Now we use (3.10) and (3.11), and the Schwartz Formula [3] to specify the function Φ(z):

Φ(z) = −\frac{1}{2πi} ∫_{γ} Re ω_{2}^{κ}(t, 0)\frac{t + z}{t − z} \frac{dt}{t} + i C_{1}.

where

C_{1} = −Im[ω_{2}^{κ}(p′, 0) + \frac{1}{2πi} ∫_{γ} Re ω_{2}^{κ}(t, 0)\frac{t + z}{t − z} \frac{dt}{t}], for z = p_{1} + ip_{2}.

Therefore, we have

u_{1}^{κ}(x) = Im [ω_{2}^{κ}(x) + Φ(z) exp(iλ_{κ}x_{3})]  (3.12)
u_{2}^{κ}(x) = Re [ω_{2}^{κ}(x) + Φ(z) exp(iλ_{κ}x_{3})] .  (3.13)

Then we arrive to the following result.

Theorem 3.2. The components of the vector value eigen-function u^{κ} of problem (1.1)-(1.2) associated to a positive eigen-value λ_{κ} expressed by (2.3) are given by (3.12), (3.13) and real (or imaginary) part of (2.4) respectively. Moreover, if we replace λ_{κ} by −λ_{κ} in (3.3),..., (3.12), (3.13), we obtain the eigen-function u_{−}^{κ} of problem (1.1)-(1.2) associated to a negative eigen-value −λ_{κ}.

Thus, we have shown: on one side for any solution (λ_{κ}, u^{κ}) of the problem (1.1)- (1.2) a pair (ν_{κ}, v^{κ}) is a solution of the problem (2.1) -(2.2), where ν_{κ} = λ_{κ}^{2} and v^{κ} = (u^{κ}, e_{3}) = u_{3}^{κ} is a projection of the vector-function u^{κ} on the ax of cylinder e_{3}. Evidently, if the pair (−λ_{κ}, u_{−}^{κ}) is another solution of the problem (1.1)-(1.2) a pair (ν_{κ}, v_{−}^{κ}) is also a solution of the problem (2.1)- (2.2) (with same ν_{κ} = λ_{κ}^{2} and v_{−}^{κ} = (u_{−}^{κ}, e_{3}) ≠ v^{κ} in general case).

On other side, for any solution (ν_{κ}, v^{κ}) of the problem (2.1)- (2.2) (with a real function v^{κ}), we have constructed two solutions (λ_{κ}, u^{κ}) and (−λ_{κ}, u_{−}^{κ}) of the problem (1.1)-(1.2) such that λ_{κ} = \sqrt{ν_{κ}} and (u^{κ}, e_{3}) = (u_{−}^{κ}, e_{3}) = v^{κ}.

Now Theorem 2.1 follows from these relations and properties of eigen-values of the problem (2.1)-(2.2) (see [7], f.e.). Remark. Later (in 2004) we have calculated the components of eigen-functions u^{κ} and u_{−}^{κ} directly using the series representation of (2.4), which in variables (x_{1}, x_{2}, x_{3}) has the form

u_{3}^{κ} = a_{κ}(x_{1} + ix_{2})^{k} sin(\frac{mπ}{l} x_{3}) ∑_{p=0}^{∞} \frac{(−1)^{p}ρ_{k,j}^{2p}(x_{1}^{2} + x_{2}^{2})^{p}}{(2R)^{2p}p!(p + k)!} .  (3.14)

These results will be published.

A short review of the physical background for the eigenvalue problem of the curl operator in a 3-dimensional bounded domain, with another boundary condition, can be found in [8].

References

[1] Chandrasekhar S., Kendall P. C.; On force-free magnetic fields, Astrophys J. 126 (1957) p. 457-460.

[2] Montgomery D., Turnel L., Vahala G.; Three-dimensional magnetohydrodynamic turbulence in cylindrical geometry, Phys. Fluids 21 (1978), 757-764 .

[3] Muskhelishvili N. I.; Singular integral equations, Noordhoff, Groningen - Holland (1953).

[4] Saks R. S.; On the spectrum of the operator 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).

[5] Taylor J. B.; Relaxation of toroidal and generation of reverse magnetic fields, Phys. Rev. Lett. 33 (1974), No. 19, 1139-1141.

[6] Tutschke W.; Complex methods in the theory of initial value problems, Complex Methods for Partial Differential Equations, Kluwer Academic Publishers, Dordrecht-Boston-London (1999).

[7] Vladimirov V. S., Equations of mathematical Physics, Marcel Dekker, INC., New York (1971).

[8] Yoshida Z., Giga Y.; Remarks on spectra of operator rot, Math. Z. 204 (1990), 235-245 .

Romen Saks Bashkir State University, 32, Frunze street, 450074 Ufa, Russia

   E-mail address: saks@ic.bashedu.ru

Carmen Judith Vanegas Department of Mathematics, Universidad Simón Bolívar, Sartenejas-Edo. Miranda, P. O. Box 89000, Venezuela

   E-mail address: cvanegas@usb.ve