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So I have a problem which (in one case) leads me to the following vector Laplace equation:

$\nabla^2 \mathbf{A} = 0$

with $\mathbf{A}$ the magnetic vector potential, whereon I have imposed the Coulomb gauge $\nabla\cdot\mathbf{A} = 0$. The problem contains cylindrical symmetry, which is why I want to solve this in cylindrical coordinates. Boundary conditions are to follow from solving the homogeneous Helmholtz equation and demanding continuity at a cylindrical surface.
However, I need to solve the Laplace equation first, which I don't know how to do. If the coordinates were cartesian, the vector equation would just be equivalent to three scalar equations, but now it's not so easy. Does anyone feel like explaining to me how I would best go about solving this vector Laplace equation in cylindrical coordinates? Greatly appreciated!

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    Your vector Laplace equations are three independent equations (without the gauge condition). So I would take the solutions of the scalar Laplace equations and then see what kind of relation you obtain due to the gauge condition.2012-12-07
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    I'm pretty sure you can get a nice Fourier series solution using separation of variables.2012-12-07
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    @Fabian: the gauge condition is necessary to obtain this Laplace equation and I didn't plan on using it anywhere else, I just thought I'd mention it to be complete. Unfortunately, according to this wolfram page (http://mathworld.wolfram.com/VectorLaplacian.html) the equations for the radial and the angular component of the vector are not independent. They are coupled, which is what's making things difficult for me.2012-12-08
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    @Matt: are you sure about that? Again referring to the same wolfram page (http://mathworld.wolfram.com/VectorLaplacian.html), the equations for the radial and the angular component of the vector are coupled. So separation of variables is not possible, right?2012-12-08
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    @Wouter: yes they are coupled by the gauge condition. But still I would argue that you should solve the problem without the gauge condition (which you can do explicitly) and then see what kind of relations between the different modes you get due to the gauge condition.2012-12-08
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    @Fabian: let me be more clear about what I did. Originally the equation was rotrot(A) = 0. So I used the gauge condition to get to this Laplace equation. I don't see how it is coupling the equations?2012-12-08
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    The solution of this problem depends on the boundary conditions. If the cylinder is infinitely long then you have the problem of a waveguide. In the waveguide the vector nature of the problem is solve by knowing that one can solve the problem using [TE and TM modes](http://en.wikipedia.org/wiki/Transverse_mode). The whole solution is rather cumbersome and I would like to refer you to the literature. Jackson discusses the problem of a cylindrical waveguide and cavity in chaper 8 of his book `classical electrodynamics'.2012-12-08
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    @Fabian: I'll take a look at Jackson's book, thanks for the reference. But perhaps I should have said the external magnetic field is along the z-axis and the cylinder lies along the z-axis as well, so it's not really a waveguide problem. The physical context is that of a superconducting nanowire along z in an external magnetic field along z, which I am describing using the Ginzburg-Landau equations. This Laplace equation is derived from the second GL equation with the assumption that the order parameter is zero outside and constant inside the nanowire (and the regime is stationary).2012-12-08

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