How to prove uniqueness (or non-uniqueness) of solution

[JustinLevy]

I've only learned differential equations for use in physics, and never took a rigorous math course on all their amazing features. So I'm hoping someone can teach me a bit here, in the context of this question:

Consider Maxwell's equations in vacuum, units don't matter here so I'll get rid of all constants:
$$\nabla \cdot \vec{E} = 0$$
$$\nabla \cdot \vec{B} = 0$$
$$\nabla \times \vec{E} = - \frac{\partial}{\partial t} \vec{B}$$
$$\nabla \times \vec{B} = \frac{\partial}{\partial t} \vec{E}$$

Now consider a finite region of space, with the boundary condition that the fields and their derivatives are zero on the boundary at time 0<=t<T. What solutions are there for the fields in the region during this time?

One obvious solution is: E=0, B=0 everywhere.

Is this question well posed enough to prove that this solution is unique?
If so, how? If not, what is missing?

 

[JustinLevy]

Okay, I came up with another solution.

If we define, at t=0, an E field with no divergence, and let B=0. Then I can use Maxwell's equations to evolve the time dependence, right? So the problem is reduced to finding a finite volume E field with no divergence, which I don't see why that is a problem.

Using cylindrical coordinates, I can define:
$$\mathbf{E}(t_0) = \hat{\phi} f(r)g(z)$$
This field has no divergence.

Now looking at the time dependence
$$-\frac{\partial \mathbf{B}}{\partial t} = -\hat{r} f(r) \frac{\partial}{\partial z}g(z) + \hat{z} \frac{1}{r} g(z) \frac{\partial}{\partial}[r f(r)]$$
So B will have r and z components. But these components only depend r and z. So the curl of B will only have
$$\frac{\partial \mathbf{E}}{\partial t} = \nabla \times \mathbf{B} = \hat{\phi}(\frac{\partial B_r}{\partial z}- \frac{\partial B_z}{\partial r})$$
So E will remain in the phi direction, and so on for all time.

This is true for any function f(r) and g(z). So I can just choose a solution initially confined enough that it doesn't have time to propagate to the boundary.

Does this look correct?
To do this I'd need f(r) to be non-analytic (since it needs to be identically zero for a region of r). Is that somehow a problem?