Exotic Drums and the Helmholtz Equation

Suppose we have a drum whose membrane is a connected subset M of R^2 with
a piecewise smooth boundary, and we assume that the displacement of the
membrane f(x,y,t) obeys a lossless, dispersionless wave equation:

lap f = (1/v^2) @^2 f / @t^2

where "lap" is the 2-dimensional Laplacian. The boundary condition is
that f is zero on the boundary of M.

If the drum has any pure tones, there will be solutions for f that
separate into the form:

f(x,y,t) = F(x,y) cos(omega t)

and F will be a solution of the Helmholtz equation:

lap F = -k^2 F

where k = omega / v. F will be zero on the boundary of M.

Now, I know how to find analytical solutions of the Helmholtz equation
when the boundary is "nice" -- when it consists of curves of constant
coordinate values, for some coordinate system in which the equation is
separable. So rectangles, circles, annuli, and wedges are all easy to
find solutions for.

In those nice cases, there is a very simple pattern to the modes: for
each coordinate, there can be any positive integer number of peaks in the
standing wave across each coordinate direction. So a rectangular
membrane has modes consisting of n peaks in one direction and m peaks in
the other, with the lowest-frequency mode containing just a single peak,
and F(x,y) is zero only on the boundary.

My two questions are:

(1) Is there an existence theorem which says that there *must* be
solutions of the Helmholtz equation for an arbitarily-shaped membrane?
In other words, are there always eigenfunctions of the Laplacian with
real, negative eigenvalues which are zero on any given piecewise smooth
simple closed curve? Or is it possible to build a drum which (even under
our idealised assumptions) is physically incapable of producing a pure
tone?

(2) If/when solutions exists, must there always be one solution which is
zero only on the boundary (as is certainly the case for "nice"
boundaries)?

Thanks.

.

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