part1

First draft(s) in attachment(s)

Part of the story is providing results for the fundamental mode analogous to an earlier paper on the elastic torsion problem

{1993c}

(with A. McNabb) The elastic torsion problem: solutions in convex domains. N.Z. Journal of Mathematics, 22, 43-64.

and, some results are recoverable from those of an even earlier paper

{1985b}

Power concavity of solutions of some semilinear elliptic boundary-value problems. Bull. Australian Math. Soc., 31, 181-184.

doi:10.1017/S0004972700004676

(The ANU-CMA Research Report has the details of the calculations, and this is more useful in the Helmholtz eq case than the summary that is in the journal article.)

Of course, there are umpteen related papers by others, and some of these I have checked.