  An analytical perturbative method is suggested for solving the Helmholtz
equation (\bigtriangledown^{2} + k^{2}){\psi} = 0 in two dimensions where
{\psi} vanishes on an irregular closed curve. We can thus find the energy
levels of a quantum mechanical particle confined in an infinitely deep
potential well in two dimensions having an irregular boundary or the vibration
frequencies of a membrane whose edge is an irregular closed curve. The method
is tested by calculating the energy levels for an elliptical and a
supercircular boundary and comparing with the results obtained numerically.
Further, the phenomenon of level crossing due to shape variation is also
discussed.
