Let us assume we have a unit disk $D\subset\mathbb{R}^2$ s.t. $\vec{0}\in D$. To obtain the eigenfrequencies and eigenmodes (or eigenvalues and -functions if you like) we must solve $\Delta \psi + \lambda^2 \psi = 0$ with appropriate boundary conditions.
This can be done analytically using the ansatz $\psi(r,\phi)=R(r)\Phi(\phi)$ to obtain Bessel's equation. After a long but simple derivation one obtains the roots of Bessel's functions of the first kind as eigenvalues.
Here's my problem then:
Assuming we start to rotate the disk in the plane around the origin will result in a moving boundary.
Can we still determine the eigenvalues of the unit disc analytically?
I know that one can obtain the eigenvalues employing coordinate transformations which will modify the equation, transforming it into a coordinate frame where the boundary is at rest and hence Dirichlet conditions may be used. But I'm interested specifically in an analytic solution where the differential equation is not modified.
Does such a solution exist?