I would like to solve the Biharmonic Equation
$$\Delta^2 \phi=0$$ on an elliptical domain $c_1<\frac{x^2}{a^2}+\frac{y^2}{b^2}<c_2$.
I converted the Biharmonic equation into $(R,\omega)$ coordinates, where $$R\cos(\omega)=\frac{x}{a},$$ $$R\sin(\omega)=\frac{y}{b}$$ In this new coordinate system the boundary conditions of the PDE correspond to $R=c_1$ and $R=c_2$ and are related the derivatives of $\phi$ with respect to $R$ and $\omega$.
In this new coordinate system I had a PDE where every term had the form $$\frac{C_{k,m}(\omega)}{R^{4-k}}\frac{\partial^{m+k}\phi}{\partial R^k\partial \omega^m}.$$
This looked like a Cauchy-Euler equation in $R$, so I guessed a solution of the form $$\phi(R,\omega)=R^nH(\omega).$$ When I made this substitution the PDE simplified to an ODE of the form
$$P_4(cos(2\omega))H^{(4)}+P_3(\cos(2\omega),\sin(2\omega))H'''+P_2(\cos(2\omega),\sin(2\omega))H''$$ $$+P_1(\cos(2\omega),\sin(2\omega))H' +P_0(\cos(2\omega))H=0,$$
where $$P_4(x)=c_{41}x+c_{42}x^2,$$ $$P_3(x,y)=c_{31}y+c_{32}xy,$$ $$P_2(x,y)=c_{21}+c_{22}y+c_{23}x+c_{33}xy+c_{34}x^2+c_{35}y^2,$$ $$P_1(x,y)=c_{11}y+c_{12}xy,$$ $$P_0(x)=c_{01}+c_{02}x+c_{03}x^2$$ and the coefficients depend on $n$, $a$, and $b$. If I can find $H$ in terms of $n$ and then I am hoping that I can write $$\phi(R,\omega)=\sum_{n\in K}\left(A_nR^nH_n(\omega)\right)$$ for some set $K$ (or maybe it might be an integral rather than a sum if there is an uncountable number of values of $n$ that I would use) and then I would solve for the $A_n$'s using my boundary conditions.
In summary, I want to know how to solve the biharmonic equation on an elliptical domain with a hole in it. Either I would like to know how to start from scratch if my attempt won't get me there, how to solve for $H$ in the ODE I wrote down, or if there is a source where they already solved my problem and where I can find their paper.
EDIT: I tried using elliptical coordinates as mattos suggested and I got the PDE
$$\left(\frac{\partial^4\phi}{d\mu^4}+\frac{\partial^4\phi}{\partial\mu^2d\nu^2}+\frac{\partial^4\phi}{\partial\nu^4}\right)\left(\cosh(2\mu)-\cos(2\nu)\right)-4\sinh(2\mu)\left(\frac{\partial^3\phi}{\partial\mu^3}+\frac{\partial^3\phi}{\partial\mu\partial\nu^2}\right)$$ $$-4\sin(2\nu)\left(\frac{\partial^3\phi}{\partial\mu^2\partial\nu}+\frac{\partial^3\phi}{\partial\nu^3}\right)$$ $$+4(\cosh(2\mu)+\cos(2\nu))\left(\frac{\partial^2\phi}{\partial\mu^2}+\frac{\partial^2\phi}{\partial\nu^2}\right)=0.$$ I can't use separation of variables because of the mixed derivative terms and the method of characteristics doesn't work well for elliptic PDEs so I'm not sure how to proceed (the only two methods for solving PDEs that I know of are separation of variables and the method of characteristics)