Consider below equation in an ellipsoid:
$\nabla^2 f(x,y) = (1 + f(x,y))$
Knowing how to write this equation in polar coordinate with symmetries of a circle:
$\partial_r f(r)/r + \partial_r^2 f(r) = (1 + f(r))$
But I don't know in the case of an ellipse with elliptical symmetries, how can this equation be written. Could any one help? I know that laplacian in elliptical coordinate can be written as:
$\frac{(\partial_u^2+\partial_v^2)}{a^2(sinh^2 u + sin^2 v)}$
But I don't know how to apply symmetry of an ellipse to simplify it. Any help is highly appreciated.
In elliptic coordinates ($a>0$ fixed, $u\geq0$ and $0\leq v\leq2\pi$), the equation $\nabla^2 f(x,y) = (1 + f(x,y))$ can be written
$$\frac{1}{a^2(\sinh^2 u + \sin^2 v)}\frac{\partial^2 f}{\partial u^2} + \frac{1}{a^2(\sinh^2 u + \sin^2 v)}\frac{\partial^2 f}{\partial v^2} = 1 + f,$$
using the formula for the Laplacian in elliptic coordinates that you correctly identified. This cannot be simplified for general $f=f(u,v)$, except in special cases such as $f=f(u)$, $f=f(v)$, or $\frac{\partial^2 f}{\partial u^2} + \frac{\partial^2 f}{\partial v^2} = 0$.