Maybe this is trivial. Let assume that U and W are polynomials (in the reals with real coeffs) defined in the unit disk, can we apply the Green´s second identity on the unit disk?, are not U and W non differentiable at the boundary?
Let say, W is one of the Zernike polynomials, i.e., orthonormal polynomials defined in the unit disk ($\rho\leq 1$), and are not defined for $\rho>1$. So has W a discontinuity at the boundary and wouldn't it be non differentiable?
U can be another one from a set of polynomials also defined in the unit disk, and zero beyond it. Wouldn't it be also non differentiable at the disk boundary?
Thanks for your time.