A-harmonic equation and cavitation

Suppose that f is a homeomorphism from the punctured unit disk D \ {0} onto the annulus A(r & PRIME;) = {r & PRIME; < |z| < 1}, r & PRIME; & GE; 0, and f is quasiconformal in every A(r), r > 0, but not in D. If r & PRIME; > 0 then f has cavitation at 0 and no cavitation if r & PRIME; = 0. The singular factorization problem is to find harmonic functions h in A(r & PRIME;) such that h degrees f satisfies the elliptic PDE associated with f with a singularity at 0. Sufficient conditions in terms of the dilatation Kf-1(z) together with the properties of h are given to the factorization problem, to the continuation of h degrees f to 0 and to the regularity of h degrees f. We also give sufficient conditions for cavitation and non-cavitation in terms of the complex dilatation of f and demonstrate both cases with several examples.