A-harmonic equation and cavitation

Abstract

Suppose that $f$ is a homeomorphism from the punctured unit disk $D\setminus \{0\}$ 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 $h\circ f$ satisfies the elliptic PDE associated with $f$ with a singularity at $0$. Sufficient conditions in terms of the dilatation $K_{f^{-1}}(z)$ together with the properties of $h$ are given to the factorization problem, to the continuation of $h\circ f$ to $0$ and to the regularity of $h\circ f$. 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.