A-harmonic equation and cavitation

Abstract

Suppose that \(f\) is a homeomorphism from the punctured unit disk \(D \setminus \{0\}\) onto the annulus \(A(r') = \{r' < |z| <1 \}\), \(r' \geq 0\), and \(f\) is quasiconformal in every \(A(r)\), \(r> 0\), but not in \(D\). If \(r' > 0\) then \(f\) has cavitation at \(0\) and no cavitation if \(r' = 0\). The singular factorization problem is to find harmonic functions \(h\) in \(A(r')\) 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.