A-harmonic equation and cavitation


  • Vladimir Gutlyanskii NAS of Ukraine, Institute of Applied Mathematics and Mechanics
  • Olli Martio University of Helsinki, Department of Mathematics and Statistics
  • Vladimir Ryazanov NAS of Ukraine, Institute of Applied Mathematics and Mechanics


Cavitation, harmonic factorization, quasiconformal maps with singularity


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.



How to Cite

Gutlyanskii, V., Martio, O., & Ryazanov, V. (2023). A-harmonic equation and cavitation. Annales Fennici Mathematici, 48(1), 277–297. https://doi.org/10.54330/afm.127639