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.
Keywords:
Cavitation, harmonic factorization, quasiconformal maps with singularity
Affiliations des auteurs :
Vladimir Gutlyanskii
1
;
Olli Martio
2
;
Vladimir Ryazanov
1
1
NAS of Ukraine, Institute of Applied Mathematics and Mechanics
2
University of Helsinki, Department of Mathematics and Statistics
Vladimir Gutlyanskii; Olli Martio; Vladimir Ryazanov. A-harmonic equation and cavitation. Annales Fennici Mathematici, Tome 48 (2023) no. 1, pp. 277-297. doi: 10.54330/afm.127639
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title = {A-harmonic equation and cavitation},
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