Geodesic
A-harmonic equation and cavitation
Vladimir Gutlyanskii1 ; Olli Martio2 ; Vladimir Ryazanov1
1 NAS of Ukraine, Institute of Applied Mathematics and Mechanics
2 University of Helsinki, Department of Mathematics and Statistics
Annales Fennici Mathematici, Tome 48 (2023) no. 1, pp. 277-297.

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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.
DOI : 10.54330/afm.127639
Keywords: Cavitation, harmonic factorization, quasiconformal maps with singularity

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 
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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. http://geodesic.mathdoc.fr/articles/10.54330/afm.127639/

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