TY - JOUR
AU - Gutlyanskii, Vladimir
AU - Martio, Olli
AU - Ryazanov, Vladimir
PY - 2023/03/09
Y2 - 2023/05/31
TI - A-harmonic equation and cavitation
JF - Annales Fennici Mathematici
JA - Ann. Fenn. Math.
VL - 48
IS - 1
SE - Articles
DO - 10.54330/afm.127639
UR - https://afm.journal.fi/article/view/127639
SP - 277-297
AB - <pre>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.</pre>
ER -