×

Regularity of a boundary having a Schwarz function. (English) Zbl 0728.30007

Summary: In his book, P. J. Davis discussed various interesting aspects concerning a Schwarz function. It is a holomorpic function S defined in a neighbourhood of a real analytic arc satisfying \(S(\zeta)={\bar \zeta}\) on the arc, where \({\bar \zeta}\) denotes the complex conjugate of \(\zeta\). The author defines a Schwarz function for a portion of a boundary of an arbitrary open set and shows the regularity of the portion of the boundary. More precisely, let \(\Omega\) be an open set of the unit disk B such that the boundary \(\partial \Omega\) contains the origin 0 and let \(\Gamma =(\partial \Omega)\cap B\). A function S defined on \(\Omega\cup \Gamma\) is called the Schwarz function of \(\Omega\cup \Gamma\) if (i) S is holomorphic in \(\Omega\), (ii) S is continuous on \(\Omega\cup \Gamma\) and (iii) \(S(\zeta)={\bar \zeta}\) on \(\Gamma\). The author gives a classification of a boundary having a Schwarz function. The main theorem asserts that there are four types of the boundary if 0 is not an isolated boundary point of \(\Omega\) : 0 is a regular, nonisolated degenerate, double or cusp point of the boundary.

MSC:

30C35 General theory of conformal mappings
30C99 Geometric function theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ahlfors, L. V. &Sario, L.,Riemann Surfaces. Princeton Univ. Press, Princeton, 1960.
[2] Caffarelli, L. A. &Rivière, N. M., Smoothness and analyticity of free boundaries in variational inequalities.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 289–310. · Zbl 0363.35009
[3] – Asymptotic behaviour of free boundaries at their singular points.Ann. of Math. (2), 106 (1977), 309–317. · Zbl 0364.35041
[4] Collingwood, E. F. &Lohwater, A. J.,The Theory of Cluster Sets. Cambridge Univ. Press, Cambridge, 1966. · Zbl 0149.03003
[5] Davis, P. J.,The Schwarz Function, and its Applications. Carus Math. Monographs, No. 17. Math. Assoc. America, Washington, D.C., 1974. · Zbl 0293.30001
[6] Fuchs, W. H. J., A Phragmén-Lindelöf theorem conjectured, by D. J. Newman.Trans. Amer. Math. Soc., 267 (1981), 285–293. · Zbl 0472.30025
[7] Nevanlinna, R.,Eindeutige analytische Funktionen. 2nd ed., Springer, Berlin, 1953.
[8] Sario, L. &Nakai, M.,Classification Theory of Riemann Surfaces. Springer, Berlin, 1970. · Zbl 0199.40603
[9] Schaeffer, D. G., Some examples of singularities in a free boundary.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4 (1977), 133–144. · Zbl 0354.35033
[10] Shapiro, H. S., Unbounded quadrature domains. InComplex Analysis Vol. I Lecture Notes in Math. Vol. 1275. Springer, Berlin, 1987, pp. 287–331. · Zbl 0634.30037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.