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Pontryagin’s Maximum Principle for the Loewner Equation in Higher Dimensions

Published online by Cambridge University Press:  20 November 2018

Oliver Roth
Oliver Roth*
Affiliation:
Department of Mathematics, University of Würzburg, Emil Fischer Straβe 40, 97074 Würzburg, Germany e-mail: roth@mathematik.uni-wuerzburg.de
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Abstract

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In this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagin’s maximum principle from optimal control theory for the Loewner equation in several complex variables. Based on recent work of Arosio, Bracci, and Wold, we then apply our version of the Pontryagin maximum principle to obtain first-order necessary conditions for the extremal mappings for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball in ${{\mathbb{C}}^{n}}$.

Keywords

32H0230C5549K15Univalent functionsLoewner’s equation
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 4 , 01 August 2015 , pp. 942 - 960
Copyright
Copyright © Canadian Mathematical Society 2015

References

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