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Pontryagin's maximum principle for the Loewner equation in higher dimensions

• Oliver Roth,
Department of Mathematics, University of Würzburg, Emil Fischer Straße 40, 97074 Würzburg, Germany
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Abstract

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: univalent function, Loewner's equation
 MSC Classifications: 32H02 - Holomorphic mappings, (holomorphic) embeddings and related questions 30C55 - General theory of univalent and multivalent functions 49K15 - Problems involving ordinary differential equations