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Pontryagin’s Maximum Principle for the Loewner Equation in Higher Dimensions
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- Journal:
- Canadian Journal of Mathematics / Volume 67 / Issue 4 / 01 August 2015
- Published online by Cambridge University Press:
- 20 November 2018, pp. 942-960
- Print publication:
- 01 August 2015
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- Article
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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, andWold, we then apply our version of the Pontryagin maximum principle to obtain first-order necessaryconditions for the extremal mappings for a wide class of extremal problems over the set of normalizedbiholomorphic mappings on the unit ball in
${{\mathbb{C}}^{n}}$ .
