The string equation for some rational functions
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Publication:2419335
DOI10.1007/978-3-030-02650-9_11zbMATH Open1418.30007arXiv1803.03020OpenAlexW2911741393MaRDI QIDQ2419335
Publication date: 13 June 2019
Abstract: For conformal maps defined in the unit disk one can define a certain Poisson bracket that involves the harmonic moments of the image domain. When this bracket is applied to the conformal map itself together with its conformally reflected map the result is identically one. This is called the string equation, and it is closely connected to the governing equation, the Polubarinova-Galin equation, for the evolution of a Hele-Shaw blob of a viscous fluid (or, by another name, Laplacian growth). In the present paper we investigate to what extent the string equation makes sense and holds for non-univalent analytic functions. We give positive answers in two cases: for polynomials and for a special class of rational functions.
Full work available at URL: https://arxiv.org/abs/1803.03020
Poisson bracketstring equationHele-Shaw flowPolubarinova-Galin equationharmonic momentsnon-univalent rational functions
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