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Shape Holomorphy of the Calderón Projector for the Laplacian in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^2$$\end{document}

Authors
  • Henríquez, Fernando1
  • Schwab, Christoph1
  • 1 Seminar for Applied Mathematics ETH Zurich, Raemistrasse 101, Zurich, 8092, Switzerland , Zurich (Switzerland)
Type
Published Article
Journal
Integral Equations and Operator Theory
Publisher
Springer International Publishing
Publication Date
Jul 03, 2021
Volume
93
Issue
4
Identifiers
DOI: 10.1007/s00020-021-02653-5
Source
Springer Nature
Keywords
Disciplines
  • Article
License
Yellow

Abstract

We establish the holomorphic dependence of the boundary integral operators (BIOs) comprising the Calderón projector for the Laplacian in two dimensions on the boundary shape. More precisely, we show that the Calderón projector, as an element of the Banach space of bounded linear operators satisfying suitable mapping properties, depends holomorphically on a set of boundaries given by a collection of C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {C}}^2$$\end{document}–regular Jordan curves in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^2$$\end{document}. In turn, this result implies that the solution of a well-posed first or second kind boundary integral equation (BIE) arising from the boundary reduction of the Laplace problem set on a domain of class C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {C}}^2$$\end{document} in two spatial dimensions depends holomorphically on the shape of the boundary, provided that the corresponding right-hand side does so as well. This property of shape holomorphy is of crucial significance to mathematically justify the construction of sparse parametric shape surrogates of polynomial chaos type, and to prove dimension-independent convergence rates for the approximation of parametric solution families of BIEs in forward and inverse computational shape uncertainty quantification.

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