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Shape Holomorphy of the Calderón Projector for the Laplacean in R2

by F. Henriquez and Ch. Schwab

(Report number 2019-43)

Abstract
We establish the holomorphic dependence of the boundary integral operators (BIOs) comprising the Calderón projector for Laplacean in two dimensions on the shape of the boundary. 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 \(\mathscr{C}^2\)-smooth Jordan curves in \(\mathbb{R}^2\). 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 \(\mathscr{C}^2\) 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 surrogates of polynomial chaos type, and for dimension-independent convergence rates for the approximation of parametric solution families of BIEs in forward and inverse computational shape uncertainty quantification.

Keywords: Shape Holomorphy, Boundary Integral Operators, Boundary Integral Equations, Uncertainty Quantification

@Techreport{HS19_847,
  author = {F. Henriquez and Ch. Schwab},
  title = {Shape Holomorphy of the Calderón Projector for the Laplacean in R2},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-43},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-43.pdf },
  year = {2019}
}

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