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Shape Holomorphy of the Calderón Projector on Lipschitz Surfaces

by F. Henriquez and J. Dölz and J. Pinto and Ch. Schwab

(Report number 2026-42)

Abstract
We establish the holomorphic dependence of the Calderón projector for the Laplace operator upon shape boundary deformations of a family of bounded Lipschitz domains. More precisely, after transporting the four boundary integral operators (BIOs) contained in the Calderón projector for the Laplace problem to a fixed reference boundary, we prove that the map from the domain transformation to the resulting transported Calderón projector admits a bounded holomorphic extension, a property usually referred to as shape holomorphy. Previous approaches relied on intrinsic arguments for BIOs under boundary regularity of class \(\mathscr{C}^{1,1}\) or higher. Our argument traces the holomorphic dependence on the shape by transferring strong ellipticity of a differential operator to the boundary through fundamental solutions in ambient space and via holomorphic shape-dependence of the trace and conormal derivative operators. This allows us to consider all BIOs in the Calderón-projector as operators between their natural fractional Sobolev trace-spaces and enables Lipschitz-regular domain and boundary deformations. The holomorphic dependence on the Lipschitz-boundary provides the analytic foundation for applying high-dimensional approximation of the transported BIOs, their Galerkin discretizations, and the resulting boundary densities as well as underlying quantities of interest.

Keywords:

@Techreport{HDPS26_1197,
  author = {F. Henriquez and J. D\"olz and J. Pinto and Ch. Schwab},
  title = {Shape Holomorphy of the Calderón Projector on Lipschitz Surfaces},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-42},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-42.pdf },
  year = {2026}
}

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