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Variational forms for the inverses of integral logarithmic operators over an interval

by C. Jerez-Hanckes and J.-C. Nédélec

(Report number 2010-21)

Abstract
We present explicit and exact variational formulations for the weakly singular and hypersingular operators over an open interval as well as for their corresponding inverses. Contrary to the case of a closed curve, these operators no longer map fractional Sobolev spaces in a dual fashion but degenerate into different subspaces depending on their extensibility by zero. We show that a symmetric and antisymmetric decomposition leads to precise coercivity results in fractional Sobolev spaces and characterize the mismatch occurring between associated functional spaces in this limiting case. Moreover, we naturally define Calderón-type identities in each case with potential use as preconditioners.

Keywords: Open surface problems; Laplace equation; integral logarithmic equations; boundary integral equations; Calderón projectors

@Techreport{JN10_427,
  author = {C. Jerez-Hanckes and J.-C. Nédélec},
  title = {Variational forms for the inverses of integral logarithmic operators over an interval},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2010-21},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-21.pdf },
  year = {2010}
}

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