Explicit Variational Forms for the Inverses of Integral Logarithmic Operators Over an Interval

We introduce 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 degene...

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Bibliographic Details
Published in:SIAM journal on mathematical analysis 2012-01, Vol.44 (4), p.2666-2694
Main Authors: Jerez-Hanckes, Carlos, Nedelec, Jean-Claude
Format: Article
Language:English
Subjects:
Coercive force
Copyright
Dirichlet problem
Extensibility
Integral equations
Integrals
Intervals
Mathematical analysis
Mathematics
Numerical Analysis
Operators
Sobolev space
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Summary:We introduce 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 an average and jump decomposition leads to precise coercivity results and characterize the mismatch occurring between associated functional spaces. Through this setting, we naturally define Calderon-type identities with their potential use as preconditioners. Moreover, we provide an interesting relation between the logarithmic operators and one-dimensional Laplace Dirichlet and Neumann problems. This work is a detailed and extended version of the article "Variational Forms for the Inverses of Integral Logarithmic Operators over an Interval" by Jerez-Hanckes and Nedelec [C.R. Acad. Sci. Paris Ser. I, 349 (2011), pp. 547--552]. [PUBLICATION ABSTRACT]
ISSN:0036-1410
1095-7154
DOI:10.1137/100806771