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The edge algebra structure of the Zaremba problem

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Abstract

We study mixed boundary value problems, here mainly of Zaremba type for the Laplacian within an edge algebra of boundary value problems. The edge here is the interface of the jump from the Dirichlet to the Neumann condition. In contrast to earlier descriptions of mixed problems within such an edge calculus, cf. (Harutjunjan and Schulze, Elliptic mixed, transmission and singular crack problems, 2008), we focus on new Mellin edge quantisations of the Dirichlet-to-Neumann operator on the Neumann side of the boundary and employ a pseudo-differential calculus of corresponding boundary value problems without the transmission property at the interface. This allows us to construct parametrices for the original mixed problem in a new and transparent way.

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Acknowledgments

The first draft of this paper was initiated when the authors visited the National Center for Theoretical Sciences, Hsinchu, Taiwan during January, 2013. They would like to express their profound gratitude to the Director of NCTS, Professor Winnie Li for her invitation and for the warm hospitality extended to them during their stay in Taiwan.

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Authors and Affiliations

  1. Department of Mathematics and Department of Computer Science, Georgetown University, Washington, DC, 20057, USA

    Der-Chen Chang

  2. Department of Mathematics, Fu Jen Catholic University, Taipei, 242, Taiwan, ROC

    Der-Chen Chang

  3. Institute of Mathematics, University of Potsdam, 14469 , Potsdam, Germany

    Nadia Habal & B.-W. Schulze

Authors
  1. Der-Chen Chang
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  2. Nadia Habal
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  3. B.-W. Schulze
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Correspondence to B.-W. Schulze.

Additional information

D.-C. Chang is partially supported by an NSF grant DMS-1203845 and Hong Kong RGC Competitive Earmarked Research Grant \(\#\)601410 and a multi-year research grant MYRG115(Y1-L4)-FST13-QT at University of Macau.

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Chang, DC., Habal, N. & Schulze, BW. The edge algebra structure of the Zaremba problem. J. Pseudo-Differ. Oper. Appl. 5, 69–155 (2014). https://doi.org/10.1007/s11868-013-0088-7

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  • DOI: https://doi.org/10.1007/s11868-013-0088-7

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