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Compactness, or Lack Thereof, for the Harmonic Double Layer

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From Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 285))

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Abstract

Ahlfors regular domains which are infinitesimally flat, in a scale-invariant fashion, constitute the most general geometric context in which the harmonic double layer potential as well as other similar singular integral operators are compact in the framework of Lebesgue spaces. We review recent progress in this direction and, working with a drop-shaped domain, we give a direct, self-contained proof of the fact that the harmonic double layer potential fails to be compact on Lebesgue spaces in the presence of a single corner (or conical) singularity.

The authors are delighted to dedicate this article to Lance Littlejohn, whose warm friendship they much value and cherish, on the occasion of his 70-th birthday.

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Acknowledgements

The authors gratefully acknowledge partial support from the Simons Foundation (through grants # 426669, # 637481), as well as NSF (grant # 1900938).

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Authors and Affiliations

  1. Department of Mathematics, Baylor University, Waco, TX, USA

    Dorina Mitrea & Marius Mitrea

  2. Department of Mathematics, Temple University, Philadelphia, PA, USA

    Irina Mitrea

Authors
  1. Dorina Mitrea
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  2. Irina Mitrea
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  3. Marius Mitrea
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Correspondence to Dorina Mitrea or Marius Mitrea .

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Editors and Affiliations

  1. Department of Mathematics, Baylor University, Waco, TX, USA

    Fritz Gesztesy

  2. Department of Mathematics, Baylor University, Waco, TX, USA

    Andrei Martinez-Finkelshtein

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Mitrea, D., Mitrea, I., Mitrea, M. (2021). Compactness, or Lack Thereof, for the Harmonic Double Layer. In: Gesztesy, F., Martinez-Finkelshtein, A. (eds) From Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory . Operator Theory: Advances and Applications, vol 285. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-75425-9_17

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