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On the Solvability of the Zaremba Problem in Infinite Sectors and the Invertibility of Associated Singular Integral Operators

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Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

This paper is concerned with the study of invertibility properties of a singular integral operator naturally associated with the Zaremba problem for the Laplacian in infinite sectors in two dimensions, when considering its action on an appropriate Lebesgue scale of p integrable functions, for 1 < p < ∞. Concretely we consider the case in which a Dirichlet boundary condition is imposed on one ray of the sector, and a Neumann boundary condition is imposed on the other ray. In this geometric context, using Mellin transform techniques, we identify the set of critical integrability indexes p for which the invertibility fails, and we establish an explicit characterization of the L p spectrum of this operator for each p ∈ (1, ∞). This analysis, along with a divergence theorem with non-tangential trace, are then used to establish well-posedness of the Zaremba problem with L p data in infinite sectors in \(\mathbb{R}^{2}\).

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Acknowledgements

This work was supported in part by the SQuaRE program at the American Institute of Mathematics, by the NSF DMS grant 1458138, and by the Simons Foundation grant 318658.

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Authors and Affiliations

  1. Department of Mathematics, Temple University, 1805 N. Broad St., Philadelphia, PA, 19122, USA

    Hussein Awala & Irina Mitrea

  2. Department of Mathematics, Bates College, 2 Andrews Road, Lewiston, ME, 04240, USA

    Katharine Ott

Authors
  1. Hussein Awala
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  2. Irina Mitrea
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  3. Katharine Ott
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Correspondence to Hussein Awala .

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Editors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, Pennsylvania, USA

    Isaac Pesenson

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia

    Quoc Thong Le Gia

  3. Department of Mathematics, The Graduate Center, CUNY, New York, New York, USA

    Azita Mayeli

  4. Institute of Mathematical Sciences, Claremont Graduate University, Claremont, California, USA

    Hrushikesh Mhaskar

  5. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

    Ding-Xuan Zhou

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Awala, H., Mitrea, I., Ott, K. (2017). On the Solvability of the Zaremba Problem in Infinite Sectors and the Invertibility of Associated Singular Integral Operators. In: Pesenson, I., Le Gia, Q., Mayeli, A., Mhaskar, H., Zhou, DX. (eds) Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55556-0_10

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