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Invertibility Properties of Singular Integral Operators Associated with the Lamé and Stokes Systems on Infinite Sectors in Two Dimensions

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Abstract

In this paper we establish sharp invertibility results for the elastostatics and hydrostatics single and double layer potential type operators acting on \(L^p(\partial \Omega )\), \(1<p<\infty \), whenever \(\Omega \) is an infinite sector in \({\mathbb {R}}^2\). This analysis is relevant to the layer potential treatment of a variety of boundary value problems for the Lamé system of elastostatics and the Stokes system of hydrostatics in the class of curvilinear polygons in two dimensions, such as the Dirichlet, the Neumann, and the Regularity problems. Mellin transform techniques are used to identify the critical integrability indices for which invertibility of these layer potentials fails. Computer-aided proofs are produced to further study the monotonicity properties of these indices relative to parameters determined by the aperture of the sector \(\Omega \) and the differential operator in question.

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Authors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, PA, 19122, USA

    Irina Mitrea

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

    Katharine Ott

  3. Department of Mathematics, Uppsala University, Uppsala, Sweden

    Warwick Tucker

Authors
  1. Irina Mitrea
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  2. Katharine Ott
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  3. Warwick Tucker
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Correspondence to Irina Mitrea.

Additional information

This work was supported in part by the SQuaRE program at the American Institute of Mathematics, by the NSF DMS Grant 1458138, the Simons Foundation Grant 318658, and the Swedish Research Council Grant 2008-7510.

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Mitrea, I., Ott, K. & Tucker, W. Invertibility Properties of Singular Integral Operators Associated with the Lamé and Stokes Systems on Infinite Sectors in Two Dimensions. Integr. Equ. Oper. Theory 89, 151–207 (2017). https://doi.org/10.1007/s00020-017-2396-4

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