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Integral Equations and Operator Theory

, Volume 89, Issue 2, pp 151–207 | Cite as

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

  • Irina Mitrea
  • Katharine Ott
  • Warwick Tucker
Article
  • 60 Downloads

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.

Keywords

Lamé system Stokes system Mellin transform Hardy kernel operator Single layer potential Double layer potential Conormal derivative Pseudo-stress conormal derivative Infinite sector Interval analysis Computer-aided proof Validated numerics 

Mathematics Subject Classification

Primary 35J25 35J47 42A38 Secondary 45E05 45B05 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsTemple UniversityPhiladelphiaUSA
  2. 2.Department of MathematicsBates CollegeLewistonUSA
  3. 3.Department of MathematicsUppsala UniversityUppsalaSweden

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