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

Publication Title

Integral Equations and Operator Theory

Document Type

Article

Department or Program

Mathematics

Publication Date

2017

Keywords

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

Abstract

In this paper we establish sharp invertibility results for the elastostatics and hydrostatics single and double layer potential type operators acting on Lp(∂Ω) , 1 < p< ∞, whenever Ω is an infinite sector in R2. 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 Ω and the differential operator in question.

Comments

Original version is available from the publisher at: https://doi.org/10.1007/s00020-017-2396-4

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