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Invertibility properties of singular integral operators associated with the Lamé and Stokes systems on infinite sectors in two dimensions. (English) Zbl 1381.35032

Authors’ 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.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35J47 Second-order elliptic systems
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
45E05 Integral equations with kernels of Cauchy type
45B05 Fredholm integral equations

Software:

INTLAB; filib++; C-XSC
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References:

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