Layer Potentials for Elastostatics and Hydrostatics in Curvilinear Polygonal Domains

Abstract

ABSTRACT. The symbolic calculus of pseudodifferential operators of Mellin type is applied to study layer potentials on a plane domain 0+ whose boundary &0+ is a curvilinear polygon. A "singularity type" is a zero of the determinant of the matrix of symbols of the Mellin operators and can be used to calculate the "bad values" of p for which the system is not Fredholm on LP(&O+),Using the method of layer potentials we study the singularity types of the system of elastostaticsin a plane domain 0+ whose boundary &0+ is a curvilinear polygon. Here fJ > 0 and -fJ :S 2 :S +00 . When 2 = +00 , the system is the Stokes system of hydrostatics. For the traction double layer potential, we show that all singularity types in the strip 0 < Re z < I lie in the interval (1, I) so that the system of integral equations is a Fredholm operator of index 0 on LP(&O+) for all p, 2 :S p < 00. The explicit dependence of the singularity types on 2 and the interior angles (J of &0+ is calculated; the singularity type of each corner is independent of 2 iff the corner is nonconvex.

On the Spectra of Elastostatic and Hydrostatic Layer Potentials on Curvilinear Polygons

“…Similar results for the linearized Stokes system of hydrostatics have been obtained in [10]. Using a symbolic calculus of pseudodifferential operators of Mellin type, it has been shown in [17] that I + K Lame is Fredholm with index zero on L p (∂ ) for each 2 ≤ p < ∞ whenever is a (bounded) curvilinear polygonal domain in R 2 . Consider, for example, I + K Lame , the double layer potential operator relevant for the solution of the Dirichlet problem for the Lamé system.…”

“…We follow closely the notation in [17]. We drop the subscripts L and S. Parameterize ∂ and rewrite the operator K in the corresponding parametric coordinates as we did in Section 7.…”

The Oblique Derivative Problem for the Laplace Equation in a Plain Domain

“…[36]. If p ≥ p 0 we can repeat the reasoning from the proof of Lemma 24 using results in [29], analogical to the result in [52], used in the proof. Therefore, if p ≥ p 0 then 1 2 I + K * is not a Fredholm operator with index 0 on L p (H).…”

Interval analysis techniques for boundary value problems of elasticity in two dimensions

“…Traditionally, the major theoretical tools involved in the study of the spectra of singular integral operators are Calderón-Zygmund theory (for layer potential operators on Lipschitz domains; see, e.g., the work of Fabes, Kenig, Verchota and Escauriaza in [4,6,28]) and Mellin transform techniques (for layer potentials on domains with isolated singularities; see, e.g., the work of Elschner, Fabes, Lewis, Maz'ya and collaborators and Shelepov in [5,7,8,11,12,15,16,24]). The main novel aspect of our present work is the realization that the use of interval analysis and rigorous computations (employed by Tucker in [26,27] to prove Smale's 14th problem concerning the existence of the Lorenz attractor) can play a significant role in the study of spectral problems of the type described above.…”