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

https://doi.org/10.1016/j.jde.2006.10.010Get rights and content
Under an Elsevier user license
open archive

Abstract

In this paper we prove that the L2 spectral radius of the traction double layer potential operator associated with the Lamé system on an infinite sector in R2 is within 10−2 from a certain conjectured value which depends explicitly on the aperture of the sector and the Lamé moduli of the system. This type of result is relevant to the spectral radius conjecture, cf., e.g., Problem 3.2.12 in [C.E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Reg. Conf. Ser. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1994]. The techniques employed in the paper are a blend of classical tools such as Mellin transforms, and Calderón–Zygmund theory, as well as interval analysis—resulting in a computer-aided proof.

MSC

primary
45E05
47A05
secondary
35J25
42B20

Keywords

Lamé system
Layer potentials
Traction conormal derivative
Spectral radius
Interval analysis
Computer-aided proof

Cited by (0)

1

Supported in part by the NSF Grant DMS 0547944 and a University of Virginia FEST Grant.

2

Research fellow of the Royal Swedish Academy of Sciences.