$L^p$ Spectral Radius Estimates for the Lamé System on an Infinite Sector

Abstract

We prove, using interval analysis methods, that the $L^2$, $L^4$, and $L^8$ spectral radii of the traction double layer potential operator associated with the Lam\'e system on an infinite sector in $\mathbb R}^2$ are within $2.5 x 10^-3$, $10^-2$, and $10^-2$, respectively, from a certain conjectured value that depends explicitly on the aperture of the sector and the Lamé moduli of the system. We also indicate how to extend these results to $L^p$ for entire intervals of $p$, $p\geq2$.