Some counterexamples for the spectral-radius conjecture

Abstract

The goal of this paper is to produce a series of counterexamples for the $L^p$ spectral radius conjecture, $1 <p <\infty$, for double-layer potential operators associated to a distinguished class of elliptic systems in polygonal domains in $\mathbb R^2$. More specifically the class under discussion is that of second-order elliptic systems in two dimensions whose coefficient tensor (with constant real entries) is symmetric and strictly positive definite. The general techniques employed are those of the Mellin transform and Calderón-Zygmund theory. For the case $p\in(1,4)$, we construct a computer-aided proof utilizing validated numerics based on interval analysis.