Differential and Integral Equations
- Differential Integral Equations
- Volume 16, Number 12 (2003), 1409-1439.
Some counterexamples for the spectral-radius conjecture
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.
Differential Integral Equations, Volume 16, Number 12 (2003), 1409-1439.
First available in Project Euclid: 21 December 2012
Permanent link to this document
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 45E05: Integral equations with kernels of Cauchy type [See also 35J15]
Secondary: 31A15: Potentials and capacity, harmonic measure, extremal length [See also 30C85] 35J25: Boundary value problems for second-order elliptic equations 45P05: Integral operators [See also 47B38, 47G10] 47A10: Spectrum, resolvent 47G10: Integral operators [See also 45P05]
Mitrea, Irina; Tucker, Warwick. Some counterexamples for the spectral-radius conjecture. Differential Integral Equations 16 (2003), no. 12, 1409--1439. https://projecteuclid.org/euclid.die/1356060495