Differential and Integral Equations

Some counterexamples for the spectral-radius conjecture

Irina Mitrea and Warwick Tucker

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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.

Article information

Source
Differential Integral Equations, Volume 16, Number 12 (2003), 1409-1439.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060495

Mathematical Reviews number (MathSciNet)
MR2029908

Zentralblatt MATH identifier
1073.45500

Subjects
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]

Citation

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


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