Diagonal symmetrizers for hyperbolic operators with triple characteristics

TL;DR

This paper constructs diagonal symmetrizers for hyperbolic operators with triple characteristics and proves Ivrii's conjecture for such operators with time-dependent coefficients, advancing understanding of hyperbolic PDEs.

Contribution

It introduces a method to diagonalize symmetrizers for hyperbolic operators with triple characteristics and proves Ivrii's conjecture in the time-dependent coefficient case.

Findings

Diagonal symmetrizers are constructed via Bezoutian matrix diagonalization.

Ivrii's conjecture is proved for operators with coefficients depending on time.

The approach advances the analysis of hyperbolic operators with multiple characteristics.

Abstract

Symmetrizers for hyperbolic equations are obtained by diagonalizing the Bezoutian matrix of hyperbolic symbols. Such diagonal symmetrizers are applied to the Cauchy problem for hyperbolic operators with triple characteristics. In particular, the V.Ivrii's conjecture concerned with triple effectively hyperbolic characteristics is proved for differential operators with coefficients depending on the time variable.