Cauchy problem for operators with triple effectively hyperbolic characteristics-Ivrii's conjecture-

TL;DR

This paper proves Ivrii's conjecture for operators with triple effectively hyperbolic characteristics, extending previous results that were limited to double characteristics, thereby establishing $C^{ abla}$ well-posedness in these cases.

Contribution

The paper confirms Ivrii's conjecture for operators with triple effectively hyperbolic characteristics, filling a gap in the understanding of well-posedness for such operators.

Findings

Proves $C^{ abla}$ well-posedness for operators with triple effectively hyperbolic characteristics.

Extends Ivrii's conjecture validation from double to triple characteristic cases.

Provides a comprehensive analysis of critical points with triple hyperbolic behavior.

Abstract

Ivrii's conjecture asserts that the Cauchy problem is $C^{\infty}$ well-posed for any lower order term if every critical point of the principal symbol is effectively hyperbolic. Effectively hyperbolic critical point is at most triple characteristic. If every characteristic is at most double this conjecture has been proved in 1980's. In this paper we prove the conjecture for the remaining cases, that is for operators with triple effectively hyperbolic characteristics.