## Kyoto Journal of Mathematics

- Kyoto J. Math.
- Volume 55, Number 2 (2015), 281-297.

### A simple proof of the existence of tangent bicharacteristics for noneffectively hyperbolic operators

#### Abstract

The behavior of orbits of the Hamilton vector field ${H}_{p}$ of the principal symbol $p$ of a second-order hyperbolic differential operator is discussed. In our previous paper, assuming that $p$ is noneffectively hyperbolic on the doubly characteristic manifold $\Sigma $ of $p$, we have proved that if ${H}_{S}^{3}p=0$ on $\Sigma $ with the Hamilton vector field ${H}_{S}$ of some specified $S$, then there exists a bicharacteristic landing on $\Sigma $ tangentially. The aim of this paper is to provide a much more simple proof of this result since the previous proof was fairly long and rather complicated.

#### Article information

**Source**

Kyoto J. Math., Volume 55, Number 2 (2015), 281-297.

**Dates**

Received: 3 September 2013

Revised: 9 January 2014

Accepted: 5 March 2014

First available in Project Euclid: 11 June 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1433982756

**Digital Object Identifier**

doi:10.1215/21562261-2871758

**Mathematical Reviews number (MathSciNet)**

MR3356074

**Zentralblatt MATH identifier**

1320.35183

**Subjects**

Primary: 35L15: Initial value problems for second-order hyperbolic equations 35L10: Second-order hyperbolic equations

Secondary: 35L80: Degenerate hyperbolic equations

#### Citation

Nishitani, Tatsuo. A simple proof of the existence of tangent bicharacteristics for noneffectively hyperbolic operators. Kyoto J. Math. 55 (2015), no. 2, 281--297. doi:10.1215/21562261-2871758. https://projecteuclid.org/euclid.kjm/1433982756