Abstract
Symmetrizers for hyperbolic operators are obtained by diagonalizing the Bézoutian matrix of the principal symbols and its derivatives. Such diagonal symmetrizers are applied to the Cauchy problem for hyperbolic operators with triple characteristics. In particular, the Ivrii’s conjecture concerned with strongly hyperbolic operators with triple effectively hyperbolic characteristics is proved for differential operators with time dependent coefficients, also for third order differential operators with two independent variables with analytic coefficients.
Highlights
This paper is devoted to the Cauchy problem Dtm u + m−1 j =0|α|+ j≤m a j,α(t, x )Dxα Dtj u = 0, Dtj u(0, x) = u j (x), j = 0, . . . , m − 1 (1.1)where t ≥ 0, x ∈ Rn and the coefficients a j,α(t, x) are real valued C∞ functions in a neighborhood of the origin of R1+n and Dx = (Dx1, . . . , Dxn ), Dx j = (1/i )(∂/∂ x j ) and Dt = (1/i)(∂/∂t)
The problem is C∞ well-posed near the origin for t ≥ 0 if one can find a δ > 0 and a neighborhood U of the origin of Rn such that (1.1) has a unique solution u ∈ C∞([0, δ) × U ) for any u j (x) ∈ C∞(Rn)
We assume that the principal symbol p is hyperbolic for t ≥ 0, that is Communicated by Y
Summary
Re (Op(S)U , U ) ≥ −C D −1/2U 2 which is, in general, too weak to study the Cauchy problem for general weakly hyperbolic operator P, in particular the well posed Cauchy problem with loss of derivatives, applying this symmetrizer many interesting results are obtained by several authors, see for example [1,5,6,16,19], [29]. In these works one of the main points is. In the last section we show that the same idea is applicable to hyperbolic operators with general triple characteristics, utilizing a homogeneous third order operator with two independent variables (Theorem 6.1)
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