By Paolo Boggiatto, Luigi Rodino, Joachim Toft, M. W. Wong
ISBN-10: 3764375132
ISBN-13: 9783764375133
ISBN-10: 3764375140
ISBN-13: 9783764375140
This quantity includes articles in keeping with lectures given on the overseas convention on Pseudo-differential Operators and comparable themes at Växjö collage in Sweden from June 22 to June 25, 2005. 16 refereed articles through specialists from Canada, Denmark, England, Italy, Japan, Mexico, Russia, Serbia and Montenegro, and Sweden are dedicated to pseudo-differential operators and similar issues. They hide a huge spectrum of themes corresponding to partial differential equations, Wigner transforms, Weyl transforms on Euclidean areas and Lie teams, mathematical physics, time-frequency research, frames and stochastic techniques.
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Example text
7) Degenerate Hyperbolic Operators 25 Notice that for a = a(t, ξ) independent of x, the effective hyperbolicity is equivalent to a(t, ξ) = 0 ⇒ ∂t2 a(t, ξ) > 0 t ∈ [0, T ], |ξ| = 1, that can be expressed also as follows 2 |∂tj a(t, ξ)| = 0, t ∈ [0, T ], |ξ| = 1. 7) is satisfied with γ = 0 (no Levi condition). On the other hand for γ = 1/2 one can take k = ∞ that means that under the Levi condition it is not necessary to assume that a(t, ξ) has only zeros of finite order. 8) P = Dt2 − t2 Dx2 + tν Dx is well-posed in C ∞ if and only if ν ≥ − 1, see [15].
While the function M = M (x, ξ) is locally bounded, it is allowed to grow at infinity. 2) to hold with M = 1. In the strictly hyperbolic case we may set M = 0. The case M = 2 corresponds to the case when aj and ak define glancing hypersurfaces (as considered by Melrose). Here we also assume that aj and ak are not identically the same at (x, ξ). Let us now give some examples where this property holds while the tranversality condition (M = 1) fails. Example 1. In scalar equations with Levi conditions studied by Chazarain 1974, Mizohata-Ohya 1971, Zeman, one assumed that {aj , ak } = Cjk (aj − ak ).
4] S. Coriasco and L. Rodino, Cauchy problem for SG-hyperbolic equations with constant multiplicities, Ricerche di Matematica, Suppl. Vol. XLVIII (1999), 25–43. V. -W. Schulze, Pseudo-Differential Operators, Singularities, Applications, Operator Theory: Advances and Applications, 93, Birkh¨ auser Verlag, Basel, 1997. E. E. Marsden, The Einstein evolution equations as a first order quasi-linear symmetric hyperbolic system, Comm. Math. Phys. 28 (1972), 1–38 . [7] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch.