A note on reduced forms of effectively hyperbolic operators and energy integrals. (English) Zbl 0564.35068
Consider a hyperbolic operator with principal symbol P. At a double characteristic point, the coefficient matrix of the Hamiltonian system associated with the quadratic form leading the Taylor expansion in the cotangent bundle is called the fundamental matrix. If the eigenvalues of the matrix are non-zero and real, then the operator is called effectively hyperbolic.
The author presents a method, using certain homogeneous canonical transformations, of reducing effectively hyperbolic operators to certain standard forms. He works out some examples which show connections with energy integrals. However the energy integral estimates appear elsewhere.
The author presents a method, using certain homogeneous canonical transformations, of reducing effectively hyperbolic operators to certain standard forms. He works out some examples which show connections with energy integrals. However the energy integral estimates appear elsewhere.
Reviewer: S.G.Krantz
MSC:
35L25 | Higher-order hyperbolic equations |
35L40 | First-order hyperbolic systems |
35S05 | Pseudodifferential operators as generalizations of partial differential operators |
35A30 | Geometric theory, characteristics, transformations in context of PDEs |