The cauchy problem for hyperbolic operators : multiple characteristics. micro-local approach
著者
書誌事項
The cauchy problem for hyperbolic operators : multiple characteristics. micro-local approach
（Mathematical topics, v. 12）
Akademie Verlag, c1997
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注記
Includes bibliographical references and indexes
内容説明・目次
内容説明
A construction of the fundamental solution to the Cauchy problem for hyperbolic operators with multiple characteristics is the target of this book. Investigations of the problem in various functional spaces and a propagation of singularities of the solutions are also presented. For operators with multiple characteristics, so-called Levy conditions play a crucial rule. Levy conditions described in the book allow the construction of fundamental solutions. A turning point theory for ordinary differential equations is the starting area of the treatment. An approach is also given which is available to the turning points of infinite and higher order equations. Applications to the problem are given for partial differential equations (Cauchy problem, local solvability and hypoellipticity) with multiple characteristics and to some problems of quantum mechanics. The approach represented in the book is essentially based on the zeros of the complete symbol of the operator. For operators with variable coefficients, hyperbolicity conditions are formulated by means of these zeros similarly to Hadamard's conditions for operators with constant coefficient.
This approach needs Fourier integral operators with inhomogeneous phase functions. The required knowledge on these is given.
目次
- Fourier integral operators with inhomogeneous phase functions
- representation theorem for solutions of equations with turning point
- Gevrey asymptotic representation of solutions of equations with turning point
- WBK solutions of equations with infinite order turning point
- the cauchy problem in Gevrey classes
- tunneling phenomena
- problems of hypoellipticity and local solvability
- hyperbolicity
- in the zeros of the complete symbol
- Hamiltonian vector fields
- exponential functions of certain pseudodifferential operators
- Egorov's theorem
- ideals of fourier integral operators
- Garding's asymptotic inequality
- Hadamard's example in light of Lyapunov stability theory of ordinary differential equations
- lower bound for the energy
- necessary correctness conditions for the cauchy problem.
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