Titlebook: Analytic Partial Differential Equations; François Treves Book 2022 The Editor(s) (if applicable) and The Author(s), under exclusive licens

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Distributions in Euclidean SpaceIn the first section of this chapterwe transition from functions to generalized functions of the most commonly used kind, the . in an open subset Ω of ., the elements of the complex vector space . (Ω).

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Analytic Tools in Distribution TheoryThis chapter is the true introduction, at a level as elementary as possible, to the core of the book. In the first section the reader will find the needed definitions and basic results that can be translated to the . category from the last section of the preceding chapter.

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Analyticity of Solutions of Linear PDEs. Basic ResultsIn the late 1930s I.G. Petrowski proved that all classical solutions of a linear PDE with constant coefficients are analytic if and only if the equation is . (he extended this result to a class of systems of linear PDEs with constant coefficients).

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The Cauchy–Kovalevskaya TheoremThe bulk of this chapter is devoted to the fundamental theorem in analytic PDE theory and one of the most important mathematical discoveries of the XIXth century: that

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Hyperfunctions in Euclidean SpaceHyperfunctions are defined in real space ., where analytic functionals are localizable, i.e., have a uniquely defined support (Corollary 6.3.12).

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Hyperdifferential OperatorsHyperdifferential operators act on analytic functionals in . and decrease their carriers; as a consequence, they act on hyperfunctions in domains in . and decrease their supports.

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Elements of Differential GeometryThe purpose of this chapter is to recall a number of basic definitions in differential geometry needed for the transition from analysis in Euclidean phase-space Ω × . to analysis in the cotangent bundle of a . manifold ..

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Elements of Symplectic GeometryThe first section of this chapter is a primer on symplectic algebra and geometry in Euclidean spaces, real or complex. Concepts and results are extended to manifolds in Section 13.3 and revolve around the Darboux Theorem 13.3.20, which asserts the existence of local Darboux coordinates in an arbitrary symplectic manifold.