Geometry of finite-order contact structures and the classical theory of symmetries of partial differential equations

We discuss the integration problem for systems of partial differential equations in one unknown function and special attention is given to the first order systems. The Grassmannian contact structures are the basic setting for our discussion and the major part of our considerations inquires on the nature of the Cauchy characteristics in view of obtaining the necessary criteria that assure the existence of solutions. In all the practical applications of partial differential equations, what is mostly needed and what in fact is hardest to obtains are the solutions of the system or, occasionally, some specific solutions. This work is based on four most enlightening Mémoires written by Élie Cartan in the beginning of the last century.

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Foundations of the Classical Theory of Partial Differential Equations

10.1007/978-3-642-58093-2 ◽

1998 ◽

Cited By ~ 24

Author(s):

Yu. V. Egorov ◽

M. A. Shubin

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Classical Theory ◽

Partial Differential

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Semiclassical fundamental solutions

Abstract and Applied Analysis ◽

10.1155/aaa.2005.45 ◽

2005 ◽

Vol 2005 (1) ◽

pp. 45-57 ◽

Cited By ~ 1

Author(s):

Patrick Guidotti

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Classical Theory ◽

Analytical Approach ◽

Special Functions ◽

Fundamental Solutions ◽

Partial Differential ◽

Solution Of Equations ◽

Explicit Representations

It is the aim of this paper to show how the classical theory, based on fundamental solutions and explicit representations, via special functions can be combined with the functional analytical approach to partial differential equations, to produce semiclassical representation formulae for the solution of equations in cylinder-like domains.

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Distributions of anisotropic order and applications to H-distributions

Analysis and Applications ◽

10.1142/s0219530520500165 ◽

2020 ◽

pp. 1-43

Author(s):

Nenad Antonić ◽

Marko Erceg ◽

Marin Mišur

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Finite Order ◽

Partial Differential ◽

Schwartz Kernel ◽

Main Motivation ◽

Radon Measures ◽

Central Result ◽

Continuous Operators ◽

Kernel Theorem

We define distributions of anisotropic order on manifolds, and establish their immediate properties. The central result is the Schwartz kernel theorem for such distributions, allowing the representation of continuous operators from [Formula: see text] to [Formula: see text] by kernels, which we prove to be distributions of order [Formula: see text] in [Formula: see text], but higher, although still finite, order in [Formula: see text]. Our main motivation for introducing these distributions is to obtain the new result that H-distributions (Antonić and Mitrović), a recently introduced generalization of H-measures are, in fact, distributions of order 0 (i.e. Radon measures) in [Formula: see text], and of finite order in [Formula: see text]. This allows us to obtain some more precise results on H-distributions, hopefully allowing for further applications to partial differential equations.

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