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An introduction to the theory of partial differential equations. (Une introduction à la théorie des équations aux dérivées partielles.) (French) Zbl 0756.35001

Montréal: Centre de Recherches Mathématiques, Université de Montréal. xi, 138 p. (1989).
The book is devoted mainly to second order partial differential equations. After examples it starts with classification of a partial differential equation with constant coefficients and then concentrates at elliptic, parabolic and hyperbolic types. For the elliptic case the author proves the fundamental properties of harmonic functions and studies the uniqueness of classical solutions of Dirichlet, Neumann and mixed boundary value problems. The properties of the spectrum of an elliptic operator are given and frequently used in the method of separation of variables, for example to deduce the Poisson integral.
For heat and wave equations the author deals with the Cauchy problem as well as with the initial-boundary value problem and shows the important features of the solutions. Two chapters are devoted to a short introduction to semigroups and to the solution of an evolution equation in one space variable in this frame.
At the end of the book the definition of a weak (Sobolev) solution is given and the existence theorems for Dirichlet and Neumann boundary value problems for an elliptic equation are proved.
An interesting feature of the book is that besides giving to readers a survey on fundamental results about classical solutions, it is also explaining, at every suitable place, the notions of distributional and weak (Sobolev) solutions.
Reviewer: J.Stará (Praha)

MSC:

35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35Dxx Generalized solutions to partial differential equations
35A15 Variational methods applied to PDEs

Citations:

Zbl 0703.35001
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