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Degenerate elliptic equations. (English) Zbl 0786.35063

Mathematics and its Applications (Dordrecht). 258. Dordrecht: Kluwer Academic Publishers. xi, 431 p. (1993).
The book consists of 11 chapters in which three main circles of subjects are studied:
(1) Investigation of the Fredholm property, a priori estimates and analogues of the Gårding inequality for wide classes of degenerate elliptic differential operators. (Chapters 2-6).
(2) The same questions for hypoelliptic pseudodifferential operators \((\psi DO)\) (Chapters 7-8).
(3) Computation of the main terms of spectral asymptotics (Chapters 9- 11).
Chapter 1 may be considered as an auxiliary one. In this chapter the author formulates definitions and theorems of the Weyl-Hörmander calculus of \(\psi DO\) with operator symbols in the form suggested by the author in his previous book [S. Levendorskij, Asymptotic distribution of eigenvalues of differential operators (Kluwer 1990; Zbl 0721.35049)]. This calculus allows to study differential operators in the case of strong degeneration on the boundary by means of the methods of general theory of \(\psi DO\) and to reduce the investigation of wide classes of degenerate operators to that of four model classes.
The chapter 2 is devoted to the investigation of these model classes of degenerate elliptic operators. Type 1 are the strongly degenerate operators, type 2 are the elliptic operators along the boundary and strongly degenerate ones in normal direction, type 3 are the elliptic operators along the boundary and Euler operators in normal direction, type 4 are the operators which require boundary and coboundary conditions.
In Chapters 3-4 the general classes of degenerate elliptic operators are studied. Chapter 5 is devoted to \(L_ p\)-theory for degenerate elliptic operators. The methods of this chapter are based on the Weyl-Hörmander calculus in spaces connected with \(L_ p\)-spaces.
In chapter 6 the problem of coerciveness of degenerate quadratic forms is considered. Chapter 7 is devoted to some classes of hypoelliptic pseudodifferential operators with multiple characteristics on closed manifolds. The Boutet-de-Monvel algebras of boundary value problems for the class of \(\psi DO\)’s which change order on the boundary are studied in chapter 8. Fredholm properties, a priori estimates and index formula are proved in this chapter.
Chapter 9 is devoted to the presentation of the approximate spectral projection method for the study of spectral asymptotis.
In chapters 10-11 this method is used for obtaining Weyl’s formula for degenerate elliptic operators and for hypoelliptic operators with multiple characteristics.

MSC:

35J70 Degenerate elliptic equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
65H10 Numerical computation of solutions to systems of equations
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35S05 Pseudodifferential operators as generalizations of partial differential operators

Citations:

Zbl 0721.35049
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