0.1 The partial differential equation (1) (Au)(x) = L aa(x)(Dau)(x) = f(x) m lal9 is called elliptic on a set G, provided that the principal symbol a2m(X,€) = L aa(x)€a lal=2m of the operator A is invertible on G X (~n \ 0); A is called elliptic on G, too. This definition works for systems of equations, for classical pseudo differential operators ("pdo), and for operators on a manifold n. Let us recall some facts concerning elliptic operators. 1 If n is closed, then for any s E ~, is Fredholm and the following a priori estimate holds (2) 1 2 Introduction If m > 0 and A: C=(O; C') -+ L (0; C') is formally self - adjoint 2 with respect to a smooth positive density, then the closure Ao of A is a self - adjoint operator with discrete spectrum and for the distribu tion functions of the positive and negative eigenvalues (counted with multiplicity) of Ao one has the following Weyl formula: as t -+ 00, (3) n 2m = t / II N±(1,a2m(x,e))dxde T·O\O (on the right hand side, N±(t,a2m(x,e))are the distribution functions of the matrix a2m(X,e): C' -+ CU).
Produkteigenschaften
- Artikelnummer: 9789048142828
- Medium: Buch
- ISBN: 978-90-481-4282-8
- Verlag: Springer Netherlands
- Erscheinungstermin: 15.12.2010
- Sprache(n): Englisch
- Auflage: 1. Auflage. Softcover version of original hardcover Auflage 1993
- Serie: Mathematics and its Applications
- Produktform: Kartoniert, Paperback
- Gewicht: 674 g
- Seiten: 436
- Format (B x H x T): 155 x 235 x 25 mm
- Ausgabetyp: Kein, Unbekannt