Degenerate Elliptic Equations

By Serge Levendorskii

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).

• Format: Paperback
• Language: English
• ISBN: 9048142822
• ISBN13: 9789048142828
• Release Date: December 2010
• Publisher: Springer
• Length: 436 Pages
• Weight: 1.35 lbs.
• Dimensions: 1.1" x 6.0" x 9.0"