Degenerate Elliptic Equations

di Serge Levendorskii (Autore)

Springer, 2010

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).

• Autore: Serge Levendorskii
• Editore: Springer
• Collana: Mathematics and Its Applications
• Anno: 2010
• Rilegatura: Paperback / softback
• Pagine: 436 p.
• Dimensioni: 297 x 210 mm
• EAN: 9789048142828

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