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).