Quasielliptic operators and Sobolev type equations. II

Abstract

We consider matrix quasielliptic operators on the whole space. Under the quasihomogeneity condition for symbols, we establish the isomorphism theorem for these operators in the special scales of Sobolev spaces. In particular, this result implies a series of available isomorphism theorems for elliptic operators and theorems about the unique solvability of the initial value problem for a broad class of systems of Sobolev type.We continue the study of the matrix quasielliptic operatorson the whole space R n . The class of operators in question lies in the class of quasielliptic operators which was introduced by Volevich [1] and includes, in particular, homogeneous elliptic operators, Petrovskiȋ elliptic and parabolic operators, elliptic operators in the Douglis-Nirenberg sense, and so on. For these operators we establish the isomorphism theorem in the special scales of Sobolev spaces W l p,σ (R n ), yielding a series of available isomorphism theorems for elliptic and parabolic operators on R n . The isomorphism theorem ensures the unique solvability of the initial value problem for a broad class of the system of Sobolev typeAs was noted in the first part of the article (see [2]), the isomorphism theorems have numerous applications to the theory of partial differential equations. However, the statements of these theorems are not obvious even in the case of scalar differential operators. In the case of matrix differential operators, the formulations are much more complicated. This relates in particular to the elliptic operators in the Douglis-Nirenberg sense. Stating the isomorphism theorems for the operators, we involve the special scales of weighted Sobolev spaces with the different exponents of smoothness and weight (see [2][3][4][5]). In this article we continue the research of [2][3][4][5][6]. Some close questions on the properties of differential operators are considered in [7][8][9][10][11][12]. § 2. Statement of the Main ResultsWe start with pointing out the conditions on the class (1.1) of matrix differential operators.Condition 1 0 . We assume that the matrix ν×ν-differential operator L (D x ) has constant coefficients and its symbol L (iξ) = (l j,r (iξ)) satisfies the following conditions.