Quasielliptic operators and Sobolev type equations

Abstract

We consider a class of matrix quasielliptic operators on the n-dimensional space. For these operators, we establish the isomorphism properties in some special scales of weighted Sobolev spaces. Basing on these properties, we prove the unique solvability of the initial value problem for a class of Sobolev type equations.Keywords: quasielliptic operator, weighted Sobolev space, isomorphism, Sobolev type equations § 1. IntroductionIn this article we consider a class of matrix quasielliptic operators,on the whole space R n . For these operators, we establish the isomorphism properties in some special scales of weighted Sobolev spaces W l p,σ and apply these properties to proving the solvability of the initial value problem for the class of Sobolev type equations(1.2)The theorems on the isomorphism properties of differential operators have numerous applications in the theory of partial differential equations. However, in many cases their statements are far from obvious. For instance, the differential operator Δ − εI : W 2 p (R n ) → L p (R n ), 1 < p < ∞, for ε > 0 establishes an isomorphism, but this is false for the Laplace operator Δ : W 2 p (R n ) → L p (R n ), 1 < p < ∞. The situation is similar for the polyharmonic operator Δ m : W 2m p (R n ) → L p (R n ), 1 < p < ∞. This operator is not an isomorphism for any m. In particular, using the results of [1, Chapter 12], we can show that the solvability of the equationsin the Sobolev space W 2m p (R n ) with p ≤ n n−2m requires that the right-hand side f (x) satisfy the orthogonality conditions of the form R n f (x)x α dx = 0, |α| ≤ s.

Quasielliptic operators and Sobolev type equations. II

“…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.…”

“…As 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.…”

“…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].…”

“…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].…”

“…Note that the class of differential operators satisfying the above conditions in the case of s 1 = · · · = s ν = 0 is discussed in [2]. This class contains, in particular, the homogeneous quasielliptic operators, parabolic and elliptic operators in the Petrovskiȋ sense.…”