W^2,2 A Priori Bounds for a Class of Elliptic Operators

Abstract

We obtain some W 2,2 a priori bounds for a class of uniformly elliptic second-order differential operators, both in a no-weighted and in a weighted case. We deduce a uniqueness and existence theorem for the related Dirichlet problem in some weighted Sobolev spaces on unbounded domains.

A Priori Bounds in and in for Solutions of Elliptic Equations

“…Let us start collecting some preliminary results concerning the existence and uniqueness of the solution of problem (22), as well as some a priori estimates. For the case where assumptions (ℎ 1 )-(ℎ 3 ) are taken into account and for = 2, we refer to [2] while for ≥ 3 details can be found in [3].…”

“…with the following conditions on the leading coefficients: In view of Theorem 1, under the assumptions (ℎ 0 )-(ℎ 3 ), the operator : 2, (Ω) → (Ω) is bounded. The first application is contained in Theorem 3.2 and Corollary 3.3 of [7] (see also [22] where the case = 2 is considered) and reads as follows.…”

Dirichlet Problem for Elliptic Equations in Weighted Sobolev Spaces

“…This section is devoted to a class of weighted Sobolev spaces recently introduced in [18], where all the details concerning the properties stated below can be found. Once the definition is given, we recall a fundamental lemma allowing to use no-weighted results in order to pass to the weighted contest.…”

“…• W 1,p s (Ω) and L p s (Ω) are some classes of weighted Sobolev and Lebesgue spaces recently introduced in [18], where a complete account of their properties can be found. In these spaces, the weight ρ s is a power of a function ρ of class C 2 (Ω) such that ρ :Ω → R + and As an example, one can think of the function ρ(x) = (1 + |x| 2 ) t , t ∈ R\{0}.…”

A W2,p-ESTIMATE FOR A CLASS OF ELLIPTIC OPERATORS

“…In particular, for bounded domains we recall for p = 2 the results of [1], [12] and [15] and for p > 1 the very general case considered in [13] and [14], where the a ij have vanishing mean oscillation (VMO), that is a kind of continuity in the average sense and not in the pointwise sense. Concerning unbounded domains, hypotheses similar to those of [18] have been considered for instance in [25], [26], [5], [6] and [19] for p = 2 and in [7] for p > 1. Assumptions as those of [13] and [14] have been taken into account in [8], and in [3] and [4] in a weighted case.…”