The Theory of Weights and the Dirichlet Problem for Elliptic Equations

Abstract

In this work we consider questions from the area of partial equations and the related questions from harmonic analysis. Specifically, we will study the Dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients. Roughly speaking, we consider the following question: Suppose that we are given two operators Lo and LI, and that we have good estimates for the Dirichlet problem for Lo. (Good in this context could mean several different things. For instance, it might mean that there exists p, 1 < p < oo, so that the Dirichlet problem is solvable, in a suitable sense, with Lp data.) What are the optimal conditions on the difference between the coefficients of Lo and L1 which guarantee that the Dirichlet problem for LI also has good estimates? This question has been extensively studied recently (cf. [F-J-K], [R.F], and [D4]). Our work uses in a crucial way the results in [R.F]. The main results in [D4] and [R.F] are proved using an ingenious differential for a family of harmonic measures, introduced in [D4]. Another of our goals in this article is to give new, direct proofs of the results in [D4] and [R.F] without the use of this inequality (Theorems 2.4, 2.5 and 2.18). These direct approaches have already had important applications [K-P2]. An interesting class of examples for the theorems quoted above arises when one pulls back the Laplacian via a quasiconformal mapping of the plane into itself. Specializing our theorem to this case suggests a new characterization of the class of Am weights, introduced in connection with several problems in harmonic analysis by Muckenhoupt [Mul] and Coifman-C. Fefferman [Co-F]. We shall present (in Theorem 2.3) a criterion, in terms of Carleson measures, which is very close to log w E BMO and is necessary and sufficient for w E Ad.