Generalized Carleson perturbations of elliptic operators and applications

TL;DR

This paper broadens the concept of Carleson perturbations for elliptic operators, introducing new types and applying the theory to diverse domains and problems, including free-boundary issues and measure characterizations.

Contribution

It introduces scalar-multiplicative and antisymmetric Carleson perturbations, extending the classical theory to more general boundary differences and domain types.

Findings

Unified approach to Carleson perturbation theory across various domains

New characterizations of $A_{ extinfty}$ among elliptic measures

Applications to Dahlberg-Kenig-Pipher operators and free-boundary problems

Abstract

We extend in two directions the notion of perturbations of Carleson type for the Dirichlet problem associated to an elliptic real second-order divergence-form (possibly degenerate, not necessarily symmetric) elliptic operator. First, in addition to the classical perturbations of Carleson type, that we call additive Carleson perturbations, we introduce scalar-multiplicative and antisymmetric Carleson perturbations, which both allow non-trivial differences at the boundary. Second, we consider domains which admit an elliptic PDE in a broad sense: we count as examples the 1-sided NTA (a.k.a. uniform) domains satisfying the capacity density condition, the 1-sided chord-arc domains, the domains with low-dimensional Ahlfors-David regular boundaries, and certain domains with mixed-dimensional boundaries; thus our methods provide a unified perspective on the Carleson perturbation theory of elliptic operators. Our proofs do not introduce sawtooth domains or the extrapolation method. We also present several applications to some Dahlberg-Kenig-Pipher operators, free-boundary problems, and we provide a new characterization of $A_{\infty}$ among elliptic measures.