Title. Additive and scalar-multiplicative Carleson perturbations of elliptic operators on domains with low dimensional boundaries.

Abstract. At the beginning of the 90s, Fefferman, Kenig and Pipher (FKP) obtained a rather sharp (additive) perturbation result for the Dirichlet problem of divergence form elliptic operators. Without delving into details, the point is that if the (additive) disagreement of two operators satisfies what is known as a Carleson measure condition, then quantitative absolute continuity of the elliptic measure is transferred from one operator to the other, if one of the operators already possesses this property. Their (additive) perturbation result has since then been generalized to increasingly weaker geometric and topological assumptions on boundaries of co-dimension 1, by multiple authors.

This talk will consist of two main parts.  In the first part, we will see an extension of the FKP result to the degenerate elliptic operators of David, Feneuil and Mayboroda, which were developed to study geometric and analytic properties of sets with boundaries whose co-dimension is higher than 1. These operators are of the form -div A∇ , where A is a degenerate elliptic matrix crafted to weigh the distance to the low-dimensional boundary in a way that allows for the nourishment of an elliptic theory. When this boundary is a d-Alhfors-David regular set in R^n with d in [1, n-1), and n≥ 3, we prove that the membership of the elliptic measure in A_∞  is preserved under (additive) Carleson measure perturbations of the matrix of coefficients, yielding in turn that the L^p-solvability of the Dirichlet problem is also stable under these perturbations (with possibly different p).  If the Carleson measure perturbations are suitably small, we establish solvability of the Dirichlet problem in the same L^p space. One of the corollaries of our results together with a previous result of David, Engelstein and Mayboroda, is that, given any d-ADR boundary Γ  with d in [1, n-2), n≥ 3, there is a family of degenerate operators of the form described above whose elliptic measure is absolutely continuous with respect to the  d-dimensional Hausdorff measure on Γ.  Our method of proof uses the method of Carleson measure extrapolation, as developed by Lewis and Murray, and adapted to a dyadic setting by Hofmann and Martell in the past decade. This is joint work with Svitlana Mayboroda.

In the second part of the talk, we will adopt a slightly different perspective than has been customary in the literature of these perturbation results, by considering scalar-multiplicative Carleson perturbations, as communicated to us by Joseph Feneuil and inspired by the work on equations with drift terms of Hofmann and Lewis, and Kenig and Pipher, at the start of the 21st century. Essentially,  if we may write A=bA_0 with b a scalar function bounded above and below by a positive number, and ∇b·dist(· ,Γ) satisfying a Carleson measure condition, then we still retain the transference of the quantitative absolute continuity of the elliptic measure for -div A∇, if -div A_0∇ already has this property. By way of examples in the setting of low dimensional boundaries, we will see that one ought to consider these two types of perturbations (namely, additive and scalar-multiplicative) to reckon a more complete picture of the absolute continuity of elliptic measure. This is joint work with Joseph Feneuil.