# Carleson perturbations of elliptic operators on domains with low dimensional boundaries

@article{Mayboroda2020CarlesonPO, title={Carleson perturbations of elliptic operators on domains with low dimensional boundaries}, author={Svitlana Mayboroda and Bruno Poggi}, journal={arXiv: Analysis of PDEs}, year={2020} }

We prove an analogue of a perturbation result for the Dirichlet problem of divergence form elliptic operators by Fefferman, Kenig and Pipher, for 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 $-\text{div} A\nabla$, where $A$ is a weighted elliptic matrix crafted to weigh the distance to the high co-dimension boundary…

## 6 Citations

Generalized Carleson perturbations of elliptic operators and applications

- Mathematics
- 2020

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…

Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition

- Mathematics
- 2019

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (aka uniform domain), that is, a set which satisfies the interior Corkscrew and Harnack chain…

Harmonic measure is absolutely continuous with respect to the Hausdorff measure on all low-dimensional uniformly rectifiable sets

- Mathematics
- 2020

It was recently shown that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a domain with an $n-1$ dimensional uniformly rectifiable boundary, in the presence of…

A change of variable for Dahlberg-Kenig-Pipher operators.

- Mathematics
- 2021

In the present article, we purpose a method to deal with Dahlberg-Kenig-Pipher (DPK) operators in boundary value problems on the upper half plane. We give a nice subclass of the weak DKP operators…

Absolute continuity of degenerate elliptic measure

- Mathematics
- 2021

Let Ω ⊂ R be an open set whose boundary may be composed of pieces of different dimensions. Assume that Ω satisfies the quantitative openness and connectedness, and there exist doubling measures m on…

The Green function with pole at infinity applied to the study of the elliptic measure.

- Mathematics
- 2020

In $\mathbb R^{d+1}_+$ or in $\mathbb R^n\setminus \mathbb R^d$ ($d<n-1$), we study the Green function with pole at infinity introduced by David, Engelstein, and Mayboroda. In two cases, we deduce…

## References

SHOWING 1-10 OF 126 REFERENCES

Elliptic theory in domains with boundaries of mixed dimension

- Mathematics
- 2020

Take an open domain $\Omega \subset \mathbb R^n$ whose boundary may be composed of pieces of different dimensions. For instance, $\Omega$ can be a ball on $\mathbb R^3$, minus one of its diameters…

Regularity theory for solutions to second order elliptic operators with complex coefficients and the L Dirichlet problem

- MathematicsAdvances in Mathematics
- 2019

We establish a new theory of regularity for elliptic complex valued second order equations of the form $\mathcal L=$div$A(\nabla\cdot)$, when the coefficients of the matrix $A$ satisfy a natural…

Square function estimates, the BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets

- MathematicsAnalysis & PDE
- 2019

In the recent work [DFM1, DFM2] G. David, J. Feneuil, and the first author have launched a program devoted to an analogue of harmonic measure for lower-dimensional sets. A relevant class of partial…

Layer potentials and boundary value problems for elliptic equations with complex $L^{\infty}$ coefficients satisfying the small Carleson measure norm condition

- Mathematics
- 2013

We consider divergence form elliptic equations $Lu:=\nabla\cdot(A\nabla u)=0$ in the half space $\mathbb{R}^{n+1}_+ :=\{(x,t)\in \mathbb{R}^n\times(0,\infty)\}$, whose coefficient matrix $A$ is…

BMO Solvability and $$A_{\infty }$$A∞ Condition of the Elliptic Measures in Uniform Domains

- Mathematics
- 2018

We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO…

Critical Perturbations for Second Order Elliptic Operators. Part I: Square function bounds for layer potentials

- Mathematics
- 2020

This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $-\mathop{\operatorname{div}} A \nabla$ by first and zero order terms,…

Perturbations of elliptic operators in chord arc domains

- Mathematics
- 2012

We study the boundary regularity of solutions to divergence form operators which are small perturbations of operators for which the boundary regularity of solutions is known. An operator is a small…

The Dirichlet problem in domains with lower dimensional boundaries

- Mathematics
- 2018

The present paper pioneers the study of the Dirichlet problem with $L^q$ boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in…

Perturbations of elliptic operators in 1-sided chord-arc domains. Part II: Non-symmetric operators and Carleson measure estimates

- Mathematics
- 2019

We generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative…

Dahlberg's theorem in higher co-dimension

- MathematicsJournal of Functional Analysis
- 2019

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension $n-1$ in $\mathbb…