Generalized Carleson perturbations of elliptic operators and applications

Feneuil, Joseph; Poggi, Bruno

doi:10.1090/tran/8635

Cited by 8 publications

(14 citation statements)

References 43 publications

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“…We would like to mention that after an initial version of this work was posted on arXiv [1], Feneuil and Poggi in [13] obtained results related to ours, compare for instance Theorem 1.3 with [13,Theorem 1.27]. Also, the recent work [4] complements this paper and its companion [2], see for instance [4,Corollary 1.4].…”

supporting

confidence: 61%

Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Akman¹,

Hofmann²,

Martell³

et al. 2021

Preprint

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Let Ω ⊂ R n+1 , n ≥ 2, be a 1-sided non-tangentially accessible domain (aka uniform domain), that is, Ω satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that Ω satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider L0u = −div(A0∇u), Lu = −div(A∇u), two real (non-necessarily symmetric) uniformly elliptic operators in Ω, and write ωL 0 , ωL for the respective associated elliptic measures. The goal of this program is to find sufficient conditions guaranteeing that ωL satisfies an A∞-condition or a RHq-condition with respect to ωL 0 . In this paper we are interested in obtaining square function and non-tangential estimates for solutions of operators as before. We establish that bounded weak null-solutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak null-solution, the associated square function can be controlled by the non-tangential maximal function in any Lebesgue space with respect to the associated elliptic measure. These results extend previous work of Dahlberg-Jerison-Kenig and are fundamental for the proof of the perturbation results in [2].

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supporting

confidence: 61%

Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Akman¹,

Hofmann²,

Martell³

et al. 2021

Preprint

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“…This work can be particularized to our setting and contains some results which overlap with ours. First, [24,Theorem 1.22] corresponds to (c) =⇒ (a) in Theorem 1.1. It should be mentioned that both arguments use the ideas originated in [42] (see also [43]) which present some problems when extended to the 1-sided NTA setting.…”

Section: Introductionmentioning

confidence: 99%

On the condition for elliptic operators in 1-sided nontangentially accessible domains satisfying the capacity density condition

Cao

Domínguez

Martell

et al. 2022

Forum of Mathematics, Sigma

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Let $\Omega \subset \mathbb {R}^{n+1}$ , $n\ge 2$ , be a $1$ -sided nontangentially accessible domain, that is, a set which is quantitatively open and path-connected. Assume also that $\Omega $ satisfies the capacity density condition. Let $L_0 u=-\mathop {\operatorname {div}}\nolimits (A_0 \nabla u)$ , $Lu=-\mathop {\operatorname {div}}\nolimits (A\nabla u)$ be two real (not necessarily symmetric) uniformly elliptic operators in $\Omega $ , and write $\omega _{L_0}, \omega _L$ for the respective associated elliptic measures. We establish the equivalence between the following properties: (i) $\omega _L \in A_{\infty }(\omega _{L_0})$ , (ii) L is $L^p(\omega _{L_0})$ -solvable for some $p\in (1,\infty )$ , (iii) bounded null solutions of L satisfy Carleson measure estimates with respect to $\omega _{L_0}$ , (iv) $\mathcal {S}<\mathcal {N}$ (i.e., the conical square function is controlled by the nontangential maximal function) in $L^q(\omega _{L_0})$ for some (or for all) $q\in (0,\infty )$ for any null solution of L, and (v) L is $\mathrm {BMO}(\omega _{L_0})$ -solvable. Moreover, in each of the properties (ii)-(v) it is enough to consider the class of solutions given by characteristic functions of Borel sets (i.e, $u(X)=\omega _L^X(S)$ for an arbitrary Borel set $S\subset \partial \Omega $ ). Also, we obtain a qualitative analog of the previous equivalences. Namely, we characterize the absolute continuity of $\omega _{L_0}$ with respect to $\omega _L$ in terms of some qualitative local $L^2(\omega _{L_0})$ estimates for the truncated conical square function for any bounded null solution of L. This is also equivalent to the finiteness $\omega _{L_0}$ -almost everywhere of the truncated conical square function for any bounded null solution of L. As applications, we show that $\omega _{L_0}$ is absolutely continuous with respect to $\omega _L$ if the disagreement of the coefficients satisfies some qualitative quadratic estimate in truncated cones for $\omega _{L_0}$ -almost everywhere vertex. Finally, when $L_0$ is either the transpose of L or its symmetric part, we obtain the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for $\omega _{L_0}$ -almost every vertex.

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“…Many of the results have been extended to complex valued elliptic operators and elliptic systems ( [HKMP15a], [DP19], [DP20], and [DHM21]). For an interested reader, who is new to this area, a nice and detailled discussion on those topics can be found in the introduction of [FP21].…”

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confidence: 99%

“…In our article, we take one of the most advanced existing results for the regularity problem, which is the stability of the regularity problem under Carleson perturbations [KP95] on a ball, and we prove that we can extend it in several directions: first we consider operators which are not necessarily symmetric, second we extend the geometric setting to uniform domains -which are domains with non-tangential access and Ahlfors regular boundaries, and third, we allow low dimensional boundaries, which were studied for the Dirichlet problem by Guy David, Zihui Zhao, Bruno Poggi, and the two last authors (see [DFM21b], [DFM19], [MZ19], [DFM20], [MP20], [FMZ21], [FP21], [DM20], and [Fen20]). Combined with another paper under preparation ( [DFM21a]), we ultimately prove the solvability of the regularity problem on the complement of a Lipschitz graph of lower dimension.…”

mentioning

confidence: 99%

“…The stability of the solvability of the Dirichlet problem under Carleson perturbations was established in [FKP91] (when Ω is a ball), [CHM19] and [CHMT20] (when Ω is a uniform domain in with n − 1-dimensional boundary), [MP20] (when Ω is uniform with lower dimensional boundary), and [FP21] (1-sided NTA domains and enough basic bounds on the harmonic measure, a setting that includes all the previous ones and more). These results are as follows:…”

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confidence: 99%

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Carleson Perturbations for the Regularity Problem

Dai¹,

Feneuil²,

Mayboroda³

2022

Preprint

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We prove that the solvability of the regularity problem in L q (∂Ω) is stable under Carleson perturbations. If the perturbation is small, then the solvability is preserved in the same L q , and if the perturbation is large, the regularity problem is solvable in L r for some other r ∈ (1, ∞). We extend an earlier result from Kenig and Pipher to very general unbounded domains, possibly with lower dimensional boundaries as in the theory developed by Guy David and the last two authors. To be precise, we only need the domain to have non-tangential access to its Ahlfors regular boundary, together with a notion of gradient on the boundary.

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Generalized Carleson perturbations of elliptic operators and applications

Cited by 8 publications

References 43 publications

Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

On the condition for elliptic operators in 1-sided nontangentially accessible domains satisfying the capacity density condition

Carleson Perturbations for the Regularity Problem

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