Carleson perturbations of elliptic operators on domains with low dimensional boundaries

Mayboroda, Svitlana; Poggi, Bruno

doi:10.1016/j.jfa.2021.108930

Cited by 11 publications

(4 citation statements)

References 51 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then, analogously to (9.61), we can finish the argument. We remark that the elliptic measure of L on ∂Ω F fits the hypothesis and, in particular, is doubling by [MP,DFM4].…”

Section: And Otherwise Replace It With An Ancestorsupporting

confidence: 72%

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

David¹,

Mayboroda²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

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 now well understood additional topological constraints. The topological restrictions, while mild, are necessary, as the counterexamples of C. Bishop and P. Jones show, and no analogues of these results have been available for higher co-dimensional sets.In the present paper we show that for any d < n − 1 and for any domain with a ddimensional uniformly rectifiable boundary the elliptic measure of an appropriate degenerate elliptic operator is absolutely continuous with respect to the Hausdorff measure of the boundary. There are no topological or dimensional restrictions contrary to the aforementioned results.Résumé en Franc ¸ais. On sait que la mesure harmonique associée à un domaine de R n dont a frontière est uniformément rectifiable de dimension n − 1 est absolument continue par rapport à la mesure de surface, sous des conditions topologiques récemment bien comprises. Ces conditions, bien que faibles, sont nécessaires, comme l'ont montré des contre exemples de C. Bishop and P. Jones. On ne disposait pas jusqu'ici de résultats analogues lorsque la frontière est de codimension plus grande.On démontre dans cet article que lorsque la frontière est uniformément rectifiable de dimension d < n − 1, la mesure elliptique associée à des opérateurs elliptiques dégénérés appropriés est absolument continue par rapport à la mesure de Hausdorff, sans avoir besoin de condition topologique supplémentaire.Key words/Mots clés. Harmonic measure, elliptic measure, uniform rectifiability, domains with large co-dimensional boundaries/ mesure harmonique, mesure elliptique, uniforme rectifiabilité, domaines à frontière de grande co-dimension.

show abstract

“…Then, analogously to (9.61), we can finish the argument. We remark that the elliptic measure of L on ∂Ω F fits the hypothesis and, in particular, is doubling by [MP,DFM4].…”

Section: And Otherwise Replace It With An Ancestorsupporting

confidence: 72%

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

David¹,

Mayboroda²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Indeed, it is not clear to us that sawtooth domains of PDE friendly domains are themselves PDE friendly. Even if they were, the construction of, and verification of PDE friendly axioms on sawtooth domains of some rough domains are long and difficult tasks [HM14,MP]. Our method resembles loosely that of the recent paper [CHMT], where an analogue of Theorem 1.22 is used to extend the FKP (additive) perturbation theory to the case of 1-sided chord-arc domains, but they also use sawtooth domains.…”

Section: History Of the Carleson Perturbationsmentioning

confidence: 99%

Generalized Carleson perturbations of elliptic operators and applications

Feneuil¹,

Poggi²

2020

Preprint

View full text Add to dashboard Cite

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 ∞ among elliptic measures.

show abstract

“…We would like to note that the arguments of [18,38,7] are written explicitly only in the case of real symmetric coefficients, but we would expect that similar arguments could be carried over to the non-symmetric case as well. We also mention [8], where the nonsymmetric case is also considered by using a different method, as well as [37], where perturbation theory for certain degenerate elliptic operators is developed in the setting of domains with lower dimensional boundaries.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

et al. 2023

View full text Add to dashboard Cite

Let Ω ⊂ ℝ n + 1 {\Omega\subset\mathbb{R}^{n+1}} , n ≥ 2 {n\geq 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 L 0 ⁢ u = - div ⁢ ( A 0 ⁢ ∇ ⁡ u ) {L_{0}u=-\mathrm{div}(A_{0}\nabla u)} , L ⁢ u = - div ⁢ ( A ⁢ ∇ ⁡ u ) {Lu=-\mathrm{div}(A\nabla u)} , two real (non-necessarily symmetric) uniformly elliptic operators in Ω, and write ω L 0 {\omega_{L_{0}}} , ω L {\omega_{L}} for the respective associated elliptic measures. The goal of this program is to find sufficient conditions guaranteeing that ω L {\omega_{L}} satisfies an A ∞ {A_{\infty}} -condition or a RH q {\mathrm{RH}_{q}} -condition with respect to ω L 0 {\omega_{L_{0}}} . In this paper we establish that if the discrepancy of the two matrices satisfies a natural Carleson measure condition with respect to ω L 0 {\omega_{L_{0}}} , then ω L ∈ A ∞ ⁢ ( ω L 0 ) {\omega_{L}\in A_{\infty}(\omega_{L_{0}})} . Additionally, we can prove that ω L ∈ RH q ⁢ ( ω L 0 ) {\omega_{L}\in\mathrm{RH}_{q}(\omega_{L_{0}})} for some specific q ∈ ( 1 , ∞ ) {q\in(1,\infty)} , by assuming that such Carleson condition holds with a sufficiently small constant. This “small constant” case extends previous work of Fefferman–Kenig–Pipher and Milakis–Pipher together with the last author of the present paper who considered symmetric operators in Lipschitz and bounded chord-arc domains, respectively. Here we go beyond those settings, our domains satisfy a capacity density condition which is much weaker than the existence of exterior Corkscrew balls. Moreover, their boundaries need not be Ahlfors regular and the restriction of the n-dimensional Hausdorff measure to the boundary could be even locally infinite. The “large constant” case, that is, the one on which we just assume that the discrepancy of the two matrices satisfies a Carleson measure condition, is new even in the case of nice domains (such as the unit ball, the upper-half space, or non-tangentially accessible domains) and in the case of symmetric operators. We emphasize that our results hold in the absence of a nice surface measure: all the analysis is done with the underlying measure ω L 0 {\omega_{L_{0}}} , which behaves well in the scenarios we are considering. When particularized to the setting of Lipschitz, chord-arc, or 1-sided chord-arc domains, our methods allow us to immediately recover a number of existing perturbation results as well as extend some of them.

show abstract

Carleson perturbations of elliptic operators on domains with low dimensional boundaries

Cited by 11 publications

References 51 publications

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

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

Generalized Carleson perturbations of elliptic operators and applications

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

Contact Info

Product

Resources

About