Carleson Condition Research Articles

Boundary Value Problems for Elliptic Operators Satisfying Carleson Condition

Boundary Value Problems for Elliptic Operators Satisfying Carleson Condition

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

Abstract 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.

Carleson embedding on the tri-tree and on the tri-disc

We prove a multi-parameter dyadic embedding theorem for the Hardy operator on the multi-tree. We also show that for a large class of Dirichlet spaces in the bi-disc and the tri-disc, this proves the embedding theorem of those Dirichlet spaces of holomorphic functions on the bi- and tri-disc. We completely describe the Carleson measures for such embeddings. The result below generalizes the embedding result of Arcozzi et al. from the bi-tree to the tri-tree and from the Carleson–Chang condition to the Carleson box condition. One of our embedding descriptions is similar to the Carleson–Chang–Fefferman condition, and involves dyadic open sets. On the other hand, the unusual feature is that the embedding on the bi-tree and the tri-tree turns out to be equivalent to the one box Carleson condition. This is in striking difference to works of Chang–Fefferman and the well-known Carleson quilt counterexample. Finally, we explain the obstacle that prevents us from proving our results on poly-discs of dimension four and higher.

Approximation of Green functions and domains with uniformly rectifiable boundaries of all dimensions

The present paper concerns divergence form elliptic and degenerate elliptic operators in a domain Ω⊂Rn, and establishes the equivalence between the uniform rectifiability of the boundary E=∂Ω and weak Carleson condition on the good approximation of the Green function G by affine, or distance, functions. There are two main original contexts for the results, elliptic operators in a non-tangential access domain with an n−1 dimensional boundary and degenerate elliptic operators adapted to a domain with an Ahlfors regular boundary of larger co-dimension. In both cases necessary and sufficient conditions are given, in the form of Carleson packing conditions on the collection of balls centered on E where G is not well approximated.(1)This is the first time the underlying property of the control of the Green function by affine functions, or by the distance to the boundary, in the sense of the Carleson prevalent sets, appears in the literature; some results established here are new even in the half space;(2)the results are optimal, providing a full characterization of uniform rectifiability under the (standard) mild topological assumptions;(3)to the best of the authors' knowledge, even in traditional domains with (n−1)-dimensional boundaries, this is the first free boundary result applying to all elliptic operators, without any restriction on the coefficients (the direct one assumes the standard, and necessary, Carleson measure condition);(4)this is the first free boundary result in higher co-dimensional setting and as such, the first PDE characterization of uniform rectifiability for a set of dimension d, d<n−1, in Rn. The paper offers a general way to deal with related issues considerably beyond the scope of the aforementioned theorem, including the question of approximability of the gradient of the Green function, and the comparison of the Green function to a certain version of the distance to the original set rather than distance to the hyperplanes.

Carleson estimates for the Green function on domains with lower dimensional boundaries

In the present paper we consider an elliptic divergence form operator in Rn∖Rd with d<n−1 and prove that its Green function is almost affine, in the sense that the normalized difference between the Green function with a sufficiently far away pole and a suitable affine function at every scale satisfies a Carleson measure estimate. The coefficients of the operator can be very oscillatory, and only need to satisfy some condition similar to the traditional quadratic Carleson condition.

Elliptic measures for Dahlberg–Kenig–Pipher operators: asymptotically optimal estimates

Questions concerning quantitative and asymptotic properties of the elliptic measure corresponding to a uniformly elliptic divergence form operator have been the focus of recent studies. In this setting we show that the elliptic measure of an operator with coefficients satisfying a vanishing Carleson condition in the upper half space is an asymptotically optimal $$A_\infty $$ weight. In particular, for such operators the logarithm of the elliptic kernel is in the space of (locally) vanishing mean oscillation. To achieve this, we prove local, quantitative estimates on a quantity (introduced by Fefferman, Kenig and Pipher) that controls the $$A_\infty $$ constant. Our work uses recent results obtained by David, Li and Mayboroda. These quantitative estimates may offer a new framework to approach similar problems.

Carleson Measure Estimates for the Green Function

In the present paper we consider an elliptic divergence form operator in the half-space and prove that its Green function is almost affine, or more precisely, that the normalized difference between the Green function and a suitable affine function at every scale satisfies a Carleson measure estimate, provided that the oscillations of the coefficients satisfy the traditional quadratic Carleson condition. The results are sharp, and in particular, it is demonstrated that the class of the operators considered in the paper cannot be improved.

Carleson measures induced by higher order Schwarzian derivatives and derivatives of analytic functions

Let G be a finitely generated Fuchsian group of the second kind without any parabolic element and f be a univalent analytic function in the unit diskDcompatible withG. In this paper, westudy the higher order Schwarzian derivatives: ?n+1( f ) = ?? n( f )?(n?1) f?? f???n( f ), n ? 3, where ?3(f) stands for the Schwarzian derivatives of f, and Sn(f) = (f?) n?1/2 Dn( f ?)? n?1/2 , n ? 2. For p &gt; 0, we show that if |?n(f)(z)|p(1?|z|2)p(n?1)?1dxdy (resp. |Sn( f )(z)|p(1 ? |z|2)pn?1dxdy) satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domainF ofG, then |?n( f )(z)|p(1?|z|2)p(n?1)?1dxdy (resp. |Sn( f )(z)|p(1?|z|2)pn?1dxdy) is a Carleson measure in D. Similarly, for p &gt; 0 and a bounded analytic function f in the unit disk D compatible with G, we prove that if | f ?(z)|p(1 ? |z|2)p?1dxdy satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain F of G, then |f?(z)|p(1 ? |z|2)p?1dxdy is a Carleson measure in D.

On Carleson measures induced by Beltrami coefficients being compatible with Fuchsian groups

Let \(\mu\) be a Beltrami coefficient on the unit disk, which is compatible with a finitely generated Fuchsian group \(G\) of the second kind. In this paper we show that if \(\frac{|\mu|^{2}}{1-|z|^{2}}\,dx\,dy\) satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain of \(G\), then \(\frac{|\mu|^{2}}{1-|z|^{2}}\,dx\,dy\) is a Carleson measure on the unit disk.

Carleson Measure of Harmonic Schwarzian Derivatives Associated with a Finitely Generated Fuchsian Group of the Second Kind

Let S H f be the Schwarzian derivative of a univalent harmonic function f in the unit disk D , compatible with a finitely generated Fuchsian group G of the second kind. We show that if S H f 2 1 − z 2 3 d x d y satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain F of G , then S H f 2 1 − z 2 3 d x d y is a Carleson measure in D .

The Lp Dirichlet and regularity problems for second order elliptic systems with application to the Lamé system

In a recent joint paper with Hwang and Mitrea we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domain above a Lipschitz graph, satisfying Lp -boundary data for p near 2. The main novel aspect of our result is that it applies to operators with coefficients of limited regularity and applies to operators satisfying a natural Carleson condition that has been first considered in the scalar case. In this paper we extend this result in several directions. We improve the range of solvability of the Lp Dirichlet problem to the interval for systems in dimension n = 2, 3 in the range We do this by considering solvability of the Regularity problem (with boundary data having one derivative in Lp ) in the range Secondly, we look at perturbation type-results where we can deduce solvability of the Lp Dirichlet problem for one operator from known Lp Dirichlet solvability of a “close” operator (in the sense of Carleson measure). This leads to improvement of the main result of our previous paper; we establish solvability of the Lp Dirichlet problem in the interval under a much weaker (oscillation-type) Carleson condition. A particular example of the system where all these results apply is the Lamé operator for isotropic inhomogeneous materials with Poisson ratio In this specific case further improvements of the solvability range are possible.

The 𝐿^{𝑝} Dirichlet boundary problem for second order elliptic Systems with rough coefficients

Given a domain above a Lipschitz graph, we establish solvability results for strongly elliptic second-order systems in divergence-form, allowed to have lower-order (drift) terms, with L p L^p -boundary data for p p near 2 2 (more precisely, in an interval of the form ( 2 − ε , 2 ( n − 1 ) n − 2 + ε ) \big (2-\varepsilon ,\frac {2(n-1)}{n-2}+\varepsilon \big ) for some small ε &gt; 0 \varepsilon &gt;0 ). The main novel aspect of our result is that the coefficients of the operator do not have to be constant, or have very high regularity; instead they will satisfy a natural Carleson condition that has appeared first in the scalar case. A significant example of a system to which our result may be applied is the Lamé system for isotropic inhomogeneous materials. We show that our result applies to isotropic materials with Poisson ratio ν &gt; 0.396 \nu &gt;0.396 . Dealing with genuine systems gives rise to substantial new challenges, absent in the scalar case. Among other things, there is no maximum principle for general elliptic systems, and the De Giorgi–Nash–Moser theory may also not apply. We are, nonetheless, successful in establishing estimates for the square-function and the nontangential maximal operator for the solutions of the elliptic system described earlier, and use these as alternative tools for proving L p L^p solvability results for p p near 2 2 .

Parabolic Lp Dirichlet boundary value problem and VMO-type time-varying domains

We prove the solvability of the parabolic $L^p$ Dirichlet boundary value problem for $1 < p \leq \infty$ for a PDE of the form $u_t = \mbox{div} (A \nabla u) + B \cdot \nabla u$ on time-varying domains where the coefficients $A= [a_{ij}(X, t)]$ and $B=[b_i]$ satisfy a certain natural small Carleson condition. This result brings the state of affairs in the parabolic setting up to the elliptic standard. Furthermore, we establish that if the coefficients of the operator $A,\,B$ satisfy a vanishing Carleson condition and the time-varying domain is of VMO type then the parabolic $L^p$ Dirichlet boundary value problem is solvable for all $1 < p \leq \infty$. This result is related to results in papers by Maz\'ya, Mitrea and Shaposhnikova, and Hofmann, Mitrea and Taylor where the fact that boundary of domain has normal in VMO or near VMO implies invertibility of certain boundary operators in $L^p$ for all $1 < p \leq \infty$ which then (using the method of layer potentials) implies solvability of the $L^p$ boundary value problem in the same range for certain elliptic PDEs. Our result does not use the method of layer potentials, since the coefficients we consider are too rough to use this technique but remarkably we recover $L^p$ solvability in the full range of $p$'s as the two papers mentioned above.

Perturbation Theory for Solutions to Second Order Elliptic Operators with Complex Coefficients and the Lp Dirichlet Problem

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If L0 = divA0(x)∇ + B0(x). ∇ is a p-elliptic operator satisfying the assumptions of Theorem 1.1 then the Lp Dirichlet problem for the operator L0 is solvable in the upper half-space ℝ+n. In this paper we prove that the Lp solvability is stable under small perturbations of L0. That is if L1 is another divergence form elliptic operator with complex coefficients and the coefficients of the operators L0 and L1 are sufficiently close in the sense of Carleson measures, then the Lp Dirichlet problem for the operator L1 is solvable for the same value of p. As a corollary we obtain a new result on Lp solvability of the Dirichlet problem for operators of the form L = divA(x)∇ + B(x) · ∇ where the matrix A satisfies weaker Carleson condition (expressed in term of oscillation of coefficients). In particular the coefficients of A need no longer be differentiate and instead satisfy a Carleson condition that controls the oscillation of the matrix A over Whitney boxes. This result in the real case has been established by Dindos, Petermichl and Pipher.

Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided chord-arc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness), and whose boundary $\partial\Omega$ is $n$-dimensional Ahlfors regular. Consider $L_0$ and $L$ two real symmetric divergence form elliptic operators and let $\omega_{L_0}$, $\omega_L$ be the associated elliptic measures. We show that if $\omega_{L_0}\in A_\infty(\sigma)$, where $\sigma=H^n|_{\partial\Omega}$, and $L$ is a perturbation of $L_0$ (in the sense that the discrepancy between $L_0$ and $L$ satisfies certain Carleson measure condition), then $\omega_L\in A_\infty(\sigma)$. Moreover, if $L$ is a sufficiently small perturbation of $L_0$, then one can preserve the reverse H\"older classes, that is, if for some $1<p<\infty$, one has $\omega_{L_0}\in RH_p(\sigma)$ then $\omega_{L}\in RH_p(\sigma)$. Equivalently, if the Dirichlet problem with data in $L^{p'}(\sigma)$ is solvable for $L_0$ then so it is for $L$. These results can be seen as extensions of the perturbation theorems obtained by Dahlberg, Fefferman-Kenig-Pipher, and Milakis-Pipher-Toro in more benign settings. As a consequence of our methods we can show that for any perturbation of the Laplacian (or, more in general, of any elliptic symmetric operator with Lipschitz coefficients satisfying certain Carleson condition) if its elliptic measure belongs to $A_\infty(\sigma)$ then necessarily $\Omega$ is in fact an NTA domain (and hence chord-arc) and therefore its boundary is uniformly rectifiable.

The Dirichlet boundary problem for second order parabolic operators satisfying a Carleson condition

We establish L^p , 2\le p\le\infty , solvability of the Dirichlet boundary value problem for a parabolic equation u_t- \mathrm {div}(A\nabla u) - \boldsymbol{B}\cdot\nabla u =0 on time-varying domains with coefficient matrices A=[a_{ij}] and \boldsymbol{B} = [b_{i}] that satisfy a small Carleson condition. The results are sharp in the following sense. For a given value of 1 &lt; p &lt; \infty there exists operators that satisfy Carleson condition but fail to have L^p solvability of the Dirichlet problem. Thus the assumption of smallness is sharp . Our results complements results of Hofmann, Lewis and Rivera-Noriega, where solvability of parabolic L^p (for some large p ) Dirichlet boundary value problem for coefficients that satisfy large Carleson condition was established. We also give a new (substantially shorter) proof of these results.

On the non-smoothness of the vector fields for the dynamically invariant Beltrami coefficients

Let $\Omega$ be a Carleson-Denjoy domain and $G$ be its covering group. Let $\mu$ be a Beltrami coefficient on the unit disk which is compatible with the group $G$. In this paper we show that if $\frac{|\mu|^{2}}{1-|z|^{2}}dxdy$ satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain of $G$, then $\frac{|\mu|^{2}}{1-|z|^{2}}dxdy$ is a Carleson measure on the unit disk. We also show that the above property does not hold for Denjoy domain.

Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries

Let Ω ⊂ R n + 1 \Omega \subset \mathbb {R}^{n+1} , n ≥ 2 n\geq 2 , be a 1-sided NTA domain (also known as a uniform domain), i.e., a domain which satisfies interior corkscrew and Harnack chain conditions, and assume that ∂ Ω \partial \Omega is n n -dimensional Ahlfors-David regular. We characterize the rectifiability of ∂ Ω \partial \Omega in terms of the absolute continuity of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that ∂ Ω \partial \Omega can be covered H n \mathcal {H}^n -a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains of Ω \Omega and to the fact that ∂ Ω \partial \Omega possesses exterior corkscrew points in a qualitative way H n \mathcal {H}^n -a.e. Our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally Lipschitz coefficients whose derivatives satisfy a natural qualitative Carleson condition.

A quantitative approach to weighted Carleson condition

Abstract Quantitative versions of weighted estimates obtained by F. Ruiz and J.L. Torrea [30, 31] for the operator are obtained. As a consequence, some sufficient conditions for the boundedness of Min the two weight setting in the spirit of the results obtained by C. Pérez and E. Rela [26] and very recently by M. Lacey and S. Spencer [17] for the Hardy-Littlewood maximal operator are derived. As a byproduct some new quantitative estimates for the Poisson integral are obtained.

On the geometry of rectifiable sets with Carleson and Poincar\'{e}-type conditions

A central question in geometric measure theory is whether geometric properties of a set translate into analytical ones. In 1960, E. R. Reifenberg proved that if an $n$-dimensional subset $M$ of $\mathbb{R}^{n+k}$ is well approximated by $n$-planes at every point and at every scale, then $M$ is a locally bi-Holder image of an $n$-plane. Since then, Reifenberg's theorem has been refined in several ways in order to ensure that $M$ is a bi-Lipschitz image of an $n$-plane. In this paper, we show that a Carleson condition on the oscillation of the unit normal of an $n$-Ahlfors regular rectifiable subset $M$ of $\mathbb{R}^{n+1}$ satisfying a Poincare-type inequality is sufficient to prove that $M$ is contained inside a bi-Lipschitz image of an $n$-dimensional affine subspace of $\mathbb{R}^{n+1}$. We also show that this Poincare-type inequality encodes geometrical information about $M$, namely it implies that $M$ is quasiconvex.