Green functions and smooth distances

We consider elliptic operators $L$ on a domain $\Omega\subset\mathbb{R}^{n}$. In recent years a finale of an enormous body of work brought a characterization of uniform rectifiability in terms of absolute continuity of harmonic measure (see [AHMMT], a sample of earlier articles: [DJ], [HM1], [HM2], [HMU], [AHM3TV], [Azz2], see also the related article [NTV] which proves the David-Semmes conjecture in codimension 1 and is a key step for the converse established in [AHM3TV]). It also became clear that this characterization has its restrictions, for it fails in the domains with lower dimensional boundaries and it requires, in all directions, restrictions on the coefficients – see a discussion in [DM2]. In these contexts, the Green function emerged as a more suitable object to define the relevant PDE properties. Already the work in [Azz1] and [DM2] suggested a possibility of a Green function characterization of regularity of sets. However, factually, [DM2] provided more than satisfactory “free boundary” results and only weak “direct” results (no norm control). The papers [DLM1], and [DLM2], and [DFM4] aimed at the desired quantitative version of such “direct” results but were restricted to either Lipschitz graphs or sets with lower dimensional boundaries. The primary goal of the article is to show that if $L$ is reasonably well-behaved, and $\Omega$ provides some access to its boundary, then the boundary of $\Omega$ is reasonably regular (uniformly rectifiable) if and only if the Green function behaves like a distance to the boundary.

Let us discuss some predecessors of this work, including the aforementioned ones, in more details. In [Azz1] Theorem VI, it is shown that the affine deviation of the Green function for the Laplace operator is related to the linear deviation of the boundary of the domain. In [DM2], G. David and the third author of the paper show that for a class of elliptic operators, the Green function can be well approximated by distances to planes, or by a smooth distance to $\partial\Omega$, if and only $\partial\Omega$ is uniformly rectifiable. The bounds on the Green function given in [DM2] are weak, more precisely, they carry no norm control of the sets where the Green function is close to a distance. Later, stronger and quantitative estimates on the comparison of the Green function and some distance functions are obtained in [DFM4], [DLM1], and [DLM2]. In [DLM1], a quantitative comparison between the Green function and the distance function to the boundary is given for an optimal class of elliptic operators on the upper half-space. Moreover, the proximity of the Green function and the distance function is shown to be precisely controlled by the oscillation of the coefficients of the operator. Next, [DLM2] extends the result of [DLM1] to $\mathbb{R}^{n}\setminus\mathbb{R}^{d}$ with $d$ strictly less than $n$. But the methods employed in [DLM1] and [DLM2] seem to the authors difficult to be adapted to domains whose boundaries are rougher than Lipschitz graphs. In [DFM4], a bound for the difference of the Green function and smooth distances is obtained for sets with uniformly rectifiable boundaries, but its proof, which might appear surprising, is radically dependent on the fact that the boundary is of codimension strictly larger than 1. Also, the class of operators considered in [DFM4] is not optimal. So the instant motivation for the present work is to obtain a strong estimate on the Green function for an optimal class of operators, similar to the one considered in [DLM1], in a “classical” setting: on domains with uniformly rectifiable boundaries of codimension 1. The method employed here is completely different from [DFM4] or [DLM1], and has the potential to be applicable to many other problems at the intersection of PDE and geometric measure theory.

We should also mention that in [HMT1] and [DLM1], some Carleson measure estimates on the second derivatives of the Green function have been obtained, and that in [Azz1], the second derivative of the Green function for the Laplace operator is linked to the regularity (uniform rectifiability) of the boundary of the domain. However, the result of [Azz1] is only for the Laplace operator, the class of elliptic operators considered in [HMT1] is more general but still not optimal, and the estimates obtained in [DLM1] are restricted to Lipschitz graph domains. We think that our estimates might shed some light on proving an estimate on second derivatives of the Green function for an optimal class of elliptic operators on chord-arc domains.

For the “free boundary” direction, since the weak type property of the Green function considered in [DM2] already implies uniform rectifiablity, one expects the strong estimate on the Green function that we consider in this paper to automatically give uniform rectifiability. However, linking the two conditions directly seems to be more subtle than it might appear, and we actually need to obtain uniform rectifiablity from scratch. We point out that our result also holds for bounded domains, and thus dispensing with the unboundedness assumption on the domain in [DM2].

All in all, this paper is a culmination of all of the aforementioned efforts, featuring a true equivalence (characterization) of geometry through PDEs, and an optimal class of operators.

We take a domain $\Omega\subset\mathbb{R}^{n}$ whose boundary $\partial\Omega$ is $(n-1)$-Ahlfors regular (AR for shortness), which means that there exists a measure $\sigma$ supported on $\partial\Omega$ such that

The domain $\Omega$ can be bounded or unbounded. In the unbounded case, $\operatorname{diam}\Omega=\infty$. In the rest of the paper, $\sigma$ will always be an Ahlfors regular measure on $\partial\Omega$. It is known that the Ahlfors regular measures are the ones that can be written as $d\sigma=wd\mathcal{H}^{n-1}|_{\partial\Omega}$, where $\mathcal{H}^{n-1}|_{\partial\Omega}$ is the $n-1$ dimensional Hausdorff measure on $\partial\Omega$, and $w$ is a weight in $L^{\infty}(\partial\Omega,\mathcal{H}^{n-1}|_{\partial\Omega})$ such that $C^{-1}\leq w\leq C$ for some constant $C>0$.

We shall impose more assumptions on our domain. For both the “free boundary” and the “direct” results, we will assume that $\Omega$ is a 1-sided Chord Arc Domain (see Definition 2.11). For the “direct” result, we will rely on the assumption that $\partial\Omega$ is uniformly rectifiable (see [DS1, DS2] and Section 3 below), and thus ultimately assuming that $\Omega$ is a (2-sided) Chord Arc Domain. The optimality of the assumptions on $\Omega$ is discussed in more details in the end of this subsection. Since the dimension $n-1$ plays an important role in our paper, and in order to lighten the notion, we shall write $d$ for $n-1$.

Without any more delay, let us introduce the regularized distance to a set $\partial\Omega$. The Euclidean distance to the boundary is denoted by

For $\beta>0$, we define

The fact that the set $\partial\Omega$ is $d$-Ahlfors regular is enough to have that

where the constant depends on $C_{\sigma}$, $\beta$, and $n$. The proof is easy and can be found after Lemma 5.1 in [DFM2].

The notion of Carleson measure will be central all over our paper. We say that a quantity $f$ defined on $\Omega$ satisfies the Carleson measure condition - or $f\in CM_{\Omega}(M)$ for short - if there exists $M$ such that for any $x\in\partial\Omega$ and $r<\operatorname{diam}(\Omega)$,

Our operators are in the form $L=-\mathop{\operatorname{div}}\mathcal{A}\nabla$ and defined on $\Omega$. We shall always assume that they are uniformly elliptic and bounded, that is, there exists $C_{\mathcal{A}}>1$ such that

and

A weak solution to $Lu=0$ in $E\subset\Omega$ lies in $W^{1,2}_{loc}(E)$ and is such that

If $\Omega$ has sufficient access to the boundary (and $\partial\Omega$ is $(n-1)$-Ahlfors regular), then for any ball $B$ centered on $\partial\Omega$ and any function $u$ in $W^{1,2}(B\cap\Omega)$, we have notion of trace for $u$ on $B\cap\partial\Omega$. It is well known that if $u\in W^{1,2}(B\cap\Omega)$ is such that $\operatorname{Tr}(u)=0$ on $B\cap\partial\Omega$, and if $u$ is a weak solution to $Lu=0$ on $B\cap\Omega$ with $L$ satisfying (1.6) and (1.7), then $u$ is continuous $B\cap\Omega$ and can be continuously extended by 0 on $B\cap\partial\Omega$.

In addition to (1.6) and (1.7), we assume that our operators satisfy a weaker variant of the Dahlberg-Kenig-Pipher condition. The Dahlberg-Kenig-Pipher (DKP) condition was introduced by Dahlberg and shown to be sufficient for the $L^{p}$ solvability of the Dirichlet problem for some $p>1$ by Kenig and Pipher ([KP]). It was also shown to be essentially necessary in [CFK, MM]. The DKP condition says that the coefficient matrix $\mathcal{A}$ satisfies

Our assumption, slightly weaker than the classical DKP, is as follows.

Obviously, this condition is weaker than (1.9): it allows for small Carleson perturbations and carries no supremum over the Whitney cubes. Moreover, we show in Lemma 2.1 that the weak DKP condition self improves.

We are now ready for the statement of our main result.

The equivalence $(i)\Longleftrightarrow(ii)$ is already well known, see Theorem 1.2 in [AHMNT]. Moreover, $(iii)\implies(iv)\implies(v)$ is immediate. So we need only to prove $(ii)\implies(iii)$ and $(v)\implies(i)$ in Theorem 1.12.

When the domain is unbounded, we can use the Green function with pole at infinity instead of the Green function. The Green function with pole at infinity associated to $L$ is the unique (up to a multiplicative constant) positive weak solution to $Lu=0$ with zero trace. See for instance Lemma 6.5 in [DEM] for the construction ([DEM] treats a particular case but the same argument works as long as we have CFMS estimates, see Lemma 2.18 below). So we have that:

For our proof of the “direct” result, we need the fact that, for the same operators, the $L$-elliptic measure is $A_{\infty}$-absolutely continuous with respect to $\sigma$.

The above is known for operators satisfying the DKP condition (1.9) on Chord-Arc domains. In fact, it is shown in [KP] that $\omega_{L}\in A_{\infty}(\sigma)$ for DKP operators on Lipschitz domains. But since the DKP condition is preserved in subdomains, and the Chord-Arc domains are well approximated by Lipschitz subdomains ([DJ]), the $A_{\infty}$ property can be passed from Lipschitz subdomains to Chord-Arc domains (see [JK], or [HMMTZ]). Moreover, combined with the stability of the $A_{\infty}$ property under Carleson perturbation of elliptic operators proved in [CHMT], it is also known for the elliptic operators $L=-\operatorname{div}\mathcal{A}\nabla$ for which $\mathcal{A}=\mathcal{B}+\mathcal{C}$, where

However, to the best of our knowledge, the $A_{\infty}$-absolute continuity of the elliptic measure was not proved explicitly for elliptic operators satisfying the slightly weaker condition (1.11).

We obtain Theorem 1.18 as a consequence of the following result - which is another contribution of the article - and Theorem 1.1 in [CHMT].

Notice that in this theorem, we completely dispense with the Harnack chain and corkscrew conditions (see Definitions 2.8 and 2.9) for the domain. Previously, an analogous result was obtained for bounded harmonic functions in [HMM1] (see also [GMT] for the converse) and for DKP operators in [HMM2, Theorem 7.5]. The result for elliptic operators which satisfy the weak DKP condition is again not explicitly written anywhere. However, slightly changing the proofs of a combination of papers would give the result. For instance, we can adapt Theorem 1.32 in [DFM2] to the codimension 1 case to prove Theorem 1.18 in the flat case, then extending it to Lipschitz graph by using the change of variable in [KP], and finally proving 1.18 to all complement of uniformly rectifiable domains by invoking Theorem 1.19 (iii) in [HMM2]).

Here, we claim that we can directly demonstrate Theorem 1.21 using a strategy similar to our proof of Theorem 1.12. In Section 8, we explain how to modify the proof of Theorem 1.12 to obtain Theorem 1.21. By [CHMT] Theorem 1.1, assuming that $\Omega$ is 1-sided CAD, the estimate (1.22) implies that $\omega_{L}\in A_{\infty}(\sigma)$, and therefore our Theorem 1.18 follows from Theorem 1.21. Note that the bound (1.19) is a characterization of $A_{\infty}$, see for instance Theorem 1.4.13 in [Ken].

Let us discuss in more details our assumptions for Theorem 1.12. In order to get the bound (1.13), we strongly require that the boundary $\partial\Omega$ is uniformly rectifiable and that the operator $L$ satisfies the weak DKP condition. We even conjecture that those conditions are necessary, that is, if $\partial\Omega$ is not uniformly rectifiable, then the bound (1.13) holds for none of the weak DKP operators.

The corkscrew condition and the Harnack chain condition (see Definitions 2.8 and 2.9) are only needed for the proof of Lemma 7.12 - where we used the comparison principle - and for the implication $(iii)\implies(i)$ in Theorem 1.12. However, since most of our intermediate results can be proved without those conditions and could be of interest in other situations where the Harnack chain is not assumed (like - for instance - to prove Theorem 1.21), we avoided to use the existence of Harnack chains and of corkscrew points in all the proofs and the intermediate results except for Lemma 7.12 and in Section 9, even if it occasionally slightly complicated the arguments.

These observations naturally lead to the question about the optimality of our conditions on $\Omega$, and more precisely, whether we can obtain the estimate (1.13) assuming only uniform rectifiablity. The answer is no, as we can construct a domain $\Omega$ which has uniformly rectifiable boundary but is only semi-uniform (see Definition 10.1) where (1.13) fails. More precisely, we prove in Section 10 that:

But of course, assuming $\Omega$ is a Chord-Arc Domain is not necessary for (1.13) since (1.13) obviously holds when $\Omega=\mathbb{R}^{n}\setminus\mathbb{R}^{n-1}=\mathbb{R}^{n}_{+}\cup\mathbb{R}^{n}_{-}$, because we can apply Theorem 1.12 to both $\Omega_{+}=\mathbb{R}^{n}_{+}$ and $\Omega_{-}=\mathbb{R}^{n}_{-}$ and then sum.

In this section, we present the outline of the proof of $(ii)\implies(iii)$ in Theorem 1.12. More exactly, this subsection aims to link the results of all other sections of the paper in order to provide the proof.

The approach developed in this article is new and it is interesting by itself, because it gives an alternative to proofs that use projection and extrapolation of measures. Aside from Theorems 1.12 and 1.21, we claim that our approach can be used to obtain a third proof of the main result from [Fen1] and [DM1], which establishes the $A_{\infty}$-absolute continuity of the harmonic measure when $\Omega$ is the complement of a uniformly rectifiable set of low dimension and $L$ is a properly chosen degenerate operator.

Let $\Omega$ and $L$ be as in the assumptions of Theorem 1.12, and let $\mathcal{B}$ and $\mathcal{C}$ denote the matrices in (1.11). Take $B:=B(x_{0},r)$ to be a ball centered at the boundary, that is $x_{0}\in\partial\Omega$, and then a non-negative weak solution $u$ to $Lu=0$ in $2B\cap\Omega$ such that $\operatorname{Tr}(u)=0$ on $B\cap\partial\Omega$.

We take $\{Q_{0}^{i}\}\subset\mathbb{D}_{\partial\Omega}$ as the collection of dyadic cubes that intersects $B\cap\partial\Omega$ and such that $2^{-9}r<\ell(Q_{0}^{i})\leq 2^{-8}r$. There is a uniformly bounded number of them, each of them satisfies $Q_{0}^{i}\subset\frac{3}{2}B\cap\Omega$ and $\ell(Q_{0}^{i})\leq 2^{-8}r$ and altogether, they verify

The estimate (1.13) follows by applying (1.24) to each $Q_{0}^{i}$ - using (1.4) and (1.1) when needed - and (1.26) below to $\{X\in B,\,\delta(X)>2^{-9}r\}$.

Let $\mathcal{B}(Q_{0}):=\mathcal{B}\cap\mathbb{D}_{\partial\Omega}(Q_{0})$ and then $\mathfrak{S}(Q_{0})=\{\mathcal{S}\cap\mathbb{D}_{\partial\Omega}(Q_{0})\}$. Observe that $\mathfrak{S}(Q_{0})$ contains the collection of $\mathcal{S}\in\mathfrak{S}$ such that $Q(\mathcal{S})\subset Q_{0}$ and maybe one extra element, in the case where $Q_{0}\notin\mathcal{B}\cup\bigcup_{\mathcal{S}\in\mathfrak{S}}Q(\mathcal{S})$, which is the intersection with $\mathbb{D}_{\partial\Omega}(Q_{0})$ of the coherent regime $\mathcal{S}\in\mathfrak{S}$ that contains $Q_{0}$. In any case $\mathfrak{S}(Q_{0})$ is a collection of (stopping time) coherent regimes. In addition, Lemma 3.16 shows that $\mathfrak{S}(Q_{0})$ and $\mathcal{B}(Q_{0})$ verifies the Carleson packing condition

We use this corona decomposition to decompose the sum $I$ from (1.24) as

where $W_{\Omega}(\mathcal{S})=\bigcup_{Q\in\mathcal{S}}W_{\Omega}(Q)$. For each cube $Q\in\mathcal{B}(Q_{0})$, the regions $W_{\Omega}(Q)$ are included in $\frac{7}{4}B$ and verify $\operatorname{diam}(W_{\Omega}(Q))\leq 8\delta(W_{\Omega}(Q))\leq 8\ell(Q)$, so we can use (1.26) and we obtain

by (1.1) and (1.27).

Nevertheless, what we truly want is the fact $\partial\Omega$ is well approximated by a plane - instead of a Lipschitz graph - from the standpoint of any $X\in W_{\Omega}(\mathcal{S})$, because in this case we can use Lemma 7.9. Yet, despite this slight disagreement, $\Gamma_{\mathcal{S}}$ is much better than a random uniformly rectifiable set, because $\Gamma_{\mathcal{S}}$ is the image of a plane $P$ by a bi-Lipschitz map. So we construct a bi-Lipschitz map $\rho_{\mathcal{S}}:\,\mathbb{R}^{n}\to\mathbb{R}^{n}$ that of course maps a plane to $\Gamma_{\mathcal{S}}$, but which also provides an explicit map from any point $X$ in $W_{\Omega}(\mathcal{S})$ to a plane $\Lambda(\rho_{\mathcal{S}}^{-1}(X))$ that well approximates $\Gamma_{\mathcal{S}}$ - hence $\partial\Omega$ - from the viewpoint of $X$. So morally, we constructed $\rho_{\mathcal{S}}$ such that we have a function

which, when $X\in W_{\Omega}(\mathcal{S})$, is a good approximation of

in terms of the Tolsa’s $\alpha$-numbers.

We combine the two approximations to prove (see Lemma 6.30, which is a consequence of Corollary 5.53 and our construction of $\rho_{\mathcal{S}}$) that

where $Y\to N_{\rho_{\mathcal{S}}^{-1}(X)}(Y)$ is the gradient of the distance to $\Lambda(\rho_{\mathcal{S}}^{-1}(X))$. And since the $\alpha_{\sigma,\beta}(Q)$ satisfies the Carleson packing condition, see Lemma 5.31, we deduce that

where

We are left with $I^{\prime}_{\mathcal{S}}$. We make the change of variable $(p,t)=\rho^{-1}_{\mathcal{S}}(X)$ in the integral defining $I^{\prime}_{\mathcal{S}}$ and we obtain that

where $\operatorname{Jac}(p,t)$ is the Jacobian matrix of $\rho_{\mathcal{S}}$, which is close to the identity by Lemma 6.8. We have also used that $\delta\circ\rho_{\mathcal{S}}(p,t)\approx t$ since $\delta(X)\approx\operatorname{dist}(X,\Gamma_{\mathcal{S}})$ on $W_{\Omega}(\mathcal{S})$ and the bi-Lipschitz map $\rho_{\mathcal{S}}^{-1}$ preserves this equivalence. Even if the term

looks bad, all the quantities inside are constructed by hand, and of course, we made them so that they are close to the quotient $\frac{\nabla d_{P}}{d_{P}}$, where $d_{P}$ is the distance to a plane that depends only on $\mathcal{S}$. With our change of variable, we even made it so that $P=\mathbb{R}^{n-1}\times\{0\}$, that is $\frac{\nabla d_{P}}{d_{P}}=\frac{\nabla t}{t}$. Long story short, Lemma 6.32 gives that

where

Then, in Section 4, we constructed a cut-off function $\Psi_{\mathcal{S}}$ adapted to $W_{\Omega}(\mathcal{S})$. We have shown in Lemma 4.20 that $\Psi_{\mathcal{S}}$ is 1 on $W_{\Omega}(\mathcal{S})$ and supported in $W^{*}_{\Omega}(\mathcal{S})$, on which we still have $\delta(X)\approx\operatorname{dist}(X,\Gamma_{\mathcal{S}})$. In Lemma 4.28, we proved that the support of $\nabla\Psi_{\mathcal{S}}$ is small, in the sense that implies ${\mathds{1}}_{\operatorname{supp}\nabla\Psi_{\mathcal{S}}}\in CM_{\Omega}$. What is important is that those two properties are preserved by bi-Lipschitz change of variable, and thus $\Psi_{\mathcal{S}}\circ\rho_{\mathcal{S}}$ is as in Definition 7.1.

We want the support of $\Psi_{\mathcal{S}}\circ\rho_{\mathcal{S}}$ to be included in a ball $B_{\mathcal{S}}$ such that $2B_{\mathcal{S}}$ is a subset of our initial ball $B$, and such that the radius of $B_{\mathcal{S}}$ is smaller than $C\ell(Q(\mathcal{S}))$. But those facts are an easy consequence of (1.25) and the definition of $W^{*}_{\Omega}(\mathcal{S})$ (and the fact that $\rho_{\mathcal{S}}$ is bi-Lipschitz with the Lipschitz constant close to 1).

We also want $u\circ\rho_{\mathcal{S}}$ to be a solution of $L_{\mathcal{S}}(u\circ\rho_{\mathcal{S}})=0$ for a weak-DKP operator $L_{\mathcal{S}}$. The operator $L_{\mathcal{S}}$ is not exactly weak-DKP everywhere in $\rho^{-1}_{\mathcal{S}}(\Omega)$, but it is the case on the support of $\Psi_{\mathcal{S}}$ (see Lemma 6.20), which is one condition that we need for applying Lemma 7.9.

To apply Lemma 7.9 - or more precisely for Lemma 7.12, where one term from the integration by parts argument is treated - we need to show that $\omega_{L^{*}}\in A_{\infty}(\sigma)$. This is a consequence of Theorem 1.18. Indeed, since the adjoint operator $L^{*}$ is also a weak DKP operator on $\Omega$, Theorem 1.18 asserts that $\omega_{L^{*}}\in A_{\infty}(\sigma)$, where $\sigma$ is an Ahlfors regular measure on $\partial\Omega$. A direct computation shows that for any set $E\subset\partial\Omega$ and any $X_{0}\in\Omega$,

and since the mapping $\rho_{\mathcal{S}}$ is bi-Lipschitz, $\omega_{L^{*}}\in A_{\infty}(\sigma)$ implies that the $L^{*}_{\mathcal{S}}$-elliptic measure $\omega_{L^{*}_{\mathcal{S}}}\in A_{\infty}(\widetilde{\sigma})$, where $\widetilde{\sigma}$ is an Ahlfors regular measure on the boundary of $\rho_{\mathcal{S}}^{-1}(\Omega)$. Moreover, the intrinsic constants in $\omega_{L^{*}_{\mathcal{S}}}\in A_{\infty}(\widetilde{\sigma})$ depend only on the intrinsic constants in $\omega_{L^{*}}\in A_{\infty}(\sigma)$ because the the bi-Lipschitz constants of $\rho_{\mathcal{S}}$ are bounded uniformly in $\mathcal{S}$.

All those verification made sure that we can apply Lemma 7.9, which entails that

In Section 2, we present the exact statement on our assumptions on $\Omega$, and we give the elliptic theory that will be needed in Section 7.

Sections 3 to 7 proved the implication $(ii)\implies(iii)$ in Theorem 1.12. Section 3 introduces the reader to the uniform rectifiability and present the corona decomposition that will be needed. The corona decomposition gives a collection of (stopping time) coherent regimes $\{\mathcal{S}\}_{\mathfrak{S}}$. From Subsection 3.4 to Section 6, $\mathcal{S}\in\mathfrak{S}$ is fixed. We construct in Subsection 3.4 a set $\Gamma_{\mathcal{S}}$ which is the graph of a Lipschitz function with small Lipschitz constant.

Section 4 associate a “Whitney” region $W_{\Omega}(\mathcal{S})$ to the coherent regime $\mathcal{S}$ so that from the stand point of each point of $W_{\Omega}(\mathcal{S})$, $\Gamma_{\mathcal{S}}$ and $\partial\Omega$ are well approximated by the same planes.

In Section 5, we are applying the result from Section 4 to compare $D_{\beta}$ with the distance to a plane that approximate $\Gamma_{\mathcal{S}}$.

Section 6 construct a bi-Lipschitz change of variable $\rho_{\mathcal{S}}$ that flattens $\Gamma_{\mathcal{S}}$, and we use the results from Sections 4 and 5 in order to estimate the difference

in terms of Carleson measure. Sections 3 to 6 are our arguments for the geometric side of the problem, in particular, the solutions to $Lu=0$ are barely mentionned (just to explain the effect of $\rho_{\mathcal{S}}$ on $L$).

Section 7 can be read independently and will contain our argument for the PDE side of the problem. Morally speaking, it proves Theorem 1.12 $(ii)\implies(iii)$ when $\Omega=\mathbb{R}^{n}_{+}$.

Section 8 presents a sketch of proof of Theorem 1.21. The strategy is similar to our proof of Theorem 1.12, and in particular, many of the constructions and notations from Section 3 to Section 7 are adopted in Section 8. But since we do not need to deal with the regularized distance $D_{\beta}$, the proof is much shorter.

Section 9 tackles the converse implication, proving $(v)\implies(i)$ in Theorem 1.12. The proof adapts an argument of [DM2], which states that if $G$ is sufficiently close to $D_{\beta}$, then $\partial\Omega$ is uniformly rectifiable. As mentioned earlier, we unfortunately did not succeed to link our strong estimate (1.14) directly to the weak ones assumed in [DM2], which explains why we needed to rewrite the argument.

We finish with Section 10, where we construct a semi-uniform domain and a positive harmonic solution on it for which our estimate (1.13) is false.