Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem

Abstract

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-$A_\infty$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $\Omega \subset \mathbb {R}^{n+1}$ with Ahlfors–David regular boundary, is equivalent to the solvability of the Dirichlet problem in $\Omega$, with data in $L^p(\partial \Omega )$ for some $p<\infty$. In this paper, we give a geometric characterization of the weak-$A_\infty$ property, of harmonic measure, and hence of solvability of the $L^p$ Dirichlet problem for some finite p. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors–David regularity of the boundary) that are natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors–David bounds); moreover, the examples show that the upper and lower Ahlfors–David bounds are each quantitatively sharp.

Appendix A: Some counter-examples

We shall discuss some counter-examples which show that our background hypotheses in Theorem 1.1 (namely, n-ADR and interior corkscrew condition) are natural, and in some sense in the nature of best possible. In the first two examples, $\Omega$ is a domain satisfying an interior corkscrew condition, such that $\partial \Omega$ satisfies exactly one (but not both) of the upper or the lower n-ADR bounds, and for which harmonic measure $\omega$ fails to be weak-$A_\infty$ with respect to surface measure $\sigma$ on $\partial \Omega$. In this setting, in which full n-ADR fails, there is no established notion of uniform rectifiability, but in each case, the domain will enjoy some substitute property which would imply uniform rectifiability of the boundary in the presence of full n-ADR. Moreover, these examples may be constructed in such a way that the failure of the condition (either upper or lower n-ADR) can be expressed quantitatively, with a bound that may be taken arbitrarily close to a true n-ADR bound; see (A.3) and (A.6) below.

In the last example, we construct an open set $\Omega$ with n-ADR boundary, and for which $\omega \in$ weak-$A_\infty$ with respect to surface measure, but for which the interior corkscrew condition fails, and $\partial \Omega$ is not n-UR.

Example 1

Failure of the upper n-ADR bound. In [8], the authors construct an example of a Reifenberg flat domain $\Omega \subset \mathbb {R}^{n+1}$ for which surface measure $\sigma = H^n\lfloor _{\,\partial \Omega }$ is locally finite on $\partial \Omega$, but for which the upper n-ADR bound

$$\begin{aligned} \sigma (\Delta (x,r)) \le C r^n \end{aligned}$$

(A.1)

fails, and for which harmonic measure $\omega$ is not absolutely continuous with respect to $\sigma$. Note that the hypothesis of Reifenberg flatness implies in particular that $\Omega$ and $\Omega _{ext}:= \mathbb {R}^{n+1}{\setminus } \overline{\Omega }$ are both NTA domains, hence both enjoy the corkscrew condition, so by the relative isoperimetric inequality, the lower n-ADR bound

$$\begin{aligned} \sigma (\Delta (x,r)) \ge c r^n \end{aligned}$$

(A.2)

holds. Thus, it is the failure of (A.1) which causes the failure of absolute continuity: in the presence of (A.1), the results of [21] apply, and one has that $\omega \in A_\infty (\sigma )$, and that $\partial \Omega$ satisfies a “big pieces of Lipschitz graphs” condition (see [21] for a precise statement), and hence is n-UR. We note that by a result of Badger [10], a version of the Lipschitz approximation result of [21] still holds for NTA domains with locally finite surface measure, even in the absence of the upper n-ADR condition.

In addition, given any $\varepsilon >0$, the construction in [8] can be made in such a way that (A.1) fails “within $\varepsilon$”, i.e., so that

$$\begin{aligned} \sigma (\Delta (x,r) \le C r^{n-\varepsilon }, \quad \forall \, x\in \partial \Omega ,\, r<1. \end{aligned}$$

(A.3)

Let us sketch an argument to explain why this is so; we refer the interested reader to [8] for more details.

The domain $\Omega$ in [8] is obtained by enlarging a Wolff snowflake, that we will denote here by D. Both $\Omega$ and D are $\delta$-Reifenberg flat, with $\delta$ as small as wished in the construction (recall that Wolff snowflakes can be taken $\delta$-Reifenberg flat, with $\delta$ as small as wished).

It is shown in [8, Theorem 3.1] that for all $x\in \partial \Omega$ and $r<1$,

$$\begin{aligned} H^n(B(x,r)\cap \partial \Omega ) \lesssim \max ( r^n, r^\alpha \mu (B(x,Cr)) ) \le \max ( r^n, \mu (B(x,Cr)) ) \end{aligned}$$

(A.4)

where $\mu$ is some measure supported on $\partial D$ satisfying $\mu (B(x,r)) \gtrsim r^{n-\alpha }$ for all x in some compact set $E \subset \partial \Omega \cap \partial D$, and some $\alpha >0.$ In the construction in [8], the authors take $\mu =\omega _D$, the harmonic measure for D. Further, from results of Kenig and Toro it follows that harmonic measure in a $\delta$-Reifenberg flat domain D satisfies

$$\begin{aligned} \omega _D(B(x,r)) \lesssim r^{n-\varepsilon } \omega _D(B(x,1)), \quad \forall \, x\in \partial D, \, r<1, \end{aligned}$$

with $\varepsilon \rightarrow 0$ as $\delta \rightarrow 0$ (see [40, Theorem 4.1]). As a consequence, the measure $\mu$ satisfies

$$\begin{aligned} \mu (B(x,r)) \lesssim r^{n-\varepsilon }, \quad \forall \, x\in {\mathbb {R}}^{n+1},\, r<1, \end{aligned}$$

with $\varepsilon$ as small as wished depending on $\delta$. From (A.4), it follows that

$$\begin{aligned} H^n(B(x,r)\cap \partial \Omega ) \lesssim \max ( r^n, r^{n-\varepsilon } ) \le r^{n-\varepsilon }, \quad \forall \, x\in \partial \Omega , r<1. \end{aligned}$$

Example 2

Failure of the lower n-ADR bound. In [2, Example 5.5], the authors give an example of a domain satisfying the interior corkscrew condition, whose boundary is rectifiable (indeed, it is contained in a countable union of hyperplanes), and satisfies the upper n-ADR condition (A.1), but not the lower n-ADR condition (A.2), but for which surface measure $\sigma$ fails to be absolutely continuous with respect to harmonic measure, and in fact, for which the non-degeneracy condition

$$\begin{aligned} A\subset \Delta _x:= B(x,10\delta _\Omega (x)) \cap \partial \Omega ,\quad \sigma (A)\ge (1-\eta ) \sigma (\Delta _x) \, \, \implies \quad \omega ^x(A) \ge c, \end{aligned}$$

(A.5)

fails to hold uniformly for $x\in \Omega$, for any fixed positive $\eta$ and c, and therefore $\omega$ cannot be weak-$A_\infty$ with respect to $\sigma$. We note that in the presence of the full n-ADR condition, if $\partial \Omega$ were contained in a countable union of hyperplanes (as it is in the example), then in particular it would satisfy the “BAUP” condition of [23], and thus would be n-UR [23, Theorem I.2.18, p. 36].

Moreover, given any $\varepsilon >0$, the parameters in the example of [2] can be chosen in such a way that the lower ADR bound fails “within $\varepsilon$”, i.e., so that

$$\begin{aligned} H^n(\Delta (x,r))\gtrsim \min (r^{n+\varepsilon },r^n), \quad \forall x\in \partial \Omega . \end{aligned}$$

(A.6)

To see this, we proceed as follows. We follow closely the construction in [2, Example 5.5], with some modification of the parameters. Fix $\varepsilon >0$, and set

$$\begin{aligned} c_k:= 2^{-k(n+\varepsilon )}. \end{aligned}$$

For $k\ge 1$, and $n\ge 2$, set

$$\begin{aligned} \Sigma _k:= \{(x,t)\in \mathbb {R}^{n+1}_+:\, t=2^{-k},\, x\in \overline{\Delta (0,2^{-\varepsilon k}c_k)}+ c_k {\mathbb {Z}}^n\}, \end{aligned}$$

where for $x\in \mathbb {R}^n$, $\Delta (x,r):= \{y\in \mathbb {R}^n:\, |x-y|<r\}$ is the usual n-disk of radius r centered at x. Define

$$\begin{aligned} \Omega := \mathbb {R}^{n+1}_+{\setminus } \left( \cup _{k=1}^\infty \Sigma _k \right) ,\qquad \Omega _k:= \mathbb {R}^{n+1}_+{\setminus } \Sigma _k, \end{aligned}$$

each of which is clearly open and connected. Notice that $\Omega$ satisfies the interior Corkscrew condition (since the sets $\Sigma _k$ are located at heights which are sufficently separated). Moreover, it is easy to see that $\partial \Omega$ satisfies the upper ADR condition and that $\mathbb {R}^n\times \{0\}\subset \partial \Omega$.

On the other hand, the lower ADR bound fails. To see this, let $X=(x,0)\in \partial \Omega$, and choose $\mathbf {m}_{k,x}\in {\mathbb {Z}}^n$ and $X_k=(c_k\,\mathbf {m}_{k,x}, 2^{-k})\in \Sigma _k\subset \partial \Omega$ such that $X_k\rightarrow X$. Set $B_k=B(X_k, 2^{-k-2})$, and observe that $H^n(B_k\cap \partial \Omega )/(2^{-kn})\approx 2^{-kn\varepsilon }\rightarrow 0$ as $k\rightarrow \infty$, or equivalently

$$\begin{aligned} H^n(B_k\cap \partial \Omega )\approx r_k^{n+\varepsilon '}, \end{aligned}$$

where $B_k$ has radius $r_k \approx 2^{-k}$, and $\varepsilon '=n\varepsilon$. We shall show that this behavior is in fact typical, and that (A.6) holds, with $\varepsilon '$ in place of $\varepsilon$.

Let $\omega ^{(\cdot )}:= \omega ^{(\cdot )}_\Omega$ and $\omega _k^{(\cdot )}:= \omega ^{(\cdot )}_{\Omega _k}$ denote harmonic measure for the domains $\Omega$ and $\Omega _k$ respectively.

Claim

$\omega ^{(\cdot )}(F) = 0$, with $F:=\mathbb {R}^n\times \{0\}$. Thus, in particular (A.5) fails.

It remains to verify (A.6), and the claim. As regards the former, note that for $X=(x,0)\in F$, we have the trivial standard lower n-ADR bound $H^n(\Delta (X,r))\gtrsim r^n$, whereas for $X=(x,2^{-k}) \in \Sigma _k$, we have

$$\begin{aligned} H^n\lfloor _{\partial \Omega }\big (B(X,r)\big )\ge H^n\lfloor _{\partial \Omega _k}\big (B(X,r)\big )\gtrsim \left\{ \begin{array}{ll} r^n, &{} r<2^{-\varepsilon k} c_k, \\ 2^{-kn\varepsilon }c_k^{n}, &{} 2^{-\varepsilon k} c_k \le r\le c_k \\ 2^{-kn\varepsilon }r^n, &{} c_k< r\le 2^{-k+1}\\ r^n,&{} r> 2^{-k+1}. \end{array} \right. \end{aligned}$$

(A.7)

The first and fourth of these estimates are of course the standard lower n-ADR bound. For $r\le c_k$, the second estimate is bounded below by $2^{-kn\varepsilon }r^n$, and in turn, with $r\lesssim 2^{-k}$, the second and third estimates are therefore bounded below by

$$\begin{aligned} 2^{-kn\varepsilon }r^n \gtrsim r^{n+n\varepsilon }= r^{n+\varepsilon '}, \end{aligned}$$

which yields (A.6) with $\varepsilon '=n\varepsilon$ in place of $\varepsilon$.

Let us now prove the claim. We first recall some definitions. Given an open set $O\subset \mathbb {R}^{n+1}$, and a compact set $K\subset O$, we define the capacity of K relative to O as

$$\begin{aligned} \text{ cap }(K,O)=\inf \left\{ \iint _O|\nabla \phi |^2\, dY:\ \phi \in C_0^\infty (O), \ \phi \ge 1 \text{ in } K\right\} . \end{aligned}$$

Also, the inhomogeneous capacity of K is defined as

$$\begin{aligned} \text{ Cap } (K)=\inf \left\{ \iint _{\mathbb {R}^{n+1}} \big (|\phi |^2+|\nabla \phi |^2\big )\, dY:\ \phi \in C_0^\infty (\mathbb {R}), \ \phi \ge 1 \text{ in } K\right\} . \end{aligned}$$

Combining [25, Theorem 2.38], [1, Theorem 2.2.7] and [1, Theorem 4.5.2] we have that if K is a compact subset of $\overline{B}$, where B is a ball with radius smaller than 1, then

$$\begin{aligned} \text{ cap }(K,2B) \gtrsim \text{ Cap }(K) \gtrsim \sup _{\mu } \mu (K) \end{aligned}$$

(A.8)

where the implicit constants depend only on n, the sup runs over all Radon positive measures $\mu$ supported on K, for which

$$\begin{aligned} W(\mu )(X):=\int _0^1 \frac{\mu (B(X,t))}{t^{n-1}}\,\frac{dt}{t} \le 1, \qquad \forall \, X\in {{\,\mathrm{supp}\,}}\mu . \end{aligned}$$

Fix $k\ge 2$, and set

$$\begin{aligned} \beta =\beta _k:= 2^{k(n-1)}c_k= 2^{k(n-1)} 2^{-k(n+\varepsilon )} = 2^{-k(1+\varepsilon )}, \end{aligned}$$

by definition of $c_k$. Our next goal is to show that

$$\begin{aligned} \mathrm{cap}\big (\overline{B(X_0,s)}\cap \Sigma _k, B(X_0,2s)\big ) \gtrsim s^{n-1},\quad X_0:=(x_0,2^{-k})\in \Sigma _k,\ \beta \le s<1. \end{aligned}$$

(A.9)

For a fixed $X_0$ and s, write $K=\overline{B(X_0,s)}\cap \Sigma _k$, set $\mu = 2^{kn\varepsilon } s^{-1} \, H^n\lfloor _K,$ and note that for $X\in K$, similarly to (A.7), we have

$$\begin{aligned} \mu \big (B(X,r)\big )\approx 2^{kn\varepsilon } s^{-1} \left\{ \begin{array}{l@{\quad }l} r^n, &{} r<2^{-\varepsilon k} c_k, \\ 2^{-kn\varepsilon }c_k^{n},&{}2^{-\varepsilon k} c_k \le r\le c_k \\ 2^{-kn\varepsilon }r^n,&{} c_k< r\le s\\ 2^{-kn\varepsilon }s^n, &{} r> s. \end{array} \right. \end{aligned}$$

(A.10)

To compute $W(\mu )(X)$ for $X\in K$ write

$$\begin{aligned} W(\mu )(X)&= \int _0^1 \frac{\mu (B(X,t))}{t^{n-1}}\,\frac{dt}{t}= \int _0^{2^{-\varepsilon k}c_k} +\, \int _{2^{-\varepsilon k}c_k}^{c_k} + \, \int _{c_k}^s +\, \int _s^1\\&=: \,I + II+III+IV. \end{aligned}$$

Then, since $s\ge \beta = 2^{k(n-1)} c_k = 2^{-k(1+\varepsilon )}$,

$$\begin{aligned} I + II \lesssim 2^{kn\varepsilon } s^{-1} \left( 2^{-\varepsilon k}c_k \, + 2^{-kn\varepsilon }c_k^{n}\int _{2^{-\varepsilon k}c_k}^\infty \frac{dt}{t^n} \right) \lesssim \, 2^{\varepsilon k(n-1)} c_k s^{-1} \lesssim 1. \end{aligned}$$

Furthermore, the last two estimates in (A.10) easily imply that $III+IV\lesssim 1$ and hence $W(\mu )(X)\lesssim 1$ for every $X\in K$. This, (A.8), and (A.10) imply as desired (A.9):

$$\begin{aligned} \mathrm{cap}\big (\overline{B(X_0,s)}\cap \Sigma _k, B(X_0,2s)\big ) \gtrsim \mu (K) \gtrsim s^{n-1}. \end{aligned}$$

Set

$$\begin{aligned} P_k:= \left\{ \left( x,2^{-k}-\beta \right) \in \mathbb {R}^{n+1}_+:\, x\in \mathbb {R}^n\right\} , \end{aligned}$$

and observe that for $X\in P_k$,

$$\begin{aligned} \beta \le \delta _k(X):= {\text {dist}}(X,\partial \Omega _k) = {\text {dist}}(X,\Sigma _k)\le 2\beta . \end{aligned}$$

Recall that $F=\mathbb {R}^n\times \{0\}$, and define

$$\begin{aligned} u(X):= \omega _k^X(F),\qquad X\in \Omega _k. \end{aligned}$$

Observe that $u\in W^{1,2}(\Omega _k)\cap C(\overline{\Omega _k})$ since $\partial \Omega _k$ is ADR (constants depend on k but we just use this qualitatively) and $\chi _F$ is a Lipschitz function on $\partial \Omega _k$. Fix $Z_0\in P_k$ and let $Z_0'\in \Sigma _k$ be such that $|Z_0-Z_0'|={\text {dist}}(Z_0, \partial \Omega _k)\le 2\beta$. Let $\Omega _{Z_0}= \Omega _k\cap B(Z_0', \frac{3}{4} 2^{-k})$, which is an open connected bounded set. We can now apply the usual capacitary estimates (see, e.g., [25, Theorem 6.18]) to find a constant $\alpha =\alpha (n)>0$ such that

$$\begin{aligned} u(Z_0) \lesssim \exp \left( -\alpha \int _{3\,\beta }^{2^{-k-2}} \frac{ds}{s} \right) \approx \big (2^k \beta \big )^\alpha =2^{-\alpha \varepsilon k}. \end{aligned}$$

where we have used (A.9), the definition of $\beta$, and the fact that $u\equiv 0$ on $\partial \Omega _k \cap B(Z_0',2^{-k-1})$. Note that the last estimate holds for any $Z_0\in P_k$ and therefore, by the maximum principle,

$$\begin{aligned} u(x,t) \lesssim 2^{-\alpha \varepsilon k}, \qquad (x,t) \in \Omega _k, \quad t > 2^{-k}-\beta . \end{aligned}$$

In particular, if we set $X_0:= (0,\dots ,0,1) \in \mathbb {R}^{n+1}_+$, then by another application of the maximum principle,

$$\begin{aligned} \omega ^{X_0}(F)\, \le \, \omega _k^{X_0}(F)= u(X_0) \, \lesssim 2^{-\alpha \varepsilon k}\, \rightarrow 0, \end{aligned}$$

as $k \rightarrow \infty$, and the claim is established.

Example 3

Failure of the interior corkscrew condition. The example is based on the construction of Garnett’s 4-corners Cantor set $\mathcal {C}\subset \mathbb {R}^2$ (see, e.g., [23, Chapter 1]). Let $I_0$ be a unit square positioned with lower left corner at the origin in the plane, and in general for each $k = 0, 1,2,\dots$, we let $I_k$ be the unit square positioned with lower left corner at the point (2k, 0) on the x-axis. Set $\Omega _0:= I_0$. Let $\Omega _1$ be the first stage of the 4-corners construction, i.e., a union of four squares of side length 1/4, positioned in the corners of the unit square $I_1$, and similarly, for each k, let $\Omega _k$ be the k-th stage of the 4-corners construction, positioned inside $I_k$. Note that ${\text {dist}}(\Omega _k, \Omega _{k+1}) = 1$ for every k. Set $\Omega := \cup _k \Omega _k$. It is easy to check that $\partial \Omega$ is n-ADR, and that the non-degeneracy condition (A.5) holds in $\Omega$ for some uniform positive $\eta$ and c, and thus by the criterion of [11], $\omega \in$ weak-$A_\infty (\sigma )$. On the other hand, the interior corkscrew condition clearly fails to hold in $\Omega$ (it holds only for decreasingly small scales as k increases), and certainly $\partial \Omega$ cannot be n-UR: indeed, if it were, then $\partial \Omega _k$ would be n-UR, with uniform constants, for each k, and this would imply that $\mathcal {C}$ itself was n-UR, whereas in fact, as is well known, it is totally non-rectifiable. One can produce a similar set in 3 dimensions by simply taking the cylinder $\Omega '=\Omega \times [0,1]$. Details are left to the interested reader.