The Neumann function and the L p Neumann problem in chord-arc domains
-
Steve Hofmann
and
Derek Sparrius
Abstract
We construct the Neumann function in a 1-sided chord-arc domain (i.e., a uniform domain with an Ahlfors regular boundary), and establish size and Hölder continuity estimates up to the boundary. We then obtain a Kenig-Pipher type theorem, in which L p solvability of the Neumann problem is shown to yield solvability in L q for 1 < q < p, and in the Hardy space H 1, in 2-sided chord-arc domains, under suitable background hypotheses.
1 Introduction, history and discussion of main results
We consider solutions of the divergence form elliptic equation Lu = 0, in a domain
and A = A(x) is a matrix of real, measurable coefficients, satisfying the uniform ellipticity condition
for some λ > 0, Λ < ∞, and for all
This paper includes two main sets of results (terminology will be defined in Section 2). First, in Section 4, in the setting of a 1-sided chord arc domain, we establish a Harnack inequality and obtain Hölder continuity, up to the boundary, for solutions with (at least locally) vanishing Neumann condition. Using these results, we construct, and prove estimates for, the Neumann function. Then in Section 5, in the setting of a 2-sided chord arc domain, we prove a result of “Calderón-Zygmund type” for solvability of the Neumann problem: we show that if (N)
p
(the Neumann problem with data in L
p
) is solvable for L, for some p > 1, then it is also solvable with data in L
q
, 1 < q < p, as well as in the Hardy space H
1(∂Ω). As background conditions, we require solvability of the Regularity problem (R)
p
for L, and the Dirichlet problem[1] (D)
p′, for the transpose operator L*, with coefficients
These results follow the program of Kenig and Pipher[2] [4], who proved analogous results in the setting of a starlike Lipschitz domain. Many of our arguments follow those of Ref. [4] rather closely, but there is one essential difference in our work, compared to that of Ref. [4]: in the latter paper, the authors obtain the aforementioned Hölder continuity up to the boundary via a reflection argument, a method that is completely unavailable in our setting, in which the boundary need not be given locally as a graph, and in which there is no distinguished direction. Instead, we adapt the program of Moser to establish a Harnack inequality up to the boundary, which then leads, as usual, to Hölder continuity.
While the present manuscript was in preparation, we learned of two different projects that overlap with (different parts of) ours, and that have been conducted independently. First, our results concerning the Harnack inequality and Hölder continuity up to the boundary, for solutions with vanishing Neumann condition, have been obtained independently in work of David, Decio, Engelstein, Mayboroda, and Michetti [5], as part of their study of the Robin boundary value problem. Second, our results concerning the “Calderón-Zygmund” theorem for solvability of the L p Neumann problem, have been obtained independently by Feneuil and Li [6]. In fact, the results in Ref. [6] are somewhat more general than ours: they are able to relax the 2-sided chord arc condition, and they are able to dispense with the background hypothesis that the L p Regularity problem is solvable (assuming merely that the Dirichlet problem for L* is solvable with data in L p′).
We remark that our results concerning existence of, and estimates for, the Neumann function have found application in recent very interesting work of Mourgoglou and Tolsa [3], who have obtained L p solvability results for the Neumann problem in chord-arc domains with very big pieces of (boundaries of) super-domains in which the Neumann problem is solvable in L q with q > p, with uniform estimates.
2 Notation, definitions, and some useful known results
Unless otherwise stated, we use the letters c, C to denote harmless positive constants, not necessarily the same at each occurrence, which depend only on dimension and the constants appearing in the hypotheses of the theorems (which we refer to as the “allowable parameters”). We shall also sometimes write a ≲ b, a ≳ b, and a ≈ b to mean, respectively, that a ≤ Cb, a ≥ cb, and 0 < c ≤ a/b ≤ C, where the constants c, C are as above, unless explicitly noted to the contrary. At times, we shall designate by M a particular constant whose value will remain unchanged throughout the proof of a given lemma or proposition, but which may have a different value during the proof of a different lemma or proposition. On the other hand, when we attach a subscript, e.g., M 1, we mean a particular constant that has been fixed henceforth.
Our ambient space is
Ω will always denote an open set in
The open n-dimensional Euclidean ball of radius r and center x will be denoted B(x, r), or sometimes B r (x) (or simply B r when the center is left implicit, or just B when both center and radius are implicit). For x ∈ ∂Ω, a surface ball is denoted Δ(x, r) ≔ B(x, r) ∩ ∂Ω (sometimes Δ r (x) or simply Δ r , or just Δ).
Given an open subset
Given a Euclidean ball B or surface ball Δ, its radius will be denoted r B or r Δ, respectively.
Given a Euclidean or surface ball B = B(x, r) or Δ = Δ(x, r), its concentric dilate by a factor of κ > 0 will be denoted κB ≔ B(x, κr) or κΔ ≔ Δ(x, κr).
Given an open set
We let
For a Borel set
Given a Borel subset
(we shall use the simplified notation f E when the measure
We shall use the letter I (and sometimes J) to denote a closed n-dimensional Euclidean dyadic cube with sides parallel to the co-ordinate axes, and we let ℓ(I) denote the side length of I.
Definition 2.1.
(ADR) (aka Ahlfors-David regular). We say that a set
where diam(E) may be infinite. Here, Δ(x, r) ≔ E ∩ B(x, r) is the surface ball of radius r, and as above,
Definition 2.3.
(Corkscrew condition). Following [7], we say that an open set
Definition 2.4.
(Harnack Chains, and the Harnack Chain condition [7]). Given two points x, x′ ∈ Ω, and a pair of numbers M, N ≥ 1, an (M, N)-Harnack Chain connecting x to x′, is a chain of open balls B
1, …, B
N
⊂ Ω, with x ∈ B
1, x′ ∈ B
N
, B
k
∩ B
k+1 ≠ ∅ and M
−1diam(B
k
) ≤ dist(B
k
, ∂Ω) ≤ Mdiam(B
k
). We say that Ω satisfies the Harnack Chain condition if there is a uniform constant M such that for any two points x, x′ ∈ Ω, there is an (M, N)-Harnack Chain connecting them, with N depending only on M and the ratio
Definition 2.5.
(1-sided NTA (aka uniform), NTA, and 2-sided NTA domains). If
Definition 2.6.
(1-sided CAD, CAD, and 2-sided CAD). We say that a connected open set
For future reference, we now recall a fundamental fact that holds in a 1-sided chord arc domain Ω. Let
Lemma 2.7
[8]. Let
The implicit constants in (2.8) depend only on ellipticity, dimension, and on the constants in the 1-sided CAD character.
The result is proved in Ref. [8] in the case that Ω is a Lipschitz domain, but the proof carries over routinely mutatis mutandis to the 1-sided chord arc setting.
3 Preliminary results
In the sequel, we shall assume that Ω is unbounded, with unbounded boundary, and at minimum satisfies the 1-sided CAD condition. In parts of Section 5, we shall further assume that Ω is a 2-sided CAD. Our arguments may be used to treat either bounded domains, or unbounded domains with unbounded boundaries, but in order to simplify the exposition, we will focus on the latter case (otherwise, we leave it to the interested reader to verify that our methods may be adapted, mutatis mutandis, to handle the case of bounded domains).
For an unbounded open set
where 2* ≔ 2n/(n − 2). We note that, when Ω is an unbounded 1-sided chord-arc domain, Y
1,2(Ω) is a homogeneous Sobolev space. Indeed, in that setting, assuming a priori that
so in particular, Y
1,2 is a Hilbert Space with the inner product
We let
Closely related to Y
1,2(Ω) is the homogeneous Sobolev space
Similarly, we define
Remark 3.1.
We observe that
As usual, W 1,2(Ω) ≔ {f ∈ L 2(Ω): ∇f ∈ L 2(Ω)} is the inhomogeneous Sobolev space with norm
and
We also define
Having defined our space modulo constants, we equip
Remark 3.3.
As was the case the for
We note for future reference that standard trace-extension theory holds in our domains. In particular, there exist extension operators
and
where in each case, the implicit constants depend only on dimension and the ADR, CS, and HC constants. A few words of explanation are in order. Since our domains are unbounded 1-sided chord arc domains with unbounded boundaries, they are in particular (ɛ, ∞) domains in the sense of Jones [9], and we may deduce the extension property (3.4) by using the extension operator constructed in Ref. [9], and following the proof of Ref. [9], Theorem 1]. The latter result applies directly to the inhomogeneous Sobolev spaces W
1,2, but the proof carries over to the homogenous Sobolev spaces
Recall that given a Borel subset
We note for future reference the following well known estimate, for any ball B (in fact, even for convex domains, see, e.g., [[12], Lemma 7.16]) and for u ∈ W 1,2(B):
Remark 3.8.
We further note for future reference that there exists a constant K
0 = K
0(CS) ≥ 2 such that, for R > 0, and for every
Proposition 3.9.
Suppose that
where the implicit constant and K depend only on dimension and the CS and HC constants.
Proof.
By HC, given y ∈ Ω R , and bearing in mind that K 0 = K 0(CS), we see that there exists a chain of balls B 0, B 1, …, B N with radii r 0, r 1, …, r N , and with B 0 ≔ B R/10(x 0) as above, such that
y ∈ B N ;
B k ∩ B k−1 ≠ ∅, 1 ≤ k ≤ N;
r k ≈ dist(B k , ∂Ω);
For each 1 ≤ k ≤ N, there is a ball
for all
The balls
By a density argument, we may assume that u ∈ C 1(Ω KR ). Since y ∈ B N , we have
where we have applied Poincaré’s inequality to the telescoping sum, and then used (3.7) in B N , in the last two steps.
By property (iv), we can estimate the last term in (3.11) as
Plugging the latter estimate into (3.11), and using properties (v) and (vi), we obtain (3.10). □
Corollary 3.12.
Suppose that
In addition,
In particular, for 2* ≔ 2n/(n − 2),
Proof.
If δ(x 1) ≥ R (so that Ω R = B R ), the result is standard (with K = 1). We therefore suppose that δ(x 1) < R. By a density argument, we may assume that u ∈ C 1(Ω KR ). To prove (3.13), we let B 0 denote the ball in Remark 3.8 and Proposition 3.9, and apply (3.10) to each summand in the inequality
To prove (3.14), we write
and then apply (3.13) to the integrand to obtain the bound
Finally, (3.15) follows immediately from Hölder’s inequality, and then (3.14) and the Hardy-Littlewood-Sobolev theorem. □
The next result is an extension of Ref. [[12], Theorem 7.21] beyond the setting of convex domains. The proof follows that in Ref. [12], using the estimates established above.
Theorem 3.16.
Suppose that
for all balls B r ⊂ B 3KR = B(x 1, 3KR). Then there exists positive constants c 0 and C depending on the allowable parameters such that
Proof.
We first observe that since (3.17) holds for all balls B r ⊂ B 3KR , we in fact have that
for every B
r
. Indeed, if B
r
⊂ B
3KR
, then (3.17) applies directly; otherwise if B
r
meets both Ω2KR
and also
Next, we use [12], Lemma 7.20] (the proof of which does not require any extra assumptions on Ω) in the domain Ω2KR , to deduce that if f satisfies
for all balls B r , then there exist constants C 1 and C 2 depending only on n such that
where
4 The Neumann function
In this section we assume
Let L = −divA∇, where A = (a ij (x)) is a matrix of bounded measurable coefficients satisfying the uniform ellipticity condition
where 0 < λ ≤ Λ < ∞. In the sequel, all implicit constants will depend only on ellipticity, dimension, and the ADR, CS, and HC constants.
We do not assume that A is a symmetric matrix. However, we then run into the issue that there is not necessarily a canonical choice of the matrix A representing the operator L = −divA∇, and hence there is not necessarily a canonical representation of the conormal derivative. The following example illustrates the idea.
Example 4.1.
Let
where L = −divA∇, the coefficient matrix A = A(x) is independent of the transverse variable t, and the conormal derivative is, at least formally, given by the formula ν ⋅ A∇u, where ν is the outer unit normal vector on ∂Ω. By Lax-Milgram, there is a unique
We now modify the matrix A in the following way. Let
We write the matrix A in the block form
where A
‖ is an n × n matrix, B is an n × 1 column vector, C is a 1 × n row vector, and D is a scalar. We now define the matrix
Observe that
and this identity may be extended by density to
i.e., there may be more than one coefficient matrix for which the associated form yields the same operator. On the other hand, if u is a weak solution of Lu = 0 in
i.e., u is also a weak solution of
To resolve the possibly ambiguity in the matrix A, we assume for the remainder of this paper that A has been fixed.
Let us introduce some further notation that we will use in the sequel.
We let A* denote the transpose (hence also the adjoint, since our coefficients are real) of the coefficient matrix A.
We reiterate that for now, we will continue to follow the presentation in Ref. [[4], Section 2], with some technical adjustments owing to the unbounded scale-invariant setting in which we work.
We recall the notational convention in Remark 3.3, namely that when it is useful to distinguish an equivalence class in
We note that
Remark 4.5.
In the sequel, when distinguishing between equivalence classes in
Lemma 4.6.
For each x ∈ Ω, elliptic measure ω
x
may be identified with an element of
Here, ⟨⋅, ⋅⟩ denotes the duality pairing of
where u is the Y
1,2(Ω) realization of the
In addition, there is a unique v x (⋅) ∈ Y 1,2(Ω) such that (recalling the notation in Remark 3.1)
In particular, for
Furthermore,
where the implicit constant depends only on the allowable parameters.
Remark 4.13.
Recall that, in order to simplify the exposition, we have assumed that Ω and ∂Ω are unbounded; if Ω were bounded, we would have had to modify the right hand side of (4.11), so that it would vanish when φ ≡ 1, and to work with a subspace of inhomogeneous W
1,2, whose elements are normalized to have mean value zero in Ω, rather than with the homogeneous Sobolev spaces
Remark 4.14.
Taking
Proof of Lemma 4.6.
We shall use the equivalence class notation in Remarks 3.1 and 3.3.
We first consider the Dirichlet problem
Here, we interpret [u]|∂Ω = [φ] in the trace sense. We construct the unique solution via the standard Lax-Milgram argument. Let
where the implicit constant depends only on n, ADR, (CS) and (HC).
Then
so by Lax-Milgram, there is a unique
Set u ≔ Φ − w, so that [u] is a weak solution of (4.15), and by (4.16) and (4.17) satisfies
Note that if [Φ1] is any other
We now fix the particular realization of
By Moser’s local boundedness estimate, for each fixed x ∈ Ω, we therefore have
Since the solution map [φ] ↦ [u] is linear, we also have that, for each x ∈ Ω, the mapping φ ↦ u(x) defines a bounded linear functional on
where we are using the convention in Remark 4.5. Thus, (4.7) holds.
We now turn to the construction of v
x
. It is enough to find
where
where in the last step we have used (4.9). We omit the routine details. Hence, we have solved the problem (4.20) with the estimate (4.12), as desired. Finally, (4.11) follows immediately from (4.10) and (4.7). □
Note that the adjoint (i.e., transpose) matrix A* = A
T
satisfies the same properties as A. Hence, Lemma 4.6 also holds with the roles of L = −divA∇ and L* = −divA*∇ reversed, with adjoint elliptic measure
Corollary 4.21.
For x, y ∈ Ω, v(x, y) = v*(y, x).
Proof.
Note that (4.10) holds for all
where in the last step we have used the adjoint version of (4.10). Since v
x
(⋅) and
Corollary 4.22.
For x, y ∈ Ω,
Proof.
Applying Moser’s local boundedness estimate to the adjoint solution v x (⋅) = v(x, ⋅), we see that
where the last inequality follows from (4.12). □
Definition 4.24.
(Neumann Function). Let G(x, y) be the Green’s function for L in Ω. The Neumann function N(x, y), x, y in Ω is defined to be N(x, y) = G(x, y) − v(x, y).
We shall sometimes use the notation N x (⋅) ≔ N(x, ⋅), and G x (⋅) ≔ G(x, ⋅), thus N x = G x − v x .
Lemma 4.25.
Let
i.e.,
Proof.
We may assume that
□
Remark 4.27.
In the sequel, we shall often work with bounded sub-domains of Ω. Given
We recall the following fact (see [13], Lemma 3.61]).
Proposition 4.28.
For each Ω R as above, with y 0 ∈ ∂Ω, there is a 1-sided chord-arc subdomain T R ⊂ Ω, and a constant κ ≥ 8, such that Ω4KR ⊂ T R ⊂ Ω κKR , where K > 1 is the constant in Proposition 3.9 and Corollary 3.12, and where κ and the 1-sided chord-arc constants (i.e., the (CS), (HC) and ADR constants) for T R depend only on n, and the ADR, (CS) and (HC) constants for Ω.
For the reader’s convenience, we mention that in the notation of Ref. [13], T R is the “Carleson tent” T Δ, with Δ = Δ(y 0, 4KR) ≔ ∂Ω ∩ B(y 0, 4KR).
Our next goal is to show that the Neumann function is Hölder continuous up to the boundary. At this point, we shall depart from the treatment in Ref. [4], as the reflection argument used there is unavailable in our setting. Instead, we shall use the methods of Moser. The following local boundedness result is known: see Ref. [14], Theorem 1.6 and its proof, and Remark 1.8].
Lemma 4.29.
Let y 0 ∈ ∂Ω, R > 0 and set Ω R = Ω R (y 0) as above. Let u ∈ W 1,2(Ω2R ). Suppose that u satisfies either
or, that u ≥ 0 in Ω2R and that
Then
where the implicit constants depend only on n, ADR, (CS), (HC) and ellipticity.
Remark 4.33.
Observe that by Lemma 4.25, we may apply Lemma 4.29 to the Neumann function u = N x , provided that x ∈ Ω\Ω2R .
Remark 4.34.
In the sequel, we let T R be the 1-sided chord-arc subdomain in Proposition 4.28. In particular, T R is a trace-extension domain, depending only on n, ADR, (CS) and (HC).
Lemma 4.35.
Fix Ω R as above. Let K be the constant in Proposition 3.9. Suppose that u ∈ W 1,2(T R ), u ≥ 0 is a local supersolution to the adjoint Neumann problem in Ω4KR , in the sense that
Then
where the implicit constants depend only on n, ADR, (CS), (HC) and ellipticity.
Proof.
Let
Since ∇u ɛ = ∇u, by ellipticity and (4.36) we therefore have
where γ ∈ (0, 1) is at our disposal. Choosing γ small enough depending only on ellipticity, we may hide the small term, to obtain
uniformly in ɛ. Letting ɛ → 0, we obtain (4.37). □
Our next goal is to establish a Harnack inequality up to the boundary for non-negative solutions with vanishing Neumann condition, and then to use that result to show that the Neumann function is locally Hölder continuous, again up to the boundary. To this end, we will follow Moser’s proof of the interior Harnack inequality. We fix R > 0 and a point y 0 ∈ ∂Ω, and let Ω R ≔ Ω ∩ B(y 0, R) as above.
Theorem 4.38.
(Weak Harnack Inequality). Let u ∈ W 1,2(T R ), u ≥ 0, be a local supersolution to the adjoint Neumann problem in Ω4KR , in the sense of Lemma 4.35, i.e., (4.36) holds. Then for some s > 0 sufficiently small,
where s and the implicit constants depend only on dimension, ADR, (CS), (HC), and ellipticity.
Proof.
We use the standard argument of Moser, following the treatment in Ref. [12], Chapter 8], adapted to our setting in which we allow estimates up to the boundary. We momentarily fix ɛ > 0, and set u
ɛ
≔ u + ɛ. We claim that, in particular, for all β > 0,
Indeed, first note that for all β > 0,
Then I ≤ 0, since
We may therefore apply Lemma 4.29 to the subsolution
i.e.,
Choosing β = s/2, and letting ɛ → 0, we obtain
To complete the proof of Theorem 4.38, it therefore suffices to show that for some s > 0 small enough,
To this end, we first note that Lemma 4.35 applies to u. In particular, setting h ≔ log u, by (4.37), we see that h ∈ W 1,2(Ω3KR ).
Let B
r
⊂ B
3KR
, and let η be a smooth bump function adapted to B
r
, i.e.,
where M depends only on the allowable parameters; i.e., (3.17) holds for all B r ⊂ B 3KR , so we may apply Theorem 3.16 to h to obtain
for some uniform constant c > 0 small enough. Taking s = c/M, we therefore have
which proves (4.42), and thus also Theorem 4.38. □
Theorem 4.43.
(Harnack Inequality). Fix Ω R = Ω R (y 0) as above, with y 0 ∈ ∂Ω. Let K be the constant in Proposition 3.9. Suppose that u ∈ W 1,2(T R ), u ≥ 0 is a local weak solution to the Neumann problem with vanishing data in Ω4KR , in the sense that
Then for all B r = B r (z) ⊂ B R/2,
where the implicit constants depend only on n, ADR, (CS), (HC) and ellipticity.
Proof.
First note that (4.32) holds for Ω r = Ω r (z), i.e.,
We now claim that
where s > 0 is the exponent in Theorem 4.38, and where the implicit constants depend only on the allowable parameters. Let us take the claim for granted momentarily, and suppose that B r ⊂ B R/2. Then B 2r ⊂ B R , so we may apply Theorem 4.38 with 2r in place of R, to obtain
Combining (4.46), (4.47), and (4.48), we obtain (4.45), as desired.
It remains only to verify (4.47). To this end, observe that (4.46) is a weak reverse Hölder inequality, holding for all B r ⊂ B R , including the case that B r is an interior ball, as well as the case that B r is centered on ∂Ω. Since weak reverse Hölder inequalities enjoy a well-known self-improvement property that allows the exponent on the right hand side to be taken arbitrarily close to zero (see, e.g., [15], Appendix B]), we find in particular that
Suppose now that B r ⊂ B R/2. Then B 2r ⊂ B R , and we may replace r by 2r in the last estimate to obtain (4.47). □
As a corollary of the Harnack inequality, we obtain local Holder continuity up to the boundary.
Corollary 4.49.
Fix Ω R = Ω R (y 0) as above, with y 0 ∈ ∂Ω. Let K be the constant in Proposition 3.9. Suppose that u ∈ W 1,2(T R ) is a local solution to the Neumann problem with vanishing data in Ω4KR , in the sense that (4.44) holds. Then there exist C < ∞ and α > 0 depending only on dimension, ellipticity, ADR, (CS) and (HC), such that for all B r ⊂ B R/2,
Proof.
We follow the standard argument, using the Harnack inequality. As in Theorem 4.43, we allow B r to be an interior ball, or to be centered on ∂Ω. Set
and set
Let B r ⊂ B R/2, and let κ be the constant in Proposition 4.28. Note that in particular, in the case that B r is centered on ∂Ω, we then have that T r/κK ⊂ Ω r . Hence, we may then apply Theorem 4.43 with Ω r/κK in place of Ω R , to obtain that for some C 0 depending only on allowable parameters,
and also
Consequently, adding (4.51) and (4.52), we obtain
i.e., with θ = (C 0 − 1)/(C 0 + 1) < 1, we have
We now obtain (4.50) by a standard iteration argument, using Lemma 4.29 to bound osc(r) in terms of the L 2 average over Ω2r . □
As an immediate consequence of Corollary 4.49, we obtain local Hölder continuity of the Neumann function, up to the boundary.
Theorem 4.53.
Fix Ω R = Ω R (y 0) as above, with y 0 ∈ ∂Ω, and suppose that x ∈ Ω\Ω κKR , where κ is the constant in Proposition 4.28, and K is the constant in Proposition 3.9. Then the Neumann function N(x, ⋅) is Hölder continuous in Ω R/2, i.e., setting N x ≔ N(x, ⋅), we have
Remark 4.55.
Note that, after possible correction of the boundary trace of v
x
on a set of σ-meaure zero on ∂Ω, we may take N
x
to be Hölder continuous all the way to the boundary, i.e., (4.54) holds for all
Our next goal is to prove the following Theorem:
Theorem 4.56.
Let
Proof.
By Remark 4.55 (and its analogue with the roles of x and y reversed), N(x, y) extends continuously to the boundary in each variable separately, thus it is enough to prove (4.57) for x, y ∈ Ω. To this end, we recall (see Definition 4.24) that N(x, y) ≔ G(x, y) − v(x, y). By Corollary 4.22, we already know that
Moreover, it is of course well known that
Consequently, if |x − y| < min(δ(x), δ(y)), then (4.57) holds. We may therefore suppose that
We claim that the latter estimate actually self improves, to give
Indeed, suppose that, say, |x − y| ≥ δ(x). Fix
which proves (4.60).
Consequently, we may assume that (4.60) holds, otherwise there is nothing to prove. But if (4.60) holds, then
which, combined with (4.58), yields the following:
Next, we claim that
Let us now establish this claim. Again, we may suppose that (4.60) holds, otherwise we have nothing to prove. Set R ≔ |x − y|. By Lemma 4.29 and Remark 4.33, Moser’s local boundedness estimate holds up to the boundary for N(x, ⋅), hence,
since G satisfies (4.59), and v x satisfies (4.12), where we have used that −n/2* = (2 − n)/2, that
and that, by the corkscrew condition,
On the other hand, by Corollary 4.21, N(x, y) = N*(y, x) ≔ G*(y, x) − v*(y, x), where G*(y, x) = G(x, y) is the adjoint Green function. Clearly, all of our estimates for N(x, y) also apply to the adjoint Neumann function N*(y, x), with the roles of x and y reversed, and therefore, since condition (4.60) is symmetric with respect to x and y, we may reverse the roles of x and y in the previous argument to get
which proves (4.62), as claimed.
With (4.62) in hand, for x ∈ Ω fixed, and for γ > 0 (small), set
Note that
for each γ > 0, by virtue of (4.62).
Thus, our goal is to show that K γ (x) ≤ C 0, uniformly for all x ∈ Ω, and for all small γ > 0; in that case, we then obtain (4.57) by letting γ → 0.
In fact, we shall prove that for all suitably small ɛ > 0, for all small γ > 0, and for arbitrary x ∈ Ω,
with C depending only on the allowable parameters, and with C ɛ depending on these and on ɛ. Choosing ɛ ≤ 1/(4C), we may hide the small term on the left hand side of the inequality, and we are done.
Let us now prove (4.64). Fix x ∈ Ω, and γ > 0. We may then choose y ∈ Ω\{x} such that
where we have set R ≔ |x − y|. We may suppose that δ(y) < R/10, otherwise (4.62) trivially implies (4.57).
Set
Define also
Observe that by the Corkscrew condition, for ɛ small enough we have
Note also that by [13], Lemma 5.3],
Now, by Lemma 4.29 and Remark 4.33,
Since
Furthermore, by (4.67), (4.63), and the definition of K γ (x),
Dividing through by R 2−n + γR (2−n)/2 δ(x)(2−n)/2, and using (4.65), we obtain (4.64), as desired. □
Corollary 4.68.
Let
Proof.
As above, let κ and K be the constants in Propositions 4.28 and 3.9, respectively. Observe that it is enough to prove the result when 2κK|y − y 0| < |x − y 0|, because in the range 2|y − y 0| < |x − y 0| ≤ 2κK|y − y 0|, we have |y − y 0| ≈ |x − y 0| ≈ |x − y|, and the desired bound follows trivially from the pointwise estimate in Theorem 4.56.
Thus, we now suppose that 2κK|y − y 0| < |x − y 0|. Define R so that |x − y 0| = κKR, hence x ∈ Ω\Ω κKR (y 0), and |y − y 0| < R/2. Then by Theorem 4.53, Remark 4.55, and Theorem 4.56,
where we used that
5 The Neumann problem
In this section, we show that L p solvability of the Neumann problem, for some p > 1, paired with L p solvability of the regularity problem, and L p ′ solvability of the Dirichlet problem[4] for the adjoint operator L*, implies L q solvability of the Neumann problem for 1 < q < p. As before, we let L = −divA∇, and we interpret Lu = 0 in the weak sense. As above, for the sake of lightening the exposition, we shall continue to focus on the case that Ω and ∂Ω are unbounded. We leave it to the interested reader to make the appropriate adjustments needed to treat the case of a bounded domain.
We first discuss the variational Neumann problem.
5.1
W
̇
1,2
weak solutions of the Neumann problem
We begin by constructing the weak
In the sequel, for
Theorem 5.1.
Let
i.e., there is a unique
Conversely, if
Proof.
By the trace estimate (3.6), the mapping
To establish the converse, following [4], we let
We shall continue to assume, as above, that Ω is a 1-sided CAD, until noted otherwise (i.e., until Section 5.2, at which point we shall impose the stronger 2-sided CAD condition).
Lemma 5.3.
Let u ∈ Y 1,2(Ω) ∩ C(Ω) (observe that we do not assume that u is a solution). Then
Proof. Note that by (4.11) and the standard Riesz formula (4.26), we see that (5.4) holds under the somewhat more restrictive hypothesis that
Temporarily fix x ∈ Ω, and let 0 < ɛ < δ(x)/4. Fix
We seek to show that the limit in (5.6) exists, and is equal to u(x) for a.e. x. To this end, momentarily fix γ > 2ɛ, and define
by Lemma 4.25. It therefore remains to consider
Fix a non-negative
Note that
since ψ γ (x) = 1. Then lim γ→0 lim ɛ→0 II ɛ,γ (x) = u(x) for all x ∈ Ω, since u γ → u pointwise. It therefore remains only to show that
To this end, set
and observe that
since N(x, ⋅) is a weak adjoint solution in the support of the test function (u − u γ )ψ γ (1 − ψ ɛ ).
To handle the first error term
and in turn, by definition of ψ ɛ , and then Caccioppoli’s inequality and Theorem 4.56 (the size estimate for the Neumann function), we see that
Consequently, since u γ → u uniformly on compacta in Ω (recall that u is continuous in Ω)
We now turn to the second error term
where
Let
We note that
Recall that N(x, y) ≔ G(x, y) − v
x
(y), where v
x
(y) was constructed in Lemma 4.6. In particular, there is an equivalence class
Theorem 5.10.
Let
Proof.
It is enough to prove (5.11) for
Fix x ∈ Ω, and let
uniformly for all sufficiently small ɛ, since u is the weak solution of (N). This proves (5.13), in light of (5.6). □
We next prove a localized version of the preceding result.
Lemma 5.15.
Let
Proof.
By Lemma 5.3, we have
Consequently,
Given x ∈ Ω, define
where in the last step we have used (5.9). Plugging (5.18) into the last term in (5.17), we obtain (5.16) for a.e. x ∈ Ω. □
For future reference, we now record standard boundary Caccioppoli inequalities, with either vanishing Dirichlet data, or vanishing Neumann data. Recall that by Theorem 5.1, if
Lemma 5.19.
Let r > 0, z ∈ ∂Ω, and as usual, set Ω
r
= Ω
r
(z) ≔ Ω ∩ B(z, r), and Δ
r
= ∂Ω ∩ B(z, r); similarly for Δ2r
, Ω2r
. Suppose that u is the Y
1,2(Ω) realization of a weak solution
In addition, if K is the constant in Proposition 3.9, and 2* = 2n/(n − 2), then
Proof.
Inequality (5.20) is standard in scenario (i), so we consider only scenarios (ii) and (iii). Let
In either case (ii) or (iii), uφ
2 ∈ Y
1,2(Ω), hence is the realization of
As usual, for any ɛ > 0, we have
Choosing ɛ small enough, depending only on ellipticity, we may hide the small term to obtain (5.20).
To prove (5.21), we apply first (3.15) with R = r/K, and then (5.20), to obtain
Combining this estimate with the trivial bound
5.2 L p theory
We now turn to the issue of L p solvability. We begin by defining nontangential “cones” and maximal functions, as follows. For a sufficiently large constant K 1 = K 1(HC, CS), and given x ∈ ∂Ω, set
For K 1 large enough, Γ(x) defines a nontangential approach region such that each CS point relative to any ball B(x, r) is connected to the boundary point x via a Harnack path lying within Γ(x). Given R > 0, we also define truncated versions of the cones (from above, and from below, respectively):
Following [4], we also define a variant of the nontangential maximal function as follows:
as well as its truncated versions
We are now able to define L
p
solvability for the Neumann and regularity problems for L. We recall that the space
Definition 5.27.
Given
We say that
Henceforth, we shall assume that Ω is a 2-sided CAD, unless noted explicitly to the contrary. In the latter setting, we recall, for future reference, some constructions and estimates concerning boundary Sobolev spaces presented in Ref. [16].
5.2.1 Boundary Sobolev spaces
L
̇
1
p
(
∂
Ω
)
In Ref. [16], Section 3.6], it was shown that one can make sense of tangential derivatives on ∂Ω provided that ∂Ω is ADR, and that the measure theoretic boundary ∂∗Ω coincides σ-a.e. with the topological boundary. In particular, a CAD satisfies these conditions. The tangential derivatives are defined as follows: let ν = (ν
1, …, ν
n
) be the measure theoretic outward unit normal to ∂Ω. Then for
Let
Since the restrictions of all such φ form a dense class in L
p
′(∂Ω), it follows that there is a function h ∈ L
p
(∂Ω), with
In this case, we say that g has a weak tangential derivative
provided that the tangential derivatives
and we set
We next define the Calderón-Scott [17] maximal operator
for x ∈ ∂Ω, where Δ(x, r) ≔ B(x, r) ∩ ∂Ω, and g Δ(x,r) ≔ ⨏Δ(x,r) g. In Ref. [16], it is shown (see [16], (3.6.11), (4.3.19)]) that in a 2-sided CAD,
where of course the implicit constants depend only on p, n, ADR and the CS and HC constants for Ω and Ωext. We observe that (5.30) is an extension of a result of [17], first proved in the flat setting. We caution the reader that the precise statements of [16], (3.6.11)] and [16], (4.3.19)] are inhomogeneous versions of (5.28) and (5.30), respectively, but the proof in Ref. [16] actually yields the version of (5.30) stated here.
Recall that δ(x) ≔ dist(x, ∂Ω), for x ∈ Ω. We continue to follow the approach in Ref. [4].
Theorem 5.31.
Assume
Proof.
Fix z ∈ ∂Ω with
To prove (5.32), we may assume
Note that by construction,
where we have used Moser’s local boundedness estimate, then Poincaré’s inequality, and finally that by construction, δ(z k ) ≈ δ(z k+1) ≈ θ k ɛ. Since θ < 1, it then follows that
Thus, (5.32) holds. Let g denote the nontangential limit of u. Letting y → z nontangentially in (5.32), we see that in particular, for a.e. z ∈ ∂Ω,
It remains to establish that g is a representative of an equivalence class modulo constants with
where we have used (5.33) with ɛ ≈ 1 and x = x
Δ. By Moser’s local boundedness, and our hypothesis, the last expression is finite, thus
Finally, we need to show that
where Λ∗
g is defined in (5.29). To this end, fix x ∈ ∂Ω, let r > 0, set Δ
r
≔ Δ(x, r), and let
where in the last step we have used (5.33) with ɛ = r. Taking a supremum in r, and letting
The L p bound (5.34) now follows. □
5.2.2 The regularity problem (R) p
Definition 5.35.
Fix 1 < p < ∞. The regularity problem (R)
p
, with data in
Moreover, [g] has a realization g such that u converges to g nontangentially a.e. on ∂Ω.
We shall also need the definition of the Dirichlet problem. Set
Definition 5.37.
Fix 1 < p < ∞. The Dirichlet problem (D) p′, with data in L p ′(dσ) is solvable for L* if for each f ∈ L p ′(∂Ω), there is a unique solution u satisfying
with u converging to f nontangentially a.e. on ∂Ω. Moreover, the solution u satisfies
Remark 5.38.
It is well known that in a chord arc domain
where X is any Corkscrew point with R/2 ≤ |X − z 0|≲ R ≲ δ(X), and where the various implicit constants are uniform for all z 0 ∈ ∂Ω, and all R > 0 (with R < diam(Ω) in the case of a bounded domain). We refer the reader to Ref. [18], Chapter 2]. In fact, the chord arc assumption is not needed (provided that (5.39) is suitably interpreted), and ADR alone suffices (see [19]).
It was shown in Ref. [4] that L p solvability of the Neumann problem paired with L p solvability of the regularity problem on the ball (and more generally bounded starlike Lipschitz domains) implies L q solvability of the Neumann problem for 1 < q < p. We will follow rather closely the arguments used in Ref. [4], with appropriate adjustments to allow us to treat the class of 2-sided chord arc domains.
In the next lemma, we shall use the following simple geometric fact: if M
1 is chosen sufficiently large, depending only on allowable parameters, then given a surface ball
Theorem 5.41.
(Localization of (R)
p
). Suppose that Ω is an unbounded, 2-sided chord-arc domain, with unbounded boundary. Let 1 < p < ∞ and suppose that (R)
p
is solvable for L in Ω, and (D)
p′ is solvable for L*. Let R > 0, and let ϒ
R
(z
0) = Ω2R
(z
0)\Ω
R
(z
0) be an annulus contained in Ω centered at z
0 ∈ ∂Ω. Let
Proof.
We follow the proof of [4], Theorem 5.19]. Let
Note that by (5.40), we have
By the solvability of (R) p , we have
It remains to consider the contribution of ∇F. One may write F as
where we used the fact that u is a weak solution. From our choice of φ, we see that ∇φ vanishes in Ω\ϒ R (z 0). In particular, for all x as in (5.40), F 1 is a solution in B(x, 3δ(x)/4). It then follows that for such x, we can use Caccioppoli’s inequality to obtain
Let ξ ∈ B(x, 3δ(x)/4). Using (5.40), Caccioppoli’s inequality, Harnack’s inequality, and Carleson’s estimate, we obtain
where x
R
is a corkscrew point chosen so that R ≤ |x
R
− z
0| ≲ R ≲ δ(x
R
). For
Hence, for
where
Similarly,
Consequently, a slight variation of the previous argument yields the estimate
□
We recall the definition of the truncated cones
Theorem 5.44.
(Localization of (N)
p
). Let M
1 be the constant in Theorem 5.41. Let 1 < p < ∞ and suppose that (N)
p
and (R)
p
are solvable for L in Ω, and that (D)
p′ is solvable for L*. Let R > 0 and define an annulus
where Ω R = Ω R (z).
Proof.
We follow closely the proof of [4], Theorem 6.10]. By the solvability of (N)
p
and Theorem 5.31, g = u|∂Ω exists as a nontangential limit almost everywhere on ∂Ω, and
Let
Next, note that II = II(x) = −∫Ω
N(x, ⋅) A∇u ⋅ ∇Φ dy is continuous in
It then follows that
Since
Indeed, let
The solvability of (N)
p
implies that
Since
Since (D) p′ is solvable for L*, we have
where as usual, N ∗ is the standard nontangential maximal operator defined in (5.36).
Note that the (adjoint) conormal derivative of v is
and (for future reference)
where
Further, by (5.49),
Since supp(∇Φ) ⊂ Ω2R
\Ω
R
⊂ Ψ
R
, and
where in the last step we have used (5.50) and (5.52). In turn, taking a supremum over all Lipschitz h as above (i.e., with support in Δ(z, R/2), and
where the last estimate follows from Caccioppoli’s inequality (up to the boundary), exactly like the first inequality in (5.51).
It remains to consider term I = I(x) = ∫Ω
uA∇Φ ⋅∇N(x, ⋅)dy (the first term on the right hand side of (5.16) in Lemma 5.15). We observe that I(x) is continuous in
where, exactly as before, v is the solution of the adjoint Neumann problem with data
where in the last setp we have used (5.51) and (5.52). Consequently,
Since I + II = u = g in Δ(z, R/2), we obtain (5.46), as desired. □
Before proceeding to the main result in this section, let us recall that for 1 < s ≤ ∞, an H 1 s-atom is a function a satisfying
supp(a) ⊂ Δ(y 0, r), for some surface ball Δ(y 0, r) ⊂ ∂Ω.
a ∈ L s (∂Ω), with
∫∂Ω a dσ = 0.
It is of course well known that for each s > 1, and each f ∈ H
1, there is a decomposition f = ∑
k
λ
k
a
k
, where a
k
is an s-atom for each k, and
Theorem 5.54.
Suppose that (N)
p
and (R)
p
are solvable for L, and (D)
p′ is solvable for L*, for some 1 < p < ∞. Then (N)
q
is solvable for L, for all 1 < q ≤ p, and so is
Proof.
We follow the strategy previously used in Ref. [4] (and to some extent, [20]). It suffices to establish the Hardy space estimate
for
In turn, to prove (5.55), it suffices to show that if a is an H 1 2-atom (or an H 1 p-atom, if p > 2), then
where u is the solution to (N) p with data a, and the implicit constant is uniform. To this end, we fix a 2-atom a (or a p-atom, if p > 2), adapted to a surface ball Δ(y 0, r), and let u be the solution to the Neumann problem with data a. We now make the standard decomposition into dyadic annuli:
where
We first dispose of the “local term”:
where in the last two steps we have used the assumed L p solvability, and then (5.53).
We now treat the “far away terms” ∑I
k
. Let M
1 be the constant in Theorem 5.41, and in Theorem 5.44. Let k ≥ 2, and choose R = 2
k
r/(100K
0) ≈ 2
k
r, so that S
k
may be covered by a uniformly bounded number of surface balls of the form Δ
j
≔ Δ(z
j
, R/(4M
1)), with z
j
∈ S
k
, and with the further property that Δ(y
0, r) ∩ Δ(z
j
, 10K
0
R) = ∅ for each j. We can then bound
where M = M(n, ADR, M
1, K
0). Thus, it suffices to establish the desired estimate for each individual summand in
where
Consequently, for all such x and y, Corollary 4.68 implies that
Therefore, since a has mean value zero and is supported in Δ(y 0, r),
by the case p = 1 of (5.53). Here, of course, u(x) = g(x), when
where we have used (5.57) and then (5.59) (including its boundary version) in the last two steps. Summing in k, we obtain
It remains to treat the terms
For such x, we therefore deduce from Corollary 4.68 (and Theorem 4.56) that (5.58), and hence also (5.59), continue to hold. It follows that for any y 1 ∈ Ω, with δ(y 1) ≥ R/(4M 1), by Caccioppoli’s inequality we have
By definition (see (5.25) and (5.26)), the latter bound then holds for
Since
Acknowledgments
The first author is supported by NSF grant DMS-2349846.
-
Research ethics: Not applicable.
-
Informed consent: Not applicable.
-
Author contributions: Both authors contributed to this work.
-
Use of Large Language Models, AI and Machine Learning Tools: None declared.
-
Conflict of interest: The authors state no conflict of interest.
-
Research funding: This work was supported by NSF under grant DMS-2349846.
-
Data availability: Not applicable.
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Articles in the same Issue
- Frontmatter
- Editorial
- Preface for the special issue in honor of Robert Fefferman
- Research Articles
- Sobolev extension in a simple case
- Soliton resolution and channels of energy
- Matrix weights and a maximal function with exponent 3/2
- An introduction to the distorted Fourier transform
- The Neumann function and the L p Neumann problem in chord-arc domains
- An example related to Whitney’s extension problem for L 2,p (R2) when 1 < p < 2
- Convergence rates of eigenvalue problems in perforated domains: the case of small volume
- The initial-value problem for a Gardner-type equation
- Global Schauder estimates for kinetic Kolmogorov-Fokker-Planck equations
- Maximal estimates for strong arithmetic means of Fourier series
- On the L p -boundedness of Calderón-Zygmund operators
- Current perspectives on the Halo Conjecture
- On certain trilinear oscillatory integral inequalities
Articles in the same Issue
- Frontmatter
- Editorial
- Preface for the special issue in honor of Robert Fefferman
- Research Articles
- Sobolev extension in a simple case
- Soliton resolution and channels of energy
- Matrix weights and a maximal function with exponent 3/2
- An introduction to the distorted Fourier transform
- The Neumann function and the L p Neumann problem in chord-arc domains
- An example related to Whitney’s extension problem for L 2,p (R2) when 1 < p < 2
- Convergence rates of eigenvalue problems in perforated domains: the case of small volume
- The initial-value problem for a Gardner-type equation
- Global Schauder estimates for kinetic Kolmogorov-Fokker-Planck equations
- Maximal estimates for strong arithmetic means of Fourier series
- On the L p -boundedness of Calderón-Zygmund operators
- Current perspectives on the Halo Conjecture
- On certain trilinear oscillatory integral inequalities
