The mixed problem in L^{p} for some two-dimensional Lipschitz domains

Loredana Lanzani^{∗} Department of Mathematics

University of Arkansas Fayetteville, Arkansas lanzani@uark.edu

Luca Capogna^{∗} Department of Mathematics University of Arkansas Fayetteville, Arkansas lcapogna@uark.edu

Russell M. Brown^{∗} Department of Mathematics University of Kentucky Lexington, Kentucky russell.brown@uky.edu

Abstract

We consider the mixed problem,

Δu = 0  in Ω
\frac{∂u}{∂ν} = f_{N}  on N
u = f_{D}  on D

in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data, f_{D}, has one derivative in L^{p}(D) of the boundary and the Neumann data, f_{N}, is in L^{p}(N). We find a p_{0} > 1 so that for p in an interval (1,p_{0}), we may find a unique solution to the mixed problem and the gradient of the solution lies in L^{p}.

1 Introduction

The goal of this paper is to study the mixed problem (or Zaremba’s problem) for Laplace’s equation in certain two-dimensional domains when the Neumann data comes from L^{p} and the Dirichlet problem has one derivative in L^{p}. We will consider both L^{p} with respect to arc-length, dσ and also L^{p}(wdσ) where the weight w is of the form w(x) = |x|^{ϵ}. By the mixed problem for L^{p}(wdσ), we mean the following boundary value problem

∗Research supported, in part, by the National Science Foundation.

\{ \begin{matrix}Δu = 0 & in Ω \\ u = f_{D} & on D \\ \frac{∂u}{∂ν} = f_{N} & on N \\ (∇u)^{∗} ∈ L^{p}(wdσ) \end{matrix}  (1.1)

Here, (∇u)^{∗} is the non-tangential maximal function of the gradient (see the definition in (2.1)). The domain Ω will be a Lipschitz graph domain. Thus, Ω = {(x_{1},x_{2}) : x_{2} > ϕ(x_{1})} where ϕ: R → R is a Lipschitz function with ϕ(0) = 0. We will call such domains standard Lipschitz graph domains. The sets D and N satisfy D ∪ N = ∂Ω and D ∩ N = ∅. Also, we want D to be open, so that it supports Sobolev spaces. Here, we will generally assume that f_{D} is a function with one derivative in L^{p}(D,wdσ) and that f_{N} is in L^{p}(N,wdσ). Our goal is to find conditions on the data, the exponent p and the weight w so that the problem (1.1) has a unique solution. This paper continues the work of Sykes and Brown [4, 33, 34] to provide a partial answer to problem 3.2.15 from Kenig’s CBMS lecture notes [20]. Our goal is to obtain L^{p}-results in a class of domains that is not included in the domains studied by Sykes [33].

We recall a few works which study the regularity of the mixed problem. In [1], Azzam and Kreyszig established that the mixed problem has a solution in C^{2+α} in bounded twodimensional domains provided that N and D meet at a sufficiently small angle. Lieberman [23] also gives conditions which imply that the solution is Hölder continuous. In addition, much effort has been devoted to problems in polygonal domains, see the monograph of Grisvard [16]. Savaré [29] finds solutions in the Besov space B_{2,∞}^{3/2} in smooth domains. This positive result fits quite nicely with the example below which shows that there is a solution whose gradient just misses having non-tangential maximal function in L^{2}(dσ). It is known that if a function is harmonic and the non-tangential maximal function of the gradient is in L^{2}(dσ), then the function also belong to the Besov space B_{2,2}^{3/2} (see the article of Fabes [12]). Recent work of I. Mitrea and M. Mitrea [27] extend the class of spaces where we know the mixed problem is well-posed. However, their work does not extend the class of domains beyonds those considered in [4]. Recent work of Mazya and Rossman [24, 25, 26] consider mixed boundary value problems for the Stokes system and the Navier Stokes system in polyhedral domains. It would be of interest to extend the methods presented here to other equations.

If we recall the standard tools for studying boundary value problems in Lipschitz domains, we see that the mixed problem presents an interesting technical challenge. On the one hand, the starting point for many results on boundary value problems in Lipschitz domains is the Rellich identity (see Jerison and Kenig [18], for example). This remarkable identity provides estimates at the boundary for derivatives of a harmonic function in L^{2}. On the other hand, simple and well-known examples show that the mixed problem is not solvable in L^{2}(dσ) for smooth domains. Let us recall an example in the upper half-space, R_{+}^{2} = {(x_{1},x_{2}) : x_{2} > 0}. We let N = {(x_{1},0), x_{1} > 0}, D = {(x_{1},0), x_{1} < 0}, and we consider the harmonic function

u(z) = Re (\sqrt{z}− \sqrt{z + i}), z = x_{1}+ ix_{2}.

It is easy to see that we have

| \frac{∂u}{∂ν}| ≤ C(1 + |z|)^{−3/2} on N,
| \frac{du}{dσ}| ≤ C(1 + |z|)^{−3/2} on D,

but

|∇u| ≥ \frac{C}{\sqrt{|z|}} for |z| ≤ 1/2.

Hence, we do not have (∇u)^{∗} ∈ L_{loc}^{2}(R,dσ). On the other hand we have (∇u)^{∗} ∈ L_{loc}^{p}(R,dσ) for all 1 ≤ p < 2.

We detour around this problem by establishing weighted estimates in L^{2} using the Rellich identity. This relies on an observation of Luis Escauriaza [11] that a Rellich identity holds when the components of the vector field are, respectively, the real and imaginary part of a holomorphic function. Then, we imitate the arguments of Dahlberg and Kenig [10] to establish Hardy-space estimates (with a different weight). The weights are chosen so that interpolation will give us L^{p} with respect to arc-length as an intermediate space. The weights we consider will be of the form |x|^{ϵ} restricted to the boundary of a Lipschitz graph domain. Earlier work of Shen [30] (see also §3) gives a different approach to the study of weighted estimates for the Neumann and regularity problems when the weight is a power.

In the result below and throughout this paper, we assume that Ω is a standard Lipschitz graph domain and that

D = {(x_{1},ϕ(x_{1})) : x_{1} < 0},  N = {(x_{1},ϕ(x_{1})) : x_{1} ≥ 0}.  (1.2)

We will call Ω with N and D as defined above, a standard Lipschitz graph domain for the mixed problem. The Lipschitz constant of the domain, M is defined by

M = ∥ϕ′∥_{L^{∞} (R)}.  (1.3)

Our main result is the following.

Theorem 1.1 Let Ω be a standard Lipschitz graph domain for the mixed problem, with Lipschitz constant M less than 1. There exists p_{0} = p_{0}(M) > 1 so that for 1 < p < p_{0}, if f_{N} ∈ L^{p}(N,dσ) and df_{D}/dσ ∈ L^{p}(D,dσ), then the mixed problem for L^{p}(dσ) has a unique solution. The solution satisfies

∥(∇u)^{∗}∥_{L^{p}(dσ)} ≤ C(p,M)( ∥f_{N}∥_{L^{p}(N,dσ)}+ ∥ \frac{df_{D}}{dσ} ∥_{L^{p}(D,dσ)}) .  (1.4)

2 Preliminaries

In this section, we prove uniqueness for the weighted L^{p}-Neumann and regularity problems in a Lipschitz graph domain, see (2.4) and (2.5). Here and in the sequel, we let ν denote the outer unit normal to ∂Ω. We recall that a bounded Lipschitz domain is a bounded domain whose boundary is parameterized by (finitely many) Lipschitz graphs. We begin our development with an observation that we learned from Luis Escauriaza [11].

Lemma 2.1 (L. Escauriaza) Suppose Ω is a bounded Lipschitz domain and u and α = (α_{1},α_{2}) are smooth in a neighborhood of Ω̄ with Δu = 0 and α_{1}+ iα_{2} holomorphic. Then, we have

∫_{∂Ω} |∇u|^{2}α · ν − 2\frac{∂u}{∂α} \frac{∂u}{∂ν}dσ = 0.

Proof. A calculation shows that div(|∇u|^{2}α − (2α · ∇u)∇u) = 0. Thus, the lemma is an immediate consequence of the divergence theorem.

Remark. In the plane, it is easy to see that div(|∇u|^{2}α − 2∇uα · ∇u) = 0 for all harmonic functions u if and only if the vector field α is holomorphic. In fact, we may replace harmonic by linear in the previous sentence.

Next, we recall Carleson measures and the fundamental property of these measures. The applications we have in mind are simple, but the use of Carleson measures will allow us to appeal to a well-known geometrical argument rather than invent our own.

Let σ be a measure on the boundary of Ω. A measure μ on Ω is said to be a Carleson measure with respect to σ if there is a constant A so that for each x ∈ ∂Ω and each r > 0, we have

μ(B_{r}(x) ∩ Ω) ≤ Aσ(Δ_{r}(x)).

Here, we are using B_{r}(x) to denote the ball (or disc) in R^{2} with center x and radius r and we also use Δ_{r}(x) = B_{r}(x) ∩ ∂Ω to denote a ball on the boundary of Ω.

Before we can state the next result, we need a few definitions. For θ_{0} > 0, we define Γ(0) = {re^{iθ} : r > 0, |θ − π/2| < θ_{0}} to be the sector with vertex at 0, vertical axis and opening 2θ_{0}. Then, we put Γ(x) = x + Γ(0), x ∈ ∂Ω. If Ω is a Lipschitz graph domain with constant M, then we have that Γ(x) defines a non-tangential approach region provided θ_{0} < π/2 − tan^{−1} (M). We fix such a θ_{0} and if v is a function defined on Ω, we define the non-tangential maximal function of v, v^{∗} on ∂Ω by

v^{∗}(x) = sup_{y∈Γ(x)}|v(y)|, x ∈ ∂Ω.  (2.1)

We also use these sectors to define restrictions to the boundary in the sense of non-tangential limits. For a function v in Ω, we define the restriction of v to the boundary by

v(x) = lim_{Γ(x)∋y→x} v(y), x ∈ ∂Ω  (2.2)

provided the limit exists. Finally, we recall that a measure σ is a doubling measure if there is a constant C so that for all r > 0 we have σ(Δ_{2r}(x)) ≤ Cσ(Δ_{r}(x)).

Proposition 2.2 If τ is a doubling measure on ∂Ω and μ is a Carleson measure with respect to τ with constant A, then there exists a constant C > 0 so that

|∫_{Ω} vdμ|≤ CA∫_{∂Ω} v^{∗}dτ.

This result is well-known. The proof in Stein [31, pp. 58–60] easily generalizes from Lebesgue measure to doubling measures.

A simple example of a Carleson measure that will be useful to us is the following. With R > 0 and ϵ > −1, define dσ_{ϵ} and dμ_{ϵ} by

dσ_{ϵ}(x) = |x|^{ϵ}dσ(x), x ∈ ∂Ω ; dμ_{ϵ}(y) = \frac{|y|^{ϵ}}{R}χ_{B_{R}(0)}(y)dy, y ∈ Ω,  (2.3)

where we are using dσ for arc-length measure on ∂Ω and dy for area measure in the plane. It is not hard to see that σ_{ϵ} is a doubling measure on ∂Ω (see Lemma 3.1) and μ_{ϵ} is a Carleson measure with respect to σ_{ϵ}.

Lemma 2.3 (Rellich Identity) Let Ω be a standard Lipschitz graph domain. Given ϵ > −1 and a ∈ C, we define α(z) = (Re(az^{ϵ}),Im(az^{ϵ})). Here, z = x_{1}+ ix_{2}. Let σ_{ϵ} be as in (2.3). If u is harmonic in Ω and (∇u)^{∗} ∈ L^{2}(σ_{ϵ}), then we have

∫_{∂Ω} |∇u|^{2}α · ν − 2α · ∇u\frac{∂u}{∂ν}dσ = 0.

Remark. Here and in the sequel, we let ∇u(x) denote the non-tangential limit of ∇u at x ∈ ∂Ω, see (2.2). It is well known that, with the assumptions of Lemma 2.3, the nontangential limit of ∇u exists a.e. x ∈ ∂Ω, see Dahlberg [9] or Jerison and Kenig [17].

Proof. We introduce a cut-off function η_{R} where η_{R}(y) = 1 if y ∈ B_{R}(0), η_{R}(y) = 0 if |y| > 2R, and |∇η_{R}| ≤ C/R. For τ > 0, we define a translate of u, u_{τ}, by u_{τ}(y) = u(y+τe_{2}), where e_{2} = (0,1). Note that u_{τ} is smooth in a neighborhood of \overline{Ω} . Since Ω is a graph domain and 0 ∈ ∂Ω it follows that az^{ϵ} is holomorphic in Ω. Thus, we may apply Lemma 2.1 to u_{τ} and α. The divergence theorem now yields

∫_{∂Ω} ( |∇u_{τ}|^{2}α · ν − 2α · ∇u_{τ}\frac{∂u_{τ}}{∂ν}) η_{R}dσ = ∫_{Ω} ∇η_{R}· (|∇u_{τ}|^{2}α − (2α · ∇u_{τ})∇u_{τ}) dy
≡ I_{R}.

The hypothesis ϵ > −1 is needed to justify integration by parts with the singular vector field α. Now, using the measures defined in (2.3), Proposition 2.2 and the definition of α we have

|I_{R}| ≤ C ∫_{∂Ω} f_{R}(x)dσ_{ϵ}

where f_{R}is given by

f_{R}(x) = sup_{y∈Γ(x),|y|>R} |∇u_{τ}(y)|^{2}.

Notice that for each x, lim_{R→∞} f_{R}(x) = 0. Hence, our assumption that (∇u)^{∗} is in L^{2}(dσ_{ϵ}) and the Lebesgue dominated convergence theorem imply that lim_{R→∞} I_{R}= 0.

We may now let τ → 0^{+} and use the dominated convergence theorem again to obtain the lemma.

As a step towards studying the mixed problem in weighted spaces, we consider the Neumann and regularity problems in two-dimensions. By the Neumann problem for L^{p}(wdσ), we mean the problem of finding a function u which satisfies

\{ \begin{matrix}Δu = 0 & in Ω \\ \frac{∂u}{∂ν} = f_{N}, & on ∂Ω \\ (∇u)^{∗} ∈ L^{p}(wdσ) \end{matrix}  (2.4)

where, in general, we assume that f_{N} is taken from L^{p}(wdσ). We also study the regularity problem for L^{p}(wdσ) where we look for a function u which satisfies

\{ \begin{matrix}Δu = 0 & in Ω \\ u = f_{D}, & on ∂Ω \\ (∇u)^{∗} ∈ L^{p}(wdσ) \end{matrix}  (2.5)

where, in general, we assume that df_{D}/dσ is in L^{p}(wdσ).

An important component in establishing uniqueness is the following local regularity result for solutions with zero data.

Lemma 2.4 Suppose that Δu = 0 in Ω, \frac{∂u}{∂ν} = 0 or u = 0 on ∂Ω and (∇u)^{∗} ∈ L_{loc}^{1}(∂Ω). Then, for every bounded set B ⊂ Ω, we have ∇u ∈ L^{2}(B).

Proof. We consider the case of Neumann boundary conditions. Dirichlet boundary conditions may be handled by a similar argument.

We first observe that since (∇u)^{∗} is in L_{loc}^{1}(dσ), it follows that ∇u is in L^{1}(B) for any bounded subset of Ω, B.

Fix a point x_{0} ∈ ∂Ω and let η be a smooth cutoff function which is one on B_{2r}(x_{0}) and zero outside of B_{4r}(x_{0}). We let N(z,y) be the Neumann function for Ω, thus N is a fundamental solution which satisfies homogeneous Neumann boundary conditions. Dahlberg and Kenig [10] construct the Neumann function on a graph domain by reflection. At least formally, we have the representation formula,

ηu(z) = ∫_{Ω∩B_{4r}(x_{0})} N(z,y)(u(y)Δη(y) + 2∇u(y) · ∇η(y))dy  (2.6)

−∫_{Δ_{4r}(x_{0})}N(z,t)u(t)\frac{∂η(t)}{∂ν}dσ(t), z ∈ Ω.

We now show how to estimate each term on the right-hand side of this formula. Since (∇u)^{∗}(x) is in L_{loc}^{1}(∂Ω), it follows that u is bounded on Δ_{4r}(x_{0}). As we observed above, we have ∇u ∈ L^{1}(Ω ∩ B_{4r}(x_{0})), and the Poincaré inequality gives that u is in L^{1}(Ω ∩ B_{3r}(x_{0})). We also have that the Neumann function is locally bounded when z ≠ y. Hence, the integrands in (2.6) are in L^{1} and it is a routine matter to justify this formula. Since the map y → N(z,y) is in the Sobolev space W^{1,2}(B_{r}(x_{0}) ∩ Ω), uniformly for z ∈ B_{3r}(x_{0}) \ B_{2r}(x_{0}), the estimates for u and ∇u outlined above together with (2.6) imply that ∇u ∈ L^{2}(B_{r}(x_{0}) ∩ Ω).

Next, we recall a classical fact from harmonic function theory, a Phragmen-Lindelöf theorem. This result is well-known and may be found in Protter and Weinberger [28, Section 9, Theorem 18]. We will need this theorem to control the behavior at infinity of solutions in our unbounded domains.

In the next result and below, we will let Ω_{φ} denote the sector

Ω_{φ} = {re^{iθ} : r > 0, |θ − π/2| < φ}, 0 < φ < π.  (2.7)

With this normalization, Ω_{φ} is a Lipschitz domain with constant M = |tan(π/2 − φ)|.

Theorem 2.5 (Phragmen-Lindelöf) Let φ ∈ (0,π). Suppose v is sub-harmonic in Ω_{φ}, v = 0 on ∂Ω_{φ} and

v(z) = o(|z|^{π/(2φ)}) as |z| → +∞, z ∈ Ω_{φ} .

Then v ≤ 0.

We now prove uniqueness for the regularity problem for L^{p}(wdσ), (2.5).

Lemma 2.6 Let Ω be a Lipschitz graph domain with Lipschitz constant M. Suppose that, for p ≥ 1, L_{loc}^{p}(wdσ) ⊂ L_{loc}^{1}(dσ) and that that for all surface balls Δ_{r}(x) with r > 1, we have

( ∫_{Δ_{r}(x)}w(t)^{−1/(p−1)} dσ(t))^{\frac{p−1}{p}} ≤ Cr^{π/(2β+π)} , ifp > 1;  (2.8)
( inf_{t∈Δ_{r}(x)}w(t))^{−1} ≤ Cr^{π/(2β+π)} , ifp = 1  (2.9)

where β = arctanM > 0. Under these conditions, if u is harmonic in Ω, (∇u)^{∗} ∈ L^{p}(wdσ) and u = 0 on ∂Ω, then u = 0 in Ω.

Proof. Since β = arctanM then we have Ω ⊂ Ωφ̃, with φ̃ = β + π/2. Define v by

v(x) = \{ \begin{matrix}|u|, & in Ω \\ 0, & in Ω̄^{c} . \end{matrix}

The function v is sub-harmonic in all of R^{2} and, moreover, v = 0 on ∂Ωφ̃. We verify the growth condition in the Phragmen-Lindelöf Theorem 2.5. To do this, suppose that z ∈ Ωφ̃ and set r = |z|. By Lemma 2.4 we have that ∇u is in L^{2} of each bounded subset of Ω, and the same is true for ∇v. Then, by combining the mean-value property for subharmonic functions with the Poincaré inequality, Proposition 2.2 (for the Carleson measure dμ(y) = \frac{1}{r}χ_{B_{r}(z)}dy with respect to dσ, see (2.3)) and Hölder’s inequality, in the case p > 1, we obtain

0 ≤ v(z) ≤ \frac{1}{πr^{2}} ∫_{B_{r}(z)}v(y)dy
≤ \frac{C}{r}∫_{B_{r}(z)}|∇v(y)|dy
≤ C ∫_{Δ_{2r}(x)}(∇u)^{∗}(t)dσ(t)

≤ C ( ∫_{Δ_{2r}(x)}(∇u)^{∗}(t)^{p}w(t)dσ(t))^{1/p}( ∫_{Δ_{2r}(x)}w^{−1/(p−1)} (t)dσ(t))^{\frac{p−1}{p}} .

Here, x is the projection of z onto the boundary, i.e. if z = (x_{1},ϕ(x_{1}) + t), then x = (x_{1},ϕ(x_{1})). Thus, under our assumption (2.8) it follows that

0 ≤ v(z) ≤ Cr^{\frac{π}{2φ̃}} ≤ C|z|^{\frac{π}{2φ̃}}, z ∈ Ωφ̃,

as 0 ∈ ∂Ωφ̃. We may now apply Phragmen-Lindelöf Theorem 2.5 and conclude that v = 0. The case p = 1 is treated in a similar fashion.

Remark. Using Hölder’s inequality, we see that

∫_{Δ_{s}(x)}|f(t)|dσ(t) ≤ ( ∫_{Δ_{s}(x)}|f(t)|^{p}w(t)dσ(t))^{1/p}( ∫_{Δ_{s}(x)}w(t)^{−1/(p−1)} dσ(t))^{\frac{p−1}{p}} .

Thus, we have L_{loc}^{p}(wdσ) ⊂ L_{loc}^{1}(dσ) provided w^{−1/(p−1)} is in L_{loc}^{1}(dσ). It is easy to see that this will hold for the weight w(t) = |t|^{ϵ} if ϵ < p − 1.

Finally, we give uniqueness for the weighted Neumann problem for L^{p}(wdσ), (2.4).

Lemma 2.7 Let Ω be a Lipschitz graph domain with Lipschitz constant M. Suppose w satisfies the hypotheses of Lemma 2.6. If u is a solution of the Neumann problem with (∇u)^{∗} ∈ L^{p}(wdσ), p ≥ 1, and \frac{∂u}{∂ν} = 0 a.e. ∂Ω, then u is constant.

Proof. We consider v, the conjugate harmonic function to u. Since u has normal derivative zero at the boundary, by the Cauchy-Riemann equations we have that v is constant on the boundary, and we may assume this constant is 0. Now, (∇v)^{∗} ∈ L^{p}(wdσ) (since this is the case for u) so that v satisfies the hypotheses of Lemma 2.6 and hence it is zero. But this implies that u is constant.

3 The L^{2}-Neumann and regularity problems with power weights

In this section we outline the proof of existence of solutions to the Neumann problem and the regularity problem for L^{2}(wdσ) when the weight is a power of |x|. These results are due to Shen [30, Theorems 1.7, 1.9]. Shen’s results are stated for bounded domains in three dimensions. We give a few steps of the proof to indicate the minor modifications that are needed to handle graph domains in two dimensions.

Before continuing, we record a few basic facts about the power weights |x|^{ϵ} and their relation to the Muckenhoupt class A_{p}(dσ). A weight (that is a non-negative measurable and locally integrable function) w is a member of the class A_{p}(dσ) if and only if for all surface balls Δ ⊂ ∂Ω, we have

\frac{1}{σ(Δ)} ∫_{Δ} wdσ( \frac{1}{σ(Δ)} ∫_{Δ} w^{\frac{−1}{p−1}} dσ) ^{p−1} ≤ C.

The best constant C in this inequality is called the A_{p}-constant for w. The following simple lemma tells us that |x|^{ϵ} is in A_{p}(dσ) if and only if −1 < ϵ < p − 1, see [31, p. 218].

Lemma 3.1 For ϵ > −1, the boundary measure σ_{ϵ} (see (2.3)) satisfies

σ_{ϵ}(Δ_{r}(x)) ≈ r max(|x|,r)^{ϵ}, x ∈ ∂Ω.

The proof is omitted.

We will need the following estimates from Shen [30].

Proposition 3.2 Let Ω be a standard Lipschitz graph domain and suppose that (∇u)^{∗} is in L^{2}(dσ_{ϵ}) where 0 ≤ ϵ < 1. There is a constant C = C(∥ϕ′∥_{∞}) so that

∫_{∂Ω} (∇u)^{∗}(x)^{2}dσ_{ϵ}(x) ≤ C ∫_{∂Ω} | \frac{∂u}{∂ν}|^{2} dσ_{ϵ}  (3.1)
∫_{∂Ω} (∇u)^{∗}(x)^{2}dσ_{ϵ}(x) ≤ C ∫_{∂Ω} | \frac{du}{ds}|^{2} dσ_{ϵ}  (3.2)

Proof. The arguments of Shen [30, Theorem 1.7 and 1.9] carry over with minor changes to graph domains in two dimensions. The estimates for the Neumann function in Shen’s work (see (3.2) on page 2850 of [30]) may be proven in graph domains using the ideas of Dahlberg and Kenig [10]. Also, see the argument below where we establish the estimate (6.6) for the Green’s function for the mixed problem.

Remark. Estimate (3.1) for the regularity problem holds for a larger class of weights than power weights. Estimate (3.2) for the Neumann problem is only known to hold for power weights.

We conclude this section with the following result which is a minor modification of results in [30].

Theorem 3.3 (Z. Shen) Let Ω be a standard Lipschitz graph domain with ∥ϕ′∥_{∞} < ∞. For ϵ satisfying 0 ≤ ϵ < 1 and for σ_{ϵ} as in (2.3), we have that the Neumann problem (2.4) and the regularity problem (2.5) for L^{2}(dσ_{ϵ}) are uniquely solvable.

Proof. We apply the method of layer potentials as in Verchota [35]. Note that for ϵ ∈ (−1,1), |x|^{ϵ} is an A_{2}(dσ) weight so that singular integrals are bounded on L^{2}(∂Ω,dσ_{ϵ}). The estimates of Proposition 3.2 imply that

∫_{∂Ω} | \frac{∂u}{∂ν}|^{2} dσ_{ϵ} ≈ ∫_{∂Ω} | \frac{du}{dσ}|^{2} dσ_{ϵ},  (3.3)

if u is harmonic in Ω with (∇u)^{∗} ∈ L^{2}(dσ_{ϵ}) and 0 ≤ ϵ < 1. The Neumann and regularity problems in half-space may be solved using the potential S(f)(x) = \frac{1}{π} ∫_{R}log|x − y|f(y)dσ(y). The estimate (3.3) and the method of continuity (see Brown [4, Lemma 1.16] and Gilbarg and Trudinger [15, Theorem 5.2]) now lead to the existence of a solution in general graph domains.

Uniqueness follows from Lemma 2.6 or 2.7 as it is easy to check that the measure dσ_{ϵ} satisfies the growth condition of Lemma 2.6.

4 The mixed problem in L^{2} with power weights.

The main result of this section is Theorem 4.6 where we prove existence and uniqueness for the solutions of the weighted mixed problem in L^{2} on a Lipschitz graph domain. There will be a restriction on the size of the Lipschitz constant (see Lemma 4.1). We first discuss the case when the measure is arc-length. The key estimate is obtained using the Rellich identity with a vector field α so that α · ν changes sign as we move from D to N. This technique is applied to the mixed problem in [4]. A similar technique is applied to a question about the wave equation by Lagnese [22]. This earlier work requires the domain to be smooth and thus the sets N and D must consist of connected components of the boundary.

Lemma 4.1 (Rellich Estimates for the mixed problem with w = 1) Let Ω = {x_{2} > ϕ(x_{1})}, N, D be a standard Lipschitz graph domain for the mixed problem, see (1.2). Suppose that ϕ_{x_{1}} > δ > 0 on N and ϕ_{x_{1}} < −δ < 0 on D.

Then, if u is harmonic in Ω and (∇u)^{∗} ∈ L^{2}(∂Ω,dσ), we have

∫_{D}( \frac{du}{dσ}) ^{2} dσ + ∫_{N}( \frac{∂u}{∂ν}) ^{2} dσ ≈ ∫_{N}( \frac{du}{dσ}) ^{2} dσ + ∫_{D}( \frac{∂u}{∂ν}) ^{2} dσ.  (4.1)

Proof. The proof follows from the Rellich identity with vector field α = e_{1}, see Brown [4, Lemma 1.7].

Theorem 4.2 (Mixed problem in L^{2}(dσ)) Let δ > 0 and Ω, N, D be a standard Lipschitz graph domain for the mixed problem, see (1.2). Suppose that ϕ_{x_{1}} > δ > 0 on N and ϕ_{x_{1}} < −δ < 0 on D. Then, if g_{N} ∈ L^{2}(N,dσ) and dg_{D}/dσ ∈ L^{2}(D,dσ), there exists a unique solution of the mixed problem (1.1) for L^{2}(dσ) in Ω. Moreover, the solution v satisfies

∥(∇v)^{∗}∥_{L^{2}(dσ)} ≤ C ( ∥\frac{dg_{D}}{dσ}∥_{L^{2}(D,dσ)}+ ∥g_{N}∥_{L^{2}(N,dσ)}) .  (4.2)

Remark. In particular, we have that the mixed problem with w = 1 is uniquely solvable (with estimates for (∇v)^{∗}) in all convex sectors Ω_{ψ}, 0 < ψ < π/2, see (2.7).

Proof. This is an extension of the results of Brown [4] to two dimensions and to Lipschitz graph domains. The first step is to observe that the equivalence (4.1) quickly leads to uniqueness. For if u is a solution of (1.1) with u = 0 on D and \frac{∂u}{∂ν} = 0 on N, then (4.1) implies u = 0 on ∂Ω. Lemma 2.6 now yields u = 0 in Ω.

To prove existence, we first find solutions in a sector. This is easy by symmetry and can be done as in Brown [4, Lemma 1.16]. Then we may use the method of continuity and again (4.1) to establish solutions in more general Lipschitz graph domains.

We now turn our attention to the weighted mixed problem in sectors that are not convex. For the boundary of a sector Ω_{φ} as in (2.7), we write: ∂Ω_{φ} = D_{φ} ∪ N_{φ}, with D and N as in (1.2).

Proposition 4.3 (Weighted mixed problem on sectors) Let φ ∈ (π/2,π) and suppose that 1− \frac{π}{2φ} < ϵ < 1. Let σ_{ϵ} be as in (2.3). Then, if f_{N} ∈ L^{2}(N_{φ},dσ_{ϵ}) and df_{D}/dσ ∈ L^{2}(D_{φ},dσ_{ϵ}), there exists a unique solution u of the mixed problem

\{ \begin{matrix}Δu = 0 & in Ω_{φ} \\ u = f_{D} & on D_{φ} \\ \frac{∂u}{∂ν} = f_{N} & on N_{φ} \\ (∇u)^{∗} ∈ L^{2}(∂Ω_{φ},dσ_{ϵ}). \end{matrix}  (4.3)

Moreover, u satisfies

∫_{∂Ω_{φ}} (∇u)^{∗}(x)^{2}dσ_{ϵ} ≤ C ( ∫_{N_{φ}}|f_{N}(x)|^{2} dσ_{ϵ}+ ∫_{D_{φ}} | \frac{df_{D}}{dσ} (x)|^{2} dσ_{ϵ})  (4.4)

Proof. We use a conformal map to reduce the weighted mixed problem (4.3) to the mixed problem with weight w = 1 on a convex sector, then we apply Theorem 4.2.

Let s = 1 − ϵ with ϵ as in the hypothesis, so that we have

0 < s < π/2φ < 1.  (4.5)

Define h_{s}(z) to be the conformal map

η = h_{s}(z) = i(−iz)^{s}.

Thus, h_{s} maps Ω_{φ} to Ω_{sφ}, D_{φ} to D_{sφ} and N_{φ} to N_{sφ}. Note that if we let ∂h_{s} denote the complex derivative of h_{s}, that is

∂h_{s} = \frac{1}{2} ( \frac{∂h_{s}}{∂x_{1}}+ \frac{1}{i} \frac{∂h_{s}}{∂x_{2}})  (4.6)

we have

\frac{1}{|∂h_{s}|}dσ = \frac{1}{s}dσ_{ϵ}.  (4.7)

On account of (4.5) it follows that Ω_{sφ} is a convex sector and thus Theorem 4.2 applies to Ω_{sφ}.

Given f_{D} and f_{N} as in (4.3), we define g_{D} and g_{N} on ∂Ω_{sφ} as follows:

g_{N}(η) = \frac{f_{N}}{|∂h_{s}|}∘ h_{s}^{−1} (η); g_{D}(η) = f_{D}∘ h_{s}^{−1} (η).

We verify that g_{N} and g_{D} satisfy the hypotheses of Theorem 4.2. Indeed, it is immediate to see that

∫_{N_{sφ}}|g_{N}|^{2}dσ = \frac{1}{s} ∫_{N_{φ}}|f_{N}|^{2}dσ_{ϵ}.  (4.8)

Similarly, we have

∫_{N_{sφ}} | \frac{dg_{D}}{dσ}|^{2}dσ = \frac{1}{s} ∫_{N_{φ}} | \frac{df_{D}}{dσ} |^{2}dσ_{ϵ}.  (4.9)

By Theorem 4.2 the mixed problem in L^{2}(dσ) on Ω_{sφ} has a unique solution v which satisfies (4.2). We now pull this solution back to Ω_{φ} by defining u = v ∘ h_{s}. Then, u is harmonic in Ω_{φ} and satisfies:

u = f_{D} onD_{φ}; \frac{∂u}{∂ν} = f_{N} onN_{φ} .

Moreover, on account of (4.2), (4.8) and (4.9) we have

∥(∇v)^{∗}∥_{L^{2}(∂Ω_{sφ} ,dσ)} ≤ C ∫_{N_{φ}}|f_{N}(x)|^{2} dσ_{ϵ}+ ∫_{D_{φ}} | \frac{df_{D}}{ds} (x)|^{2} dσ_{ϵ}.  (4.10)

Now we consider non-tangential maximal function estimates for ∇u. We apply the Cauchy integral formula to the complex derivative of u (which is analytic in Ω_{φ}) and obtain:

∂u(z) = \frac{1}{2πi} ∫_{∂Ω_{φ}} \frac{∂u(ζ)}{z − ζ}dζ = \frac{1}{2πi} ∫_{∂Ω_{φ}} \frac{∂h_{s}(ζ)∂v(h_{s}(ζ))}{z − ζ}dζ, z ∈ Ω_{φ} .

It follows

(∇u)^{∗}(x) = 2(∂u)^{∗}(x) ≤ ( K_{φ}(∂h_{s}· (∂v) ∘ h_{s}))^{∗}(x), a.e. x ∈ ∂Ω_{φ} ,

where K_{φ} denotes the Cauchy integral on Ω_{φ}. By the theorem of Coifman, McIntosh and Meyer [7] on the boundedness of the Cauchy integral we have

∥(K_{φ}ψ)^{∗}∥_{L^{2}(∂Ω_{φ},dσ_{ϵ})} ≤ C∥ψ∥_{L^{2}(∂Ω_{φ},dσ_{ϵ})} .

This uses that dσ_{ϵ} is in A_{2}(dσ). Combining these last two inequalities we obtain

∥(∇u)^{∗}∥_{L^{2}(∂Ω_{φ},dσ_{ϵ})} ≤ C∥∂h_{s}· (∂v) ∘ h_{s})∥_{L^{2}(∂Ω_{φ},dσ_{ϵ})} = C∥∂v∥_{L^{2}(∂Ω_{sφ},dσ)} ,  (4.11)

where the last equality was obtained by performing the change of variables h_{s}(ζ) = η, see also (4.7). This, together with (4.10) yields (4.4).

The argument is reversible, so we may conclude uniqueness in Ω_{φ} from uniqueness in Ω_{sφ}.

Remark. It is well-known that non-tangential maximal function estimates for harmonic functions behave nicely under conformal mapping, see Kenig [19] and Jerison and Kenig [17]. The previous Lemma, however, considers the non-tangential maximal function of the gradient. The estimates in this case appear to be more involved.

We now construct holomorphic vector fields which allow us to use the Rellich identity of Lemma 2.3 to obtain Rellich estimates for the weighted mixed problem.

Lemma 4.4 Suppose Ω = {x_{2} > ϕ(x_{1})}, N,D is a standard Lipschitz graph domain for the mixed problem, with Lipschitz constant M. Let β = arctanM > 0. Assume β < π/4.

Then, for 2β/(π − 2β) < ϵ < 1 there exist β_{0} = β_{0}(ϵ,M), β < β_{0} < (π − 2β)/2, and a complex number a = e^{iλ} such that the vector field α(z) = (Re(az^{ϵ}),Im(az^{ϵ})) satisfies

−|x|^{ϵ} ≤ α(x) · ν(x) < −|x|^{ϵ}sin(β_{0}− β),  x ∈ N;  (4.12)
|x|^{ϵ} ≥ α(x) · ν(x) > |x|^{ϵ}sin(β_{0}− β),  x ∈ D.  (4.13)

Proof. The outer unit normal ν lies in {(cosφ,sinφ) : −π/2 − β ≤ φ ≤ −π/2 + β}. On account of (1.2) and (1.3) we have that N is contained in the sector {x = re^{iθ} : −β < θ < β}, whereas D is contained in {x = re^{iθ} : π − β < θ < π + β}. Thus, in order to have α · ν < 0 on N we need α/|α| = (cosψ,sinψ) for some ψ ∈ (−2π + β,−π − β), whereas α · ν > 0 on D requires ψ ∈ (−π + β,−β). To obtain a strictly negative upper bound for α · ν on N and a strictly positive lower bound for α · ν on D, we pick β_{0} (to be selected later) so that β < β_{0} < π/2 and then, with the same notations as above, require that, for x = re^{iθ} ∈ N, ψ(θ) lie in [β_{0},π − β_{0}] whereas, for x ∈ D, we require that ψ(θ) lie in [π + β_{0},2π − β_{0}]. To this end, given ϵ as in the hypothesis, we let let ψ(θ) be a linear function with slope ϵ, and we choose β_{0} so that ψ(β) = π − β_{0} and ψ(π − β) = π + β_{0}. Writing ψ(θ) = ϵθ + λ, we define α(z) = e^{iλ}z^{ϵ}, that is

α(re^{iθ}) = r^{ϵ}e^{iψ(θ)}

where λ = π − πβ_{0}/(π − 2β) and ϵ = 2β_{0}/(π − 2β) (note that the latter defines β_{0}). This construction, however, grants that α · ν has the desired sign only near the endpoints θ = β (for N) and θ = π − β (for D). In order to make sure that α · ν keeps the desired sign all the way through the two other endpoints we need to restrict the range for the selection of β_{0} to:

β < β_{0} < \frac{π}{2} − β.

Then the angle between α(x) and ν(x) will lie in the intervals (−3π/2 + (β_{0}− β),−π/2 − (β_{0}−β)), for x ∈ N, and in (−π/2+(β_{0}−β),π/2−(β_{0}−β)), for x ∈ D, so that (4.12) and (4.13) hold.

Proposition 4.5 (Weighted Rellich estimates for the mixed problem) Let Ω, N, D be a standard Lipschitz graph domain for the mixed problem (1.1), with Lipschitz constant M < 1. Let β = arctanM. Then, for σ_{ϵ} as in (2.3), for ϵ in the range 2β/(π − 2β) < ϵ < 1 and for u harmonic with (∇u)^{∗} ∈ L^{2}(dσ_{ϵ}), we have

∫_{∂Ω} (∇u)^{∗}(x)^{2}dσ_{ϵ} ≤ C [∫_{N}( \frac{∂u}{∂ν}) ^{2} dσ_{ϵ}+ ∫_{D}( \frac{du}{dσ}) ^{2} dσ_{ϵ}] .

Proof. The identity of Lemma 2.3 together with the vector field constructed in Lemma 4.4 and standard manipulations involving the boundary terms as in Brown [4, Lemma 1.7] yield the estimate

∫_{∂Ω} |∇u|(x)^{2}dσ_{ϵ} ≤ C [∫_{N}( \frac{∂u}{∂ν}) ^{2} dσ_{ϵ}+ ∫_{D}( \frac{du}{dσ}) ^{2} dσ_{ϵ}] .

The key point is that since α · ν changes sign as we pass from N to D, we can estimate the full gradient of u on the boundary by the data for the mixed problem.

In order to obtain the estimate for the non-tangential maximal function of ∇u, we argue as in Proposition 4.3. We may represent the holomorphic function ∂u using the Cauchy kernel

∂u(x) = \frac{1}{2πi} ∫_{∂Ω} \frac{∂u(y)}{x − y}dy.  (4.14)

Since σ_{ϵ} is an A_{2} weight with respect to σ for −1 < ϵ < 1, the theorem of Coifman, McIntosh and Meyer [7] implies the non-tangential maximal estimate ∥(∂u)^{∗}∥_{L^{2}(σ_{ϵ})} ≤ C∥∂u∥_{L^{2}(σ_{ϵ})}. The estimate of the proposition follows from this and the corresponding estimate for ¯∂u.

Theorem 4.6 (Mixed problem in L^{2}(σ_{ϵ})) Suppose Ω, N and D is a standard Lipschitz graph domain for the mixed problem with Lipschitz constant M < 1. Let β = arctan(M). Given ϵ which satisfies

2β/(π − 2β) < ϵ < 1,

there is a unique solution u to the L^{2}(dσ_{ϵ}) mixed problem (1.1). Moreover, u satisfies

∫_{∂Ω} (∇u)^{∗}(x)^{2}dσ_{ϵ} ≤ ( ∫_{D} | \frac{df_{D}}{dσ} |^{2} dσ_{ϵ}+ ∫_{N}|f_{N}|^{2}dσ_{ϵ}) .

Proof. Using the estimate of Proposition 4.5, the existence result for sectors in Proposition 4.3, and the method of continuity (see Brown [4, Lemma 1.16] and Gilbarg and Trudinger [15, Theorem 5.2]), we obtain the existence of a solution to the mixed problem with data in L^{2}(dσ_{ϵ}).

Next, we consider uniqueness. If u is a solution of the mixed problem with zero data, then the Rellich estimates or Proposition 4.5 imply that u is a solution of the regularity and Neumann problems for L^{2}(dσ_{ϵ}) with zero data. By the results in Section 2 it follows that u is zero.

5 The regularity and Neumann problems in H^{1}(dσ_{ϵ}).

In this section we assert the existence of solutions for the Neumann problem when the data is in H^{1}(dσ_{ϵ}), and for the regularity problem when the data has one derivative in H^{1}(dσ_{ϵ}). The proof of these results follows the work of Dahlberg and Kenig [10]; thus, we shall be brief. We first recall the definition of the Hardy spaces H^{1}(dσ_{ϵ}) and H^{1,1}(dσ_{ϵ}).

Let ϵ > −1. We say that a is an H^{1}(dσ_{ϵ})-atom for ∂Ω if a is supported on a surface ball Δ_{s}(x), ∫ adσ = 0 and ∥a∥_{∞} ≤ [σ_{ϵ}(Δ_{s}(x))]^{−1} . We remark that for ϵ ≤ 0, L^{1}(dσ_{ϵ}) ⊂ L_{loc}^{1}(dσ). Thus we define the space H^{1}(dσ_{ϵ}), for ϵ ≤ 0, to be the collection of functions that are represented as

f(x) = ∑_{j=1}^{∞} λ_{j}a_{j}(x)  (5.1)

where {a_{j}}_{j∈N} is a sequence of H^{1}(dσ_{ϵ})-atoms for ∂Ω and the coefficients {λ_{j}} satisfy ∑ _{j=1}^{∞} |λ_{j}| < ∞. The sum for f in (5.1) converges in L^{1}(dσ_{ϵ}). the sum converges in L^{1}(dσ_{ϵ}). The H^{1}(dσ_{ϵ})-norm is defined by

∥f∥_{H^{1}(dσ_{ϵ})} = inf{∑ |λ_{j}|}  (5.2)

where the infimum is taken over all possible representations of f. Note that while H^{1}(dσ_{ϵ})- atoms are defined for ϵ > −1, we consider (and need for our application in Theorem 7.2) the space H^{1}(dσ_{ϵ}) only for ϵ ≤ 0. This allows us to avoid having to define spaces of distributions on Lipschitz graph domains. (See Coifman and Weiss [8] or Stromberg and Torchinsky [32] for a discussion of these spaces.)

Let ϵ ≤ 0. We say that A is an H^{1,1}(dσ_{ϵ})-atom for ∂Ω if for some x_{0} in ∂Ω

A(x) = ∫_{x_{0}}^{x}a(t)dσ(t), x ∈ ∂Ω

where a is an H^{1}(dσ_{ϵ})-atom, and ∫_{x_{0}}^{x} denotes integration along the portion of the boundary ∂Ω with endpoints x_{0} and x. Given an H^{1}(dσ_{ϵ})-atom a, the integral above defines A uniquely up to an additive constant. For ϵ ≤ 0, we define the space H^{1,1}(dσ_{ϵ}) to be sums of the form

F (x) = ∑_{j=1}^{∞} λ_{j}A_{j}(x)

where the coefficients satisfy ∑ _{j=1}^{∞} |λ_{j}| < ∞. The norm of F in H^{1,1}(σ_{ϵ′}), ∥F ∥_{H^{1,1}(dσ_{ϵ})}, is defined to be the infimum of ∑ _{j=1}^{∞} |λ_{j}| over all possible representations of F as sums of atoms. If we choose the base point x_{0} for each atom to be zero, then we have that the sum defining F converges in the norm given by

sup|x|^{−ϵ} |F (x)|.

Furthermore, it is easy to see that we have

∥F ∥_{H^{1,1}(dσ_{ϵ})} = ∥\frac{dF}{dσ}∥_{H^{1}(dσ_{ϵ})}.

By the Neumann problem for H^{1}(dσ_{ϵ}) we mean the Neumann problem for L^{1}(dσ_{ϵ}), see (2.4), where the data f_{N} is now taken from H^{1}(dσ_{ϵ}). By the regularity problem for H^{1,1}(dσ_{ϵ}) we mean the regularity problem for L^{1}(dσ_{ϵ}), see (2.5), where the data f_{D} is taken from H^{1,1}(dσ_{ϵ}).

For future reference, we also define spaces on subsets of the boundary. A function f is in H^{1}(N,dσ_{ϵ}) if and only if f is the restriction to N of a function in H^{1}(dσ_{ϵ}). Such functions can be written as sums of atoms that are restrictions to N of H^{1}(dσ_{ϵ})- atoms for ∂Ω. The space H^{1,1}(D,dσ_{ϵ}) is defined in a similar fashion. See Chang, Krantz and Stein [6], Sykes [33], and Sykes and Brown [34] for additional works that study Hardy spaces on domains.

The main result of this section is the following theorem. We omit the proof since it is quite similar to the argument for the mixed problem in the following section (Theorem 7.2).

Theorem 5.1 Let Ω be a standard Lipschitz graph domain with Lipschitz constant M and let ϵ_{0}(M) < ϵ′≤ 0 be small. Then the H^{1}(dσ_{ϵ′})-Neumann and the H^{1,1}(dσ_{ϵ′})-regularity problems are uniquely solvable. The solutions satisfy, respectively,

∥u∥_{H^{1,1}(dσ_{ϵ′})}+ ∫_{∂Ω} (∇u)^{∗}(x)dσ_{ϵ′} ≤ C ∥ \frac{∂u}{∂ν}∥_{H^{1}(dσ_{ϵ′})}
∥ \frac{∂u}{∂ν}∥_{H^{1}(dσ_{ϵ′})}+ ∫_{∂Ω} (∇u)^{∗}(x)dσ_{ϵ′} ≤ C∥u∥_{H^{1,1}(dσ_{ϵ′})}.

Remark. We do not assert uniqueness when ϵ′> 0. In the rest of this paper, we will only use the existence of a solution for the H^{1}(dσ_{ϵ′})-Neumann problem in the case when ϵ′≤ 0.

(In order to fully treat the case ϵ′> 0, one needs to give a different definition of the normal derivative at the boundary. For ϵ′> 0 the H^{1}(dσ_{ϵ′})-boundary data may not be in L_{loc}^{1}(dσ) and hence fail to be a function. See Brown [5] and Fabes, Mendez and Mitrea [13] for a treatment of the Neumann problem with data which is not locally integrable).

6 The mixed problem in H^{1} with power weights.

By the mixed problem for H^{1}(dσ_{ϵ′}) we mean the mixed problem for L^{1}(dσ_{ϵ′}), see (1.1), where the Dirichlet data f_{D} is taken from H^{1,1}(D,dσ_{ϵ′}) and the Neumann data f_{N} is in H^{1}(N,dσ_{ϵ′}). In this section, we consider the mixed problem where the Neumann data is an H^{1}(N,dσ_{ϵ′}) atom and the Dirichlet data is zero. Since atoms lie in L^{2}(dσ_{ϵ}) for ϵ > −1, Theorem 4.6 yields existence and uniqueness of the solution to the mixed problem for L^{2}(dσ_{ϵ}) with these data. Our first goal is to show that the gradient of the solution also has non-tangential maximal function in L^{1}(dσ_{ϵ′}), for ϵ′near zero.

Theorem 6.1 Suppose Ω, N, D is a standard Lipschitz graph domain for the mixed problem with Lipschitz constant M < 1 and set β = tan^{−1} M. Then, there is δ = δ(M) with 1 > δ > 0 so that, for ϵ′satisfying \frac{4β−π}{2(π−2β)} < ϵ′< δ we may solve the mixed problem (1.1) for H^{1}(dσ_{ϵ′}) with zero Dirichlet data and with Neumann data an H^{1}(N,dσ_{ϵ′})-atom, a. The solution u satisfies the estimate

∫_{∂Ω} (∇u)^{∗}(x)dσ_{ϵ′}≤ C(M,ϵ′).  (6.1)

In addition, we have

∥u∥_{H^{1,1}(dσ_{ϵ′})}+ ∥ \frac{∂u}{∂ν}∥_{H^{1}(dσ_{ϵ′})} ≤ C(M,ϵ′).  (6.2)

As a step towards the proof of Theorem 6.1, we construct a Green’s function for the mixed problem using the method of reflections–an old idea that was used by Dahlberg and Kenig [10] to obtain a similar result for the Neumann and regularity problems. The estimates for the Green’s function are a consequence of the Hölder regularity of weak solutions of divergence-form equations with bounded measurable coefficients.

We begin by constructing a bi-Lipschitz map, Φ : R^{2} → R^{2} with Φ(Ω) = Q where Q is the first quadrant, Q = {(x_{1},x_{2}) : x_{1} > 0,x_{2} > 0}. We also require that Φ(N) = {(x_{1},x_{2}) : x_{1} ≥ 0,x_{2} = 0} and Φ(D) = {(x_{1},x_{2}) : x_{1} = 0,x_{2} > 0}. On Q we now define the operator L = divA∇ where the coefficient matrix A has bounded entries (these are first-order derivatives of Φ), so that u satisfies

Δu = 0,  in Ω
\frac{∂u}{∂ν} = f_{N},  on N
u = f_{D},  on D

if and only if the function v defined by v = u ∘ Φ^{−1} satisfies

Lv = 0,  in Q
A∇v · ν = f_{N}∘ Φ^{−1} on Φ(N)
v = f_{D}∘ Φ^{−1}  on Φ(D)

Now, we extend the coefficients of L by reflection, so that Lv = 0 if and only if L(v∘R_{j}) = 0 where R_{1} and R_{2} are the reflections R_{1}(x_{1},x_{2}) = (−x_{1},x_{2}) and R_{2}(x_{1},x_{2}) = (x_{1},−x_{2}). Next, letting G denote the Green’s function for L in Q, we set

N (z,w) = G(z,w) − G(z,R_{1}w) + G(z,R_{2}w) − G(z,R_{1}R_{2}w), z,w ∈ Q

and observe that N is a Green’s function for the mixed problem for L in Q. Recall that we may find a Green’s function for L in Q which satisfies

|G(z,w)| ≤ C(1 + |log|z − w||), z,w ∈ Q,

see Kenig and Ni [21]. We then observe that if |z−ζ| = 1 and |ζ−w| < 1/2, then |G(z,ζ)− G(z,w)| ≤ C|ζ − w|^{δ}, where 0 < δ < 1, δ = δ(M). This is a standard estimate of Hölder continuity for solutions of divergence-form elliptic operators. Finally, by rescaling, we obtain

|G(z,ζ) − G(z,w)| ≤ C ( \frac{|ζ − w|}{|z − ζ|} ) ^{δ},  if |ζ − w| < \frac{1}{2}|z − ζ|.  (6.3)

The latter immediately implies the same estimate for N in Q, namely

|N (z,ζ) − N (z,w)| ≤ C ( \frac{|ζ − w|}{|z − ζ|} ) ^{δ} if z,ζ,w ∈ \overline{Q} and |ζ − w| < \frac{1}{2}|z − ζ|  (6.4)

We will need an additional estimate for N (z,ζ) when ζ ∈ Q is near Φ(D). Since we have N (z,x) = 0 if z ∈ Q and x ∈ Φ(D), it follows by continuity that N is small near Φ(D). More precisely, let ζ ∈ Q and suppose that x̂is a point on Φ(D) for which |x̂− ζ| = dist(ζ,Φ(D)). For z ∈ Q, we have N (z,x̂) = 0 and (6.4) implies

|N (z,ζ)| ≤ C \frac{|ζ|^{δ}}{|z − ζ|^{δ}} , if z,ζ ∈ Q anddist(ζ,D) < \frac{1}{2}|z − ζ|,  (6.5)

(here we have used that 0 ∈ ∂Q). Finally, we define the Green’s function for the mixed problem by

M (x,y) = N (Φ(x),Φ(y)).

Since Φ is bi-Lipschitz the estimates (6.4) and (6.5) imply similar results for M in Ω:

|M (x,ζ) − M (z,w)| ≤ C ( \frac{|ζ − w|}{|z − ζ|} ) ^{δ},
if z,ζ,w ∈ Ω̄ with |ζ − w| < \frac{1}{2}|z − ζ|  (6.6)

|M (z,ζ)| ≤ C \frac{|ζ|^{δ}}{|z − ζ|^{δ}}

ifz,ζ ∈ Ω anddist(ζ,D) < \frac{1}{2}|z − ζ|.  (6.7)

These estimates are a key ingredient in the study of the behavior of the solution of the mixed problem with atomic Neumann data.

Lemma 6.2 Assume ϵ′> −1. Let u be a solution of the mixed problem (1.1) for L^{1}(dσ_{ϵ′}), where the Neumann data is an H^{1}(N,dσ_{ϵ′})-atom, a, and the Dirichlet data is zero. Then, for any integer k ≥ 1, u satisfies

|u(z)| ≤ C \frac{ρ^{δ}}{|z − x_{a}|^{δ}} \frac{σ(Δ_{ρ}(x_{a}))}{σ_{ϵ′}(Δ_{ρ}(x_{a}))} , z ∈ Ω, |z − x_{a}| ≥ 2^{k}ρ.

Here, 0 < δ < 1 is as in the estimate for the Green’s function for the mixed problem (6.3) and Δ_{ρ}(x_{a}) is the surface ball where a is supported.

Proof. We consider two cases: 1) Δ_{ρ}(x_{a}) ⊂ N and 2) Δ_{ρ}(x_{a}) ∩ D ≠ ∅.

In case 1), we have

u(z) = ∫_{N}M (z,x)a(x)dσ(x) = ∫_{N}(M (z,x) − M (z,x_{a}))a(x)dσ(x), z ∈ Ω

where the second identity uses the fact that the atom a has mean value zero. Next, we use the continuity of M from (6.6) to obtain

|u(z)| ≤ C \frac{ρ^{δ}}{|z − x_{a}|^{δ}} ∫_{N}|a|dσ, for z ∈ Ω, |z − x_{a}| > 2^{k}ρ.  (6.8)

Finally, the normalization of a in the definition of an atom implies that

∫_{N}|a|dσ ≤ \frac{σ(Δ_{ρ}(x_{a}))}{σ_{ϵ′}(Δ_{ρ}(x_{a}))} .

This completes the proof in case 1.

In case 2), we do not have ∫_{N}adσ = 0. However, estimate (6.7) yields

|u(z)| ≤ C \frac{ρ^{δ}}{|z − x_{a}|^{δ}} ∫_{N}|a|dσ,

and then we use the normalization of a to conclude the proof.

Before proceeding, we need a few technical results. In this Lemma and below, given a point x_{a} ∈ ∂Ω, we consider a ball B_{ρ}(x_{a}) and a boundary ball Δ_{ρ}(x_{a}), and set: R_{k} = B_{2^{k+1} ρ}(x_{a})\B_{2^{k}ρ}(x_{a}), R̃_{k}= B_{2^{k+2} ρ}(x_{a})\B_{2^{k−1} ρ}(x_{a}), Δ_{k}= Δ_{2^{k}ρ}(x_{a}) and Σ_{k}= Δ_{k+1} \Δ_{k}. With these definitions, we can now state

Lemma 6.3 Let λ ∈ R and set α(z) = (Re(e^{iλ}(z)^{ϵ}),Im(e^{iλ}(z)^{ϵ})). Then, we have

∫_{R_{k}}|α|^{p}dy ≤ C2^{k}ρ∫_{Σ_{k}} |α|^{p}dσ ≤ C2^{k}ρσ_{ϵp}(Σ_{k})

provided ϵp > −1.

Proof. The proof is a computation and we omit the details.

We now take a brief detour to discuss solutions of the mixed problem in the energy sense. Let B be a ball with center in Ω. We say that u is an energy solution of the mixed problem in B,

Δu = 0  in B ∩ Ω
u = 0  on D ∩ B
\frac{∂u}{∂ν} = 0  on N ∩ B

if u lies in the Sobolev space W^{1,2}(B ∩ Ω), u vanishes on D ∩ B and for every v that lies in W^{1,2}(B ∩ Ω) and vanishes on ∂(B ∩ Ω) \ N we have

∫_{B∩Ω} ∇u · ∇vdy = 0.

Using a Carleson measure argument, see Proposition 2.2, it is not difficult to see that if u is a solution of the mixed problem in L^{2}(dσ_{ϵ}) then |∇u|^{2} is integrable on bounded subsets of Ω, provided ϵ < 1. This shows that a solution for the mixed problem in L^{2}(dσ_{ϵ}) is, in particular, an energy solution.

In the next lemma, we use −∫_{E}f dx to denote the average,

−∫_{E}f(x)dx:= |E|^{−1} ∫_{E}f(x)dx.

Lemma 6.4 Let Ω, D, N be a standard Lipschitz graph domain for the mixed problem with Lipschitz constant M. There is an exponent q_{0} = q_{0}(M) > 2 so that on any ball B with center in Ω, if u is an energy solution of

Δu = 0,  in 2B ∩ Ω
u = 0,  in D ∩ 2B
\frac{∂u}{∂ν} = 0,  on N ∩ 2B

then, for all 1 ≤ q ≤ q_{0}, we have

( − ∫_{B∩Ω} |∇u|^{q}dx) ^{1/q} ≤ \frac{C}{r}( − ∫_{2B∩Ω} |u|^{2}dx) ^{1/2},

where r is the radius of B.

Proof. This follows from the Caccioppoli inequality and a reverse Hölder inequality as in Giaquinta [14].

The next estimate gives the decay at infinity of a solution to the mixed problem with atomic data.

Lemma 6.5 Let Ω, D, N be a standard Lipschitz graph domain for the mixed problem (1.1). Let ϵ′> −1 and −1+2/q_{0} < ϵ < 1, where q_{0} is as in Lemma 6.4. If u is a solution of the mixed problem for L^{2}(dσ_{ϵ}) with zero Dirichlet data and with Neumann data a σ_{ϵ′}-atom, a, which is supported in Δ_{ρ}(x_{a}), then we have

∫_{Σ_{k}} |∇u|^{2}dσ_{ϵ} ≤ C \frac{σ_{ϵ}(Σ_{k})}{(2^{k}ρ)^{2}} [ sup_{∪_{|j|≤1} R̃_{k+j}} |u|]^{2} , for allk ≥ 2.

Proof. Let η_{k} ≥ 0 be a smooth cutoff function which equals one on R_{k}, is zero outside R̃_{k}, and satisfies |∇η| ≤ C/(2^{k}ρ). We let α denote a vector field of the form α(z) = (Re(e^{iλ}z^{ϵ}),Im(e^{iλ}z^{ϵ})), for some λ ∈ R and for ϵ as in the hypothesis.

We apply the Rellich identity with vector field αη_{k} where α is as in Lemma 4.4 to conclude that for k ≥ 2, we have

∫_{Σ_{k}} |∇u|^{2}dσ_{ϵ} ≤ \frac{C}{2^{k}ρ} ∫R̃_{k}|∇u|^{2}|α|dy.  (6.9)

(this uses that the mixed data of u is zero on the support of η_{k} when k ≥ 2).

Applying Hölder’s inequality we obtain

∫R̃_{k}|∇u|^{2}|α|dy ≤ ( ∫R̃_{k}|∇u|^{2p}dx) ^{1/p}( ∫R̃_{k}|α|^{p′}dy) ^{1/p′},  (6.10)

where 1/p+1/p′= 1 and p lies in the interval 1/(1−|ϵ|) < p ≤ q_{0}/2 (if −1+2/q_{0} < ϵ <

0) or, if ϵ > 0, 1 ≤ p ≤ q_{0}/2. These conditions grant that Lemmata 6.4, 6.3 and 3.1 apply in what follows. We now cover R̃_{k}with a (fixed) number of balls B_{k,n} (each centered at a point in Ω), n = 1,...,m, such that diamB_{k,n}≈ 2^{k}ρ and

R̃_{k}⊆ ∪_{n=1}^{m} B_{k,n}⊆ ∪_{|j|≤2} R_{k+j} = ∪_{|j|≤1} R̃_{k+j} .

By Lemma 6.4, for p as above, we have

( ∫R̃_{k}|∇u|^{2p}dy) ^{1/p} ≤ C(2^{k}ρ)^{\frac{2}{p}−4} ∑_{|j|≤1} ∫R̃_{k+j} |u|^{2}dy.

Moreover, Lemma 6.3 and Lemma 3.1, also for p as above, imply

( ∫R̃_{k}|α|^{p′}dy) ^{1/p′}≤ C(2^{k}ρ)^{1− \frac{2}{p}} σ_{ϵ}(Σ_{k}).  (6.11)

Combining (6.9) to (6.11), we obtain

∫_{Σ_{k}} |∇u|^{2}dσ_{ϵ} ≤ \frac{C}{2^{k}ρ}(2^{k}ρ)^{\frac{2}{p}−4} (2^{k}ρ)^{1− \frac{2}{p}} σ_{ϵ}(Σ_{k}) ∑_{|j|≤1} ∫R̃_{k+j} |u|^{2}dx

≤ Cσ_{ϵ}(Σ_{k})(2^{k}ρ)^{−2} [ sup_{x∈∪_{|j|≤1} R̃_{k+j}} |u|]^{2} .

Lemma 6.6 Let Ω,N,D be a standard Lipschitz graph domain for the mixed problem. Suppose u is the solution of the mixed problem for L^{2}(dσ_{ϵ}), see (1.1), with zero Dirichlet data and with Neumann data an H^{1}(N,dσ_{ϵ′})-atom, a, supported in a surface ball Δ_{ρ}(x_{a}). Let ϵ′> −1 and −1+2/q_{0} < ϵ < 1 (with q_{0} as in Lemma 6.4). Then, for all integers k ≥ 5 we have

∫_{Σ_{k}}(∇u)^{∗}(x)^{2}dσ_{ϵ} ≤ \frac{Cσ_{ϵ}(Σ_{k})}{2^{2k(1+δ)}(σ_{ϵ′}(Δ_{ρ}(x_{a})))^{2}} ,

where δ is as in (6.3).

Proof. By Lemmata 6.2 and 6.5, we have

∫_{Σ_{k}} |∇u|^{2}dσ_{ϵ} ≤ \frac{C}{(2^{k}ρ)^{2}}σ_{ϵ}(Σ_{k})( \frac{ρ^{δ}}{(2^{k}ρ)^{δ}} \frac{σ(Δ_{ρ}(x_{a}))}{σ_{ϵ′}(Δ_{ρ}(x_{a}))} ) ^{2}

= \frac{C}{(2^{k}ρ)^{2}}σ_{ϵ}(Σ_{k})2^{−2kδ} ( \frac{ρ}{σ_{ϵ′}(Δ_{ρ}(x_{a}))} ) ^{2}.  (6.12)

In order to pass from the estimate above for ∇u to an estimate for (∇u)^{∗} , we use the Cauchy kernel to represent ∂u, the complex derivative of u, see (4.6). To carry out this argument, we consider a cutoff function η_{k} that is supported in B_{2^{k+4} ρ}(x_{a})\B_{2^{k−3} ρ}(x_{a}), equals one on B_{2^{k+3} ρ}(x_{a}) \ B_{2^{k−2} ρ}(x_{a}) and, furthermore, satisfies: |∂η_{k}| ≤ C/2^{k}ρ. On account of the analyticity of ∂u, the Cauchy-Pompeiu formula, see e.g. Bell [2], yields

η_{k}(z)∂u(z) = K_{∂Ω} (η_{k}∂u)(z) + I(z), z ∈ Ω,  (6.13)

where we have set

K_{∂Ω} (η_{k}∂u)(z) = \frac{1}{2πi} ∫_{∂Ω} \frac{1}{z − ζ} η_{k}(ζ)∂u(ζ)dζ, z ∈ Ω,  (6.14)

(here, dζ denotes complex line integration) and

I(z) = \frac{1}{π} ∫_{R_{k−2} ∪R_{k+3}} \frac{1}{z − y} ¯∂η_{k}(y)∂u(y)dy,  z ∈ Ω  (6.15)

(here, dy denotes area measure in the plane).

Next, for x ∈ Σ_{k} we decompose the sector Γ(x) = Γ^{n}(x) ∪ Γ^{f} (x) where Γ^{n}(x) = Γ(x) ∩ B_{κ2^{k}ρ}(x_{a}), Γ^{f} (x) = Γ(x)\B_{κ2^{k}ρ}(x_{a}), and the constant κ is chosen so that Γ^{n}(x) ⊂ ∪_{|j|≤1} R_{k+j} for x ∈ Σ_{k}. If we let v_{n}^{∗}(x) denote the supremum of |v| on Γ_{n}(x) and similarly for v_{f}^{∗}, using (6.13) and the theorem of Coifman, McIntosh and Meyer [7] we obtain

∫_{Σ_{k}}(∇u)_{n}^{∗}(x)^{2}dσ_{ϵ}(x) = ∫_{Σ_{k}}(∂u)_{n}^{∗}(x)^{2}dσ_{ϵ}(x)  (6.16)
≤ ∑_{|j|≤3} ∫_{Σ_{k+j}} |∂u(x)|^{2}dσ_{ϵ}(x)

+σ_{ϵ}(Σ_{k})[ sup_{x∈Σ_{k}}( sup_{z∈∪_{|j|≤1} R_{k+j}} |I(z)|) ^{2}] ,

where we have used that suppη_{k} ⊂ ∪_{|j|≤3} R_{k+j} .

We now estimate the term I(z): by the Cauchy-Schwartz inequality we have

|I(z)| ≤ \frac{C}{ρ2^{k−1}} ( ∫_{R_{k−3} ∪R_{k+3}} |∇u|^{2}dy) ^{\frac{1}{2}} , z ∈ ∪_{|j|≤1} R_{k+j} .

On account of the vanishing boundary conditions on u and ∂u/∂ν we may now apply Caccioppoli inequality and conclude

I(z) ≤ \frac{1}{2^{k}ρ} sup_{y∈∪_{|j|≤4} R_{k+j}} |u(y)| for allz ∈ ∪_{|j|≤1} R_{k+j} andx ∈ Σ_{k}.

Using interior estimates, we also obtain

(∇u)_{f}^{∗}(x) ≤ \frac{1}{2^{k}ρ} sup_{z∈Ω\B_{2^{k−1} ρ}} |u(z)|, x ∈ Σ_{k}.

The conclusion now follows from (6.12), (6.16) and Lemma 6.2.

We are ready to prove Theorem 6.1.

Proof of Theorem 6.1. We fix ϵ > 0 such that 2β/(π − 2β) < ϵ < 1 so that, for ϵ′ as in the hypotheses we have: (ϵ − 1)/2 < ϵ′. The proof is based on the following elementary observations: given any ϵ′> −1 and ϵ > −1, if a is a σ_{ϵ′}-atom then a lies in L^{2}(N,dσ_{ϵ}); by Theorem 4.6 we may solve the mixed problem (1.1) for L^{2}(dσ_{ϵ}) with Neumann data a and zero Dirichlet data (provided β and ϵ are as in the hypothesis). We let u denote such a solution and show that u satisfies (6.1) and (6.2). We begin by studying ∇u near the support of a. By the Cauchy-Schwarz inequality, together with the normalization of a and the estimate for the L^{2}(dσ_{ϵ})-mixed problem (see Theorem 4.6), we have

∫_{Δ_{2^{10}ρ}(x_{a})}(∇u)^{∗}(x)dσ_{ϵ′} ≤ ( ∫_{Δ_{2^{10}ρ}(x_{a})}(∇u)^{∗}(x)^{2}dσ_{ϵ}) ^{1/2}σ_{2ϵ′−ϵ}(Δ_{2^{10}ρ}(x_{a}))^{1/2}
≤ C∥a∥_{L^{2}(σ_{ϵ})}σ_{2ϵ′−ϵ}(Δ_{2^{10}ρ}(x_{a}))^{1/2}
≤ C \frac{σ_{ϵ}(Δ_{ρ}(x_{a}))^{1/2}σ_{2ϵ′−ϵ}(Δ_{2^{10}ρ}(x_{a}))^{1/2}}{σ_{ϵ′}(Δ_{ρ}(x_{a}))} ,

provided 2ϵ′− ϵ > −1, that is ϵ′> (ϵ − 1)/2 (note that the latter is bounded below by (4β − π)/2(π − 2β) > −1). Lemma 3.1 now grants

\frac{σ_{ϵ}(Δ_{ρ}(x_{a}))^{1/2}σ_{2ϵ′−ϵ}(Δ_{2^{10}ρ}(x_{a}))^{1/2}}{σ_{ϵ′}(Δ_{ρ}(x_{a}))} ≤ C.

Next, we consider ∇u away from the support of a: we will show that there is η > 0 so that

∫_{Σ_{k}}(∇u)^{∗}(x)dσ_{ϵ′}≤ C2^{−ηk} .  (6.17)

Summing over k will give the estimate for (∇u)^{∗}.

We begin with the Cauchy-Schwarz inequality and then use the estimate of Lemma 6.6 (note that we have ϵ > 2β/(π − 2β) > 0 > −1 + 2/q_{0}) to obtain

∫_{Σ_{k}}(∇u)^{∗}(x)dσ_{ϵ′}(x) ≤ ( ∫_{Σ_{k}}(∇u)^{∗}(x)^{2}dσ_{ϵ}(x)) ^{1/2}σ_{2ϵ′−ϵ}(Σ_{k})^{1/2}
≤ C \frac{σ_{2ϵ′−ϵ}(Σ_{k})^{1/2}σ_{ϵ}(Σ_{k})^{1/2}}{σ_{ϵ′}(Δ_{ρ}(x_{a}))2^{k(1+δ)}}

= C2^{−δk} \frac{max(|x_{a}|,2^{k}ρ)^{ϵ′}}{max(|x_{a}|,ρ)^{ϵ′}}

where the last inequality follows from Lemma 3.1. We have

\frac{max(|x_{a}|,2^{k}ρ)}{max(|x_{a}|,ρ)} = \{ \begin{matrix}1, & \frac{|x_{a}|}{ρ} < 1 \\ |x_{a}|/ρ, & 1 < \frac{|x_{a}|}{ρ} < 2^{k} \\ 2^{k}, & \frac{|x_{a}|}{ρ} > 2^{k}. \end{matrix}  (6.18)

In particular, we have

1 ≤ \frac{max(|x_{a}|,2^{k}ρ)}{max(|x_{a}|,ρ)} ≤ 2^{k}.

Thus, letting η:= δ − max{ϵ′,0} > 0, we obtain (6.17). Summing over k yields estimate (6.1).

Now we indicate why ∂u/∂ν lies in H^{1}(∂Ω,dσ_{ϵ′}). A similar, but simpler, argument shows that u lies in H^{1,1}(∂Ω,dσ_{ϵ′}). We first prove the vanishing moment condition:

∫_{∂Ω} \frac{∂u}{∂ν}dσ = 0.  (6.19)

To this end, we observe that, since ∫_{∂Ω} adσ = 0, we may proceed as in the proof of Lemma 6.5 and inequality (6.17) to obtain

∫_{∂Ω} (∇u)^{∗}dσ < +∞.  (6.20)

On account of (6.20) we may now apply the divergence theorem to obtain (6.19). To continue, we follow the arguments of Coifman and Weiss [8]. We begin by writing

\frac{∂u}{∂ν} = ∑_{k=0}^{∞} b_{k}

where b_{0} := χ_{Δ_{0}}(\frac{∂u}{∂ν}−∫_{Δ_{ρ}} − \frac{∂u}{∂ν}dσ)and, for k ≥ 1, b_{k}:= χ_{Σ_{k}} \frac{∂u}{∂ν}+χ_{Δ_{k−1}} ∫_{Δ_{k−1}} − \frac{∂u}{∂ν}dσ−χ_{Δ_{k}} ∫_{Δ_{k}} − \frac{∂u}{∂ν}dσ. The estimate of Lemma 6.6 and arguments used above imply that

∫_{∂Ω} |b_{k}|dσ_{ϵ′}≤ ( ∫_{Δ_{k}} |b_{k}|^{2}dσ_{ϵ}) ^{1/2}σ_{2ϵ′−ϵ}(Δ_{k})^{1/2} ≤ 2^{−ηk} .

where again η:= δ − max(ϵ′,0) > 0. The one tricky point in this argument is that we must use (6.19) to obtain that ∫_{Δ_{k}} \frac{∂u}{∂ν}dσ = −∫_{∂Ω\Δ_{k}} \frac{∂u}{∂ν}dσ. Thus we have that 2^{ηk}b_{k} is normalized in L^{1}(dσ_{ϵ′}). In order to prove that b_{k} is in H^{1}(dσ_{ϵ′}), one now proceeds as in Strömberg and Torchinsky [32, Theorem 1, p. 111] to show that the grand maximal function is in L^{1}(dσ_{ϵ′}) and then find an atomic decomposition. This argument gives estimate (6.2) for ∂u/∂ν in H^{1}(dσ_{ϵ′}); the corresponding estimate for u is obtained in a similar fashion.

7 Conclusion

In this section, we give the final arguments to prove existence and uniqueness for the mixed problem in L^{p}(dσ). This is the result stated in Theorem 1.1.

In section 4, we obtained existence and uniqueness for the solution of the L^{2}(dσ_{ϵ})- mixed problem for ϵ in an interval which includes positive values of ϵ and does not include 0. In this section, we will study solutions which have atomic data and show that the nontangential maximal function for such solutions will lie in L^{1}(dσ_{ϵ′}) for ϵ′small.

Our results are restricted to domains with Lipschitz constant M that is less than 1. This restriction is inherited from the previous sections (Lemma 4.4). We do not make an effort to find the largest value of p_{0} (nor do we expect that the restriction M < 1 is essential). However, as the example discussed in the introduction indicates, we cannot expect to always have p_{0} ≥ 2, even in the case of smooth domains. Furthermore, in Brown [4] the mixed problem is solved in L^{2}(dσ) for certain Lipschitz domains with arbitrarily large Lipschitz constant; thus, it is more than the Lipschitz constant that governs the solvability of this problem.

We begin with our uniqueness result.

Lemma 7.1 Let δ > 0 be as in Lemma 6.2. Suppose that ϵ′and p ≥ 1 satisfy: −δ < ϵ′≤ 0, 1/p′− ϵ′/p < δ. Under these hypotheses, the solution of the mixed problem (1.1) for L^{p}(dσ_{ϵ′}) is unique.

Proof. Suppose that u solves the mixed problem for L^{p}(dσ_{ϵ′}) with zero data, that is

\{ \begin{matrix}Δu = 0 & in Ω \\ u = 0 & on D \\ \frac{∂u}{∂ν} = 0 & on N \\ (∇u)^{∗} ∈ L^{p}(dσ_{ϵ′}) \end{matrix}  (7.1)

We will show that u solves the regularity problem (2.5) with zero data and then use Lemma 2.6 to conclude that u = 0. To see that u vanishes on N ⊂ ∂Ω, we will show that there is η > 0 such that for any H^{1}(dσ_{η})-atom, a, we have

∫_{N}uadσ = 0.  (7.2)

This means that u is constant almost everywhere. In order to prove (7.2), we fix η > 0 so that

0 < \frac{1}{p′}− \frac{ϵ′}{p}< η < δ

(where 1/p + 1/p′= 1) and we let a be a σ_{η}-atom supported in a boundary ball Δ_{r}(x_{a}). According to (the proof of) Theorem 6.1, the mixed problem in H^{1}(dσ_{η}) with Neumann data a and with zero Dirichlet data has a solution v that satisfies

|v(z)| ≤ C|z − x_{a}|^{−δ} , if |z − x_{a}| ≥ 2ρ ;  (7.3)

moreover, for ϵ as in Theorem 4.6, we have

(∇v)^{∗} ∈ L^{2}(dσ_{ϵ}) ∩ L^{1}(dσ_{η}).  (7.4)

We will need a pointwise estimate for u; to this end, given z in Ω, we consider the path in Ω from 0 to z given by: γ_{z}(t) = (tz_{1},ϕ(tz_{1}) + t(z_{2} − ϕ(z_{1}))). Recall that ϕ is the function whose graph gives ∂Ω. We define the following Carleson measure with respect to dσ:

dμ_{z}(y) := χ_{B_{2|z|}}(y)dH_{1}|_{γ_{z}}, y ∈ Ω,

where H_{1} denotes 1−dimensional Hausdorff measure. By the Fundamental Theorem of Calculus, properties of Carleson measures (Proposition 2.2) and Hölder’s inequality, we have

|u(z)| ≤ ∫_{γ_{z}}|∇u||dγ_{z}|
≤ C ∫_{Δ_{2|z|}(0)}(∇u)^{∗}dσ
≤ |z|^{\frac{1}{p′}− \frac{ϵ′}{p}} ∥(∇u)^{∗}∥_{L^{p}(dσ_{ϵ′})}.  (7.5)

Note that by a similar argument we may show that u is locally bounded on ∂Ω.

With these estimates collected, we now proceed to the main part of the argument. Let R be large and let ψ_{R}be a cutoff function which is equal to 1 on B_{R}(0), zero outside B_{2R}(0) and such that |∇ψ_{R}| ≤ C/R. We apply Green’s second identity to the pointwise products vψ_{R}and uψ_{R}and obtain

∫_{∂Ω} ψ_{R}^{2}( v\frac{∂u}{∂ν}− u\frac{∂v}{∂ν}) dσ = 2∫_{Ω} ψ_{R}(v∇u · ∇ψ_{R}− u∇v · ∇ψ_{R})dy.  (7.6)

Concerning the left-hand side of (7.6), on account of the boundary conditions satisfied by u and v, we have

∫_{∂Ω} ψ_{R}^{2}( v\frac{∂u}{∂ν}− u\frac{∂v}{∂ν}) dσ = −∫_{N}ψ_{R}^{2}uadσ.

Since a is compactly supported and u is locally bounded (see (7.5) and comment thereafter) we may apply the Lebesgue dominated convergence theorem and conclude

lim_{R→∞} ∫_{∂Ω} ψ_{R}^{2}( v\frac{∂u}{∂ν}− u\frac{∂v}{∂ν}) dσ = −∫_{N}uadσ.  (7.7)

Concerning the right-hand side of (7.6), we will show that

lim_{R→∞} ∫_{Ω} ψ_{R}v∇u · ∇ψ_{R}dy = lim_{R→∞} ∫_{Ω} ψ_{R}u∇v · ∇ψ_{R}dy = 0.  (7.8)

Indeed, if R is large, then from Lemma 6.2, for z in the support of ∇ψ_{R}, we have

|v(z)| ≤ CR^{−δ}  (7.9)

and it follows that

∫_{Ω} ψ_{R}v∇u · ∇ψ_{R}dy ≤ CR^{−δ} ∫_{Ω} |∇ψ_{R}||∇u|dy.

By Hölder inequality we have

∫_{Ω} |∇ψ_{R}||∇u|dy ≤ R^{− \frac{ϵ′}{p}}∫_{B_{R}(0)}|∇ψ_{R}||∇u||y|^{\frac{ϵ′}{p}}dy

≤ R^{\frac{1}{p′}− \frac{ϵ′}{p}} ( ∫_{Ω} |∇u|^{p}χ_{B_{R}(0)} \frac{|y|^{ϵ′}}{R}dy) ^{\frac{1}{p}} .

By Carleson Theorem, see (2.3) and Proposition 2.2, the last term in the inequalities above is bounded by

CR^{\frac{1}{p′}− \frac{ϵ′}{p}}∥(∇u)^{∗}∥_{L^{p}(dσ_{ϵ′})}.

We conclude

|∫_{Ω} ψ_{R}(v∇u · ∇ψ_{R})dy| ≤ CR^{\frac{1}{p′}− \frac{ϵ′}{p}−δ} ∥(∇u)^{∗}∥_{L^{p}(dσ_{ϵ′})};  (7.10)

our hypotheses on η, ϵ′and p imply that the exponent of R is negative, so the first integral in (7.8) vanishes as R → ∞. To handle the second integral in (7.8) we apply (7.5); using the fact the ∇ψ_{R}is supported in the annulus R < |z| < 2R we obtain

|∫_{Ω} ψ_{R}u∇ψ_{R}· ∇vdy| ≤ CR^{\frac{1}{p′}− \frac{ϵ′}{p}} ∥(∇u)^{∗}∥_{L^{p}(∂Ω.dσ_{ϵ′})} R^{−η} ∫_{R<|y|<2R} |∇v|\frac{|y|^{η}}{R}dy.

By applying Proposition 2.2 on Carleson measures one more time, we see that the latter is bounded by

CR^{\frac{1}{p′}− \frac{ϵ′}{p}−η} ∥(∇u)^{∗}∥_{L^{p}(dσ_{ϵ′})}∫_{∂Ω} (∇v)^{∗}dσ_{η}.

It follows that the second integral in (7.8) also vanishes as R tends to infinity. This completes the proof of (7.8) and of this Lemma.

Theorem 7.2 Suppose Ω, N, D is a standard Lipschitz graph domain for the mixed problem, with Lipschitz constant M < 1 and set β = arctan(M). There is δ = δ(M) satisfying 0 < δ < 1 so that, for max\{−δ, \frac{4β−π}{2(π−2β)} \} < ϵ′≤ 0 the mixed problem (1.1) for L^{1}(dσ_{ϵ′}) with Dirichlet data in H^{1,1}(D,dσ_{ϵ′}) and with Neumann data in H^{1}(N,dσ_{ϵ′}) is uniquely solvable. The solution u satisfies the following estimates

∫_{∂Ω} (∇u)^{∗}(x)dσ_{ϵ′} ≤ C(M,ϵ′) ( ∥f_{N}∥_{H^{1}(N,dσ_{ϵ′})}+ ∥f_{D}∥_{H^{1,1}(D,dσ_{ϵ′})}) ; (7.11)
∥u∥_{H^{1,1}(dσ_{ϵ′})}+ ∥ \frac{∂u}{∂ν}∥_{H^{1}(dσ_{ϵ′})}≤ C(M,ϵ′)( ∥f_{N}∥_{H^{1}(N,dσ_{ϵ′})}+ ∥f_{D}∥_{H^{1,1}(D,dσ_{ϵ′})}) . (7.12)

Proof. We first observe that we may use the solution of the regularity problem from Theorem 5.1 to reduce to the case where the Dirichlet data is zero. More precisely, we consider the (unique) solution v of the regularity problem (2.5) with data f̃_{D} ∈ H^{1,1}(dσ_{ϵ′}) (here f̃_{D} denotes an extension of f_{D} to ∂Ω). By Theorem 5.1 it follows that ∂v/∂ν ∈ H^{1}(∂Ω,dσ_{ϵ′}), and it is easy to see that u is the unique solution of the mixed problem with data f_{D} and f_{N} if and only if u − v solves the mixed problem with zero Dirichlet data and with Neumann data g_{N}:= f_{N}− ∂v/∂ν.

To prove existence we consider any atomic decomposition for g_{N}, namely

g_{N}(x) = ∑_{j=1}^{∞} λ_{j}a_{j}(x), ∑_{j=1}^{∞} |λ_{j}| < +∞,  (7.13)

see (5.1) and (5.2). By Theorem 6.1 it follows that for each j the mixed problem:

\{ \begin{matrix}Δh_{j} = 0 & in Ω \\ h_{j} = 0 & on D \\ \frac{∂h_{j}}{∂ν} = a_{j} & on N \\ (∇h_{j})^{∗} ∈ L^{1}(dσ_{ϵ′}) \end{matrix}  (7.14)

has a solution h_{j} that satisfies:

∥h_{j}∥_{H^{1,1}(dσ_{ϵ′})}+ ∥ \frac{∂h_{j}}{∂ν} ∥_{H^{1}(dσ_{ϵ′})} ≤ C(M,ϵ′).  (7.15)

Thus, the function

h:= ∑_{j=1}^{∞} λ_{j}h_{j}

is a solution of the mixed problem with zero Dirichlet data and with Neumann data g_{N}, and it satisfies:

∥ \frac{∂h}{∂ν}∥_{H^{1}(dσ_{ϵ′})} ≤ C(M,ϵ′) ∑_{j=1}^{∞} |λ_{j}|.

By Lemma 7.1, h is unique, (i.e. h is independent of the choice of the atomic decomposition for g_{N}); taking the infimum over all atomic decompositions of g_{N} now yields (7.11). This proves existence; uniqueness follows from Lemma 7.1.

Next, we recall a few well known results concerning the the complex interpolation spaces of weighted L^{p} and Hardy spaces, see Bergh and Löfstrom [3, Theorem 5.5.3, Corollary 5.5.4], Strömberg and Torchinsky [32, Theorem 3, pg. 179]. For weights w_{0} and w_{1}, and exponents p_{0},p_{1} we set

\frac{1}{p_{θ}} = \frac{1 − θ}{p_{0}} + \frac{θ}{p_{1}}, w_{θ} = w0^{\frac{(1−θ)p_{θ}}{p_{0}}} w1^{\frac{θp_{θ}}{p_{1}}} , 0 ≤ θ ≤ 1.  (7.16)

We let [A,B]_{θ} denote the complex interpolation space of index θ as defined in the monograph of Bergh and Löfstrom [3, Chapt. 4].

Theorem 7.3 Suppose w_{0} and w_{1} are weights, then we have

[L^{1}(w_{0}dσ),L^{2}(w_{1}dσ)]_{θ} = L^{p_{θ}}(w_{θ}dσ).

If in addition the weights w_{j} are in A_{∞} (dσ) for j = 0,1, then we have

[H^{1}(w_{0}dσ),L^{2}(w_{1}dσ)]_{θ} = L^{p_{θ}}(w_{θ}dσ).

Moreover, if a linear operator T is bounded:

T : H^{1}(w_{0}dσ) → L^{1}(w_{0}dσ),
T : L^{2}(w_{1}dσ) → L^{2}(w_{1}dσ)

with norms M_{0} and M_{1} respectively, then T is bounded:

L^{p_{θ}}(w_{θ}dσ) → L^{p_{θ}}(w_{θ}dσ)

with norm M satisfying

M ≤ CM_{0}^{1−θ} M_{1}^{θ}.

Remark. The constant C in the estimate for the operator norm is 1 when we consider Lebesgue spaces. It may not be one for Hardy spaces, see Strömberg and Torchinsky [32].

We will focus on the case: w_{0}dσ = dσ_{ϵ′}, with ϵ′< 0 as in Theorem 7.2, and w_{1}dσ = dσ_{ϵ}, where ϵ > 0 is as in Theorem 4.6. We are now ready to give the proof of our main result, Theorem 1.1.

Proof of Theorem 1.1. We first use a result of Dahlberg and Kenig [10, Theorem 3.8] to reduce to the case where the Dirichlet data in the mixed problem is zero. (While Dahlberg and Kenig only discuss n ≥ 3 in their work, one can extend their results to two dimensions.)

We will use interpolation to establish existence of solutions satisfying the estimate (1.4). Because the complex method applies to linear operators, we employ a standard technique to obtain the non-tangential maximal function as a supremum of linear operators. Fix {y_{j}}_{j∈N} , a dense subset of the sector Γ(0) and let E = {E_{j}} be decomposition of ∂Ω into disjoint measurable subsets, E_{j}. We define a linear operator:

f_{N} ∈ H^{1}(dσ_{ϵ′}) + L^{2}(dσ_{ϵ}) → T_{E}(f_{N})(x) := ∑_{j} χ_{E_{j}}(x)∇u(x + y_{j}), x ∈ ∂Ω,

where u is the solution of the H^{1}(dσ_{ϵ′})-mixed problem (resp. the L^{2}(dσ_{ϵ})-mixed problem) for data f_{N} and f_{D} = 0. Note that for a suitable sequence of decompositions {E_{k}}, we have

(∇u)^{∗}(x) = lim_{k→∞} |T_{E_{k}}f_{N}(x)| a.e. x ∈ ∂Ω.

Now complex interpolation, (see Theorem 7.3) implies that the operator T_{E} is bounded on the intermediate spaces L^{p_{θ}}(w_{θ}dσ) with a norm independent of E. Fatou’s lemma then yields boundedness for the non-tangential maximal function. It is easy to see that for the spaces L^{2}(dσ_{ϵ}), (2β)/(π − 2β) < ϵ < 1 and H^{1}(dσ_{ϵ′}), max(−δ,(ϵ − 1)/2) < ϵ′≤ 0, the intermediate spaces include L^{p}(dσ) for 1 < p < p_{1}(M) where

p_{1}(M) = \frac{2(π − 2β)min(δ, \frac{π−4β}{2(π−2β)} ) + 2β}{(π − 2β)min(δ, \frac{π−4β}{2(π−2β)} ) + 2β} .

In addition, in Lemma 7.1, uniqueness was established for p in the range: 1 < p < p_{2}(M), p_{2}(M) = 1/(1 − δ(M)). Thus, we have existence and uniqueness for p ∈ (1,p_{0}(M)) with p_{0} = min(p_{1}(M),p_{2}(M)).

We close by listing a few open questions.

• Can we remove the restriction that the Lipschitz constant of the domain is at most one?

• Can we obtain similar results in higher dimensions? The obvious problem here is that our two-dimensional weighted estimates rely on a Rellich identity which is based on complex function theory.

• Can we study domains where the boundary between D and N is more interesting? For example in R^{3}, let Ω = {x: x_{3} > c(|x_{1}| + |x_{2}|)} and let D = ∂Ω ∩ {x: x_{1}x_{2} > 0} and then N = ∂Ω\D. Can we solve the mixed problem for some L^{p} space in this domain?

• Can we extend the solution of the mixed problem to general Lipschitz domains, rather than only Lipschitz graph domains?

References

[1] A. Azzam and E. Kreyszig. On solutions of elliptic equations satisfying mixed boundary conditions. SIAM J. Math. Anal., 13(2):254–262, 1982.

[2] S.R. Bell. The Cauchy transform, potential theory, and conformal mapping. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.

[3] J. Bergh and J. Löfström. Interpolation spaces. Springer-Verlag, 1976.

[4] R.M. Brown. The mixed problem for Laplace’s equation in a class of Lipschitz domains. Comm. Partial Diff. Eqns., 19:1217–1233, 1994.

[5] R.M. Brown. The Neumann problem on Lipschitz domains in Hardy spaces of order less than one. Pac. J. Math., 171(2):389–407, 1995.

[6] D.C. Chang, S.G. Krantz, and E.M. Stein. H^{p} theory on a smooth domain in R^{n} and elliptic boundary value problems. J. Funct. Anal., 114(2):286–347, 1993.

[7] R.R. Coifman, A. McIntosh, and Y.Meyer. L’intégrale de Cauchy définit un opérateur borné sur L^{2} pour les courbes lipschitziennes. Ann. of Math., 116:361–387, 1982.

[8] R.R. Coifman and G. Weiss. Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc., 83:569–645, 1976.

[9] B.E.J. Dahlberg. Estimates of harmonic measure. Arch. Rational Mech. Anal., 65(3):275–288, 1977.

[10] B.E.J. Dahlberg and C.E. Kenig. Hardy spaces and the Neumann problem in L^{p} for Laplace’s equation in Lipschitz domains. Ann. of Math., 125:437–466, 1987.

[11] L. Escauriaza. Personal communication, 2000.

[12] E.B. Fabes. Layer potential methods for boundary value problems on Lipschitz domains. In Potential theory—surveys and problems, volume 1344 of Lecture notes in math., pages 55–80. Springer Verlag, 1988.

[13] E.B. Fabes, O. Mendez, and M. Mitrea. Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains. J. Funct. Anal., 159(2):323–368, 1998.

[14] M. Giaquinta. Multiple integrals in the calculus of variations and nonlinear elliptic systems, volume 105 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1983.

[15] D. Gilbarg and N.S. Trudinger. Elliptic partial differential equations of second order. Springer-Verlag, Berlin, 1983.

[16] P. Grisvard. Elliptic problems in nonsmooth domains. Pitman, 1985.

[17] D.S. Jerison and C.E. Kenig. Hardy spaces, A_{∞}, and singular integrals on chord-arc domains. Math. Scand., 50(2):221–247, 1982.

[18] D.S. Jerison and C.E. Kenig. The Neumann problem on Lipschitz domains. Bull. Amer. Math. Soc., 4:203–207, 1982.

[19] C.E. Kenig. Weighted H^{p} spaces on Lipschitz domains. Amer. J. Math., 102(1):129– 163, 1980.

[20] C.E. Kenig. Harmonic analysis techniques for second order elliptic boundary value problems. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1994.

[21] C.E. Kenig and W.M. Ni. On the elliptic equation Lu−k+K exp[2u] = 0. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 12(2):191–224, 1985.

[22] John Lagnese. Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differential Equations, 50(2):163–182, 1983.

[23] G. M. Lieberman. Mixed boundary value problems for elliptic and parabolic differential equations of second order. J. Math. Anal. Appl., 113(2):422–440, 1986.

[24] V.G. Maz’ya and J. Rossman. L_{p} estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains. Preprint, 2006.

[25] V.G. Maz’ya and J. Rossman. Mixed boundary value problems for the Navier-Stokes system in polyhedral domains. Preprint, 2006.

[26] V.G. Maz’ya and J. Rossman. Pointwise estimates for Green’s kernel of a mixed boundary value problem to the Stokes system in a polyhedral cone. Preprint, 2006.

[27] I. Mitrea and M. Mitrea. The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in nonsmooth domains. To appear, Trans. Amer. Math. Soc.

[28] M.H. Protter and H.F. Weinberger. Maximum principles in differential equations. Prentice-Hall Inc., Englewood Cliffs, N.J., 1967.

[29] G. Savaré. Regularity and perturbation results for mixed second order elliptic problems. Comm. Partial Diff. Eqns., 22:869–899, 1997.

[30] Z. Shen. Weighted estimates in L^{2} for Laplace’s equation on Lipschitz domains. Trans. Amer. Math. Soc., 357:2843–2870, 2005.

[31] E.M. Stein. Harmonic Analysis: Real-variable methods, orthogonality and oscillatory integrals. Princeton, 1993.

[32] J.O. Strömberg and A. Torchinsky. Weighted Hardy spaces. Springer-Verlag, Berlin, 1989.

[33] J.D. Sykes. L^{p} regularity of solutions of the mixed boundary value problem for Laplace’s equation on a Lipschitz graph domain. PhD thesis, University of Kentucky, 1999.

[34] J.D. Sykes and R.M. Brown. The mixed boundary problem in L^{p} and Hardy spaces for Laplace’s equation on a Lipschitz domain. In Harmonic analysis and boundary value problems (Fayetteville, AR, 2000), volume 277 of Contemp. Math., pages 1–18. Amer. Math. Soc., Providence, RI, 2001.

[35] G.C. Verchota. Layer potentials and regularity for the Dirichlet problem for Laplace’s equation on Lipschitz domains. J. Funct. Anal., 59:572–611, 1984.

August 3, 2006