The Neumann problem in graph Lipschitz domains in the plane

Abstract

We study new aspects of the solvability of the classical Neumann boundary value problem in a graph Lipschitz domain in the plane. When the domain is the upper half-plane, the boundary data is assumed to belong to weighted Lebesgue or weighted Lorentz spaces; we show that the solvability of the Neumann problem in these settings may be characterized in terms of Muckenhoupt weights and related weights, respectively. For a general graph Lipschitz domain $\Omega$, as proved in an unpublished work by E. Fabes and C. Kenig, there exists $\varepsilon _\Omega >0$ such that the Neumann problem is solvable with data in $L^p(\partial \Omega )$ for $1<p<2+\varepsilon _\Omega ;$ we review the proof of this result and show that the Neumann problem is solvable at the endpoint $2+\varepsilon _\Omega$ with data in the Lorentz space $L^{2+\varepsilon _\Omega ,1}(\partial \Omega ).$ We present examples of our results in Schwarz–Christoffel Lipschitz domains and related domains.

Appendix A

In this appendix, we state a number of known results used in the proofs of Theorems 1.4 and 1.5.

For the following statements, $\Omega ,$ $\Lambda$, $\Phi$ and L are as defined in Sect. 1 (see (1.2)).

Lemma A.1

(Lemma 1.13 in Kenig [16]) There exist $0<\alpha <\arctan (1/L)$ and $0<\beta <\pi /2,$ which depend only on L,  such that for almost every $\xi \in \Lambda$ with respect to ds, $\Phi ^{-1}(\Gamma _\alpha (\xi ))\subset \Gamma _\beta (\Phi ^{-1}(\xi )).$ If $L<1,$ there exist $0<\alpha <\pi /2$ and $0<\beta <\arctan (1/L)$ such that for almost every $x\in {\mathbb {R}}$ with respect to Lebesgue measure, $\Phi (\Gamma _\alpha (x))\subset \Gamma _\beta (\Phi (x))\subset \Omega .$

Definition A.2

(Definition 1.14 in Kenig [16]) Given $0\le \lambda \le 1,$ define $\Phi _\lambda$ such that $\Phi _\lambda '(z)=(\Phi '(z))^{\lambda }.$ Note that $\Phi '_\lambda$ is never zero and $|\arg (\Phi _\lambda '(z))|\le \lambda \arctan (L)<\pi /2;$ then $\Phi _\lambda$ is one-to-one. We define $\Omega _\lambda =\Phi _\lambda ({\mathbb {R}}^2_+)$ and note that $\Omega _\lambda$ is a Lipschitz domain of the same type as $\Omega$ and $\Phi _\lambda$ is a conformal mapping from ${\mathbb {R}}^2_+$ onto $\Omega _\lambda$ that satisfies the same properties as $\Phi .$ Set $\Lambda _\lambda =\partial \Omega _\lambda$ and denote its arc length measure by $ds_\lambda .$ Also observe that $\Lambda _\lambda$ can be parametrized by $\xi _\lambda (x)=x+i\gamma _\lambda (x)$ with $\Vert \gamma _\lambda '\Vert _{L^\infty }\le \tan (\lambda \arctan (L)).$

Lemma A.3

(Lemma 1.15 in Kenig [16]) For $0\le \lambda _1\le 1$ and $0\le \lambda _2\le 1$ and using the notation introduced in Definition A.2, consider $\sigma _{\lambda _1,\lambda _2}:\Omega _{\lambda _2}\rightarrow \Omega _{\lambda _1}$ given by $\sigma _{\lambda _1,\lambda _2}=\Phi _{\lambda _1}\circ (\Phi _{\lambda _2})^{-1}.$ There exist $\varepsilon >0,$ $0<\alpha <\arctan (1/L)$ and $0<\beta <\arctan (1/L),$ depending only on L,  such that if $\lambda _2<\lambda _1$ and $\lambda _1-\lambda _2<\varepsilon$ then for almost every $z \in \Lambda _{\lambda _2}$ with respect to $ds_{\lambda _2},$ it holds that

$$\begin{aligned} \sigma _{\lambda _1,\lambda _2}(\Gamma _\alpha (z))\subset \Gamma _\beta (\sigma _{\lambda _1,\lambda _2}(z))\subset \Omega _{\lambda _1}. \end{aligned}$$

Lemma A.4

(Lemma 2.3 in Kenig [16]) If $0<\alpha <\arctan (1/L),$ $0<\beta <\arctan (1/L)$ and $\nu \in A_\infty (\Lambda )$ then

$$\begin{aligned} \nu (\{\xi \in \Lambda : {\mathcal {M}}_\alpha h (\xi )>\lambda \})\sim \nu (\{\xi \in \Lambda : {\mathcal {M}}_\beta h (\xi )>\lambda \}),\quad \forall \lambda >0, \end{aligned}$$

where the implicit constant depends only on $\nu ,$ L,  $\alpha$ and $\beta .$

Theorem A.5

(Theorem 2.8 in Kenig [16]) Let $0<p<\infty$ and $\nu \in A_\infty (\Lambda ).$ Then $h\in H^p(\Omega , \nu )$ if and only if $h\circ \Phi \in H^p({\mathbb {R}}^2_+,\Phi (\nu )),$ with equivalence of norms.

Definition A.6

(Definition 2.10 in Kenig [16]) Let $\nu \in A_\infty (\Lambda ).$ The space $AE(\nu )$ is the class of functions h that are analytic and different from 0 on $\Omega ,$ have a non-tangential limit almost everywhere in $\Lambda$ with respect to ds that satisfies $|h(\xi )|=\nu (\xi )$ for almost every $\xi \in \Lambda ,$ and there exists $m\ge 0$ such that $(h\circ \Phi )(z) \Phi '(z)/(i+z)^m\in H^1({\mathbb {R}}^2_+,dx).$

Theorem A.7

(Theorem 2.13 in Kenig [16]) Let $0<p<\infty$ and $\nu \in A_\infty (\Lambda ).$ If h is analytic in $\Omega ,$ the following conditions are equivalent:

• (a) $h\in H^p(\Omega ,\nu ),$

• (b) $\sup _{t>0}(\int _\Lambda |h(\xi +it)|^p d\nu (\xi ))^{1/p}<\infty ,$

• (c) for any $k\in AE(\nu ),$ $h \,k^{1/p}\in H^p(\Omega ,ds).$

Moreover, the corresponding norms are equivalent.

For the next two results, consider curves $\Lambda _1$ and $\Lambda _2$ in the complex plane, given parametrically by $\xi _1(x)=x+i \gamma _1(x)$ and $\xi _2(x)=x+i \gamma _2(x)$ for $x\in {\mathbb {R}},$ respectively, where $\gamma _1$ and $\gamma _2$ are Lipchitz functions. Denote $\Omega _j=\{z\in \mathbb {C}: z=x+iy \text { and } y>\gamma _j(x)\}$ for $j=1,2.$ Consider a conformal mapping $\sigma _{1,2}:\Omega _2\rightarrow \Omega _1$ such that $\sigma _{1,2}(\infty )=\infty$ (then $\sigma _{1,2}$ extends as a homeomorphism from $\overline{\Omega _2}$ onto $\overline{\Omega _1}$). If $\nu$ is a measure on $\Lambda _1,$ then $\sigma _{1,2}(\nu )$ denotes the measure on $\Lambda _2$ defined by $\sigma _{1,2}(\nu )(U)=\nu (\sigma _{1,2}(U))$ for any measurable set $U\subset \Lambda _2.$

Lemma A.8

(Lemma 1.16 in Kenig [16]) If $\nu \in A_{\infty }(\Lambda _1),$ then $\sigma _{1,2}(\nu )\in A_\infty (\Lambda _2).$

Definition A.9

(Definition 2.11 in Kenig [16]) Consider the corresponding conformal mappings $\Phi _1:{\mathbb {R}}^2_+\rightarrow \Omega _1$ and $\Phi _2:{\mathbb {R}}^2_+\rightarrow \Omega _2,$ and let $\nu \in A_\infty (\Lambda _1)$ and $\mu \in A_\infty (\Lambda _2).$ The space $AE(\nu ,\mu )$ is the class of functions h that are analytic and different from 0 on $\Omega _2,$ have a non-tangential limit almost everywhere in $\Lambda _2$ with respect to arc length measure $ds_2$ that satisfies $|h(\xi )|= (d\sigma _{1,2}(\nu )/d\mu )(\xi )$ for almost every $\xi \in \Lambda _2,$ and there exists $m\ge 0$ and $k\in AE(\mu )$ such that $(h(\Phi _2(z))k(\Phi _2(z)) \Phi _2'(z))/(i+z)^m\in H^1({\mathbb {R}}^2_+,dx).$

Theorem A.10

(Corollary 2.18 in Kenig [16]) Let $0<p<\infty ,$ $\nu \in A_\infty (\Lambda _1),$ $\mu \in A_\infty (\Lambda _2)$ and $k\in AE(\nu ,\mu ).$ Then $h\in H^p(\Omega _1,\nu )$ if and only if $(h\circ \sigma _{1,2})\,k^{1/p}\in H^p(\Omega _2,\mu ),$ with equivalent norms.

We end this appendix by stating two lemmas in the setting of ${\mathbb {R}}^2_+.$

Lemma A.11

(Corollary 4.3 in Garnett [11]) Let $0<p,r<\infty$ and $h\in H^p({\mathbb {R}}^2_+,dx).$ If the non-tangential limit of h belongs to $L^r({\mathbb {R}}),$ then $h\in H^r({\mathbb {R}}^2_+,dx).$

Lemma A.12

(Lemma I.10 in García–Cuerva [10]) Let $1<q<\infty ,$ $w\in A_q({\mathbb {R}})$ and s(x, t) be a non-negative subharmonic function on ${\mathbb {R}}^2_+$ which is in $L^q({\mathbb {R}}, w)$ uniformly in $t>0.$ Then s has a least harmonic majorant which is the Poison integral of some function $s_0\in L^q({\mathbb {R}},w).$ Moreover, $s_0$ is the pointwise limite of $s(\cdot ,t)$ as $t\rightarrow 0$ if such limit exits.