Abstract
We introduce a space of \(L^2\) vector fields with bounded mean oscillation whose “normal” component to the boundary is well-controlled. We establish its Helmholtz decomposition in the case when the domain is a perturbed \(C^3\) half space in \({\textbf{R}}^n\) \((n \ge 3)\) with small perturbation.
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1 Introduction
This is a continuation of our paper [14]. It is well known that the Helmholtz decomposition plays a key role in analyzing the Stokes and the Navier–Stokes equations [25]. Such decomposition is well studied for \(L^p\) space with \(1<p<\infty \). It is a topological direct sum decomposition
of the \(L^p\) vector fields in a domain \(\Omega \subseteq {\textbf{R}}^n\). Here, \(L^p_\sigma (\Omega )\) denotes the \(L^p\)-closure of the space of all smooth div-free vector fields that are compactly supported and \(G^p\) denotes the space of all \(L^p\) gradient fields. If \(p=2\), such decomposition holds for any domain \(\Omega \). It is an orthogonal decomposition and often called Weyl’s decomposition. For \(1<p<\infty \), the decomposition still holds for various domain including the whole space \({\textbf{R}}^n\), the half space \({\textbf{R}}^n_+\), a perturbed half space, a bounded smooth domain [10] and an exterior smooth domain. However, there are smooth unbounded domains which do not admit \(L^p\)-Helmholtz decomposition; see e.g. a nice book of Galdi [11].
If \(p=\infty \), such a decomposition does not hold even when \(\Omega ={\textbf{R}}^n\) since the projection operator is a kind of the Riesz operator which is unbounded in \(L^\infty \), though it is bounded in \(L^p\) (\(1<p<\infty \)). We replace \(L^\infty \) by BMO space. It turns out that it is convenient to consider a subspace vBMO of BMO to have the Helmholtz decomposition, at least for a half space [12] and a bounded domain [14]. Our goal is to extend such a result to a perturbed half space. Unfortunately, it seems that a direct extension is difficult because the behavior at space infinity is not well controlled. Thus we consider the \(L^2\) intersection of this space. For \(L^p\) space, Farwig, Kozono, and Sohr [7] established the Helmholtz decomposition of \(L^p\cap L^2\) (\(p\ge 2\)) and \(L^p+L^2\) (\(1<p<2\)) for arbitrary uniformly \(C^2\) smooth domain in \({\textbf{R}}^3\). (Later, it is extended to arbitrary dimension [8].) Although we consider \(vBMO\cap L^2\) in a slightly perturbed half space in the present paper, our results extend to any uniformly \(C^3\) domain. This will be discussed in a separate forthcoming paper.
Let us recall the BMO space of vector fields introduced in [13, 14]. For \(\mu \in (0,\infty ]\), we recall a BMO seminorm. For a locally integrable function f in \(\Omega \), i.e., \(f\in L^1_{loc}(\Omega )\), we set
here \(f_B\) denotes the average over B and \(B_r(x)\) denotes the open ball of radius r centered at x and \(|B |\) denotes the Lebesgue measure of B. For \(\nu \in (0,\infty ]\), we also use a seminorm
where \(\Gamma :=\partial \Omega \) denotes the boundary of \(\Omega \). The space
is essentially introduced in [2] and well studied in [5]. The Stokes semigroup in such spaces was studied [2, 4] and it is useful to prove that the analyticity of the Stokes semigroup still holds in some unbounded domains which do not admit \(L^p\)-Helmholtz decomposition [3].
Our space vBMO requires a control only on the normal component. Let \(d_\Gamma \) denote the distance function from \(\Gamma \). We set
where \(\cdot \) denotes the standard inner product. This is a seminorm. If \(\Omega ={\textbf{R}}^n_+\), this is not a norm unless \(n=1\), \(\nu =\infty \). However, if \(\Gamma \) has a fully curved part in the sense of [13, Definition 7], then \([\cdot ]_{vBMO^{\mu ,\nu }(\Omega )}\) becomes a norm [13, Lemma 8]. In particular, when \(\Omega \) is a bounded \(C^2\) domain, this is a norm. In this paper, we consider the case where \(\Omega \) is a perturbed \(C^k\) half space
where \(h\not \equiv 0\) is in \(C^k_c({\textbf{R}}^{n-1})\), i.e., h is a compactly supported \(C^k\) function in \({\textbf{R}}^{n-1}\); here \(x'=(x_1,\ldots ,x_{n-1})\) for \(x\in {\textbf{R}}^n\). A perturbed \(C^k\) (\(k\ge 2\)) half space \({\textbf{R}}^n_h\) is said to be of type (K) if
where \(\nabla ':= (\partial _1, \partial _2,\ldots , \partial _{n-1})\). We note that the perturbed \(C^k\) half space has a fully curved part so that \([v]_{vBMO^{\mu ,\nu }(\Omega )}\) is a norm. By definition, there always exists \(R_h>0\) such that the support \({\text {supp}} h \subseteq \overline{B_{R_h}(0')}\). We say that the perturbed \(C^k\) half space \({\textbf{R}}^n_h\) has small perturbation if
where \(C^*(n)\) is a specific constant depending only on the space dimension n,
To simplify the behavior near the infinity, we consider
Note that this space is independent of the choice of \(\mu ,\nu \).
The purpose of this paper is to establish the Helmholtz decomposition for the space \(vBMOL^2\big ( {\textbf{R}}_h^n \big )\) in the case where \({\textbf{R}}_h^n\) is a perturbed \(C^3\) half space that has small perturbation with \(n \ge 3\). Here is our main theorem.
Theorem 1
Let \({\textbf{R}}_h^n\) be a perturbed \(C^3\) half space of type (K) that has small perturbation with \(n \ge 3\) and \(\Gamma = \partial {\textbf{R}}_h^n\). Then for any \(v \in vBMOL^2\big ( {\textbf{R}}_h^n \big )\), there exists a unique decomposition \(v = v_0 + \nabla q\) with
satisfying the estimate
where \(C(K,R_*,R_h)\) is a constant that depends only on the constant K which controls the second order derivative of h, the reach of the boundary \(R_*\) and the size \(R_h\) which characterizes the support of h. In particular, the Helmholtz projection \(P_{vBMOL^2}\), defined by \(P_{vBMOL^2}(v) = v_0\), is a bounded linear map on \(vBMOL^2\big ( {\textbf{R}}_h^n \big )\) with range \(vBMOL^2_\sigma \big ( {\textbf{R}}_h^n \big )\) and kernel \(GvBMOL^2\big ( {\textbf{R}}_h^n \big )\).
Roughly speaking, the reach \(R_*\) represents the size of a small neighborhood of the boundary within which every point has a unique projection on the boundary. Here we would like to direct the readers to Sect. 2.1 for its precise definition.
Our strategy to prove Theorem 1 follows from the potential theoretic strategy we used to establish the Helmholtz decomposition in a bounded \(C^3\) domain [14]. Let E represents the fundamental solution of \(-\Delta \) in \({\textbf{R}}^n\). By [21], we see that as long as the boundary \(\Gamma \) is uniformly \(C^2\), the space \(BMO^\infty (\Omega ) \cap L^2(\Omega )\) allows the standard cut-off, i.e., we can decompose v into two parts \(v=v_1+ v_2\) with \(v_2 = \varphi v\) and \(v_1 = v - v_2\) with some \(\varphi \in C^\infty ({\textbf{R}}^n)\) that is supported within a small neighborhood of \(\Gamma \). Thus, the support of \(v_2\) lies in a small neighborhood of \(\Gamma \) whereas the support of \(v_1\) is away from \(\Gamma \). For \(v_1\), by extending \(v_1\) as zero outside \(\Omega \), we just set \(q^1_1 = E * {\text {div}} v_1\). Then, the \(L^\infty \) bound for \(\nabla q^1_1\) is well controlled near \(\Gamma \), which yields a bound for \(b^\nu \) seminorm. To estimate \(v_2\), we use a normal coordinate system near \(\Gamma \) and reduce the problem to the half space. We extend \(v_2\) to \({\textbf{R}}^n\) so that the normal part \((\nabla d \cdot \overline{v_2}) \nabla d\) is odd and the tangential part \(\overline{v_2} - (\nabla d \cdot \overline{v_2}) \nabla d\) is even in the direction of \(\nabla d\) with respect to \(\Gamma \). In such type of coordinate system, the minus Laplacian can be transformed as
where \(\eta _n\) is the normal direction to the boundary so that \(\{ \eta _n>0\}\) is the half space. We then use a freezing coefficient method to construct volume potential \(q_1^{\textrm{tan}}\) and \(q_1^{\textrm{nor}}\), which corresponds to the contribution from the tangential part \(\overline{v_2}^{\textrm{tan}}\) and the normal part \(\overline{v_2}^{\textrm{nor}}\), respectively. Since the leading term of \({\text {div}} \overline{v_2}^{\textrm{nor}}\) in normal coordinate consists of the differential of \(\eta _n\) only, if we extend the coefficient \(b_{ij}\) even in \(\eta _n\), \(q_1^{\textrm{nor}}\) is constructed so that the leading term of \(\nabla d \cdot \nabla q_1^{\textrm{nor}}\) is odd in the direction of \(\nabla d\). On the other hand, as the leading term of \({\text {div}} \overline{v_2}^{\textrm{tan}}\) in normal coordinate consists of the differential of \(\eta ' = (\eta _1,\ldots , \eta _{n-1})\) only, the even extension of \(b_{ij}\) in \(\eta _n\) gives rise to \(q_1^{\textrm{tan}}\) so that the leading term of \(\nabla d \cdot \nabla q_1^{\textrm{tan}}\) is also odd in the direction of \(\nabla d\). Disregarding lower order terms and localization procedure, \(q_1^{\textrm{tan}}\) and \(q_1^{\textrm{nor}}\) are constructed as
One is able to arrange \(BA^{-1}\) small by working in a small neighborhood of a boundary point. Then \((I-BA^{-1})^{-1}\) is given as the Neumann series \(\sum ^\infty _{m=0}(BA^{-1})^m\). The BMO-BMO estimates for \(\nabla q_1^{\textrm{tan}}\) and \(\nabla q_1^{\textrm{nor}}\) follow from [9]. Since the leading term of \(\nabla d \cdot (\nabla q_1^{\textrm{tan}} + \nabla q_1^{\textrm{nor}})\) is odd in the direction of \(\nabla d\) with respect to \(\Gamma \), the BMO bound implies \(b^\nu \) bound. The \(L^2\) estimates for \(\nabla q_1^{\textrm{tan}}\) and \(\nabla q_1^{\textrm{nor}}\) hold as in the localization procedure, the partition of unity we consider for a small neighborhood of \(\Gamma \) is locally finite. As a result, setting \(q_1=q^1_1+q_1^{\textrm{tan}}+q_1^{\textrm{nor}}\) would give us our desired volume potential corresponding to \({\text {div}} v\). Some results needed for the construction of volume potential \(q_1\) have already been established in [14, Section 3] and [21], for these parts we omit their proofs and recall them directly.
Theorem 2
(Construction of a suitable volume potential). Let \(\Omega \subset {\textbf{R}}^n\) be a uniformly \(C^3\) domain of type \((\alpha ,\beta ,K)\) with \(n \ge 2\). Let \(R_*\) be the reach of the boundary \(\Gamma = \partial \Omega \). Then, there exists a bounded linear operator \(v\longmapsto q_1\) from \(vBMOL^2(\Omega )\) to \(L^\infty (\Omega )\) such that
and that there exists a constant \(C=C(\alpha ,\beta ,K,R_*)>0\) satisfying
In particular, the operator \(v\longmapsto \nabla q_1\) is a bounded linear operator in \(vBMOL^2(\Omega )\).
Here \((\alpha ,\beta ,K)\) are parameters that characterize the regularity of \(\Gamma \). We would like to direct the readers to Sect. 2.1 for their precise definitions. Although the construction of the suitable volume potential works for arbitrary uniformly \(C^3\) domain in \({\textbf{R}}^n\) with \(n \ge 2\), for the rest of the theory we need to focus back on perturbed half space in \({\textbf{R}}^n\) with \(n \ge 3\). For \(v \in vBMOL^2\big ( {\textbf{R}}_h^n \big )\), we observe that \(w=v-\nabla q_1\) is divergence free in \({\textbf{R}}_h^n\). Unfortunately, this w may not fulfill the trace condition \(w \cdot {\textbf{n}}=0\) on the boundary \(\Gamma = \partial {\textbf{R}}_h^n\). We construct another potential \(q_2\) by solving the Neumann problem
We then set \(q=q_1+q_2\). Since \(\partial q_2/\partial {\textbf{n}}=\nabla q_2\cdot {\textbf{n}}\), \(v_0=v-\nabla q\) gives the Helmholtz decomposition. To complete the proof of Theorem 1, it suffices to control \(\Vert \nabla q_2\Vert _{vBMOL^2\big ( {\textbf{R}}_h^n \big )}\) by \(\Vert v\Vert _{vBMOL^2\big ( {\textbf{R}}_h^n \big )}\).
Lemma 3
(Estimate of the normal trace). Let \({\textbf{R}}_h^n\) be a perturbed \(C^{2+\kappa }\) half space of type (K) with \(\kappa \in (0,1)\), \(n \ge 3\) and \(\Gamma = \partial {\textbf{R}}_h^n\). Then, there is a constant \(C = C(K,R_*,R_h)>0\) such that
for all \(w \in vBMOL^2\big ( {\textbf{R}}_h^n \big )\) with \({\text {div}} w = 0\) in \({\textbf{R}}_h^n\).
Here \({\dot{H}}^{-\frac{1}{2}}(\Gamma )\) is a Hilbert space that is isomorphic to \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\), which turns out to be the dual space of the homogeneous fractional Sobolev space \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\). Here, the homogeneous Sobolev space of order \(s \in {\textbf{R}}\) is defined as
where \({\mathcal {S}}'({\textbf{R}}^{n-1})\) denotes the space of Schwartz’s tempered distributions and \({\widehat{f}}\) denotes the Fourier transform of f, see e.g. [1, Section 1.3]. Unfortunately, the space \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) is complete if and only if \(n \ge 3\). Thus, we assume that \(n \ge 3\) when the \({\dot{H}}^{-\frac{1}{2}}\) norm appears. The basic idea to establish Lemma 3 is as follows. The \(L^\infty \) estimate of \(w \cdot {\textbf{n}}\) follows directly from the trace theorem established in [13, Theorem 22]. For the \({\dot{H}}^{-\frac{1}{2}}\) estimate for \(w \cdot {\textbf{n}}\), we split the boundary into the straight part and the curved part. Since we have the \(L^\infty \) estimate for \(w \cdot {\textbf{n}}\) and the curved part is compact, the contribution in the \({\dot{H}}^{-\frac{1}{2}}\) estimate for \(w \cdot {\textbf{n}}\) that comes from the curved part can be estimated by the \(L^\infty \) norm of \(w \cdot {\textbf{n}}\). For the contribution in the \({\dot{H}}^{-\frac{1}{2}}\) estimate of \(w \cdot {\textbf{n}}\) that comes from the straight part, we invoke the \({\dot{H}}^{-\frac{1}{2}}\) estimate of \(w \cdot {\textbf{n}}\) in the case of the half space. We finally need the estimate for the Neumann problem.
Lemma 4
(Estimate for the Neumann problem). Let \({\textbf{R}}_h^n\) be a perturbed \(C^2\) half space of type (K) that has small perturbation with \(n \ge 3\). Let \(R_*\) be the reach of \(\Gamma = \partial {\textbf{R}}_h^n\). For any \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\), there exists a unique (up to constant) solution \(u \in L_{loc}^2\big ( {\textbf{R}}_h^n \big )\) to the Neumann problem
such that the operator \(g \longmapsto u\) is linear. Moreover, there exists a constant \(C = C(K,R_*,R_h)>0\) such that
To establish Lemma 4, the basic strategy is the same as in [14], we firstly show that
can be controlled for any \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) where
represents the single layer potential for g. Since the boundary \(\Gamma = \partial {\textbf{R}}_h^n\) is curved and not compact, we appeal a perturbation argument. For \(g \in L^\infty (\Gamma )\), we decompose g into the curved part \(g_1\) and the straight part \(g_2\) by setting \(g_1\big ( y',h(y') \big ):= 1'_{B_{2R_h}(0')}(y') g\big ( y',h(y') \big )\) for \(y' \in {\textbf{R}}^{n-1}\) and \(g_2:= g - g_1\), where \(1'_{B_{2R_h}(0')}\) represents the characteristic function for the open ball \(B_{2R_h}(0')\) in \({\textbf{R}}^{n-1}\). Since \(g_2\) vanishes in the curved part of \(\Gamma \), we can define \(g_2^H \in L^\infty ({\textbf{R}}^{n-1}) \cap {\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\) by setting \(g_2^H(y',0) = g_2\big ( y',h(y') \big )\) for any \(y' \in {\textbf{R}}^{n-1}\). Note that
for any \(x \in {\textbf{R}}^n\). By setting \(g_2^{{\textbf{R}}^n}(y',y_n):= g_2^H(y',0)\) for any \((y',y_n) \in {\textbf{R}}^n\), we can deduce the BMO estimate of \(\nabla E *\big ( \delta _\Gamma \otimes g_2 \big )\) by applying the \(L^\infty -BMO\) estimate for singular integral operator [19, Theorem 4.2.7] to \(\nabla \partial _{x_n} E *1_{{\textbf{R}}_+^n} g_2^{{\textbf{R}}^n}\). Since \(g_1\big ( \cdot ', h(\cdot ') \big )\) is compactly supported in \({\textbf{R}}^{n-1}\), we may extend \(g_1\) to some \(g_{1,\textrm{c}}^\textrm{e} \in L^\infty ({\textbf{R}}^n)\) such that \(g_{1,\textrm{c}}^\textrm{e}\) is compactly supported and \(\nabla d \cdot \nabla g_{1,\textrm{c}}^\textrm{e} = 0\) within a small neighborhood of \(\Gamma \). Then, we can rewrite \(\nabla E *\big ( \delta _\Gamma \otimes g_1 \big )\) as
with some compactly supported continuous function \(f_{\theta ,\rho _0/4}\) (see Proof of Lemma 25 (i)) that is independent of g. The BMO estimate for the first term on the right hand side of (3) can be controlled using the \(L^\infty -BMO\) estimate for singular integration operator. The second term on the right hand side of (3) can be controlled by the \(L^\infty \) norm as \(\nabla E (x)\) is a locally integrable kernel and \(f_{\theta ,\rho _0/4}\) has compact support. We thus obtain the BMO estimate for \(\nabla E *\big ( \delta _\Gamma \otimes g \big )\) for \(g \in L^\infty (\Gamma )\).
For the \(b^\nu \) estimate of the normal component of \(\nabla E *\big ( \delta _\Gamma \otimes g \big )\) with \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\), we also decompose g into the curved part \(g_1\) and the straight part \(g_2\). It is possible to show that
where \(\Gamma ^{{\textbf{R}}^n}_{\rho _0}:= \{ x \in \Omega \,\vert \,d(x) < \rho _0 \}\) denotes the \(\rho _0\)-neighborhood of \(\Gamma \) in \(\Omega \). Since \(g_1\big ( \cdot ',h(\cdot ') \big )\) is compactly supported in \({\textbf{R}}^{n-1}\), estimate (4) allows us to show that
for any \(x \in \Gamma ^{{\textbf{R}}^n}_{\rho _0}\) with some constant \(C=C(K,R_h,R_*)>0\). On the other hand, for x close to the curved part of \(\Gamma \), \(\nabla d(x)\) is not necessarily \((0,\ldots ,0,1)\), in this case
would contain contributions from \(\nabla ' \big ( E *(\delta _\Gamma \otimes g_2) \big )\), which cannot be estimated by the \(L^\infty \) bound of \(g_2\) (see Proposition 26). As a result, we introduce the \({\dot{H}}^{-\frac{1}{2}}\) bound of \(g_2\). Since \(g_2\big ( \cdot ',h(\cdot ') \big )\) is supported in \(B_{2R_h}(0')^\textrm{c}\), \(\nabla d(x) \cdot \nabla E (x - \cdot ')\) can be viewed as an element of \({\dot{H}}^{\frac{1}{2}}(\Gamma )\) for any x close to the curved part of \(\Gamma \). Hence, by the \({\dot{H}}^{-\frac{1}{2}}-{\dot{H}}^{\frac{1}{2}}\) duality we can deduce that
with some constant \(C=C(K,R_h,R_*)>0\). We thus obtain an \(L^\infty \) estimate for the normal component of \(\nabla E *\big ( \delta _\Gamma \otimes g \big )\) within a small neighborhood of \(\Gamma \). The \(b^\nu \) estimate naturally follows.
For \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\), we prove that the trace of
is of the form
where \(S: L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma ) \rightarrow L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) is a bounded linear operator satisfying
with \(C^*(n)\) denoting a specific fixed constant which depends on dimension n only. Therefore, if \({\textbf{R}}_h^n\) is a perturbed \(C^2\) half space that has small perturbation with \(n \ge 3\), the inverse of \(I - 2S\) is well-defined as a bounded linear map from \(L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) to \(L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) by the Neumann series
The solution to the Neumann problem (2) is formally given by
We finally need the \(L^2\) estimate for \(\nabla u\) in \({\textbf{R}}_h^n\). In the case of the half space, for \(g \in {\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\), the single layer potential \(E *\big ( \delta _{\partial {\textbf{R}}_+^n} \otimes g \big )\) is exactly half of the solution u to the Neumann problem. By directly considering the partial Fourier transform of \(E(x',x_n)\) with respect to \(x'\), we could easily deduce that
In the case that \({\textbf{R}}_h^n\) is a perturbed \(C^2\) half space with \(n \ge 3\), for \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\), we still decompose g into the curved part \(g_1\) and the straight part \(g_2\). Since \(L^{\frac{2n-2}{n}}(\Gamma )\) is continuously embedded in \({\dot{H}}^{-\frac{1}{2}}(\Gamma )\) and the curved part \(g_1\) has compact support in \(\Gamma \), \(g \in L^\infty (\Gamma )\) would imply that both \(g_1,g_2 \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\). Since for any \(x \in {\textbf{R}}^n\) we have that
and for any \(x = (x',x_n) \in {\textbf{R}}_+^n\) we have that
the \(L^2\) estimate of \(\nabla E *\big ( \delta _\Gamma \otimes g_2 \big )\) in \({\textbf{R}}^n\) follows from the \(L^2\) estimate of \(\nabla E *\big ( \delta _{\partial {\textbf{R}}_+^n} \otimes g_2^H \big )\) in \({\textbf{R}}_+^n\), i.e., we have that
For the curved part \(g_1\), we recall the argument which establishes the \(vBMO^{\infty ,\nu }\) norm for \(\nabla E *\big ( \delta _\Gamma \otimes g \big )\). We extend \(g_1\) to \(g_{1,\textrm{c}}^\textrm{e} \in L^\infty ({\textbf{R}}^n)\) and consider equation (3). Since \(\nabla {\text {div}} E\) is \(L^p\) integrable for any \(1<p<\infty \), see e.g. [18, Theorem 5.2.7 and Theorem 5.2.10], the \(L^2\) norm of the first term on the right hand side of (3) can be estimated by the \(L^\infty \) norm of g. Whereas \(\nabla E(x)\) is an integration kernel that is dominated by a constant multiple of \(|x|^{-(n-1)}\), by the famous Hardy–Littlewood–Sobolev inequality [1, Theorem 1.7], we can also control the \(L^2\) norm of the second term on the right hand side of (3) by the \(L^\infty \) norm of g. Combine with the \(L^2\) estimate for the contribution from the straight part \(g_2\), we finally obtain our desired \(L^2\) estimate
for \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\). This completes the proof of Lemma 4.
When taking the normal trace and solving the Neumann problem, the reason why we need to require the dimension n to be greater than or equal to 3 is because when \(n \ge 3\), we indeed have the fact that \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) is continuously embedded in \(L^{\frac{2n-2}{n-2}}({\textbf{R}}^{n-1})\) and \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\) is the dual space of \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\), which further implies that \(L^{\frac{2n-2}{n}}({\textbf{R}}^{n-1})\) is continuously embedded in \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\). Based on these facts, we can show that in the case of any perturbed \(C^2\) half space \({\textbf{R}}_h^n\) with boundary \(\Gamma = \partial {\textbf{R}}_h^n\), \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) is isomorphic to \({\dot{H}}^{\frac{1}{2}}(\Gamma )\) and \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\) is isomorphic to \({\dot{H}}^{-\frac{1}{2}}(\Gamma )\). More importantly, we can estimate the \({\dot{H}}^{-\frac{1}{2}}\) norm of the trace operator S by its \(L^{\frac{2n-2}{n}}\) norm and do cut-offs to boundary data \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) to decompose it into the curved part \(g_1\) and the straight part \(g_2\). In the case where \(n=2\), the space \({\dot{H}}^{\frac{1}{2}}({\textbf{R}})\) is no longer complete and \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}})\) is not necessarily the dual space of \({\dot{H}}^{\frac{1}{2}}({\textbf{R}})\). The completion of \({\dot{H}}^{\frac{1}{2}}({\textbf{R}})\) cannot be embedded in the space of Schwartz’s tempered distributions. Moreover, as a limit case, \({\dot{H}}^{\frac{1}{2}}({\textbf{R}})\) is not continuously embedded in \(L^\infty ({\textbf{R}})\). As a result, at the present we lack of tools to estimate the \({\dot{H}}^{-\frac{1}{2}}\) norm of the trace operator S and we cannot do cut-offs to decompose a boundary data into the curved part and the straight part any more. This is why we focus on the case where the dimension \(n \ge 3\) in this paper. The problem of \({\dot{H}}^{\frac{1}{2}}({\textbf{R}})\) is similar to \({\dot{H}}^1({\textbf{R}}^2)\). The space \({\dot{H}}^1({\textbf{R}}^2)\) is not complete. Its completion should be the quotient space \(\{ u \in L_{loc}^1({\textbf{R}}^2) \, \vert \, \nabla u \in L^2({\textbf{R}}^2)^n \} / {\textbf{R}}\) as discussed in [11] since \({\dot{H}}^1({\textbf{R}}^2)\) includes all smooth compactly supported functions. We have to study an appropriate dual space as in [16].
This paper is organized as follows. In Sect. 2, we recall results from [21] to localize the problem and results from [14] to construct a suitable volume potential corresponding to \({\text {div}}v\). Theorem 2 is proved in this section. In Sect. 3, we take the normal trace in \({\dot{H}}^{-\frac{1}{2}}\) sense by considering isomorphisms between \({\dot{H}}^{-\frac{1}{2}}(\Gamma )\) and \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\). In Sect. 4, we establish estimates for the trace operator S of Qg. We show that S is bounded from \(L^\infty \cap {\dot{H}}^{-\frac{1}{2}}\) to \(L^\infty \cap {\dot{H}}^{-\frac{1}{2}}\). In Sect. 5, we solve Neumann problem (2) by considering the single layer potential with \(2(I-2S)^{-1} g\) and establish the \(vBMOL^2\) estimate for its gradient in \({\textbf{R}}_h^n\).
2 Volume Potential Construction in a Uniformly \(C^3\) Domain
For \(v \in vBMOL^2(\Omega )\), we shall construct a suitable potential \(q_1\) so that \(v\longmapsto \nabla q_1\) is a bounded linear operator in \(vBMOL^2(\Omega )\) as stated in Theorem 2. The construction in the case where \(\Omega \) is a uniformly \(C^3\) domain basically follows from the theory in [14], where \(\Omega \) is a bounded \(C^3\) domain.
2.1 Localization Tools
Let us recall some uniform estimates established in [21]. Let \(\Omega \) be a uniformly \(C^k\) domain in \({\textbf{R}}^n\) with \(k \in {\textbf{N}}\) and \(n \ge 2\). Let \(\Gamma = \partial \Omega \) denotes the boundary of domain \(\Omega \). There exists \(\alpha , \beta >0\) such that for each \(z_0 \in \Gamma \), up to translation and rotation, there exists a function \(h_{z_0} \in C^k\big ( B_\alpha (0') \big )\), where \(B_\alpha (0')\) denotes the open ball in \({\textbf{R}}^{n-1}\) of radius \(\alpha \) with center \(0'\), that satisfies the following properties:
-
(i)
$$\begin{aligned} K:= \sup _{0 \le s \le k} \left\Vert (\nabla ')^s h_{z_0} \right\Vert _{L^\infty \big ( B_\alpha (0') \big )} < \infty ; \quad \nabla ' h_{z_0}(0')=0, \quad h_{z_0}(0')=0, \end{aligned}$$
-
(ii)
\(\Omega \cap U_{\alpha ,\beta , h_{z_0}}(z_0)=\left\{ (x',x_n) \in {\textbf{R}}^n \,\Bigg \vert \, h_{z_0}(x')< x_n< h_{z_0}(x')+ \beta ,\ |x' |< \alpha \right\} \) where
$$\begin{aligned} U_{\alpha ,\beta , h_{z_0}}(z_0):= \left\{ (x',x_n) \in {\textbf{R}}^n \,\Bigg \vert \, h_{z_0}(x') - \beta< x_n< h_{z_0}(x') +\beta ,\ |x' |< \alpha \right\} , \end{aligned}$$ -
(iii)
\(\Gamma \cap U_{\alpha ,\beta , h_{z_0}}(z_0)=\left\{ (x',x_n)\in {\textbf{R}}^n \,\Bigg \vert \, x_n = h_{z_0}(x'),\ |x' |< \alpha \right\} \).
We say that \(\Omega \) is of type \((\alpha ,\beta ,K)\).
Let d denote the signed distance function from \(\Gamma \) which is defined by
so that \(d(x)=d_\Gamma (x)\) for \(x\in \Omega \). For a uniformly \(C^k\) domain \(\Omega \), there is \(R^\Omega >0\) such that for \(x \in \Omega \) with \(\left|d(x)\right|<R^\Omega \), there is a unique point \(\pi x \in \Gamma \) such that \(|x-\pi x|=\left|d(x)\right|\). The supremum of such \(R^\Omega \) is called the reach of \(\Gamma \) in \(\Omega \), we denote this supremum by \(R_*^\Omega \). Let \(\Omega ^{\textrm{c}}\) be the complement of \(\Omega \) in \({\textbf{R}}^n\). Similarly, there is \(R^{\Omega ^{\textrm{c}}}>0\) such that for \(x \in \Omega ^{\textrm{c}}\) with \(\left|d(x)\right|< R^{\Omega ^{\textrm{c}}}\), we can also find a unique point \(\pi x \in \Gamma \) such that \(|x-\pi x|=\left|d(x)\right|\). The supremum of such \(R^{\Omega ^{\textrm{c}}}\) is called the reach of \(\Gamma \) in \(\Omega ^{\textrm{c}}\), we denote this supremum by \(R_*^{\Omega ^{\textrm{c}}}\). We then define
which we call it the reach of \(\Gamma \). Moreover, d is \(C^k\) in the \(\rho \)-neighborhood of \(\Gamma \) for any \(\rho \in (0,R_*)\), i.e., \(d \in C^k\left( \Gamma ^{{\textbf{R}}^n}_\rho \right) \) for any \(\rho \in (0,R_*)\) with
see e.g. [20, Chap. 14, Appendix], [24, Section 4.4].
There exists \(0< \rho _0 < \min \big ( \alpha , \beta , \frac{R_*}{2}, \frac{1}{2n (K+1)} \big )\) such that for every \(z_0 \in \Gamma \),
is contained in the coordinate chart \(U_{\alpha ,\beta , h_{z_0}}(z_0)\) for any \(\rho \le \rho _0\). For \(z_0 \in \Gamma \), the normal coordinate change \(F_{z_0}: V_{\rho _0}:= B_{\rho _0}(0') \times (-\rho _0,\rho _0) \rightarrow U_{\rho _0}(z_0)\) is defined by
or shortly
Note that that for any \(z_0 \in \Gamma \), \(F_{z_0}\) is indeed a \(C^1\)-diffeomorphism between \(V_{\rho _0}\) and \(U_{\rho _0}(z_0)\). For any \(\varepsilon \in (0,1)\), there exists a constant \(c_\Omega ^\varepsilon := C(\varepsilon , K, \rho _0)>0\) and \(c_\Omega ^\varepsilon < \rho _0\) such that for any \(\rho \in (0, c_\Omega ^\varepsilon ]\) and \(z_0 \in \Gamma \), the estimates
hold simultaneously, see [21, Proposition 3].
For \(\rho \in (0, \rho _0/2)\), there exist a countable family of points in \(\Gamma \), say \({\mathcal {P}}_\Gamma := \{ x_i \in \Gamma \, \mid \, i \in {\textbf{N}} \}\), such that
Moreover, there exists a natural number \(N_*=C(n)\) such that for any \(x_i \in {\mathcal {P}}_\Gamma \), there exist at most \(N_*\) points in \({\mathcal {P}}_\Gamma \), say \(\{ x_{i_1},\ldots , x_{i_{N_*}} \} \subset {\mathcal {P}}_\Gamma \), with
for each \(1 \le l \le N_*\), see e.g. [21, Proposition 5]. Based on this open cover, a partition of unity for \(\Gamma ^{{\textbf{R}}^n}_{\rho }\) can be constructed. There exist \(\varphi _i \in C^1\big ( U_\rho (x_i) \big )\) for each \(x_i \in {\mathcal {P}}_\Gamma \) and a constant \(C(N_*,n,\rho )>0\) such that
see e.g. [21, Proposition 6].
2.2 Cut-Off and Extension
In this subsection, we assume that \(\Omega \subset {\textbf{R}}^n\) is a uniformly \(C^2\) domain of type \((\alpha , \beta , K)\) with \(n \ge 2\). Let \(\rho \in (0, \rho _0/2)\). For a function f defined in \(\Gamma ^{{\textbf{R}}^n}_\rho \cap {\overline{\Omega }}\), we define \(f_{\textrm{even}}\) to be the even extension of f to \(\Gamma ^{{\textbf{R}}^n}_\rho \) with respect to the boundary \(\Gamma \), i.e.,
and \(f_{\textrm{odd}}\) to be the odd extension of f to \(\Gamma ^{{\textbf{R}}^n}_\rho \) with respect to the boundary \(\Gamma \), i.e.,
For \(x \in \Gamma ^{{\textbf{R}}^n}_\rho \), we further define that
It is not hard to see that P(x) represents the normal projection to the direction \(\nabla d\) whereas Q(x) represents the tangential projection to the direction \(\nabla d\). For \(v\in vBMOL^2(\Omega )\) with \({\text {supp}} v \subseteq \Gamma ^{{\textbf{R}}^n}_\rho \cap \Omega \), we define \({\overline{v}}\) to be its extension of the form
for \(x \in \Gamma ^{{\textbf{R}}^n}_\rho \). Since \({\text {supp}} {\overline{v}} \subset \Gamma ^{{\textbf{R}}^n}_\rho \), \({\overline{v}}\) can be viewed as being defined in \({\textbf{R}}^n\) with \({\overline{v}}(x) = 0\) for any \(x \in {\textbf{R}}^n {\setminus } \Gamma ^{{\textbf{R}}^n}_\rho \).
Proposition 5
([21]). Let \(\Omega \subset {\textbf{R}}^n\) be a uniformly \(C^2\) domain of type \((\alpha ,\beta ,K)\) with \(n \ge 2\). Let \(\varepsilon \in (0,1)\). For any \(\rho \in (0, c_\Omega ^\varepsilon /2]\), there exists a constant \(C=C(\alpha ,\beta ,K,\rho )>0\) such that estimates
hold for all \(v \in vBMOL^2(\Omega )\) with \({\text {supp}} v \subset \Gamma ^{{\textbf{R}}^n}_\rho \) and \(\nu >0\).
Actually, we could achieve more than Proposition 5. We consider a cut off function \(\theta \in C_{\textrm{c}}^\infty ({\textbf{R}})\) such that \(0 \le \theta \le 1\), \(\theta (t) = 1\) for any \(0< |t |< 1/2\) and \(\theta (t) = 0\) for any \(|t |> 3/4\). Suppose that \(\varepsilon \in (0,1)\) and \(\rho \in (0, c_\Omega ^\varepsilon /2]\). We set \(\theta _\rho := \theta (d(x)/\rho )\). An easy check tells us that \(\theta _\rho \in C_\textrm{c}^2({\textbf{R}}^n)\) with \({\text {supp}} \theta _\rho \subseteq \overline{\Gamma ^{{\textbf{R}}^n}_\rho }\) and \(\theta _\rho (x) = 1\) for \(x \in \Gamma ^{{\textbf{R}}^n}_{\rho /2}\). Within this paper, for any subset \(D \subset {\textbf{R}}^n\) we denote \(r_D\) to be the restriction operator in D. For \(v \in vBMOL^2(\Omega )\), we let \(v_2:= (r_\Omega \theta _\rho ) v\) and \(v_1:= v - v_2\). We extend \(v_2\) to \(\overline{v_2}\) in the same way as (9) in which the normal component of \(\overline{v_2}\) is odd with respect to \(\Gamma \) and the tangential component of \(\overline{v_2}\) is even with respect to \(\Gamma \), i.e., we set
By [21, Theorem 1], we see that \({\overline{v}}:= \overline{v_2} + v_1\) is a linear extension of v to \({\textbf{R}}^n\) which satisfies
with some \(C=C(\alpha ,\beta ,K,\rho )>0\). In general, multiplication by a smooth function is not bounded in \(BMO^\infty (\Omega )\). However, since we have a bounded linear extension from \(vBMOL^2(\Omega )\) to \(BMOL^2({\textbf{R}}^n)\), such multiplication is bounded in \(vBMOL^2(\Omega )\). Since \(\rho _0\) depends on \(\alpha ,\beta ,K\) and the reach \(R_*\), by fixing an arbitrary \(\varepsilon \in (0,1)\) and an arbitrary \(\rho \in (0,c_\Omega ^\varepsilon /2]\), we can deduce the following multiplication rule.
Proposition 6
([21]). Let \(\Omega \subset {\textbf{R}}^n\) be a uniformly \(C^2\) domain of type \((\alpha ,\beta ,K)\) with \(n \ge 2\). Let \(\varphi \in C^\gamma (\Omega )\) with \(\gamma \in (0,1)\). For each \(v \in vBMOL^2(\Omega )\), the function \(\varphi v \in vBMOL^2(\Omega )\) satisfies
with some constant \(C=C(\alpha ,\beta ,K,R_*)>0\) where \(R_*\) represents the reach of \(\Gamma \).
2.3 Decomposition of Volume Potential Corresponding to v
In this subsection, we assume that \(\Omega \subset {\textbf{R}}^n\) is a uniformly \(C^3\) domain of type \((\alpha ,\beta ,K)\) with \(n \ge 2\). Let us recall some results that have already been established in [14]. There exists a constant \(\rho _*= C(\rho _0,K)>0\) such that all theories in this subsection hold for every \(\rho \in (0, \rho _*]\). Let \(\rho \in (0, \rho _*]\) and \(\theta _\rho \) be the cut-off function defined in Sect. 2.2. Since now we assume that \(\Gamma \) is uniformly \(C^3\), in this case \(\theta _\rho \in C^3_\textrm{c}({\textbf{R}}^n)\) with \({\text {supp}} \theta _\rho \subset \Gamma ^{{\textbf{R}}^n}_\rho \) and \(\theta _\rho = 1\) for any \(x \in \Gamma ^{{\textbf{R}}^n}_{\rho /2}\). Still, for \(v \in vBMOL^2(\Omega )\) we set \(v_2:= (r_\Omega \theta _\rho ) v\) and \(v_1:= v - v_2\). By Proposition 6, we see that \(v_1, v_2 \in vBMOL^2(\Omega )\) satisfying
with some constant \(C=C(\alpha ,\beta ,K,\rho )>0\).
To construct the mapping \(v\mapsto q_1\) in Theorem 2, we localize \(v_2\) by using the partition of the unity \(\{\varphi _i\}^\infty _{i=1}\) associated with the covering \(\{ U_{\rho ,i} \}^\infty _{i=1}\) of \(\Gamma ^{{\textbf{R}}^n}_\rho \). Here for each \(i \in {\textbf{N}}\), \(U_{\rho ,i}\) denotes \(U_\rho (x_i)\) with \(x_i \in {\mathcal {P}}_\Gamma \). The corresponding volume potential to \(v_1\) can be estimated directly.
Proposition 7
There exists a constant \(C(\rho )>0\) such that
for \(q^1_1= E*{\text {div}} \, v_1\) and \(v \in vBMOL^2(\Omega )\). In particular,
for any \(\nu < \rho /4\).
Proof
By the BMO-BMO estimate [9] and Proposition 6, we have the estimate
Consider \(x \in \Gamma _{\rho /4}^{\textrm{R}^n}\). Since \(\nabla q_1^1\) is harmonic in \(\Gamma _{\rho /2}^{\textrm{R}^n}\) and \(B_{\frac{\rho }{4}}(x) \subset \Gamma _{\rho /2}^{\textrm{R}^n}\), the mean value property for harmonic functions implies that
i.e., we can estimate \(|\nabla q_1^1 (x) |\) by \(C(\rho ) \Vert \nabla q_1^1 \Vert _{L^2({\textbf{R}}^n)}\). Since the convolution with \(\nabla ^2 E\) is bounded in \(L^p\) for any \(1<p<\infty \), see e.g. [18, Theorem 5.2.7 and Theorem 5.2.10], we have that
Therefore, the estimate
holds for any \(x \in \Gamma _{\rho /4}^{\textrm{R}^n}\). \(\square \)
For \(i \in {\textbf{N}}\), we extend \((r_\Omega \varphi _i) v_2\) as in Proposition 5 to get \(\overline{(r_\Omega \varphi _i) v_2}\) and set
Indeed, this extension is independent of the choice of \(\{ \varphi _i \}_{i=1}^\infty \) as long as \(\{ \varphi _i \}_{i=1}^\infty \) is a partition of unity for \(\Gamma ^{{\textbf{R}}^n}_\rho \). We next set
For \(i \in {\textbf{N}}\), we have that \(\varphi _i \in C^2(U_{\rho ,i})\) as in this case \(\Gamma \) is of regularity uniformly \(C^3\). For simplicity of notation, we denote \((r_\Omega \varphi _i)_\textrm{even}(v_2)_{\textrm{even}}\) by \(v_{2,i}\). By Proposition 5 and 6, we can easily deduce that for any \(i \in {\textbf{N}}\), \(v_{2,i} \in BMOL^2({\textbf{R}}^n)\) with \({\text {supp}} v_{2,i} \subset U_{\rho ,i}\) satisfying the estimate
We further denote \(Q \, v_{2,i}\) by \(w_i^{\textrm{tan}}\). Now, we are ready to construct the suitable potential corresponding to
Proposition 8
([14]). For every \(i \in {\textbf{N}}\), there exist bounded linear operators \(v \longmapsto p_{i,1}^{\textrm{tan}}\) and \(v \longmapsto p_{i,2}^{\textrm{tan}}\) from \(vBMOL^2(\Omega )\) to \(L^\infty ({\textbf{R}}^n)\) such that
with \(p_i^{\textrm{tan}}:= p_{i,1}^{\textrm{tan}} + p_{i,2}^{\textrm{tan}}\), \({\text {supp}} p_{i,1}^{\textrm{tan}} \subset U_{2 \rho ,i}\). Moreover, there exists a constant \(C=C(K,\rho )>0\) such that
with some \(p>n\).
Having the estimate for the volume potential near the boundary regarding its tangential component, we are left to handle the contribution from \(\overline{v_2}^{\textrm{nor}}:=\overline{v_2} - \overline{v_2}^{\tan }\). We note that \(\overline{v_2}^\textrm{nor}\) admits decomposition
For simplicity of notations, for every \(i \in {\textbf{N}}\) we denote \(\nabla d \cdot \big ( (r_\Omega \varphi _i)_\textrm{even} (v_2)_{\textrm{odd}} \big )\) by \(f_{2,i}\). In this case, for any \(i \in {\textbf{N}}\) we have that \(f_{2,i} \in BMOL^2({\textbf{R}}^n)\) with \({\text {supp}} f_{2,i} \subset U_{\rho ,i}\) satisfying the estimate
We further denote \(f_{2,i} \nabla d\) by \(w_i^{\textrm{nor}}\). With a similar idea of proof, we can establish the suitable potential corresponding to \(\overline{v_2}^{\textrm{nor}}\).
Proposition 9
([14]). For every \(i \in {\textbf{N}}\), there exist bounded linear operators \(v\longmapsto p_{i,1}^{\textrm{nor}}\) and \(v\longmapsto p_{i,2}^{\textrm{nor}}\) from \(vBMOL^2(\Omega )\) to \(L^\infty ({\textbf{R}}^n)\) such that
with \(p_i^{\textrm{nor}}:= p_{i,1}^{\textrm{nor}} + p_{i,2}^{\textrm{nor}}\), \({\text {supp}} p_{i,1}^{\textrm{nor}} \subset U_{2 \rho ,i}\). Moreover, there exists a constant \(C=C(K,\rho )>0\) such that
with some \(p>n\).
Remark 10
Specifically speaking, Proposition 8 is indeed [14, Proposition 4] whose proof is in [14, Section 3.4], Proposition 9 is indeed [14, Proposition 5] whose proof is in [14, Section 3.5]. For Proposition 8, in the local normal coordinate system at \(x_i \in {\mathcal {P}}_\Gamma \), \(p_{i,1}^\textrm{tan}\) is constructed as
with \(\theta \) denoting some cut-off function in \(V_{4\rho }\) and \(F_{x_i}: V_{\rho _0} \rightarrow U_{\rho _0}(x_i)\) is the normal coordinate change in \(U_{\rho _0}(x_i)\). Since \(\partial _{\eta _k} A^{-1} \partial _{\eta _\ell }\) is bounded in BMO [9] and in \(L^p\) (\(1<p<\infty \)) for any \(k,\ell =1,2,\ldots ,n\), see e.g. [19, Theorem 5.2.7 and Theorem 5.2.10], the \(BMO \cap L^2\) norm of \(\nabla p_{i,1}^\textrm{tan}\) is controlled by the \(BMO \cap L^2\) norm of \(v_{2,i}\). On the other hand, \(p_{i,2}^\textrm{tan}\) is constructed as the convolution of the Newton potential E with some function of \(v_{2,i}\), we can directly estimate the \(L^\infty \) norm for \(\nabla p_{i,2}^\textrm{tan}\) by H\(\ddot{\text {o}}\)lder’s inequality as \(\nabla E\) is locally \(L^p\)-integrable for p sufficiently close to 1. Similarly, for Proposition 9, \(p_{i,1}^\textrm{nor}\) is constructed as
and \(p_{i,2}^\textrm{nor}\) is constructed as the convolution of the Newton potential E with some function of \(f_{2,i}\). The \(BMO \cap L^2\) estimate for \(\nabla p_{i,1}^\textrm{nor}\) and the \(L^\infty \) estimate for \(\nabla p_{i,2}^\textrm{nor}\) can be derived by exactly the same argument as in Proposition 8. The reason why statements of Propositions 8 and 9 look different from [14, Proposition 4] and [14, Proposition 5] is because in the case that \(\Omega \) is a bounded \(C^3\) domain, the space \(vBMO(\Omega )\) is continuously embedded in \(L^1(\Omega )^n\). By the \(BMO - L^1\) interpolation inequality, the \(L^p\) norm of \(v_{2,i}\) and \(f_{2,i}\) can be controlled by their vBMO norm for any \(1<p<\infty \), see e.g. [5, Lemma 5], [23, Theorem 2.2]. If \(\Omega \) is a general uniformly \(C^3\) domain, then it is not necessary that \(vBMO^{\infty ,\infty }(\Omega )\) is continuously embedded in \(L^1(\Omega )^n\). This is why we state Propositions 8 and 9 in a different form from [14].
2.4 Volume Potential Corresponding to v
We admit Propositions 8 and 9. The corresponding volume potential to \(v_2\) can be constructed by summing up the cut-off of \(p_i^\textrm{tan}\) and \(p_i^\textrm{nor}\) to \(U_{2\rho ,i}\) for all i. We define \({\mathcal {Q}}_\Gamma := \{ U_{\rho ,i} \,\vert \, x_i \in {\mathcal {P}}_\Gamma \}\).
Proof of Theorem 2
Let \(i \in {\textbf{N}}\). We firstly consider the contribution from the tangential part. Since \(\Gamma \) is uniformly \(C^3\), there exists a cut-off function \(\theta _i \in C^2_c(U_{2\rho ,i})\) such that \(\theta _i=1\) in \(U_{\rho ,i}\), \(0 \le \theta _i \le 1\) and moreover, the estimate
holds for some constant \(C(\rho )>0\) independent of i. We next set
Note that Proposition 8 ensures that
in \(\Omega \) as \({\text {supp}}w_i^{\textrm{tan}} \subset U_{\rho ,i}\). We define that
Since \({\text {supp}} p_{i,1}^{\textrm{tan}} \subset U_{2 \rho ,i}\) for all i, by Proposition 8 we have that
Since our partition of unity for \(\Gamma ^{{\textbf{R}}^n}_\rho \) is locally finite, see Sect. 2.1, we can deduce that
where \(N_*\) is the constant which characterizes the local finiteness of \({\mathcal {Q}}_\Gamma \) in the sense that any element of \({\mathcal {Q}}_\Gamma \) can intersect for at most \(N_*\) other elements of \({\mathcal {Q}}_\Gamma \). Suppose that B is a ball of radius \(r(B) < \rho \). If B does not intersect \(\Gamma ^{{\textbf{R}}^n}_{2 \rho }\), then
for each \(i \in {\textbf{N}}\). If B intersects \(\Gamma ^{{\textbf{R}}^n}_{2 \rho }\), then B intersects at most \(N_*\) neighborhoods of \(\{ U_{2 \rho }(x_i) \,\vert \, x_i \in {\mathcal {P}}_\Gamma \}\), see [21, Lemma 11]. Hence in this case, we have that
Hence, by estimate (10) we deduce that
Note that \({\text {supp}} \theta _i p_{i,2}^{\textrm{tan}} \subset U_{2 \rho ,i}\) for any \(i \in {\textbf{N}}\) and for every \(x \in \Gamma ^{{\textbf{R}}^n}_{2\rho }\), x belongs to at most \(N_*\) elements of \({\mathcal {Q}}_\Gamma \). By Proposition 8 again we can deduce that
In addition, as
with some \(p>n\), by the \(BMO-L^2\) interpolation inequality we have that
Hence, by Proposition 5 we obtain that
Let \(f_i^{\textrm{tan}} = p_i^{\textrm{tan}} \Delta \theta _i + 2 \nabla \theta _i \cdot \nabla p_i^{\textrm{tan}}\). Since \({\text {supp}} f_i^{\textrm{tan}} \subset U_{2 \rho ,i}\), we have that
By the same argument above which proves the estimate (11), we can show that
By the \(BMO-L^1\) interpolation (cf. [5, Lemma 5]), we see that the estimate
holds for any \(1<s<\infty \). Since \(\nabla E\) is in \(L^{p'}\big ( B_{6 \rho }(0) \big )\) for any \(1<p'<n/(n-1)\), it follows that
for all \(p>n\). Thus, we have that
Since the convolution kernel \(\nabla E(x)\) is dominated by a constant multiple of \(|x |^{-(n-1)}\), by the well-known Hardy–Littlewood–Sobolev inequality, see e.g. [1, Theorem 1.7], we deduce that
Hence by estimate (12), we see that
Combine with estimate (11), we finally obtain that
By Proposition 8, the control on the boundary with respect to \(q_1^{\textrm{tan}}\) is estimated by
as the partition \({\mathcal {Q}}_\Gamma \) is a locally finite open cover of \(\Gamma ^{{\textbf{R}}^n}_{\rho }\).
For the contribution coming from the normal component, we set in a similar way that
and
By making use of Proposition 9 and repeating the whole argument above that treats the case for \(q_1^{\textrm{tan}}\), we can prove that
in the same way. Then the volume potential \(q_1\) corresponding to v can be constructed as
where \(q_1^1\) is the volume potential defined in Proposition 7 corresponding to \(v_1\). By our construction, it can be easily seen that
in \(\Omega \). We are done. \(\square \)
3 Normal Trace for a \(L^2\) Vector Field with Bounded Mean Oscillation
Let \({\textbf{R}}_h^n\) be a perturbed \(C^{2+\kappa }\) half space of type (K) with \(\kappa \in (0,1)\) and \(n \ge 3\). Let \(R_*>0\) denote the reach of boundary \(\Gamma = \partial {\textbf{R}}_h^n\). We have already shown that there exists a constant \(C=C(\alpha ,\beta ,K,R_*)>0\) such that the estimate
holds for any \(w \in vBMOL^2\big ( {\textbf{R}}_h^n \big )\), see [13, Theorem 22]. In this section, we would like to further estimate the \({\dot{H}}^{-\frac{1}{2}}\) norm of the normal trace of v on \(\Gamma \) for \(w \in vBMOL^2\big ( {\textbf{R}}_h^n \big )\).
For simplicity of notations, from now on we define that for any \(\delta >0\),
and \(\Lambda _\delta \) to be the surface area of \(B'_\Gamma (\delta )\), i.e.,
Moreover, for \(h \in C_\textrm{c}^1({\textbf{R}}^{n-1})\), we define the constant \(C_s(h):= 1 + \Vert h \Vert _{C^1({\textbf{R}}^{n-1})}\).
3.1 Isomorphism Between \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) and \({\dot{H}}^{\frac{1}{2}}(\Gamma )\)
We would like to firstly give a characterization to the homogeneous fractional Sobolev space \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) before we define \({\dot{H}}^{\frac{1}{2}}(\Gamma )\). Let us recall that if \(f \in {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\), then \(f \in L_{loc}^2({\textbf{R}}^{n-1})\) and the Gagliardo seminorm
More precisely, if \(f \in {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\), then it holds that
with some constant C(n) that depends only on dimension n; see e.g. [1, Proposition 1.37]. However, the finiteness of the Gagliardo seminorm \([f]_{{\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})}\) does not imply that \(f \in {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\). Constant functions in \({\textbf{R}}^{n-1}\) are typical counterexamples. Since we are considering the case where \(n \ge 3\), we have a very important property that \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) can be continuously embedded in \(L^{\frac{2n-2}{n-2}}({\textbf{R}}^{n-1})\), i.e., there exists a constant \(C(n)>0\) such that the estimate
holds for any \(f \in {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\), see e.g. [1, Theorem 1.38]. As a result, we expect that \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) can be identified with the space
Fortunately, this expectation is indeed true. The Gagliardo seminorm \([\cdot ]_{{\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})}\) turns out to be a norm on \({\dot{G}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\). The space \({\dot{G}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) is complete with norm \([\cdot ]_{{\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})}\) and it contains \(C_\textrm{c}^\infty ({\textbf{R}}^{n-1})\) as a dense subspace; see [6, Theorem 3.1]. Since \(C_\textrm{c}^\infty ({\textbf{R}}^{n-1}) \subset {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\), we see that \(\Vert f \Vert _{{\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})}\) and \([f]_{{\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})}\) are equivalent for any \(f \in C_\textrm{c}^\infty ({\textbf{R}}^{n-1})\). Hence, the space \({\dot{G}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) coincide with the completion of \(C_\textrm{c}^\infty ({\textbf{R}}^{n-1})\) in norm \(\Vert \cdot \Vert _{{\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})}\), i.e., it holds that
The reason why the completion of \(C_\textrm{c}^\infty ({\textbf{R}}^{n-1})\) in norm \(\Vert \cdot \Vert _{{\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})}\) is indeed \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) is as follows. For \(\rho >0\), we can construct a cut-off function \(\theta _{2 \rho } \in C_\textrm{c}^\infty ({\textbf{R}}^{n-1})\) such that \(0 \le \theta _{2 \rho } \le 1\) in \({\textbf{R}}^{n-1}\), \(\theta _{2 \rho } = 1\) in \(B_\rho (0')\) and \(\theta _{2 \rho } = 0\) in \(B_{2 \rho }(0')^\textrm{c}\). We note that for any \(f \in {\mathcal {S}}({\textbf{R}}^{n-1})\) where \({\mathcal {S}}({\textbf{R}}^{n-1})\) denotes the Schwartz space, \(\theta _{2 \rho } f\) converges to f in norm \(\Vert \cdot \Vert _{{\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})}\) as \(\rho \rightarrow \infty \) (This claim will be established within the proof of Proposition 17 which appears later in Sect. 3.3). Therefore, by recalling that the space \({\mathcal {S}}_0({\textbf{R}}^{n-1})\) is dense in \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) where \({\mathcal {S}}_0({\textbf{R}}^{n-1})\) denotes the subspace of \({\mathcal {S}}({\textbf{R}}^{n-1})\) whose Fourier transform vanishes near the origin, see e.g. [1, Proposition 1.35], we can deduce that
With respect to a function f defined on \({\textbf{R}}^{n-1}\), we could define a corresponding function \(f^h\) that is defined on \(\Gamma \) by setting
Let the mapping \(f \mapsto f^h\) be denoted by \(T_h\), i.e., \(f^h = T_h(f)\). Since it is not that natural to consider the Fourier transform on a surface, we follow the characterization of \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) above to define the homogeneous fractional Sobolev space \({\dot{H}}^{\frac{1}{2}}(\Gamma )\). We say that \(f^h \in {\dot{H}}^{\frac{1}{2}}(\Gamma )\) if \(f^h \in L^{\frac{2n-2}{n-2}}(\Gamma )\) satisfies
The space \({\dot{H}}^{\frac{1}{2}}(\Gamma )\) is a Banach space (actually Hilbert space) equipped with the norm \([\cdot ]_{{\dot{H}}^{\frac{1}{2}}(\Gamma )}\). Without causing any ambiguity, we simply use the norm notation \(\Vert \cdot \Vert _{{\dot{H}}^{\frac{1}{2}}(\Gamma )}\) to denote \([\cdot ]_{{\dot{H}}^{\frac{1}{2}}(\Gamma )}\). Finally, we would like to note that
i.e., \(f \in L^{\frac{2n-2}{n-2}}({\textbf{R}}^{n-1})\) if and only if \(f^h \in L^{\frac{2n-2}{n-2}}(\Gamma )\).
Lemma 11
The mapping \(T_h: {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1}) \rightarrow {\dot{H}}^{\frac{1}{2}}(\Gamma )\) is an isomorphism.
Proof
Let \(f \in {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\). We naturally have that
Hence, the \({\dot{H}}^{\frac{1}{2}}(\Gamma )\) estimate for \(f^h\) reads as
Let \(f^h \in {\dot{H}}^{\frac{1}{2}}(\Gamma )\). With respect to \(f^h\), we define that \(f(y'):= f^h\big ( y',h(y') \big )\) for any \(y' \in {\textbf{R}}^{n-1}\). By the mean value theorem, we see that the estimate \(\left|h(x') - h(y')\right|\le \Vert \nabla ' h \Vert _{L^\infty ({\textbf{R}}^{n-1})} |x'-y' |\) holds for any \(x',y' \in {\textbf{R}}^{n-1}\). As a result, for any \(x,y \in \Gamma \) we have that
Hence, we deduce that
Therefore, we obtain that
\(\square \)
As an application of Lemma 11, we have the following embedding result.
Corollary 12
\({\dot{H}}^{\frac{1}{2}}(\Gamma )\) is continuously embedded in \(L^{\frac{2n-2}{n-2}}(\Gamma )\).
Proof
Let \(f^h \in {\dot{H}}^{\frac{1}{2}}(\Gamma )\) and \(f = T_h^{-1}(f^h)\). Since \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) is continuously embedded in \(L^{\frac{2n-2}{n-2}}({\textbf{R}}^{n-1})\), by estimate (15) we see that
\(\square \)
3.2 Isomorphism Between \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\) and \({\dot{H}}^{-\frac{1}{2}}(\Gamma )\)
Let us further recall that a tempered distribution g is said to belong to the homogeneous Sobolev space \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\) if \({\widehat{g}} \in L_{loc}^1({\textbf{R}}^{n-1})\) satisfies
Since \(\frac{1}{2} < \frac{n-1}{2}\) for all \(n \ge 3\), \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\) is characterized as the dual space of \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\), i.e., it holds that
see e.g. [1, Proposition 1.36]. In order to establish an isomorphism between the dual spaces of \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) and \({\dot{H}}^{\frac{1}{2}}(\Gamma )\), we need the following multiplication rule for \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\).
Proposition 13
Let \(\rho >0\) and \(\zeta \in C^1({\textbf{R}}^{n-1})\) satisfies that \(\zeta \) is identically a constant in \(B_\rho (0')^\textrm{c}\). Then for any \(f \in {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\), we have that \(\zeta f \in {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) satisfying
Proof
Let \(f \in {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\). For \(x',y' \in {\textbf{R}}^{n-1}\), by the triangle inequality we see that
Since \(\zeta \) is identically a constant in \(B_\rho (0')^\textrm{c}\), it is sufficient to estimate
For \(|x' |\ge 2\rho \) and \(|y' |<\rho \), we have that \(|x'-y' |\ge |x' |- \rho \). Hence, by H\(\ddot{\text {o}}\)lder’s inequality and the embedding of \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) in \(L^{\frac{2n-2}{n-2}}({\textbf{R}}^{n-1})\), see estimate (13), we deduce that
Similarly, for \(|y'|\ge 2\rho \) and \(|x'|<\rho \), we also have that \(|x'-y'|\ge |y'|- \rho \). By H\(\ddot{\text {o}}\)lder’s inequality and estimate (13) again, we see that
By the mean value theorem, it holds that \(\left|\zeta (x') - \zeta (y') \right|\le \Vert \nabla ' \zeta \Vert _{L^\infty ({\textbf{R}}^{n-1})} |x'-y' |\). Therefore, by H\(\ddot{\text {o}}\)lder’s inequality and estimate (13) once more, we finally have that
\(\square \)
In accordance with the duality relation between \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\) and \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\), we define the homogeneous Sobolev space \({\dot{H}}^{-\frac{1}{2}}(\Gamma )\) to be the dual space of \({\dot{H}}^{\frac{1}{2}}(\Gamma )\) with
for \(g^h \in {\dot{H}}^{-\frac{1}{2}}(\Gamma )\). Based on the fact that \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) is isomorphic to \({\dot{H}}^{\frac{1}{2}}(\Gamma )\), we can show that their dual spaces are isomorphic with each other as well.
Lemma 14
The mapping \(T_h: {\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1}) \rightarrow {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) is an isomorphism.
Proof
Let \(g \in {\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\) and \(g^h = T_h(g)\). For any \(f^h \in {\dot{H}}^{\frac{1}{2}}(\Gamma )\), we have that
Since \(\big ( 1 + \left|\nabla ' h(y') \right|^2 \big )^{\frac{1}{2}} \in C^1({\textbf{R}}^{n-1})\) and \(\big ( 1 + \big |\nabla ' h(y') \big |^2 \big )^{\frac{1}{2}} = 1\) in \(B_{R_h}(0')^\textrm{c}\), by Proposition 13 we deduce that
where
By controlling \(\Vert f \Vert _{{\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})}\) by \(\Vert f^h \Vert _{{\dot{H}}^{\frac{1}{2}}(\Gamma )}\) using estimate (15), we obtain that
Let \(g^h \in {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) and \(g = T_h^{-1}(g^h)\). For any \(f \in {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\), we have that
Since we define \({\dot{H}}^{-\frac{1}{2}}(\Gamma )\) to be the dual space of \({\dot{H}}^{\frac{1}{2}}(\Gamma )\), the duality relation says that
Note that for any \(f_*^h \in {\dot{H}}^{\frac{1}{2}}(\Gamma )\), we have that
Let \(f_*= T_h^{-1}(f_*^h)\). By estimate (14), we see that
Since \(\big ( 1 + \big |\nabla ' h(y') \big |^2 \big )^{-\frac{1}{2}} \in C^1({\textbf{R}}^{n-1})\) and \(\big ( 1 + \big |\nabla ' h(y') \big |^2 \big )^{-\frac{1}{2}}\) is also identically 1 in \(B_{R_h}(0')^\textrm{c}\), by Proposition 13 again we deduce that
Hence, by estimate (15) we have that
Finally, by controlling the \({\dot{H}}^{\frac{1}{2}}(\Gamma )\) norm of \(f^h\) by the \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) norm of f using estimate (14) again, we obtain from estimate (17) that
i.e.,
\(\square \)
Since \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\) is the dual space of \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\), the fact that \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) is continuously embedded in \(L^{\frac{2n-2}{n-2}}({\textbf{R}}^{n-1})\) would imply that \(L^{\frac{2n-2}{n}}({\textbf{R}}^{n-1})\) is continuously embedded in \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\), i.e., there exists a constant \(C(n)>0\) such that the estimate
holds for any \(g \in {\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\). By the isomorphism between \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\) and \({\dot{H}}^{-\frac{1}{2}}(\Gamma )\), we deduce the following embedding result.
Corollary 15
\(L^{\frac{2n-2}{n}}(\Gamma )\) is continuously embedded in \({\dot{H}}^{-\frac{1}{2}}(\Gamma )\).
Proof
We consider \(g^h \in {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) and \(f^h \in {\dot{H}}^{\frac{1}{2}}(\Gamma )\) with \(\Vert f^h \Vert _{{\dot{H}}^{\frac{1}{2}}(\Gamma )} \le 1\). Let \(g = T_h^{-1}(g^h)\) and \(f = T_h^{-1}(f^h)\). By Proposition 13, estimate (19) and estimate (15), we can deduce that
Since \(\Vert g \Vert _{L^{\frac{2n-2}{n}}({\textbf{R}}^{n-1})} \le \Vert g^h \Vert _{L^{\frac{2n-2}{n}}(\Gamma )}\), we obtain that
\(\square \)
3.3 \({\dot{H}}^{-\frac{1}{2}}\) Estimate for the Normal Trace \(w \cdot {\textbf{n}}\)
Since we are considering the case where \(n \ge 3\), similar to the characterization of \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\), the space \({\dot{H}}^1({\textbf{R}}^n)\) has the characterization that \(u \in {\dot{H}}^1({\textbf{R}}^n)\) if and only if \(u \in L^{\frac{2n}{n-2}}({\textbf{R}}^n)\) such that \(\nabla u \in L^2({\textbf{R}}^n)\). The space \({\dot{H}}^1({\textbf{R}}^n)\) is complete with norm \(\Vert \nabla u \Vert _{L^2({\textbf{R}}^n)}\). Moreover, it contains \(C_\textrm{c}^\infty ({\textbf{R}}^n)\) as a dense subspace; see e.g. [6, Theorem 3.1], [11, 16].
Proposition 16
There exists a bounded linear lifting operator \(\ell _{\textbf{n}}: {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1}) \rightarrow {\dot{H}}^1({\textbf{R}}^n)\).
Proof
For \(f \in {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\), we set
for \((x',x_n) \in {\textbf{R}}^n\), i.e., \(u_f\) is the inverse Fourier transform of \(\textrm{e}^{- |x_n||\xi '|} {\widehat{f}}(\xi ')\) with respect to \(\xi '\). By the Fourier-Plancherel formula, see e.g. [1, Theorem 1.25], we have that
for \(\xi ' \ne 0'\). Hence, we can deduce that
On the other hand, by the Fourier-Plancherel formula again, we see that
which further implies that
Therefore, we obtain that
Letting \(\ell _{\textbf{n}}(f) = u_f\) for any \(f \in {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) completes the proof of Proposition 16. \(\square \)
We start with the half space problem. If \(w \in L^2({\textbf{R}}_+^n)^n\) satisfies \({\text {div}} w = 0\) in \({\textbf{R}}_+^n\), then the normal trace \(w \cdot {\textbf{n}}\) can be taken in the \({\dot{H}}^{-\frac{1}{2}}\) sense.
Proposition 17
There exists a constant \(C(n)>0\) such that the estimate
holds for any \(w \in L^2({\textbf{R}}_+^n)^n\) with \({\text {div}} w = 0\) in \({\textbf{R}}_+^n\).
Proof
Let \(\theta _2 \in C_\textrm{c}^\infty ({\textbf{R}})\) be such that \(0 \le \theta _2 \le 1\) in \({\textbf{R}}\), \(\theta _2(z) = 1\) for any \(|z |<1\) and \(\theta _2(z) = 0\) for any \(|z |\ge 2\). We define \(\theta _{2\rho } \in C_\textrm{c}^\infty ({\textbf{R}}^n)\) by setting
For \(f \in {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\), we consider
for \(x' \in {\textbf{R}}^{n-1}\). Since \(\theta _I = 0\) in \(B_\rho (0')\), we decompose
Since \(|y'|\ge \rho \) and \(|x'|< \frac{\rho }{2}\) would imply that \(|x' - y'|\ge |y'|- \frac{\rho }{2}\), by H\(\ddot{\text {o}}\)lder’s inequality we deduce that
By symmetry, \(I_3\) follows the same estimate as \(I_2\). For \(I_1\), we follow the proof of Proposition 13 to estimate it as
Since \(\theta _I = 1\) in \(B_{2\rho }(0')^\textrm{c}\), we further decompose \(I_{1,2}\) as
By the proof of Proposition 13, we see that
Therefore, we deduce that
Since the \(L^{\frac{2n-2}{n-2}}\) norm of f is controlled by the \({\dot{H}}^{\frac{1}{2}}\) norm of f, we deduce that \(\Vert f_I \Vert _{{\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})}\) converges to zero as \(\rho \) tends to infinity, i.e., \(\theta _{2\rho }(\cdot ',0) f\) converges to f in \({\dot{H}}^{\frac{1}{2}}\) norm as \(\rho \rightarrow \infty \).
Let us note that the multiplication by a smooth function with compact support is bounded in \({\dot{H}}^1({\textbf{R}}^n)\). Indeed, by H\(\ddot{\text {o}}\)lder’s inequality and the continuous embedding of \({\dot{H}}^1({\textbf{R}}^n)\) in \(L^{\frac{2n}{n-2}}({\textbf{R}}^n)\), we see that the estimate
holds for any \(\phi \in C_\textrm{c}^\infty ({\textbf{R}}^n)\). Let \(f \in {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) and \(u_f\) be defined as in expression (21). We consider \(u_{f,2\rho }:= \theta _{2\rho } u_f\). Since \(\Vert \nabla \theta _{2\rho } \Vert _{L^n({\textbf{R}}^n)} \le C(n) \Vert \theta _2' \Vert _{L^\infty ({\textbf{R}})}\), by estimate (23) we have that
Since \({\text {supp}} u_{f,2\rho } \subseteq \overline{B_{2\rho }(0)}\), it actually holds that \(u_{f,2\rho } \in H^1({\textbf{R}}^n)\) satisfies
see e.g. [1, Proposition 1.55]. Let \(B_{2\rho }^+:= B_{2\rho }(0) \cap {\textbf{R}}_+^n\). Since \(u_{f,2\rho } \in H^1(B_{2\rho }^+)\), by the Gauss-Green formula, see e.g. [27, Lemma 1.2.3], it holds that
for any \(\rho >0\) and \(w \in L^2({\textbf{R}}_+^n)^n\) with \({\text {div}} w = 0\) in \({\textbf{R}}_+^n\). Since we have already shown above that \(\theta _{2\rho }(\cdot ',0) f\) converges to f in \({\dot{H}}^{\frac{1}{2}}\) norm as \(\rho \rightarrow \infty \), by Proposition 16 we conclude that for any \(f \in {\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) and \(w \in L^2({\textbf{R}}_+^n)\) with \({\text {div}} w = 0\) in \({\textbf{R}}_+^n\),
i.e.,
\(\square \)
Remark 18
Instead, if the domain \(\Omega \subset {\textbf{R}}^n\) that we are considering is bounded \(C^2\), then for \(w \in L^2(\Omega )^n\) satisfying \({\text {div}} w = 0\) in \(\Omega \), the normal trace \(w \cdot {\textbf{n}}\) can be taken in the \(H^{-\frac{1}{2}}\) sense, i.e., there exists a constant C, independent of w, such that
for any \(w \in L^2(\Omega )^n\) with \({\text {div}} w = 0\) in \(\Omega \); see e.g. [26, 28].
Now we are ready to consider the perturbed half space problem. For \(w \in vBMOL^2\big ( {\textbf{R}}_h^n \big )\) with \({\text {div}} w = 0\) in \({\textbf{R}}_h^n\), we show that the normal trace \(w \cdot {\textbf{n}}\) can be taken in the \(L^\infty \cap {\dot{H}}^{-\frac{1}{2}}\) sense.
Proof of Lemma 3
Let \(w \in vBMOL^2\big ( {\textbf{R}}_h^n \big )\) with \({\text {div}} w = 0\) in \({\textbf{R}}_h^n\). Let \(\varphi _*\in C_\textrm{c}^\infty ({\textbf{R}}^n)\) satisfies \(\varphi _*= 1\) in \(B_{1+\tau _1(h)}(0)\) and \({\text {supp}} \varphi _*\subseteq \overline{B_{2+\tau _1(h)}(0)}\) with \(\tau _1(h):= R_h + 4nKR_h^2\). Then we set \(w_1:= \varphi _*w\) and \(w_2:= w - w_1\). By Proposition 6, we see that \(w_1, w_2 \in vBMOL^2\big ( {\textbf{R}}_h^n \big )\) satisfying
with some constant \(C=C(\alpha ,\beta ,K,R_*)>0\). For \(x \in \Gamma \) such that \(|x' |= R_h\), we have that \(h(x') = 0\) and \(\nabla ' h (x') = 0'\). Thus, for any \(y \in B'_\Gamma (R_h)\), by picking an arbitrary \(x' \in {\textbf{R}}^{n-1}\) such that \(|x' |= R_h\) and considering the mean value theorem, we have that
As a result, we deduce that
for any \(\rho >0\). Since \(w_1 = 0\) in \({\textbf{R}}_h^n \cap B_{2+\tau _1(h)}(0)^\textrm{c}\), by Corollary 15 and the \(L^\infty \) estimate for \(w_1 \cdot {\textbf{n}}\) [13, Theorem 22], we can deduce that
On the other hand, we define that
Since \(w_2 = 0\) in \({\textbf{R}}_h^n \cap B_{1+\tau _1(h)}(0)\), we have that \(w_{2,H} = 0\) in \({\textbf{R}}_+^n \cap B_{1+\tau _1(h)}(0)\). By Proposition 17, we have that
where \({\textbf{n}}_{\partial {\textbf{R}}_+^n}\) denotes the outward normal on the boundary \(\partial {\textbf{R}}_+^n\) of the half space \({\textbf{R}}_+^n\). Since \(T_h\big ( w_{2,H} \cdot {\textbf{n}}_{\partial {\textbf{R}}_+^n} \big ) = w_2 \cdot {\textbf{n}}_\Gamma \) with \({\textbf{n}}_\Gamma \) denoting the outward normal on the boundary \(\Gamma \) of the perturbed half space \({\textbf{R}}_h^n\), by estimate (16) we obtain that
\(\square \)
4 Estimates for Some Boundary Integrals
Let E denotes the fundamental solution of \(-\Delta \) in \({\textbf{R}}^n\), i.e.,
where \(b_1(n)\) denotes the volume of the unit ball \(B_1(0)\) in \({\textbf{R}}^n\). In this section, we assume that \({\textbf{R}}_h^n\) is a perturbed \(C^2\) half space of type (K) with boundary \(\Gamma = \partial {\textbf{R}}_h^n\). The purpose of this section is to establish several estimates for the trace operator of
for \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) where \(\partial / \partial {\textbf{n}}_x\) denotes the exterior normal derivative with respect to x-variable.
Let us recall that for a perturbed \(C^2\) half space \({\textbf{R}}_h^n\) with \(h \in C_\textrm{c}^2({\textbf{R}}^{n-1})\) that is not identically zero, \(R_h>0\) represents the smallest positive real number such that \({\text {supp}} h \subseteq \overline{B_{R_h}(0')}\).
4.1 Estimate for the Normal Derivative in y of E
To be specific, we give an estimate for the boundary integral of \(\frac{\partial E}{\partial {\textbf{n}}_y}(x- \cdot )\) for \(x \in {\textbf{R}}_h^n\) that is close to the boundary \(\Gamma \). We have a compatible result with [14, Lemma 6], which deals with the case where the domain is bounded. For \(\rho \in (0,\rho _0]\), we define that \(\Gamma ^\Omega _\rho := \Gamma ^{{\textbf{R}}^n}_\rho \cap \Omega \).
Lemma 19
Let \({\textbf{R}}_h^n\) be a perturbed \(C^2\) half space of type (K) with boundary \(\Gamma = \partial {\textbf{R}}_h^n\), \(n \ge 2\) and \(\rho \in (0,\rho _0]\). Then, it holds that
-
(i)
$$\begin{aligned} \int _\Gamma \frac{\partial E}{\partial {\textbf{n}}_y} (x-y) \, d{\mathcal {H}}^{n-1}(y) = -\frac{1}{2} \quad \text {for any} \quad x \in {\textbf{R}}_h^n, \end{aligned}$$
-
(ii)
$$\begin{aligned} \sup _{x \in \Gamma ^\Omega _\rho } \, \int _\Gamma \left|\frac{\partial E}{\partial {\textbf{n}}_y} (x-y) \right|\, d{\mathcal {H}}^{n-1}(y) < C(n) C_{19}(K,h,\rho ) \end{aligned}$$
where
$$\begin{aligned} C_{19}(K,h,\rho ):= \big ( R_h^{n-1} + \rho K + \rho +1 \big ) \Vert h \Vert _{C^1({\textbf{R}}^{n-1})} + \Vert \nabla '^2 h \Vert _{L^\infty ({\textbf{R}}^{n-1})} + \rho + 1. \end{aligned}$$
Proof
-
(i)
This follows from the Gauss divergence theorem. For a bounded piecewise \(C^1\) domain \(D \subset {\textbf{R}}^n\), we have that
$$\begin{aligned} \int _{\partial D} \frac{\partial E}{\partial {\textbf{n}}_y} (x-y) \, d{\mathcal {H}}^{n-1}(y) = \int _D \Delta _y E(x-y) \, dy \end{aligned}$$for any \(x \in D\). Since \(\Delta _y E(x-y) = -\delta (x-y)\), we obtain that
$$\begin{aligned} \int _{\partial D} \frac{\partial E}{\partial {\textbf{n}}_y} (x-y) \, d{\mathcal {H}}^{n-1}(y) = -1 \end{aligned}$$for \(x \in D\). Let \(x \in {\textbf{R}}_h^n\). For \(R>0\), we define the domain \(D_R\) by
$$\begin{aligned} D_R:= \{ (y', y_n) \, \vert \, h(y')< y_n< \Vert h \Vert _{L^\infty ({\textbf{R}}^{n-1})} + 2 |x_n |, \, |y' |< R \}. \end{aligned}$$We consider \(R > R_h+ |x' |\). By applying the Gauss divergence theorem in \(D_R\), we deduce that
$$\begin{aligned} - 1&= \int _{y_n = \Vert h \Vert _{L^\infty ({\textbf{R}}^{n-1})} + 2 |x_n |, \, |y' |< R} \frac{\partial E}{\partial {\textbf{n}}_y} (x-y) \, d{\mathcal {H}}^{n-1}(y) \\&\quad + \int _{y \in \Gamma , \, |y' |< R} \frac{\partial E}{\partial {\textbf{n}}_y} (x-y) \, d{\mathcal {H}}^{n-1}(y) \\&\quad + \int _{0< y_n < \Vert h \Vert _{L^\infty ({\textbf{R}}^{n-1})} + 2 |x_n |, \, |y' |= R} \frac{\partial E}{\partial {\textbf{n}}_y} (x-y) \, d{\mathcal {H}}^{n-1}(y). \end{aligned}$$The last term tends to zero naturally as \(R \rightarrow \infty \). For the first term, since \({\textbf{n}}_y\) is pointing straightly upward but x is located below \(\{ (y',y_n) \, \vert \, y_n = \Vert h \Vert _{L^\infty ({\textbf{R}}^{n-1})} + 2 |x_n |\}\), the kernel \(\big ( \partial E / \partial {\textbf{n}}_y \big )(x-y)\) in this case is exactly the half of the Poisson kernel \(P_{\delta _{h,x_n}}(x'-y')\) with \(\delta _{h,x_n}:= \Vert h \Vert _{L^\infty ({\textbf{R}}^{n-1})} + 2 |x_n |- x_n\). Hence, the first integral on the right hand side tends to \(-\frac{1}{2}\) as \(R \rightarrow \infty \). We therefore obtain (i).
-
(ii)
Let us observe that
$$\begin{aligned} -{\textbf{n}}_y = -{\textbf{n}} \big ( y', h(y') \big ) = \big ( -\nabla ' h(y'), 1 \big )/ \omega (y') \end{aligned}$$where \(\omega (y')=\big ( 1+ \left|\nabla ' h(y')\right|^2 \big )^{1/2}\) and \(\nabla '\) is the gradient in \(y'\) variables. This implies that
$$\begin{aligned} - C(n) \frac{\partial E}{\partial {\textbf{n}}_y}(x-y) = \frac{\sigma (y')}{\omega (y') \left( |x'-y'|^2 + \big ( x_n- h(y') \big )^2 \right) ^{n/2}} \end{aligned}$$for \(y \in \Gamma \) with
$$\begin{aligned} \sigma (y'):= -\nabla ' h(y') \cdot (x'-y') + \big ( x_n - h(y') \big ) \; \; \text {where} \; \; x_n > h(x'),\ x',y' \in {\textbf{R}}^{n-1}. \end{aligned}$$We set that
$$\begin{aligned} K(x', y', x_n):= \frac{\sigma (y')}{\left( |x'-y' |^2 + \big ( x_n - h(y') \big )^2 \right) ^{n/2}}. \end{aligned}$$By the Taylor expansion, for \(|x'-y' |< 1\) we have that
$$\begin{aligned} h(x') = h(y') + \nabla ' h(y') \cdot (x'-y') + r(x',y') \end{aligned}$$with
$$\begin{aligned} r(x',y') = (x'-y')^{\textrm{T}} \cdot \int ^1_0 (1-\theta ) \Big ( \nabla '^2 h \Big ) \big ( \theta x'+(1-\theta )y' \big ) \, d\theta \cdot (x'-y'). \end{aligned}$$We obtain that
$$\begin{aligned} \sigma (y') = x_n - h(x') + r(x',y') \end{aligned}$$with an estimate
$$\begin{aligned} \left|r(x',y') \right|\le \Vert \nabla '^2 h \Vert _{L^\infty \big ( B_1(x') \big )} |x'-y' |^2. \end{aligned}$$(24)We decompose K into the sum of a leading term and a remainder term
$$\begin{aligned} K(x', y', x_n) = K_0(x', y', x_n) + R(x', y', x_n) \end{aligned}$$with
$$\begin{aligned} K_0(x', y', x_n)&:= \frac{x_n - h(x')}{\left( |x'-y' |^2 + \big ( x_n - h(y') \big )^2 \right) ^{n/2}}, \\ R(x', y', x_n)&:= \frac{r(x', y')}{\left( |x'-y' |^2 + \big ( x_n - h(y') \big )^2 \right) ^{n/2}}. \end{aligned}$$The term R is estimated as
$$\begin{aligned} \left|R(x', y', x_n) \right|\le \Vert \nabla '^2 h\Vert _{L^\infty \big ( B_1(x') \big )} |x'-y' |^{2-n} \end{aligned}$$for \(|x'-y' |<1\) by estimate (24). Hence,
$$\begin{aligned} \int _{\underset{|x'-y' |<1}{y \in \Gamma ,}} \left|\frac{R(x',y',x_n)}{\omega (y')} \right|\, d{\mathcal {H}}^{n-1}(y) \le C(n) \Vert \nabla '^2 h \Vert _{L^\infty ({\textbf{R}}^{n-1})}. \end{aligned}$$Since
$$\begin{aligned} \left|\sigma (y')\right|\le \left|\nabla ' h (y')\right|\cdot |x'-y' |+ |x_n |+ \left|h(y')\right|\end{aligned}$$for any \(y' \in {\textbf{R}}^{n-1}\), we have that
$$\begin{aligned}{} & {} \int _{\underset{|y'-x' |\ge 1}{y \in \Gamma ,}} \left|\frac{K(x',y',x_n)}{\omega (y')} \right|\, d{\mathcal {H}}^{n-1}(y) \le \int _{|y'-x' |\ge 1} \left|\nabla ' h(y')\right|\, dy' \nonumber \\{} & {} \quad + \int _{|y'-x' |\ge 1} \left|h(y')\right|\, dy' + \int _{|y'-x'|\ge 1} \frac{|x_n |}{|y'-x' |^n} \, dy'. \end{aligned}$$(25)Since \({\text {supp}} h \subseteq \overline{B_{R_h}(0')}\), the first two terms of estimate (25) can be estimated by the constant \(C(n) R_h^{n-1} \Vert h \Vert _{C^1({\textbf{R}}^{n-1})}\). On the other hand, since the estimate
$$\begin{aligned} \left|x_n - h(x') \right|\le (nK+1) \rho \Vert h \Vert _{C^1({\textbf{R}}^{n-1})} + \rho \end{aligned}$$holds for any \(x \in \Gamma ^\Omega _\rho \), The third term of estimate (25) can be controlled by the constant \(C(n) \big ( (\rho K + \rho + 1) \Vert h \Vert _{C^1({\textbf{R}}^{n-1})} + \rho \big )\). By (i), we observe that
$$\begin{aligned} \frac{C(n)}{2}&= \int _{\underset{|y'-x' |\ge 1}{y \in \Gamma ,}} \frac{K(x', y', x_n)}{\omega (y')} \, d{\mathcal {H}}^{n-1}(y) \\&\quad + \int _{\underset{|y'-x' |< 1}{y \in \Gamma ,}} \frac{K_0(x', y', x_n)}{\omega (y')} \, d{\mathcal {H}}^{n-1}(y) + \int _{\underset{|y'-x' |< 1}{y \in \Gamma ,}} \frac{R(x', y', x_n)}{\omega (y')} \, d{\mathcal {H}}^{n-1}(y). \end{aligned}$$The term \(K_0\) is very singular but it is positive for \(x \in \Gamma ^\Omega _\rho \). Hence, we have that
$$\begin{aligned} \int _{\underset{|y'-x' |<1}{y \in \Gamma ,}} \frac{K_0(x', y', x_n)}{\omega (y')} \, d{\mathcal {H}}^{n-1}(y) \le C(n) C_{19}(K,h,\rho ) \end{aligned}$$where the constant \(C_{19}(K,h,\rho )\) has the explicit expression
$$\begin{aligned} C_{19}(K,h,\rho ):= \big ( R_h^{n-1} + \rho K + \rho +1 \big ) \Vert h \Vert _{C^1({\textbf{R}}^{n-1})} + \Vert \nabla '^2 h \Vert _{L^\infty ({\textbf{R}}^{n-1})} + \rho +1. \end{aligned}$$Therefore, we finally obtain the estimate
$$\begin{aligned} \int _\Gamma \left|\frac{\partial E}{\partial {\textbf{n}}_y}(x-y) \right|\, d{\mathcal {H}}^{n-1}(y) \le C(n) C_{19}(K,h,\rho ) \end{aligned}$$which holds for any \(x \in \Gamma ^\Omega _\rho \). This completes the proof of Lemma 19.
\(\square \)
Before we end this subsection, we would like to give an estimate on the difference between gradients of the signed distance function near the boundary with explicit constant dependency on the boundary function h. This estimate plays an important role in later estimations of various boundary integrals.
Proposition 20
Let \({\textbf{R}}_h^n\) be a perturbed \(C^2\) half space with boundary \(\Gamma = \partial {\textbf{R}}_h^n\) and \(n \ge 2\). Then, for any \(x \in \Gamma ^{{\textbf{R}}^n}_{\rho _0}\) and \(y \in \Gamma \), it holds that
Proof
Let \(y \in \Gamma \), \(x \in \Gamma ^{{\textbf{R}}^n}_{\rho _0}\) and \(\pi x\) be the unique projection of x on \(\Gamma \). Note that
where \(\omega (\cdot ') =\big ( 1 + \big |\nabla ' h (\cdot ') \big |^2 \big )^{1/2}\) and \(\pi x'\) denotes the first \(n-1\) component of \(\pi x\). By rewriting
and applying the mean value theorem to both \(\nabla ' h (y') - \nabla ' h (\pi x')\) and \(\omega (\pi x') - \omega (y')\), we can deduce that
Since we must have \(|x-y|> |x - \pi x|\) for any \(y \in \Gamma \) such that \(y \ne \pi x\), by the triangle inequality \(|x - \pi x|\ge |y - \pi x|- |x-y|\), we can deduce that the inequality \(|y - \pi x|\le 2|x-y|\) holds for any \(y \in \Gamma \). Since obviously \(|\pi x' - y'|\le |\pi x - y|\), we obtain Proposition 20. \(\square \)
4.2 Criterion for a Class of Functions to be in \(H^{\frac{1}{2}}({\textbf{R}}^{n-1})\)
Let \(n \ge 2\). We say that \(f \in H^{\frac{1}{2}}({\textbf{R}}^{n-1})\) if \(f \in L^2({\textbf{R}}^{n-1})\) and
Our criterion is similar and compatible with [15, Lemma 3.2].
Proposition 21
Let \(n \ge 2\) and \(\rho >0\). Suppose that \(f \in C^1({\textbf{R}}^{n-1})\) satisfies
with some constants \(c_1\) and \(c_2\) independent of \(x' \in {\textbf{R}}^{n-1}\). Then the estimate
holds with some constant \(C(n)>0\) depending on n only.
Proof
By a direct calculation, we see that
For \(y' \in B_\rho (0')\) and \(x' \in B_{2 \rho }(0')^\textrm{c}\), we have that \(|x'-y' |\ge \rho \). Hence, we can deduce that
By symmetry, we also have that
Hence, it is sufficient to estimate
We then follow the similar idea that proves [15, Lemma 3.2]. Assume that \(|x' |\le |y' |\) and connect \(x'\) and \(y'\) by a geodesic curve in \(B_{|x' |}(0')^{\textrm{c}}\). Since the curve length is less than \((\pi /2)|x'-y' |\), by a fundamental theorem of calculus, we observe that
Since the integrand of I is symmetric with respect to \(x'\) and \(y'\), we now estimate
with
To estimate \(I_1\), we observe that
for any \(0< \delta < 1\) since \(|x'-y' |\le |x' |\). Thus,
To estimate \(I_2\), we observe that
since \(|x' |\le |y' |\). Since \(|x'-y' |\ge |x' |\) in this case, we have that
and
for any \(0<\delta <1\). Hence,
\(\square \)
4.3 \(L^\infty \) Estimate for the Trace Operator of Qg
For \(\rho \in (0,\infty )\), we let \(1'_{B_\rho (0')}\) to be the characteristic function associated with the open ball \(B_\rho (0')\) in \({\textbf{R}}^{n-1}\), i.e., we define that
For \(g \in L^\infty (\Gamma )\), we decompose g into the sum of the curved part \(g_1\) and the straight part \(g_2\) where
for any \(x' \in {\textbf{R}}^{n-1}\). Note that \(g_1, g_2 \in L^\infty (\Gamma )\). With respect to \(g_2\), we further define \(g_2^H \in L^\infty (\partial {\textbf{R}}_+^n)\) by setting
for \(x' \in {\textbf{R}}^{n-1}\).
Theorem 22
Let \({\textbf{R}}_h^n\) be a perturbed \(C^2\) half space with boundary \(\Gamma = \partial {\textbf{R}}_h^n\) and \(n \ge 3\). Moreover, let us assume that \(h \in C_\textrm{c}^2({\textbf{R}}^{n-1})\) satisfies the smallness condition
Then, the boundary trace of Qg is of the form
for \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\), where S is a bounded linear operator from \(L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) to \(L^\infty (\Gamma )\) satisfying
with some specific constant \(C^*_0(n)\) depending only on n and
Proof
For \(x \in \Gamma ^\Omega _{\rho _0}\), we decompose g into the straight part \(g_2\) and the curved part \(g_1\), i.e.,
Moreover, we further decompose \(I_2(x)\) as
Suppose that \(x \in \Gamma ^\Omega _{\rho _0}\) with \(|x' |\ge 2R_h\). Since \(|x' |\ge 2R_h\) is the straight part of \(\Gamma \), we have that
where \(P_{x_n}\) denotes the Poisson kernel. Let x tends \(x_0\) on the boundary, in this case we have that \(I_1(x)\) tends to \(\frac{1}{2} g_2^H(x_0)\), which is indeed \(\frac{1}{2} g(x_0)\). We then estimate \(I_{2,1}(x_0)\) for \(x_0 \in \Gamma \) with \(|x_0' |\ge 2R_h\). By Proposition 20, \(I_{2,1}(x_0)\) can be estimated as
If \(|x_0'|\ge 3R_h\), then we have that \(|x_0' - y'|\ge R_h\). In this case,
If \(|x_0' |< 3R_h\), then in this case we have the estimate
Hence, for \(x_0 \in \Gamma \) with \(|x_0' |\ge 2R_h\), we obtain that
Next, we estimate \(I_{2,2}(x_0)\) for \(x_0 \in \Gamma \) with \(|x_0' |\ge 2R_h\). Since \({\text {supp}} h \subseteq \overline{B_{R_h}(0')}\), for \(y \in \Gamma \) with \(R_h< |y' |<2R_h\) and \(x_0 \in \Gamma \) with \(|x_0' |\ge 2R_h\), we actually have that
Thus, for \(x_0 \in \Gamma \) with \(|x_0' |\ge 2R_h\),
By estimate (24), we have that
Since \(|x_0'|\ge 2R_h\), it holds that \(|x_0' - y'|\ge R_h\) for \(|y'|<R_h\). Hence, we obtain the estimate for \(\left|I_{2,2}(x_0) \right|\) for the case where \(|x_0'|\ge 2R_h\), i.e.,
Therefore, for \(x_0 \in \Gamma \) with \(|x_0' |\ge 2R_h\), by setting
we get that
with
Suppose now that \(x \in \Gamma ^\Omega _{\rho _0}\) with \(|x' |< 2R_h\). There exists a bounded \(C^2\) domain \(\Omega _c \subset {\textbf{R}}_h^n\) such that \(\partial \Omega _c \cap \Gamma = B'_\Gamma (2R_h)\). Let us recall a standard result concerning the double layer potential, see e.g. [22, Lemma 6.17]. Let \(f \in L^\infty (\partial \Omega _c)\), then the boundary trace of the double layer potential
is of the form
for \(w \in \partial \Omega _c\). We define \(g_c \in L^\infty (\partial \Omega _c)\) by letting
Thus, for any \(z \in \Omega _c\) we have that
Let x tends to \(x_0\) on the boundary, we deduce that
For \(x_0 \in \Gamma \) with \(|x_0' |<2R_h\), by applying Proposition 20 again, we see that \(\left|I_{2,1}(x_0) \right|\) can also be controlled by estimate (27) and (28), i.e., in this case we also have that
We now estimate \(I_{2,2}(x_0)\) for \(x_0 \in \Gamma \) with \(|x_0' |< 2R_h\). By estimate (24) again, we have that
Since in this case \(|x_0' |<2R_h\), \(|y' |<2R_h\) would imply that \(|x_0'-y' |<4R_h\), we have that
On the other hand, for \(y \in \Gamma \) such that \(|y'-x_0' |\ge 1\), we can straightforwardly estimate \(\left|\sigma (y')\right|\) in \(\partial E/ \partial {\textbf{n}}_y\) by \(\left|\nabla ' h(y')\right|\cdot |x_0'-y' |+ \left|h(x_0') \right|+ \left|h(y')\right|\). Hence, we have that
Combining these estimates together, we see that the estimate for \(\left|I_{2,1}(x_0)\right|+ \left|I_{2,2}(x_0) \right|\) reads as
In order to estimate \(I_1(x_0)\) for \(x_0 \in \Gamma \) with \(|x_0' |<2R_h\), we further decompose
for any \(y' \in {\textbf{R}}^{n-1}\) where \(r_h:= R_h^{\frac{1}{2n}}\) and
We next seek to control \(I_{1,1}(x_0)\) for \(x_0 \in \Gamma \) with \(|x_0' |<2R_h\). Let \(\theta _2 \in C_\textrm{c}^\infty ({\textbf{R}}^{n-1})\) be a cut-off function such that \(0 \le \theta _2 \le 1\) in \({\textbf{R}}^{n-1}\), \(\theta _2 = 1\) in \(B_1(0')\) and \({\text {supp}} \theta _2 \subseteq \overline{B_2(0')}\). We then set that
for \(y' \in {\textbf{R}}^{n-1}\). Since we are assuming that \(2R_h < r_h\), it holds that \(B_{2r_h}(x_0') \subset B_{3r_h}(0')\). Hence, \(1 - \theta _{2r_h,x_0} = 1\) in \(B_{2 r_h}(x_0')^\textrm{c}\) would imply that
Note that for any \(x_0,y \in \Gamma \) such that \(x_0 \ne y\), we have that
where \(\omega (x_0') = \big ( 1 + \left|\nabla ' h (x_0') \right|^2 \big )^{1/2}\). Through some simple calculations, we can deduce by the mean value theorem that the estimate
holds for any \(x_0,y \in \Gamma \) and the estimate
holds for any \(x_0,y \in \Gamma \) with \(x_0 \ne y\). In addition, for \(y \in \Gamma \) such that \(r_h<|y' - x_0'|<2r_h\), we have that
Hence, we see that the estimate
holds for any \(x_0,y \in \Gamma \). By the duality relation between \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) and \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\), we see that \(I_{1,1}(x_0)\) follows the estimate
By Proposition 13 and Proposition 21, we have that
Since \(L^{\frac{2n-2}{n}}({\textbf{R}}^{n-1})\) is continuously embedded in \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\), by estimate (18) and estimate (20) we see that
Therefore, the estimate for \(I_{1,1}(x_0)\) reads as
where
Since for \(I_{1,2}(x_0)\) and \(I_{1,3}(x_0)\), the integration region is bounded, \(I_{1,2}(x_0)\) and \(I_{1,3}(x_0)\) can be estimated in exactly the same way as \(I_{2,1}(x_0) + I_{2,2}(x_0)\) in the case where \(|x_0' |<2R_h\). As a result, here we directly give the estimate for \(I_{1,2}(x_0)\) and \(I_{1,3}(x_0)\) without going through what have already been done again. The estimate for \(I_{1,2}(x_0)\) and \(I_{1,3}(x_0)\) reads as
where
Therefore, for \(x_0 \in \Gamma \) with \(|x_0' |< 2R_h\), by setting
we obtain that
with
This completes the proof of Theorem 22. \(\square \)
4.4 \({\dot{H}}^{-\frac{1}{2}}\) Estimate for the Trace Operator S
In this subsection, we assume that \({\textbf{R}}_h^n\) is a perturbed \(C^2\) half space with boundary \(\Gamma = \partial {\textbf{R}}_h^n\) and \(n \ge 3\). We shall derive the \({\dot{H}}^{-\frac{1}{2}}\) estimate for the trace operator S from its \(L^{\frac{2n-2}{n}}\) estimate. We begin with the \(L^p\) estimate for S.
Lemma 23
Let \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\). Then, it holds that \(S g \in L^p(\Gamma )\) for any \(1 \le p < \infty \). For \(1< p < \infty \), Sg satisfies the estimate
with some specific constant \(C^*_1(n,p)>0\) that depends on n and p only. For \(p=1\), Sg satisfies the estimate
with some specific constant \(C^*_2(n)>0\) that depends on n only and
Proof
We firstly consider \(x \in \Gamma \) with \(|x'|< 3 R_h\). Since we already have the \(L^\infty \) estimate for Sg on \(\Gamma \) according to Theorem 22, the estimate
follows naturally, where the surface area \(\Lambda _{3R_h}\) is estimated by
Hence, for any \(1 \le p <\infty \), it holds that
Let \(1< p < \infty \). We then consider \(x \in \Gamma \) with \(|x'|\ge 3R_h\). For \(y \in \Gamma \) with \(|y'|< 2R_h\), the triangle inequality implies that \(|x'-y'|\ge |x'|- 2R_h\). In this case, we deduce that
Hence, we have that
where the integral on the right hand side can be estimated as
Therefore, we obtain that
For \(x \in \Gamma \) with \(|x'|\ge 3R_h\), we indeed have that \(\nabla d(x) = (0,\ldots ,0,1)\). Thus, Sg(x) has the form
Since \(|x'-y'|\ge |x'|- 2R_h\) for any \(|y'|< 2R_h\), \(\left|Sg(x)\right|\) can thus be estimated as
Hence,
\(\square \)
We are now ready to state the \({\dot{H}}^{-\frac{1}{2}}\) estimate for the trace operator S.
Corollary 24
For \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\), we have that \(S g \in {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) satisfying
with some specific constant \(C^*_3(n)>0\) that depends on n only.
Proof
Since \(L^{\frac{2n-2}{n}}(\Gamma )\) is continuously embedded in \({\dot{H}}^{-\frac{1}{2}}(\Gamma )\), by considering estimate (20) and Lemma 23 with \(p = \frac{2n-2}{n}\), we obtain Corollary 24. \(\square \)
We define constants
and \(C^*(n):= C^*_0(n) + C^*_3(n)\). We would like to emphasize that \(C^*(n)\) is a specific constant that depends on dimension n only and \(C_*(h)\) is a constant that depends on the boundary function h only. Theorem 22 and Corollary 24 guarantee that the trace operator \(S: L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma ) \rightarrow L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) is bounded linear and
Moreover, we can require \(C_*(h)\) to be arbitrarily small by taking \(R_h\) to be sufficiently small.
5 Neumann Problem with Bounded Data in a Perturbed \(C^2\) Half Space with Small Perturbation
We consider the Neumann problem for the Laplace equation in a perturbed \(C^2\) half space in \({\textbf{R}}^n\) with \(L^\infty \)-initial data for \(n \ge 3\). We shall begin with the half space problem. It is well-known that a solution of the Neumann problem
is formally given by
where N denotes the Neumann-Green function
Its exterior normal derivative \(\partial N/ \partial {\textbf{n}}_x\) for \(y_n=0\) is nothing but the Poisson kernel with the parameter \(x_n\). By symmetry we observe that
as \(x_n>0\) tends to zero. Thus u gives a solution to (29) formally. The function
is called the single layer potential of g.
For \(g \in L^\infty ({\textbf{R}}^{n-1})\), we let \({\widetilde{g}}(x',x_n):= g(x',0)\) for any \(x \in {\textbf{R}}^n\). Natrually, \({\widetilde{g}} \in L^\infty ({\textbf{R}}^n)\). Let \(1_{{\textbf{R}}_+^n}\) be the characteristic function associated with the half space \({\textbf{R}}_+^n\). In this case, we have that
Hence, by the \(L^\infty \)-BMO estimate for the singular integral operator [19, Theorem 4.2.7], we have the estimate
Moreover, since \(- \partial _{x_n} (E *(\delta _{\partial {\textbf{R}}_+^n} \otimes g))\) is the half of the Poisson integral, i.e.,
the estimate
holds for any \(g \in L^\infty ({\textbf{R}}^{n-1})\), see [14, Lemma 7]. We are able to to establish similar estimates for the case where the domain is a perturbed \(C^2\) half space.
Lemma 25
Let \({\textbf{R}}_h^n\) be a perturbed \(C^2\) half space of type (K) with boundary \(\Gamma =\partial {\textbf{R}}_h^n\) and \(n \ge 3\). Then,
-
(i)
(BMO estimate) For all \(g\in L^\infty (\Gamma )\), the estimate
$$\begin{aligned} \left[ \nabla E * \big ( \delta _\Gamma \otimes g \big ) \right] _{BMO({\textbf{R}}^n)} \le C(n) C_{25,i}(h,\rho _0) \Vert g\Vert _{L^\infty (\Gamma )} \end{aligned}$$(33)holds with
$$\begin{aligned} C_{25,i}(h,\rho _0):= C_s(h)^2 (R_h + \rho _0 +1)^n \big ( \Vert \nabla '^2 h \Vert _{L^\infty ({\textbf{R}}^{n-1})} + \rho _0^{-1} \big ). \end{aligned}$$ -
(ii)
(\(L^\infty \) estimate for normal component) For all \(g\in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\), the estimate
$$\begin{aligned} \left\| \nabla d \cdot \nabla E * \big ( \delta _\Gamma \otimes g \big ) \right\| _{L^\infty \big ( \Gamma ^\Omega _{\rho _0} \big )} \le C(n) C_{25,ii}(K,h,\rho _0) \Vert g\Vert _{L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )} \end{aligned}$$(34)holds with
$$\begin{aligned}&C_{25,ii}(K,h,\rho _0) := C_s(h) \big ( R_h^{n-1} + \rho _0 K + \rho _0 + 2 \big ) + \rho _0 \\&\quad + C_s(h)^2 (2+6R_h) \Vert \nabla '^2 h \Vert _{L^\infty ({\textbf{R}}^{n-1})} + C_s(h)^{n+10} C_1(h)^3 (1 + 3R_h)^{\frac{n}{2}}. \end{aligned}$$
For \(g \in L^\infty (\Gamma )\), the notation \(E*(\delta _\Gamma \otimes g)\) in Lemma 25 means that
5.1 BMO Estimate
We follow the idea of the proof for [14, Lemma 5 (i)], which establishes the same BMO estimate in the case where the domain is a bounded \(C^2\) doamin.
Proof of Lemma 25 (i)
For \(g \in L^\infty (\Gamma )\), we follow the setting in Sect. 4.3 to decompose g into the curved part \(g_1\) and the straight part \(g_2\) and let \(g_2^H \in L^\infty ({\textbf{R}}^{n-1})\) be defined as expression (26). Since by definition we have that
the estimate
follows from estimate (31). As we are now considering the case where the boundary \(\Gamma \) is uniformly \(C^2\), the signed distance function d is \(C^2\) in \(\Gamma ^{{\textbf{R}}^n}_{\rho _0}\), see e.g. [20, Section 14.6]. We consider \(\theta \in C_\textrm{c}^\infty ({\textbf{R}})\) such that \(0 \le \theta \le 1\), \(\theta (\sigma )=1\) for \(|\sigma |\le 1\) and \(\theta (\sigma )=0\) for \(|\sigma |\ge 2\). Note that \(\theta _d:= \theta (4d/\rho _0)\) is \(C^2\) in \({\textbf{R}}^n\). We extend \(g_1 \in L^\infty (\Gamma )\) to \(g_1^\textrm{e} \in L^\infty \big ( \Gamma ^{{\textbf{R}}^n}_{\rho _0/2} \big )\) by setting
for any \(x \in \Gamma ^{{\textbf{R}}^n}_{\rho _0/2}\) with \(\pi x\) denoting the unique projection of x on \(\Gamma \). For \(x \in \Gamma ^{{\textbf{R}}^n}_{\rho _0/2}\), by considering the normal coordinate \(x = F_{\pi x}(\eta )\) in \(U_{\rho _0/2}(\pi x)\), we have that
as \((g_1^\textrm{e})_{F_{\pi x}}(\eta ', \tau _1) = (g_1^\textrm{e})_{F_{\pi x}}(\eta ', \tau _2)\) for any \(|\eta '|< \rho _0/2\) and \(\tau _1,\tau _2 \in (- \rho _0/2, \rho _0/2)\). Here the notation \((f)_{F_{\pi x}}\) represents the composition of f and \(F_{\pi x}\), i.e., \((f)_{F_{\pi x}}:= f \circ F_{\pi x}\). Hence, we see that \(\nabla d \cdot \nabla g_1^\textrm{e} = 0\) in \(\Gamma ^{{\textbf{R}}^n}_{\rho _0/2}\).
Let us consider \(g_{1,\textrm{c}}^\textrm{e}:= \theta _d g_1^\textrm{e}\). A key observation is that
Thus,
where \(f_{\theta ,\rho _0/4}:= \theta _d \Delta d + \frac{4\theta '(4d/\rho _0)}{\rho _0}\). By the \(L^\infty \)-BMO estimate for the singular integral operator [19, Theorem 4.2.7], the first term is estimated as
Since
for \(x \in {\textbf{R}}^n\) with \(d(x,U_{\textrm{c}, \rho _0/2}) = \inf _{y \in U_{\textrm{c}, \rho _0/2}} |x-y|< 1\) we have that
where
For \(x \in {\textbf{R}}^n\) with \(d(x, U_{\textrm{c}, \rho _0/2}) = \inf _{y \in U_{\textrm{c}, \rho _0/2}} |x-y|\ge 1\), same estimate above for \(\left|I_2(x) \right|\) holds trivially as \(|x-y|^{-(n-1)} \le 1\) for any \(y \in U_{\textrm{c}, \rho _0/2}\). The proof of the first part of Lemma 25 is now complete. \(\square \)
5.2 \(L^\infty \) Estimate for Normal Component
The \(L^\infty \) estimate to the normal component of \(\nabla E *(\delta _\Gamma \otimes g)\) within a small neighborhood of \(\Gamma \) for \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) can be derived by almost the same argument as establishing the boundedness of the trace operator S from \(L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) to \(L^\infty (\Gamma )\) in Theorem 22.
Proof of Lemma 25 (ii)
Let \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) and \(x \in \Gamma ^\Omega _{\rho _0}\). Suppose firstly that \(|x'|\ge 3R_h\). By following the setting in Sect. 4.3 to decompose g into the curved part \(g_1\) and the straight part \(g_2\), we have that
where \(g_2^H = T_h^{-1}(g_2)\). By estimate (32), we see that
Since \(|x'|\ge 3R_h\), for any \(y \in B'_\Gamma (2R_h)\) we have that \(|x-y |\ge |x'-y' |\ge |x'|-|y'|\ge R_h\). Hence,
Thus, for \(x \in \Gamma ^\Omega _{\rho _0}\) with \(|x'|\ge 3R_h\), we show that
Next, we consider the case where \(|x'|< 3R_h\). In this case, we consider a modified decomposition of g. We let \(g_1^*:= 1'_{B_{1+3R_h}(0')} \cdot g\) and \(g_2^*:= g - g_1^*\) where \(1'_{B_{1+3R_h}(0')}\) is the characteristic function associated with the open ball \(B_{1+3R_h}(0')\) in \({\textbf{R}}^{n-1}\). We firstly deal with the modified curved part \(g_1^*\). Note that
By Proposition 20, we have that
On the other hand, by Lemma 19 we have that
Then, we deal with the modified straight part \(g_2^*\). Let \(\theta _*\in C_\textrm{c}^\infty ({\textbf{R}}^{n-1})\) be a cut-off function such that \(0 \le \theta _*\le 1\), \(\theta _*= 1\) in \(B_{\frac{1}{2} + 3R_h}(0')\) and \({\text {supp}} \theta _*\subseteq \overline{B_{1+3R_h}(0')}\). Let \(x \in \Gamma ^\Omega _{\rho _0}\) with \(|x'|<3R_h\). We define that
Note that
and for any \(x \in \Gamma ^\Omega _{\rho _0}\) and \(y \in \Gamma \) with \(x \ne y\), it holds that
where \(\omega (x') = \big ( 1 + \big |\nabla ' h (x') \big |^2 \big )^{\frac{1}{2}}\). By following similar calculations in the proof of Theorem 22, we can deduce that \(K_x(\cdot ') \in C^1({\textbf{R}}^{n-1})\) satisfies \({\text {supp}} K_x(\cdot ') \subset B_{\frac{1}{2} + 3R_h}(x')^{\textrm{c}}\),
Hence, by the duality relation between \({\dot{H}}^{\frac{1}{2}}({\textbf{R}}^{n-1})\) and \({\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\), Proposition 13, Proposition 21, estimate (18) and estimate (20), we can deduce that
\(\square \)
We would like to emphasize that it is insufficient to obtain an \(L^\infty \) estimate for the normal component of \(\nabla E *(\delta _\Gamma \otimes g)\) in a small neighborhood of \(\Gamma \) if we only assume that \(g \in L^\infty (\Gamma )\). Let B be a ball centered at 0 with radius \(r_B\) such that \(B'_\Gamma (2R_h) \subset B/2\). By almost the same argument as in the proof of Lemma 25 (ii), we can see that if \(x \in \Gamma _{\rho _0}^\Omega \) with \(|x'|\ge r_B/2\), then we have the estimate
In addition, if \(x \in \Gamma _{\rho _0}^\Omega \) with \(|x' |< r_B/2\), we decompose \(g = g_c + g_s\) where
for any \(x' \in {\textbf{R}}^{n-1}\). Since \(g_c\) is compactly supported, we also have that
The main barrier comes from the contribution of \(g_s\) in the case that \(|x'|< r_B/2\).
Proposition 26
For \(1 \le j \le n-1\), there does not exist a constant \(C>0\) such that
for any \(|x' |< r_B/2\) and \(g \in L^\infty ({\textbf{R}}^{n-1})\) with \({\text {supp}} g \subseteq B_{r_B}(0')^\textrm{c}\).
Proof
Let \(1 \le j \le n-1\). Note that
where \(R_j(g)\) represents the j-th Riesz transform of g. Let \(\theta _2 \in C_\textrm{c}^\infty ({\textbf{R}}^{n-1})\) be a cut-off function that satisfies \(0 \le \theta _2 \le 1\), \(\theta _2 = 1\) in \(B_1(0')\) and \({\text {supp}} \theta _2 \subseteq \overline{B_2(0')}\). We set \(\psi _{r_B/4}(z'):= \theta _2\big ( \frac{4 z'}{r_B} \big )\), \(\phi _{r_B}(z'):= 1 - \theta _2(z'/r_B)\), \(\psi _{r_B/16}(z'):= \theta _2\big ( \frac{16 z'}{r_B} \big )\) and
for any \(z' \in {\textbf{R}}^{n-1}\) and \(1 \le j \le n-1\).
We assume the contrary of Proposition 26. Suppose that there exist a constant \(C'>0\) such that
for any \(|x' |< r_B/2\) and \(g \in L^\infty ({\textbf{R}}^{n-1})\) with \({\text {supp}} g \subseteq B_{r_B}(0')^\textrm{c}\). As a consequence, the estimate
holds for any \(g \in L^\infty ({\textbf{R}}^{n-1})\) where
With respect to \(f \in L^1({\textbf{R}}^{n-1})\), we can define the adjoint operator of P by
Estimate (35) implies that \(P^*\) is a bounded linear operator which maps \(L^1({\textbf{R}}^{n-1})\) to \(L^1({\textbf{R}}^{n-1})\), i.e., it holds that
for any \(f \in L^1({\textbf{R}}^{n-1})\). For \(t>0\), we consider the Gaussian function
Since \(R_j^*(\cdot ') \psi _{r_B/4}(x' - \cdot ') \in C_\textrm{c}^\infty ({\textbf{R}}^{n-1})\) and
in the sense of distributions, we see that
Since \(\Vert f_t \Vert _{L^1({\textbf{R}}^{n-1})} = 1\) for any \(t>0\), estimate (36) implies that for any \(t>0\), it holds that
Hence, by the Bolzano–Weierstrass theorem, there exists a sequence \(\{ t_m \}_{m \in {\textbf{N}}}\) which converges to zero so that the sequence
is convergent. By Fatou’s lemma, we can then conclude that
However, for \(|x' |\ge 2r_B\) we have that \(\phi _{r_B}(x') R_j^*(x') = x_j / |x' |^n\), which is clearly not \(L^1\) integrable in the region \(\{x' \in {\textbf{R}}^{n-1} \,\vert \, |x' |\ge 2r_B \}\). We reach a contradiction. \(\square \)
5.3 \(L^2\) Estimate for the Gradient of the Single Layer Potential
We begin with the half space case.
Proposition 27
Let \(n \ge 3\). For any \(g \in {\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\), the estimate
holds with some constant \(C(n)>0\) that depends on dimension n only.
Proof
We consider the partial Fourier transform of E with respect to \(x'\), i.e., we let
Since \(E(x',x_n)\) is radial symmetric in \({\textbf{R}}^{n-1}\) for any fixed \(x_n>0\), \({\widehat{E}}'(\xi ',x_n)\) can be calculated by the Hankel transform of order \(\frac{n-3}{2}\) of the function \(r^{\frac{n-3}{2}} E(r,x_n)\) where
i.e., we have that
where \(J_{\frac{n-3}{2}}\) is the Bessel function of the first kind of order \(\frac{n-3}{2}\), see e.g. [17, Formula 6.565.2]. Then by the Fourier-Plancherel formula, we obtain that
and
\(\square \)
We then generalize this result to arbitrary perturbed \(C^2\) half space \({\textbf{R}}_h^n\).
Lemma 28
Let \({\textbf{R}}_h^n\) be a perturbed \(C^2\) half space with boundary \(\Gamma = \partial \Omega \) and \(n \ge 3\). For any \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\), the estimate
holds with
Proof
Let \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\). Following the setting in Sect. 4.3, we decompose g into the curved part \(g_1 = 1'_{B_{2R_h}(0')} g\) and the straight part \(g_2 = g - g_1\). Since \(g \in L^\infty (\Gamma )\) and \(L^{\frac{2n-2}{n}}(\Gamma )\) is continuously embedded in \({\dot{H}}^{-\frac{1}{2}}(\Gamma )\), by estimate (20) we see that \(g_1 \in {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) satisfies
Hence, it holds that both \(g_1, g_2 \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\). With respect to \(g_2\), let \(g_2^H\) be defined by expression (26). Since \(T_h(g_2^H) = g_2\) and the mapping \(T_h: {\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1}) \rightarrow {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) is an isomorphism, by estimate (18) that \(g_2^H \in {\dot{H}}^{-\frac{1}{2}}({\textbf{R}}^{n-1})\) satisfies
Next, we follow the proof of Lemma 25 (i) to estimate the \(L^2\) norm of \(\nabla E *\big ( \delta _\Gamma \otimes g_1 \big )\). We consider \(\theta \in C_\textrm{c}^\infty ({\textbf{R}})\) such that \(0 \le \theta \le 1\), \(\theta (\sigma )=1\) for \(|\sigma |\le 1\) and \(\theta (\sigma )=0\) for \(|\sigma |\ge 2\). We let \(\theta _d:= \theta (4d/\rho _0)\) where d is the signed distance function defined by expression (5). Note that this \(\theta _d\) is \(C^2\) in \({\textbf{R}}^n\). We extend \(g_1 \in L^\infty (\Gamma )\) to \(g_1^\textrm{e} \in L^\infty \big ( \Gamma ^{{\textbf{R}}^n}_{\rho _0/2} \big )\) by setting \(g_1^\textrm{e}(x):= g_1(\pi x)\) for any \(x \in \Gamma ^{{\textbf{R}}^n}_{\rho _0/2}\) with \(\pi x\) denoting the unique projection of x on \(\Gamma \). It holds that \(\nabla d \cdot \nabla g_1^\textrm{e}=0\) in \(\Gamma ^{{\textbf{R}}^n}_{\rho _0/2}\). We then set \(g_{1,\textrm{c}}^\textrm{e}:= \theta _d g_1^\textrm{e}\). For any \(x \in {\textbf{R}}^n\), we have that
where \(f_{\theta ,\rho _0/4}:= \theta _d \Delta d + \frac{4\theta '(4d/\rho _0)}{\rho _0}\). Since \(\nabla {\text {div}} E\) is bounded in \(L^p\) for any \(1<p<\infty \), see e.g. [18, Theorem 5.2.7 and Theorem 5.2.10], we can deduce that
as \({\text {supp}} g_{1,\textrm{c}}^{\textrm{e}} \subset \{ x \in \Gamma ^{{\textbf{R}}^n}_{\rho _0 /2} \,\vert \, |(\pi x)'|< 2R_h \}\). Since \(\nabla E(x)\) is an integration kernel that is dominated by \(C(n) |x |^{1-n}\) for \(x \in {\textbf{R}}^n {\setminus } \{0\}\), by the famous Hardy–Littlewood–Sobolev inequality, see e.g. [1, Theorem 1.7], we have that
where \(r= \frac{2n}{n+2}\). Since \({\text {supp}} g_1^{\textrm{e}} f_{\theta ,\rho _0 /4} \subset \{ x \in \Gamma ^{{\textbf{R}}^n}_{\rho _0 /2} \,\vert \, |(\pi x)'|< 2R_h \}\), we deduce that
Hence, we obtain the \(L^2\) estimate for \(g_1\), i.e.,
where
The \(L^2\) estimate of \(\nabla E *\big ( \delta _\Gamma \otimes g_2 \big )\) in \({\textbf{R}}_+^n\) follows directly from Proposition 27 and estimate (37). We have that
Note that for any \(x \in {\textbf{R}}^n\), it holds that
Moreover, for any \(x = (x',x_n) \in {\textbf{R}}_+^n\) we have that
Therefore, the \(L^2\) estimate of \(\nabla E *\big ( \delta _\Gamma \otimes g_2 \big )\) in \({\textbf{R}}^n\) reads as
\(\square \)
5.4 Solution to the Neumann Problem
Let \({\textbf{R}}_h^n\) be a perturbed \(C^2\) half space with boundary \(\Gamma = \partial {\textbf{R}}_h^n\) and \(n \ge 3\). We further assume that \({\textbf{R}}_h^n\) has small perturbation, i.e., we require that the boundary function \(h \in C_c^2({\textbf{R}}^{n-1})\) satisfies
where \(C^*(n)\) is a specific constant that depends on dimension n only. Under this setting, we are able to construct a solution to Neumann problem (2).
Proof of Lemma 4
Let \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\). By Corollary 24 and Theorem 22, for any \(i \in {\textbf{N}}\) we have that
Since we are now assuming that \(2 C^*(n) C_*(h) < 1\), the operator \(I - 2S\), which is bounded linear from \(L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) to \(L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\), admits a well-defined inverse constructed by the Neumann series
in the sense that \((I-2\,S)^{-1}: L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma ) \rightarrow L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\) is also bounded linear. For \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\), we have that
Hence, with respect to \(g \in L^\infty (\Gamma ) \cap {\dot{H}}^{-\frac{1}{2}}(\Gamma )\), we claim that the solution to Neumann problem (2) can be constructed as
A simple check ensures that u satisfies Neumann problem (2) formally. It is sufficient to establish the \(vBMOL^2\) estimate for \(\nabla u\). The \(vBMO^{\infty , \rho _0}\)-norm for \(\nabla u\) in \({\textbf{R}}_h^n\) is guaranteed by Lemma 25. By estimate (33) and estimate (34), we have that
where \(C_{25}(K,h,\rho _0):= C_{25,i}(h,\rho _0) + C_{25,ii}(K,h,\rho _0)\). The \(L^2\) estimate of \(\nabla u\) follows directly from Lemma 28, we have that
This completes the proof of Lemma 4. \(\square \)
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Acknowledgements
The first author was partly supported by the Japan Society for the Promotion of Science through Grants No. 19H00639 (Kiban A), No. 18H05323 (Kaitaku), No. 17H01091 (Kiban A) and by Arithmer Inc. and Daikin Industries Ltd. through a collaborative grant.
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Giga, Y., Gu, Z. The Helmholtz Decomposition of a BMO Type Vector Field in a Slightly Perturbed Half Space. J. Math. Fluid Mech. 25, 41 (2023). https://doi.org/10.1007/s00021-023-00781-z
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DOI: https://doi.org/10.1007/s00021-023-00781-z