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On the Properties of Quasi-periodic Boundary Integral Operators for the Helmholtz Equation

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Abstract

We study the mapping properties of boundary integral operators arising when solving two-dimensional, time-harmonic waves scattered by periodic domains. For domains assumed to be at least Lipschitz regular, we propose a novel explicit representation of Sobolev spaces for quasi-periodic functions that allows for an analysis analogous to that of Helmholtz scattering by bounded objects. Except for Rayleigh-Wood frequencies, continuity and coercivity results are derived to prove wellposedness of the associated first kind boundary integral equations.

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Correspondence to Carlos Jerez-Hanckes.

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This work was partially funded by Fondecyt Regular 1171491 and by grants Conicyt-PFCHA/Doctorado Nacional/2017-21171791 and 2017-21171479.

Appendices

Appendix A: Proof of Theorem 4.4

In order to prove the boundedness of the quasi-periodic Newton potential, we shall make use of the following lemma.

Lemma A.1

Let \(g\in {\mathcal {D}}({{\mathbb {R}}})\), and \(\xi \in {{\mathbb {R}}},\ \left| \xi \right| >0\). Then, for some \(C>0\), it holds that

$$\begin{aligned} \left| \int _{0}^{\infty }g(x)\sin {\left( \xi x\right) }dx\right| \le C\frac{1}{\left| \xi \right| },\qquad \left| \int _{0}^{\infty }g(x)\cos {\left( \xi x\right) }dx\right| \le C\frac{1}{\left| \xi \right| ^2}. \end{aligned}$$
(31)

Proof

Let \(R>0\) be so that the support of g(x) is contained in \([-R,R]\). Then,

$$\begin{aligned} \int _{0}^{\infty }g(x)\sin {\left( \xi x\right) }dx&=\frac{1}{\xi }\int _{0}^{\infty }g\left( \frac{t}{\xi }\right) \sin {\left( t\right) }\;{\text {d}}{t} \\&=\frac{1}{\xi }\left( \left. g\left( \frac{t}{\xi }\right) \cos {\left( t\right) } \right| _{0}^{\infty }+\int _{0}^{\infty }\frac{{\text {d}}}{{\text {d}}{t}} \left( g\left( \frac{t}{\xi }\right) \right) \cos {\left( t\right) }\;{\text {d}}{t}\right) \\&=\frac{1}{\xi }\left( g(0)+\frac{1}{\xi }\int _{0}^{\infty }g'\left( \frac{t}{\xi } \right) \cos {\left( {t}\right) }\;{\text {d}}{t}\right) , \\ \left| \int _{0}^{\infty }g(x)\sin {\left( \xi x\right) }dx\right|&= \frac{1}{|\xi |} \left| g(0)+\frac{1}{\xi }\int _{0}^{R\xi }g'\left( \frac{t}{\xi }\right) \cos {\left( t\right) }\;{\text {d}}{t}\right| \\&\le \frac{1}{|\xi |}\left( |g(0)|+R\max _{t\in [0,R]}{|g'(t)|} \right) . \end{aligned}$$

Also, the second inequality follows by integration-by-parts:

$$\begin{aligned} \int _{0}^{\infty }g(x)\cos {\left( \xi x\right) }dx&=\left. g(x)\sin {\left( \xi x\right) } \right| _{0}^\infty -\frac{1}{\xi }\int _{0}^{\infty }g'(x)\sin {\left( \xi x\right) }dx, \\&=-\frac{1}{\xi }\int _{0}^{\infty }g'(x)\sin {\left( \xi x\right) }dx. \end{aligned}$$

\(\square \)

We now prove Theorem 4.4 by adapting a strategy similar to that used in [42, Theorem 6.1].

Proof of Theorem 4.4

First, consider \(f\in {\mathcal {D}}_\theta ({\mathcal {G}})\) for which the expression

$$\begin{aligned} f(\varvec{x})=\sum _{j\in {{\mathbb {Z}}}}f_j(x_2)e^{\imath j_\theta x_1} \end{aligned}$$

holds. Since f has compact support in the \(x_2\)-direction, there exists some positive \(r \in {{\mathbb {R}}}\) such that \(f_j(x_2) = 0\) if \(\left| x_2\right| >r\), for all \(j \in {{\mathbb {Z}}}\). Fix \(R>0\) and set \(u:={\mathcal {N}}^k_\theta f\). Then, u is a quasi-periodic function on \({\mathcal {G}}\) by the quasi-periodicity of the Green’s function. Consider \(\mu \in {\mathcal {D}}({{\mathbb {R}}})\) such that \(\mu (t)=1\), for all \(t\in [0,r+R]\). We define a modified version of u as

$$\begin{aligned} u_\mu (\varvec{x}):=\int _{{\mathcal {G}}}G^k_\theta (\varvec{x},\varvec{y})\mu (\left| x_2-y_2\right| )f(\varvec{y})d\varvec{y}. \end{aligned}$$

Notice that for \(\varvec{x}\in {\mathcal {G}}^R := {\mathcal {G}}\cap \left\{ \left| x_2\right| < R\right\} \), \(u_\mu (\varvec{x}) = u(\varvec{x})\). Hence, \(u_\mu \) is an extension of u and, from the norm definition for \(H^s_\theta \left( {\mathcal {G}}^R\right) \), we find that \(\left\Vert u\right\Vert _{H^s_\theta \left( {\mathcal {G}}^R\right) }\le \left\Vert u_\mu \right\Vert _{H^s_\theta ({\mathcal {G}})}\). We now prove the boundedness of \(\left\Vert u_\mu \right\Vert _{H^s_\theta ({\mathcal {G}})}\). Since \(u_\mu \) is also \(\theta \)-quasi-periodic, it holds

$$\begin{aligned} u_{\mu ,j}(x_2)&=\int _{0}^{2\pi }u_{\mu }(x_1,x_2)e^{-\imath j_\theta x_1}\;{\text {d}}x_1,\\ {\widehat{u}}_{\mu ,j}(\xi )&=\int _{{{\mathbb {R}}}}e^{-\imath 2\pi x_2\xi }\int _{0}^{2\pi }u_{\mu }(x_1,x_2)e^{-\imath j_\theta x_1}\;{\text {d}}x_1{\text {d}}x_2 \\&=\int _{{{\mathbb {R}}}}\int _{0}^{2\pi }\int _{{{\mathbb {R}}}}\int _0^{2\pi }e^{-\imath 2\pi x_2\xi }e^{-\imath j_\theta x_1}G^k_{\theta }(\varvec{x},\varvec{y})\mu (\left| x_2-y_2\right| )f(\varvec{y})\;{\text {d}}y_1{\text {d}}y_2{\text {d}}x_1{\text {d}}x_2. \end{aligned}$$

Since \(\mu \) and f have compact support, we can exchange the integration order so as to write

$$\begin{aligned}&{\widehat{u}}_{\mu ,j}(\xi )\\&\quad =\int _{{{\mathbb {R}}}}\int _{0}^{2\pi }\int _{{{\mathbb {R}}}}\int _0^{2\pi }e^{-\imath 2\pi x_2\xi }e^{-\imath j_\theta x_1}G^k_{\theta }(\varvec{x},\varvec{y})\mu (\left| x_2-y_2\right| )f(\varvec{y})\;{\text {d}}x_1{\text {d}}x_2 {\text {d}}y_1{\text {d}}y_2 \\&\quad =\int _{{{\mathbb {R}}}}\int _{0}^{2\pi }\int _{{{\mathbb {R}}}}\int _{-y_1}^{2\pi -y_1}e^{-\imath 2\pi (z_2+y_2)\xi }e^{-ij_\theta (z_1+y_1)}G^k_{\theta }(\varvec{z},\varvec{0})\mu (\left| z_2\right| )f(\varvec{y})\; {\text {d}}z_1{\text {d}}z_2{\text {d}}y_1{\text {d}}y_2 \\&\quad =\int _{{{\mathbb {R}}}}\int _{0}^{2\pi }\int _{{{\mathbb {R}}}}\int _{0}^{2\pi }e^{-\imath 2\pi (z_2+y_2)\xi }e^{-\imath j_\theta (z_1+y_1)}G^k_{\theta }(\varvec{z},\varvec{0})\mu (\left| z_2\right| )f(\varvec{y})\;{\text {d}}z_1 {\text {d}}z_2{\text {d}}y_1{\text {d}}y_2, \end{aligned}$$

and where we used the periodicity of \(e^{-\imath j_\theta z_1}G^k_\theta (\varvec{z},0)\). Then, replacing \(G^k_{\theta }\) by its expansion (Proposition 4.2) yields

$$\begin{aligned} {\widehat{u}}_{\mu ,j}(\xi )={\widehat{f}}_j(\xi )\frac{1}{i\beta _j}\int _{{{\mathbb {R}}}}e^{-\imath 2\pi z_2\xi }e^{\imath \beta _j\left| z_2\right| }\mu (\left| z_2\right| )\;{\text {d}}z_2. \end{aligned}$$

Observe that

$$\begin{aligned} \frac{1}{\imath \beta _j}\int _{{{\mathbb {R}}}}e^{-\imath 2\pi z_2\xi }e^{i\beta _j\left| z_2\right| }\mu (\left| z_2\right| )\;{\text {d}}z_2=\frac{2}{\imath \beta _j}\int _{0}^{\infty }e^{\imath \beta _j{z_2}}\mu ({z_2})\cos {\left( 2\pi \xi z_2\right) }\;{\text {d}}z_2 \end{aligned}$$

and consider \(j_\theta \) such that \(\beta _j\in {{\mathbb {R}}}\), i.e. \(j_\theta ^2<k^2\). From Lemma A.1, we get

$$\begin{aligned} \left| \frac{2}{\imath \beta _j}\int _{0}^{\infty }e^{\imath \beta _j\left| z_2\right| }\mu (\left| z_2\right| )\cos {\left( 2\pi \xi z_2\right) }\;{\text {d}}z_2\right| \le C_k\frac{1}{\left| \xi \right| ^2+1}. \end{aligned}$$

Furthermore, since \(\beta _j\) is real for a finite number of j, depending only on k and \(\theta \), then for all \(j\in {{\mathbb {Z}}}\) such that \(j_\theta <k^2\), yields

$$\begin{aligned} \left| \frac{2}{\imath \beta _j}\int _{0}^{\infty }e^{\imath \beta _j\left| z_2\right| }\mu ({z_2})\cos {\left( 2\pi \xi z_2\right) }\;{\text {d}}z_2\right| \le C_{k,\theta }\frac{1}{1+\left| \xi ^2\right| +j_\theta ^2}. \end{aligned}$$

Now, let us take \(j_\theta ^2>k^2\) so that \(\beta _j\) is imaginary and \(e^{\imath \beta _j\left| z_2\right| }\) decays as \(\left| z_2\right| \) increases. Since

$$\begin{aligned} \frac{{\text {d}}}{{\text {d}} z_2}\left( e^{\imath \beta _j z_2}\frac{\xi \sin {(\xi z_2)}+\imath \beta _j\cos {(\xi z_2)}}{\xi ^2+\beta _j^2} \right) =e^{\imath \beta _j{z_2}}\cos {\left( 2\pi \xi z_2\right) }, \end{aligned}$$

integration-by-parts gives

$$\begin{aligned}&\frac{2}{\imath \beta _j}\int _{0}^{\infty }e^{\imath \beta _j{z_2}}\mu ({z_2})\cos {\left( 2\pi \xi z_2\right) }\;{\text {d}}z_2\\&\quad =\frac{2}{\imath \beta _j}\left. \left( \mu (z_2) e^{\imath \beta _j z_2}\frac{\xi \sin {(\xi z_2)}+\imath \beta _j\cos {(\xi z_2)}}{\xi ^2+\beta _j^2} \right) \right| ^\infty _0\\&\qquad -\frac{2}{\imath \beta _j}\int _{0}^{\infty } \mu '(z_2) e^{\imath \beta _j z_2}\frac{\xi \sin {(\xi z_2)}+\imath \beta _j\cos {(\xi z_2)}}{\xi ^2+\beta _j^2}\;{\text {d}}z_2 \\&\quad =\frac{2}{\xi ^2+\beta _j^2}\left( 1-\frac{1}{\imath \beta _j}\int _{0}^{\infty } \mu '(z_2) e^{\imath \beta _j z_2}\left( {\xi \sin {(\xi z_2)}+\imath \beta _j\cos {(\xi z_2)}}\right) \;{\text {d}}z_2\right) . \end{aligned}$$

By Lemma A.1, we deduce that

$$\begin{aligned} \frac{1}{\imath \beta _j}\int _{0}^{\infty } \mu '(z_2) e^{\imath \beta _j z_2}\left( {\xi \sin {(\xi z_2)}+\imath \beta _j\cos {(\xi z_2)}}\right) dz_2 \end{aligned}$$

is bounded for all \(\xi \in {{\mathbb {R}}}\), \(j\in {{\mathbb {Z}}}\). Hence,

$$\begin{aligned}&\left| \frac{2}{\xi ^2+\beta _j^2}\left( 1-\int _{0}^{\infty } \mu '(z_2) e^{\imath \beta _j z_2}\left( {\xi \sin {(\xi z_2)}+\imath \beta _j\cos {(\xi z_2)}}\right) dz_2\right) \right| \end{aligned}$$
(32)
$$\begin{aligned}&\quad \le C\frac{1}{\left| \left| \xi \right| ^2+j_\theta ^2-k^2\right| } \le C\frac{1}{1+\left| \xi \right| ^2+j_\theta ^2}, \end{aligned}$$
(33)

where C depends only on k and \(\mu \). Thus, there exists \(C>0\) depending only on \(k,k_1,\) and \(\mu \) such that for all \(s\in {{\mathbb {R}}}\),

$$\begin{aligned} (1+j_\theta ^2+\left| \xi \right| ^2)^{\frac{s}{2}}|{\widehat{u}}_{\mu ,j}(\xi )|\le C|{\widehat{f}}_j(\xi )|(1+j_\theta ^2+\left| \xi \right| ^2)^{\frac{s}{2}-1}. \end{aligned}$$
(34)

Taking the squared \(L^2\)-norm of both sides of (34) and adding over \(j\in {{\mathbb {Z}}}\), we obtain

$$\begin{aligned} \left\Vert u_\mu \right\Vert _{H^s_\theta ({\mathcal {G}})}^2\le C\left\Vert f\right\Vert _{H^{s-2}_\theta ({\mathcal {G}})}^2\quad \forall \ s\in {{\mathbb {R}}}. \end{aligned}$$

Since \({\mathcal {D}}_\theta ({\mathcal {G}})\) is dense in \(H^{s-2}_\theta ({\mathcal {G}})\) (cf. Proposition 2.8), the result is proven. \(\square \)

Appendix B: Regularity of Solutions and Continuity of BIOs

We extend the main results in [30, Chapter 4], introduced by Nečas [32], to the periodic case. We highlight changes needed to replicate the arguments. Our starting point is the result presented in Sect. 3. Recall that \(\theta \in [0,1)\), \(\Gamma \) a periodic curve in \({\mathcal {G}}:=[0,2\pi ]\times {{\mathbb {R}}}\) and \(\Omega \) as the open domain above \(\Gamma \) (see Fig. 2).

Lemma B.1

(Lemma 2.3 in [31], and 3.2 in [41]) Let \(u\in H^1_\theta (\Omega ^H)\) be such that

$$\begin{aligned} \left\{ \begin{array}{l} (-\Delta - k^2)u=0\ \text {on}\ \Omega ^H,\\ \gamma ^{i}_{0}u=0\ \text {or}\ \gamma ^{i}_{1}u=0\ \text {on}\ \Gamma , \\ \gamma ^{i}_{1}u={\mathcal {T}}(k_1,k)\gamma ^i_0u,\ \text {on}\ \Gamma ^H,\\ \end{array}\right. \end{aligned}$$

with \({\mathcal {T}}(k_1,k)\) being the DtN operator from Definition 3.6. Then, the Fourier coefficients \(u_{j}=0\) for all j in \(J^{-}_{k_1}:=\lbrace j\in {{\mathbb {Z}}}\ |\ k^2>j_{\theta }^2\rbrace \).

Proof

We proceed as in [31, Lemma 2.3],

$$\begin{aligned} 0&=\int _{\Omega ^H}(\Delta u{\overline{u}}\ - u\overline{\Delta u})\;{\text {d}}\varvec{x}=\int _{\Gamma \cup \Gamma _H}(\gamma ^i_1 u\;\overline{\gamma ^i_0 u}-\overline{\gamma ^i_1u}\;\gamma ^i_0u)\;{\text {d}}S_{\varvec{x}}. \end{aligned}$$

The integral over \(\Gamma \) vanishes due to either condition: \(\gamma ^{i}_{0} u=0\) or \(\gamma ^{i}_{1} u=0\). Hence, we only need to consider the integration on \(\Gamma ^H\),

$$\begin{aligned} 0= & {} \int _{\Gamma _H}(\gamma ^{i}_{1}u\ \overline{\gamma ^{i}_0 u}-\overline{\gamma ^{i}_{1}u}\ \gamma ^{i}_{0}u)\;{\text {d}}S_{\varvec{x}}\nonumber \\= & {} \int _{\Gamma _H}({\mathcal {T}}(k_1,k)\gamma ^{i}_{0}u\ \overline{\gamma ^{i}_0 u}-\overline{{\mathcal {T}}(k_1,k)\gamma ^{i}_0u}\ \gamma ^{i}_0u)\;{\text {d}}S_{\varvec{x}}. \end{aligned}$$
(35)

Recall the Fourier series for u and the DtN operator,

$$\begin{aligned}&u(x_1,H)=\sum _{j\in {{\mathbb {Z}}}}u_j(H)e^{\imath j_{\theta }x_1},\quad {\mathcal {T}}(k_1,k)u(x_1,H)=\sum _{j\in {{\mathbb {Z}}}}\imath \beta _ju_j(H)e^{\imath j_{\theta }x_1}. \end{aligned}$$

Hence,

$$\begin{aligned} \int _{\Gamma _H}({\mathcal {T}}(k_1,k)\gamma ^{i}_{0}u\ \overline{\gamma ^{i}_{0} u})\;{\text {d}}S_{\varvec{x}}=\sum _{j\in {{\mathbb {Z}}}} \imath \beta _j\left| u_j(H)\right| ^2, \end{aligned}$$

and (35) becomes,

$$\begin{aligned} \int _{\Gamma _H}({\mathcal {T}}(k_1,k)\gamma ^{i}_{0}u\ \overline{\gamma ^{i}_{0} u}-\overline{{\mathcal {T}}(k_1,k)\gamma ^{i}_{0}u}\ \gamma ^i_0u)\;{\text {d}}S_{\varvec{x}}=\sum _{j\in {{\mathbb {Z}}}}{\left( \imath \beta _j- \overline{\imath \beta _j}\right) }\left| u_j(H)\right| ^2=0. \end{aligned}$$

For \(j\not \in J^{-}_{k_1}\), we have \(\imath \beta _j\in {{\mathbb {R}}}\) and \(\overline{\imath \beta _j}-\imath \beta _j=0\). Thus,

$$\begin{aligned} 0=\sum _{j\in J^{-}_{k_1}}\left( \overline{\imath \beta _j}-\imath \beta _j\right) \left| u_j(H)\right| ^2=\sum _{j\in J^{-}_{k_1}}2\beta _j\left| u_j(H)\right| ^2. \end{aligned}$$

Since \(\beta _j>0\) for all \(j\in J^{-}_{k_1}\), \(\left| u_j(H)\right| =0\), for all \(j\in J^{-}_{k_1}.\)\(\square \)

Proposition B.2

Let \(k >0\) and \(f \in {\widetilde{H}}^{-1}_{\theta }(\Omega )\) with compact support.

  1. (i)

    Let \(g \in H^{\frac{1}{2}}_\theta (\Gamma )\) and \(k \notin K^{(\text {TM})}_{sing}\). Then, there is a unique \(u \in H^1_{\theta , {\mathrm {loc}}}(\Omega )\) that satisfies

    $$\begin{aligned} \left\{ \begin{array}{l} (-\Delta -k^2)u(\varvec{x}) = f(\varvec{x})\ \text {on}\ \Omega , \\ \gamma ^i_0 u = g\ \text {on}\ \Gamma ,\\ u\text { satisfies radiation conditions at infinity}. \end{array}\right. \end{aligned}$$

    Moreover, the solution depends continuously on the data

    $$\begin{aligned} \left\Vert u\right\Vert _{H^1_\theta (\Omega ^R)} \lesssim \left\Vert f\right\Vert _{{\widetilde{H}}^{-1}_{\theta }(\Omega )} +\left\Vert g\right\Vert _{H^{\frac{1}{2}}_\theta (\Gamma )}. \end{aligned}$$
  2. (ii)

    Let \(w \in H^{-\frac{1}{2}}_\theta (\Gamma )\) and \(k \notin K^{(\text {TE})}_{sing}\). Then, there is a unique \(u\in H^1_{\theta ,{\mathrm {loc}}}(\Omega )\) that satisfies

    $$\begin{aligned} \left\{ \begin{array}{l} (-\Delta -k^2)u(\varvec{x}) = f(\varvec{x})\ \text {on}\ \Omega , \\ \gamma ^i_1 u = w\ \text {on}\ \Gamma ,\\ u\text { satisfies radiation conditions at infinity}. \end{array}\right. \end{aligned}$$

    Also, it holds

    $$\begin{aligned} \left\Vert u\right\Vert _{H^1_\theta (\Omega ^R)} \lesssim \left\Vert f\right\Vert _{{\widetilde{H}}^{-1}_{\theta }(\Omega )} + \left\Vert w\right\Vert _{H^{-\frac{1}{2}}_\theta (\Gamma )}. \end{aligned}$$
  3. (iii)

    Let \(g \in H^{\frac{1}{2}}_\theta (\Gamma )\) and \(k \notin K^{(\text {TM})}_{sing}\). Then, there is a unique \(u \in H^1_{\theta , {\mathrm {loc}}}(\Omega )\) that satisfies

    $$\begin{aligned} \left\{ \begin{array}{l} (-\Delta -k^2)u(\varvec{x}) = f(\varvec{x})\ \text {on}\ \Omega , \\ \gamma ^i_0 u = g\ \text {on}\ \Gamma ,\\ u\text { satisfies the adjoint radiation condition at infinity}. \end{array}\right. \end{aligned}$$

    The next bound holds

    $$\begin{aligned} \left\Vert u\right\Vert _{H^1_\theta (\Omega ^R)} \lesssim \left\Vert f\right\Vert _{{\widetilde{H}}^{-1}_{\theta }(\Omega )} +\left\Vert g\right\Vert _{H^{\frac{1}{2}}_\theta (\Gamma )}. \end{aligned}$$
  4. (iv)

    Let \(w \in H^{-\frac{1}{2}}_\theta (\Gamma )\). If \(k \notin K^{(\text {TE})}_{sing}\), there is a unique \(u\in H^1_{\theta ,{\mathrm {loc}}}(\Omega )\) that satisfies

    $$\begin{aligned} \left\{ \begin{array}{l} (-\Delta -k^2)u(\varvec{x}) = f(\varvec{x})\ \text {on}\ \Omega , \\ \gamma ^i_1 u = w\ \text {on}\ \Gamma ,\\ u\text { satisfies the adjoint radiation condition at infinity}. \end{array}\right. \end{aligned}$$

    Moreover, the solution is bounded by the data

    $$\begin{aligned} \left\Vert u\right\Vert _{H^1_\theta (\Omega ^R)} \lesssim \left\Vert f\right\Vert _{{\widetilde{H}}^{-1}_{\theta }(\Omega )} +\left\Vert w\right\Vert _{H^{-\frac{1}{2}}_\theta (\Gamma )}. \end{aligned}$$

Proof

For the standard radiation condition (Definition 3.2), items (i) and (ii) follow from the Fredholm alternative and Theorem 3.11 (see [41, Theorems 3.3 and 3.4]). The same strategy holds if the adjoint radiation condition (see Definition 3.5) is used: we just need to show that the equations in items (iii) and (iv) have the same eigenvalues as those in (i) and (ii), which follows from noticing that one can build solutions of the equations with one radiation condition from the other. \(\square \)

The last proposition motivates the definition of solution operators. We consider two different cases. Let \(k \notin K^{(\text {TM})}_{sing}\) and \(g \in H^{\frac{1}{2}}_\theta (\Gamma )\), we set

$$\begin{aligned} {\mathcal {U}}_k :={\left\{ \begin{array}{ll} H^{\frac{1}{2}}_\theta (\Gamma ) &{}\rightarrow H^1_{\theta ,{\mathrm {loc}}}(\Omega ) \\ g &{}\mapsto u \end{array}\right. }, \end{aligned}$$

where u is the only element in \(H^1_{\theta ,{\mathrm {loc}}}(\Omega )\) that satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta -k^2)u(\varvec{x}) = 0\ \text {on}\ \Omega , \\ \gamma ^i_0 u = g\ \text {on}\ \Gamma ,\\ u \text { satisfies radiation conditions at infinity}. \end{array}\right. } \end{aligned}$$

The corresponding adjoint version is

$$\begin{aligned} {\mathcal {V}}_k :={\left\{ \begin{array}{ll} H^{\frac{1}{2}}_\theta (\Gamma ) &{}\rightarrow H^1_{\theta ,{\mathrm {loc}}}(\Omega ) \\ g &{}\mapsto v \end{array}\right. }, \end{aligned}$$

where v is the only element in \(H^1_{\theta ,{\mathrm {loc}}}(\Omega )\) that satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta -k^2)v(\varvec{x}) = 0\ \text {on}\ \Omega , \\ \gamma ^i_0 v = g\ \text {on}\ \Gamma ,\\ v \text { satisfies the adjoint radiation condition at infinity}. \end{array}\right. } \end{aligned}$$

We also consider Steklov-Poincaré operators defined as

$$\begin{aligned} \gamma _1^i {\mathcal {U}}_k : H^{\frac{1}{2}}_\theta (\Gamma ) \rightarrow H^{-\frac{1}{2}}_\theta (\Gamma ), \quad \gamma _1^i {\mathcal {V}}_k : H^{\frac{1}{2}}_\theta (\Gamma ) \rightarrow H^{-\frac{1}{2}}_\theta (\Gamma ). \end{aligned}$$

For a given domain \({\mathcal {O}}\subset {\mathcal {G}}\), \(k >0\) and a pair of functions \(u,v \in H^1_\theta ({\mathcal {O}})\), we define the following sesquilinear form:

$$\begin{aligned} \Phi ^k_{\mathcal {O}}(u,v) := \int _{\mathcal {O}}(\nabla u(\varvec{x}) \cdot \nabla \overline{v(\varvec{x})} -k^2u(\varvec{x})\overline{v(\varvec{x})}) \ {\text {d}}\varvec{x}. \end{aligned}$$
(36)

Lemma B.3

For \(g_1\), \(g_2 \in H^{\frac{1}{2}}_\theta (\Gamma ) \) we have that

$$\begin{aligned} \langle \gamma _1^i {\mathcal {U}}_k g_1, g_2 \rangle _{\Gamma } = \langle g_1, \gamma _1^i {\mathcal {V}}_k g_2 \rangle _{\Gamma }. \end{aligned}$$

Proof

From the radiation conditions, there is an \(R>0\) such that for \(x_2 \ge R\), it holds

$$\begin{aligned} {\mathcal {U}}_k g_1 (\varvec{x}) = \sum _{j \in {{\mathbb {Z}}}} a_j e^{\imath \beta _j (x_2-R)}e^{i j_\theta x_1}, \quad {\mathcal {V}}_k g_2 (\varvec{x}) = \sum _{j \in {{\mathbb {Z}}}} b_j e^{\imath {\widetilde{\beta }}_j (x_2-R)}e^{i j_\theta x_1}, \end{aligned}$$

with \(\beta _j\) and \({\widetilde{\beta }}_j\) as in (20) and (23), respectively. Using Lemma 2.33 and the definitions of \({\mathcal {U}}_k\) and \({\mathcal {V}}_k\) leads to

$$\begin{aligned} \Phi ^k_{\Omega ^R}({\mathcal {U}}_k g_1 ,{\mathcal {V}}_k g_2)&= \langle \gamma _1^i{\mathcal {U}}_k g_1, \gamma _0^i {\mathcal {V}}_{k} g_2 \rangle _{\Gamma \cup \Gamma ^R},\\ \Phi ^k_{\Omega ^R}({\mathcal {U}}_k g_1 ,{\mathcal {V}}_k g_2)&= \langle \gamma _0^i {\mathcal {U}}_{k} g_1,\gamma _1^i{\mathcal {V}}_k g_2 \rangle _{\Gamma \cup \Gamma ^R}, \end{aligned}$$

with \(\Phi ^k_{\Omega ^R}\) as in (36) and \(\Gamma ^R:=\{\varvec{x}\in {\mathcal {G}}\ |\ x_2=R\}\). Subtracting these last equations, we get

$$\begin{aligned} \langle \gamma _1^i{\mathcal {U}}_k g_1, g_2 \rangle _{\Gamma } = \langle g_1,\gamma _1^i{\mathcal {V}}_k g_2 \rangle _{\Gamma } + \langle \gamma _0^i {\mathcal {U}}_{k} g_1,\gamma _1^i{\mathcal {V}}_k g_2 \rangle _{\Gamma ^R}- \langle \gamma _1^i{\mathcal {U}}_k g_1, \gamma _0^i {\mathcal {V}}_{k} g_2 \rangle _{\Gamma ^R}. \end{aligned}$$

In \(\Gamma ^R\) we can use the expansions given by the radiation conditions:

$$\begin{aligned} \langle \gamma _0^i {\mathcal {U}}g_1,\gamma _1^i{\mathcal {V}}_k g_2 \rangle _{\Gamma ^R}- \langle \gamma _1^i{\mathcal {U}}_k g_1, \gamma _0^i {\mathcal {V}}g_2 \rangle _{\Gamma ^R} = -\sum _{j \in {{\mathbb {Z}}}} a_j \overline{b_j} (\imath \overline{{\widetilde{\beta }}}_j+\imath \beta _j). \end{aligned}$$

Then, for j such that \(\beta _j \) is a real number we have that \({\widetilde{\beta }}_j = - \beta _j\). Hence, \((\imath \overline{{\widetilde{\beta }}}_j+\imath \beta _j) = \imath (-\beta _j + \beta _j) = 0\). On the other hand, if \(\beta _j\) is pure imaginary we have that \({\widetilde{\beta }}_j = \beta _j\) and \((\imath \overline{{\widetilde{\beta }}}_j+\imath \beta _j) = \imath (-\beta _j + \beta _j) = 0\). Thus, the duality products over \(\Gamma ^R\) cancel each other out, yielding

$$\begin{aligned} \quad \langle \gamma _1^i{\mathcal {U}}_k g_1, g_2 \rangle _{\Gamma } = \langle g_1,\gamma _1^i{\mathcal {V}}_k g_2 \rangle _{\Gamma }, \quad \langle \gamma _1^i{\mathcal {U}}_k g_1, g_2 \rangle _{\Gamma ^R} = \langle g_1,\gamma _1^i{\mathcal {V}}_k g_2 \rangle _{\Gamma ^R}.\qquad \end{aligned}$$

\(\square \)

Following [30, Chapter 4], we now focus on establishing regularity properties of solutions in \(\Omega \). If u is a \(\theta \)-quasi-periodic function defined in \(\Omega \), we denote by \(u^p\) its \(\theta \)-quasi-periodic extension. For \(h\in {{\mathbb {R}}}\) with \(\left| h\right| < \pi \), we define the following estimators for the partial derivatives

$$\begin{aligned} \Delta _h^1 u(\varvec{x}) :=h^{-1}\left( u^p(\varvec{x}+h\varvec{e_1})-u^p(\varvec{x})\right) ,\ \Delta _h^2 u(\varvec{x}) := h^{-1}\left( u(\varvec{x}+h\varvec{e_2})-u(\varvec{x})\right) . \end{aligned}$$

The properties of \(\Delta _h^2\) are established in [30, Lemmas 4.13 to 4.15], where \(L_2({{\mathbb {R}}}^d)^m\) and \({\mathcal {D}}({{\mathbb {R}}}^d)^m\) have to be replaced by \(L^2_\theta ({\mathcal {G}})\), and \({\mathcal {D}}_\theta ({\mathcal {G}})\), respectively. There are, however, slight differences in the proofs for \(\Delta _h^1\), which are exposed when proving Lemmas B.4 and B.5.

Lemma B.4

(Lemma 4.13 in [30]) For \(\theta \in [0,1)\), let u be a \(\theta \)-quasi-periodic function. Then, for \(i = 1,2\), it holds

  1. (a)

    If \(\partial _i u \in L^2_\theta ({\mathcal {G}})\), then \(\left\Vert \Delta ^i_h u\right\Vert _{L^2_\theta ({\mathcal {G}})} \le \left\Vert \partial _i u\right\Vert _{L^2_\theta ({\mathcal {G}})}\) and \(\left\Vert \Delta ^i_h u - \partial _i u\right\Vert _{L^2_\theta ({\mathcal {G}})} \rightarrow 0\) as \(h \rightarrow 0\).

  2. (b)

    If there is a constant M such that \(\left\Vert \Delta ^i_h u\right\Vert _{L^2_\theta ({\mathcal {G}})} \le M\), then, for h small, we have that \(\partial _i u \in L^2_\theta ({\mathcal {G}})\), and \(\left\Vert \partial _i u\right\Vert _{L^2_\theta ({\mathcal {G}})} \le M\).

Proof

The proof for (b) follows Lemma 4.13 in [30]. Similarly for (a) for \(i = 2\) whereas for \(i=1\), we observe that

$$\begin{aligned} \left| \Delta ^1_hu(\varvec{x})\right| ^2\le \int _0^1\left| \partial _1u^p(\varvec{x}+th\varvec{e}_1)\right| ^2\;{\text {d}}t. \end{aligned}$$

Integrating over \({\mathcal {G}}\) yields,

$$\begin{aligned}&\left\Vert \Delta ^1_hu(\varvec{x})\right\Vert _{L^2_\theta ({\mathcal {G}})}^2\nonumber \\&\quad \le \int _{{\mathcal {G}}}\left( \int _0^1 \left| \partial _1u^p(\varvec{x}+th\varvec{e}_1)\right| ^2\;{\text {d}}t\right) \;{\text {d}}\varvec{x}\nonumber \\&\quad =\int _0^1\left( \int _{{\mathcal {G}}} \left| \partial _1u^p(\varvec{x}+th\varvec{e}_1)\right| ^2\;{\text {d}}\varvec{x}\right) \;{\text {d}}t =\int _0^1\left( \int _{{\mathcal {G}}+{\varvec{e}_1(th)}} \left| \partial _1u^p(\varvec{y})\right| ^2\;{\text {d}}\varvec{y}\right) \;{\text {d}}t\nonumber \\&\quad =\int _{0}^{1}\left( \int _{\{{\mathcal {G}}+{\varvec{e}_1(th)}\}\cap {\mathcal {G}}} \left| \partial _1u(\varvec{y})\right| ^2\;{\text {d}}\varvec{y}+\int _{\{{\mathcal {G}}+{\varvec{e}_1(th)}\}\setminus {\mathcal {G}}} \left| \partial _1u^p(\varvec{y})\right| ^2\;{\text {d}}\varvec{y}\right) \;{\text {d}}t\nonumber \\&\quad =\int _0^1\left( \int _{\{{\mathcal {G}}+{\varvec{e}_1(th)}\}\cap {\mathcal {G}}} \left| \partial _1u(\varvec{y})\right| ^2\;{\text {d}}\varvec{y}+\int _{{\mathcal {G}}\setminus \{{\mathcal {G}}+{\varvec{e}_1(th)}\}} \left| \partial _1u(\varvec{y})\right| ^2\;{\text {d}}\varvec{y}\right) \;{\text {d}}t\nonumber \\&\quad =\int _0^1\left( \int _{{\mathcal {G}}}\left| \partial _1u(\varvec{y})\right| ^2\;{\text {d}}\varvec{y}\right) \;{\text {d}}t= \left\Vert \partial _1u\right\Vert _{L^2({\mathcal {G}})}^2, \end{aligned}$$
(37)

where (37) follows from the periodicity of \(\left| \partial _1u(\varvec{y})\right| \). \(\square \)

Lemma B.5

(Lemma 4.15 in [30]) Let u and v belong to \(L^2_\theta ({\mathcal {G}})\), \(h \in {{\mathbb {R}}}\) such that \(\left| h\right| < \pi \). Moreover, let \(k >0 \) and \({\mathcal {O}}\subset {\mathcal {G}}\) be an open bounded set whose boundary is given by two disjoint periodic curves. Assume further that \({\text {supp}}u \subset {\mathcal {O}}\cap ({\mathcal {O}}-h\varvec{e}_2)\) and \({\text {supp}}v \subset {\mathcal {O}}\cap ({\mathcal {O}}+h\varvec{e}_2)\). Then,

  1. (a)

    if \(u,v \in L^2_\theta ({\mathcal {O}})\), then \(\left( \Delta ^i_h u, v \right) _{L^2_\theta ({\mathcal {O}})} = - \left( u, \Delta ^i_{-h} v\right) _{L^2_\theta ({\mathcal {O}})}\), \(i=1,2\).

  2. (b)

    if \(u,v \in H^1_\theta ({\mathcal {O}})\), then \(\Phi ^k_{{\mathcal {O}}}\left( \Delta ^i_h u,v\right) = -\Phi ^k_{{\mathcal {O}}}\left( u,\Delta ^i_{-h}v\right) \), \(i=1,2\).

Proof

For \(i=2\) the result follows verbatim from [30] whereas for \(i=1\), this is deduced directly from the definition of \(\Delta ^1_h\) and the quasi-periodicity property. \(\square \)

Theorem B.6

(Thm. 4.16 in [30]) Let \({\mathcal {O}}\subset \Omega \) be a bounded open set, whose boundary is given by two periodic curves and such that \({\overline{{\mathcal {O}}}}^{\mathcal {G}}\subset \Omega \). For \(r \ge 0\) and \(k>0\), let \(f \in H^r_\theta (\Omega )\) and \(u \in H^1_{\theta ,{\mathrm {loc}}}(\Omega )\) be such that

$$\begin{aligned} (-\Delta -k^2) u = f\ \text {in}\ \Omega . \end{aligned}$$

Then, \(u \in H^{r+2}_\theta ({\mathcal {O}})\) and for any \(R>0\) such that \({\overline{{\mathcal {O}}}}^{\mathcal {G}}\subset \Omega ^R\), we have that

$$\begin{aligned} \Vert u\Vert _{H_\theta ^{r+2}({\mathcal {O}})} \lesssim \Vert u \Vert _{H_\theta ^1(\Omega ^R)}+\Vert f\Vert _{H^r_\theta (\Omega )}. \end{aligned}$$

Proof

We take similar steps to those in the proof of Theorem 4.16 in [30]. Set \(r=0\) and consider a function \(\chi \in {\mathcal {D}}_\theta (\Omega ^R)\) such that \(\chi = 1\) in \({\mathcal {O}}\). Define

$$\begin{aligned} f_1 := (-\Delta -k^2)(\chi u). \end{aligned}$$

By direct computation, we obtain that \(\left\Vert f_1\right\Vert _{L^2_\theta (\Omega ^R)} \lesssim \left\Vert u\right\Vert _{H^1_\theta (\Omega ^R)}+\left\Vert f\right\Vert _{L^2_\theta (\Omega ^R)}\), so \(f_1 \in L^2_\theta (\Omega ^R)\). Let \(v \in H^1_{\theta ,{\mathrm {loc}}}(\Omega )\) with null trace in \(\partial ^{\mathcal {G}}\Omega ^R\). Using (11), we have that

$$\begin{aligned} \Phi ^k_{\Omega ^R}(\chi u , v) = (f_1 , v)_{L^2_\theta (\Omega ^R)}. \end{aligned}$$

Also, by Lemma B.5, we have that for \(i=1,2\), if \(\overline{{\text {supp}}v}^{{\mathcal {G}}}\subset \Omega ^R\) and h is sufficiently small, it holds

$$\begin{aligned} |\Phi ^k_{\Omega ^R}(\Delta ^i_h( \chi u) , v)| = | \Phi ^k_{\Omega ^R}( \chi u , \Delta ^i_{-h}v)|. \end{aligned}$$

Hence,

$$\begin{aligned} |\Phi ^k_{\Omega ^R}(\Delta ^i_h( \chi u) , v)| = |(f_1,\Delta ^i_{-h}v)|. \end{aligned}$$

By Lemma B.4 and norm definitions, we have that

$$\begin{aligned} |\Phi ^k_{\Omega ^R}(\Delta ^i_h( \chi u) , v)| \lesssim \Vert f_1\Vert _{L^2_\theta (\Omega ^R)}\Vert v\Vert _{H^1_\theta (\Omega ^R)}. \end{aligned}$$
(38)

On the other hand, by the coercivity of the Helmholtz operator, we get

$$\begin{aligned} \left\Vert \Delta ^i_h (\chi u)\right\Vert _{H^1_\theta ({\Omega ^R})}^2 \lesssim \left\Vert \Delta ^i_h (\chi u)\right\Vert _{L^2_\theta ({\Omega ^R})}^2 + \Phi ^k_{\Omega ^R}(\Delta ^i_h (\chi u), \Delta ^i_h (\chi u)). \end{aligned}$$

Taking \(v =\Delta ^i_h (\chi u)\) in (38) leads to

$$\begin{aligned} \Vert \Delta ^i_h (\chi u) \Vert ^2_{H^1_\theta ({\Omega ^R})} \lesssim \Vert \Delta ^i_h (\chi u)\Vert ^2_{L^2_\theta ({\Omega ^R}) } + \Vert f_1\Vert _{L^2_\theta (\Omega ^R)}\Vert \Delta ^i_h (\chi u)\Vert _{H^1_\theta (\Omega ^R)}. \end{aligned}$$

Here, we use the inequality \(ab \le \frac{1}{2}(\epsilon a +\epsilon ^{-1}b^2)\) for a small \(\epsilon \) to obtain

$$\begin{aligned} \Vert \Delta ^i_h (\chi u) \Vert ^2_{H^1_\theta ({\Omega ^R})} \lesssim \Vert \Delta ^i_h (\chi u)\Vert ^2_{L^2_\theta ({\Omega ^R}) } + \Vert f_1\Vert ^2_{L^2_\theta (\Omega ^R)}. \end{aligned}$$

Again, by Lemma B.4, \(\left\Vert \Delta ^i_h (\chi u)\right\Vert _{L^2_\theta ({\Omega ^R}) }\lesssim \left\Vert u\right\Vert _{H^1_\theta ({\Omega ^R})}\) and, by the bound for the norm of \(f_1\), we retrieve

$$\begin{aligned} \left\Vert \Delta ^i_h (\chi u)\right\Vert _{H^1_\theta ({\Omega ^R})}^2 \lesssim \left\Vert u\right\Vert _{H^1_\theta ({\Omega ^R})}^2+\left\Vert f\right\Vert _{L^2_\theta ({\Omega ^R})}^2. \end{aligned}$$

Finally, by recalling the norm definition on a subset \({\mathcal {O}}\subset \Omega ^R\) and Lemma B.4, it holds

$$\begin{aligned} \left\Vert u\right\Vert _{H^2_\theta ({\mathcal {O}})}^2 \lesssim \left\Vert u\right\Vert _{H^1_\theta ({\Omega ^R})}^2+\left\Vert f\right\Vert _{L^2_\theta ({\Omega ^R})}^2. \end{aligned}$$

The proof is then achieved by induction, analogously to that of Theorem 4.16 in [30]. \(\square \)

Now, we establish regularity results up to the boundary.

Theorem B.7

(Thm. 4.18 [30]) Assume \(\Omega \) to be a \({\mathcal {C}}^{r-1,1}\)-domain, with \(r\ge 2\). Let \({\mathcal {O}}\subset \Omega \) be a bounded subset whose boundary is composed of two periodic curves, one of them being \(\Gamma =\partial ^{{\mathcal {G}}}\Omega \). Moreover, let the wavenumber \(k>0\), \(f \in H^{r-2}_\theta (\Omega )\) and \(u \in H^1_{\theta , {\mathrm {loc}}}(\Omega )\) be such that

$$\begin{aligned} (-\Delta -k^2)u = f\ \text {on}\ \Omega . \end{aligned}$$

Then, the following bounds hold

  1. (i)

    If \(\gamma ^i_0u \in H_\theta ^{r-\frac{1}{2}}(\Gamma )\), then \(u \in H_\theta ^{r}({\mathcal {O}})\) and

    $$\begin{aligned} \left\Vert u\right\Vert _{H_\theta ^{r}({\mathcal {O}})} \lesssim \left\Vert u\right\Vert _{H_\theta ^{1}(\Omega ^R)}+\left\Vert \gamma ^i_0u\right\Vert _{H_\theta ^{r-\frac{1}{2}}(\Gamma )}+\left\Vert f\right\Vert _{H^{r-2}_\theta (\Omega ^R)}. \end{aligned}$$
  2. (ii)

    If \(\gamma _1^iu \in H_\theta ^{r-\frac{3}{2}}(\Gamma )\), then \(u \in H_\theta ^{r}({\mathcal {O}})\) and

    $$\begin{aligned} \left\Vert u\right\Vert _{H_\theta ^{r}({\mathcal {O}})} \lesssim \left\Vert u\right\Vert _{H_\theta ^{1}(\Omega ^R)}+\left\Vert \gamma ^i_1u\right\Vert _{H_\theta ^{r-\frac{3}{2}}(\Gamma )}+\left\Vert f\right\Vert _{H^{r-2}_\theta (\Omega ^R)}. \end{aligned}$$

for all \(R>0\) such that \({\mathcal {O}}\subset \Omega ^R\).

Proof

We bound the derivative \(\partial _1 u\) as in Theorem B.6 while bounds for \(\partial _2 u\) may be obtained from the boundary value problem:

$$\begin{aligned} -\partial _2^2u=f+k^2u+\partial _1^2u. \end{aligned}$$

The remainder of the proof follows that of [30, Theorem 4.18], requiring only minor modifications to the periodic setting. \(\square \)

Corollary B.8

(Thm. 4.21 [30]) Assume that \(\Omega \) is a \({\mathcal {C}}^{r-1,1}\)-domain and \(r\ge 2\). Then, for \(k \notin K^{(\text {TM})}_{sing}\), we have that

  1. (i)

    For \(0\le s \le r-1\),

    $$\begin{aligned} {\mathcal {U}}_k : H^{s+\frac{1}{2}}_\theta (\Gamma ) \rightarrow H^{s+1}_{\theta ,{\mathrm {loc}}}(\Omega ), \quad {\mathcal {V}}_k : H^{s+\frac{1}{2}}_\theta (\Gamma ) \rightarrow H^{s+1}_{\theta ,{\mathrm {loc}}}(\Omega ). \end{aligned}$$
  2. (ii)

    For \(-r+1\le s \le r-1\),

    $$\begin{aligned} \gamma _1^i {\mathcal {U}}_k : H^{s+\frac{1}{2}}_\theta (\Gamma ) \rightarrow H^{s-\frac{1}{2}}_{\theta }(\Gamma ),\quad \gamma _1^i {\mathcal {V}}_k : H^{s+\frac{1}{2}}_\theta (\Gamma ) \rightarrow H^{s-\frac{1}{2}}_{\theta }(\Gamma ). \end{aligned}$$

Proof

We begin by proving (i). The case \(s=0\) is direct from Proposition B.2, while the result for \(s=r+1\) follows from Theorem B.7. For \(0<s<r-1\), the result is derived by interpolation [30, Appendix B]—interpolation of quasi-periodic spaces in the boundary \(\Gamma \) follows from their definition, inducing an isomorphism to regular Sobolev spaces on closed boundaries [24, Chapter 8]). For (ii), the result for positive s is deduced by similar arguments as those used for (i) whereas the result for \(s<0\) is due to the duality pairing in Lemma B.3. \(\square \)

Theorem B.9

(Theorem 4.24 in [30]) Assume \(\Omega \) to be Lipschitz. Let \(k>0\), \(f \in L^2_\theta (\Omega )\) and \(u \in H^1_{\theta ,{\mathrm {loc}}}(\Omega )\) such that

$$\begin{aligned} (-\Delta -k^2) u = f\ \text {on}\ \Omega . \end{aligned}$$

If \(\gamma _0^iu \in H^1_\theta (\Gamma ) \) then \(\gamma _1^iu \in L_\theta ^2(\Gamma )\) and, for R such that \(\Gamma \subset {\mathcal {G}}^R\), we have that

$$\begin{aligned} \Vert \gamma _1^i u \Vert _{ L_\theta ^2(\Gamma )} \lesssim \Vert \gamma _0^iu \Vert _{ H^1_\theta (\Gamma )}+ \Vert u\Vert _{H_\theta ^1(\Omega ^R)}+\Vert f\Vert _{L^2_\theta (\Omega )}. \end{aligned}$$

Proof

First, we assume that \(u \in H^2_{\theta ,{\mathrm {loc}}}(\Omega )\) and, following the proof for [30, Theorem 4.24], it can be shown that

$$\begin{aligned} \Vert \gamma _1^i u \Vert _{ L_\theta ^2(\Gamma \cup {\Gamma ^R})} \lesssim \Vert \gamma _0^iu \Vert _{ H^1_\theta (\Gamma \cup {\Gamma ^R})}+ \Vert u\Vert _{H_\theta ^1(\Omega ^R)}+\Vert f\Vert _{L^2_\theta (\Omega )}. \end{aligned}$$

Now consider a bounded open set \({\mathcal {O}}\subset \Omega ^R\) such that \({\overline{{\mathcal {O}}}}^{\mathcal {G}}\subset \Omega ^R\), with \(\partial ^{\mathcal {G}}{\mathcal {O}}\) composed of two periodic curves, one of them being \(\Gamma ^R\). By Theorem 2.29 and the definition of the Neumann trace for smooth functions, we have that

$$\begin{aligned} \Vert \gamma _0^i u \Vert _{H^1_\theta (\Gamma ^R)} \lesssim \Vert u\Vert _{H^{\frac{3}{2}}_\theta ({\mathcal {O}})}\le \left\Vert u\right\Vert _{H^{2}_\theta ({\mathcal {O}})}, \end{aligned}$$

and, for \(0<\epsilon <\frac{1}{2}\), it holds

$$\begin{aligned} \Vert \gamma _1^i u \Vert _{L^2_\theta (\Gamma ^R)}\le \left\Vert \gamma _1^iu\right\Vert _{H^{\epsilon }_\theta (\Gamma ^R)} \lesssim \Vert u\Vert _{H^{\frac{3}{2}+\epsilon }_\theta ({\mathcal {O}})}\le \left\Vert u\right\Vert _{H^{2}_\theta ({\mathcal {O}})}. \end{aligned}$$

Then, by Theorem B.6, we derive

$$\begin{aligned} \Vert u\Vert _{H^{2}_\theta ({\mathcal {O}})} \lesssim \Vert u\Vert _{H^1_\theta (\Omega ^R)}+\Vert f\Vert _{L^2_\theta (\Omega )}. \end{aligned}$$

We now take \(u\in H^1_{\theta , {\mathrm {loc}}}(\Omega )\) and assume that \(\Gamma \) can be parametrized as \((x,\zeta (x))\) with \(x \in (0,2\pi )\). Consider a sequence of smooth functions \(\{\zeta _n\}_{n \in {{\mathbb {N}}}}\) such that

$$\begin{aligned} \zeta _n \rightarrow \zeta \text{ in } L^\infty ((0,2\pi )),\quad \nabla \zeta _n \rightarrow \nabla \zeta \text{ in } L^p((0,2\pi )) \text{ for } 1\le p <\infty ,\\ \nabla \zeta _n \text{ is } \text{ uniformly } \text{ bounded },\quad \zeta _n(x) \ge \zeta (x) \text{ for } x \in (0,2\pi ). \end{aligned}$$

Then, \(\Omega = \{\varvec{x}\in {\mathcal {G}}\ |\ x_2 > \zeta (x)\}\). Define

$$\begin{aligned} \Omega _{n}:=\{ \varvec{x}\in {\mathcal {G}}\ |\ x_2 > \zeta _n(x_1)\},\quad \Gamma _n := \{ \varvec{x}\in {\mathcal {G}}\ |\ x_2 = \zeta _n(x_1)\}=\partial ^{\mathcal {G}}\Omega _{n}. \end{aligned}$$

Following the proof of [30, Theorem 4.24], let us define \(g(\varvec{x}):= \gamma _0^i u(x_1,\zeta (x_1))\), where the trace is taken over \(\Gamma \). Finally, we consider \(\lambda >k^2\), \(R>0\) such that \(\Gamma \subset {\mathcal {G}}^R\) and a sequence \(\{u_n\}_{n\in {{\mathbb {N}}}}\) where each \(u_n\in H^1_\theta (\Omega _n)\) satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta -k^2+\lambda )u_n = f+\lambda u\ \text {on}\ \Omega _n^R, \\ \gamma _0^i u_n = \gamma _0 g\ \text {on}\ \Gamma _n,\\ \gamma _0^i u_n = \gamma _0 u\ \text {on}\ \Gamma ^R. \end{array}\right. } \end{aligned}$$

The elements of the sequence \(\{u_n\}_{n \in {{\mathbb {N}}}}\) are well defined in \(H^1_\theta (\Omega _{n}^R)\) since the domain is bounded and the associated operator is elliptic. Since \(\Gamma _n\) is smooth, by Theorem B.7, we have that each \(u_n\) belongs to \(H^2_\theta (\Omega _{n}^R)\) for all \(n\in {{\mathbb {N}}}\). Hence, we can use the result for elements of \(H^2_{\theta , {\mathrm {loc}}}(\Omega )\). To conclude, we need to show that a proper extension of \(u_n\) converges to u in \(H^1_\theta (\Omega ^R)\), which is done in [30, Theorem 4.24] for regular Sobolev spaces and extended to quasi-periodic Sobolev spaces with only minor modifications. \(\square \)

Corollary B.10

(Theorem 4.25 in [30]) Assume \(\Omega \) to be Lipschitz. Let \(k \notin K^{(\text {TM})}_{sing}\). For \(\left| s\right| \le \frac{1}{2}\), it holds

$$\begin{aligned} \gamma _1^i {\mathcal {U}}_k : H^{s+\frac{1}{2}}_\theta (\Gamma ) \rightarrow H^{s-\frac{1}{2}}_\theta (\Gamma ),\quad \gamma _1^i {\mathcal {V}}_k : H^{s+\frac{1}{2}}_\theta (\Gamma ) \rightarrow H^{s-\frac{1}{2}}_\theta (\Gamma ). \end{aligned}$$

Proof

The case \(s=0\) is given by Proposition B.2, \(s=\frac{1}{2}\) is given by Theorem B.9, and \(s=-\frac{1}{2}\) is obtained by the duality relation in Lemma B.3. For all other \(\left| s\right| <\frac{1}{2}\), the result follows by interpolation. \(\square \)

In order to prove the mapping properties of the double layer potential, we need one more auxiliary result. For \(k>0\), we denote by \({\mathcal {U}}^-_k\) the solution operator in \(\Omega ^- := {\mathcal {G}}\setminus {\overline{\Omega }}^{\mathcal {G}}\), and \({\mathcal {U}}^+_k := {\mathcal {U}}_k\). Given \(\lambda \in {{\mathbb {R}}}\) such that \(k^2-\lambda >0\), we set \({\mathcal {U}}^\pm _{k,\lambda } := {\mathcal {U}}^\pm _{\sqrt{k^2-\lambda }}\).

Lemma B.11

Let \(\Omega \) be Lipschitz and set \(k >0\). Then, there exists \(\lambda \in {{\mathbb {R}}}\) such that \(k^2-\lambda >0\) and \({\mathcal {U}}^+_{k,\lambda }\) as well as \({\mathcal {U}}^-_{k,\lambda }\) are well defined in \(H^{\frac{1}{2}}_\theta (\Gamma )\). For \(\left| s\right| <\frac{1}{2}\), we also have that

$$\begin{aligned} {\mathcal {U}}^+_{k,\lambda } : H^{s+\frac{1}{2}}_{\theta }(\Gamma ) \rightarrow H^{s+1}_{\theta ,{\mathrm {loc}}}(\Omega ). \end{aligned}$$

Proof

Since the eigenvalues of the problem in \(\Omega \) and \(\Omega ^-\) are numerable we can find \(\lambda \) such that \(k^2-\lambda >0\) and \(\left| \theta +j\right| \ne \sqrt{k^2-\lambda }\), for every \(j \in {{\mathbb {Z}}}\). Then, the following sets of equations

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta -k^2+\lambda ) u = 0\ \text {on}\ \Omega ,\\ \gamma ^i_0 u = g\ \text {on}\ \Gamma , \\ u \text { satisfies radiation conditions at infinity}, \end{array}\right. }\\ {\left\{ \begin{array}{ll} (-\Delta -k^2+\lambda ) u = 0\ \text {on}\ \Omega ^-,\\ \gamma ^i_0 u = g\ \text {on}\ \Gamma , \\ u \text { satisfies radiation conditions at infinity}, \end{array}\right. } \end{aligned}$$

are satisfied by only one element of \(H^1_{\theta ,{\mathrm {loc}}}(\Omega )\) and \(H^1_{\theta ,{\mathrm {loc}}}(\Omega ^-)\), respectively. Then, consider w defined as

$$\begin{aligned} w := {\left\{ \begin{array}{ll} {\mathcal {U}}^+_{k,\lambda }g \text { on } \Omega , \\ {\mathcal {U}}^-_{k,\lambda }g \text { on } \Omega ^-. \end{array}\right. } \end{aligned}$$

Thanks to the properties of the solution operators, we have

$$\begin{aligned} w|_{\Omega } \in H^1_{\theta ,{\mathrm {loc}}}(\Omega ),\quad w|_{\Omega ^-} \in H^1_{\theta ,{\mathrm {loc}}}(\Omega ^-),\quad w \in L^2_{\theta ,{\mathrm {loc}}}({\mathcal {G}}). \end{aligned}$$

By Theorem 4.9, it holds

$$\begin{aligned} w = -{\mathsf {S}}{\mathsf {L}}^{\sqrt{k^2-\lambda }}_\theta ([\gamma _1 w]), \end{aligned}$$

with \([\gamma _1 w]:=\gamma ^i_1w-\gamma ^e_1w\). Then, by the continuity of the single layer potential and Corollary B.10, it holds

$$\begin{aligned} \left\Vert {\mathsf {S}}{\mathsf {L}}^{\sqrt{k^2-\lambda }}_\theta [\gamma _1 w ]\right\Vert _{H^{s+1}_\theta (\Omega ^R)} \lesssim \Vert [\gamma _1 w] \Vert _{H^{s-\frac{1}{2}}_\theta (\Gamma )} \lesssim \left\Vert g\right\Vert _{H^{s+\frac{1}{2}}_\theta (\Gamma )}. \end{aligned}$$

Thus, we can conclude that

$$\begin{aligned} \Vert {\mathcal {U}}_{k,\lambda }^+g \Vert _{H^{s+1}_\theta (\Omega ^R)} = \Vert w\Vert _{H^{s+1}_\theta (\Omega ^R)} \lesssim \Vert g\Vert _{H^{s+\frac{1}{2}}_\theta (\Gamma )}, \end{aligned}$$

from where the result follows. \(\square \)

We define operators \({\mathcal {V}}^\pm _{k,\lambda }\) in a similar fashion to \({\mathcal {U}}^\pm _{k,\lambda }\) by using the adjoint radiation condition (cf. Definition 3.5) and repeating the steps presented above. It is easy to check that both operators have the same properties.

Proof of Theorem 4.10

Results for \({\mathsf {S}}{\mathsf {L}}_\theta ^{k}\) and \({{\mathsf {V}}}_\theta ^{k}\) can be established directly from their definitions and Theorems 2.29 and 4.7. Now, consider \(\eta \), \(\mu \in {\mathcal {D}}_\theta (\Gamma )\) and let \(\lambda \in {{\mathbb {R}}}\) be such that \(k^2-\lambda >0\) and \({\mathcal {V}}^+_{k,\lambda }\) is well defined. By the mapping properties of \({\mathsf {S}}{\mathsf {L}}_\theta ^k\), we have that

$$\begin{aligned} \left( (-\Delta -k^2+\lambda ){\mathsf {S}}{\mathsf {L}}_\theta ^k \eta , {\mathcal {V}}^+_{k,\lambda } \mu \right) _{L^2_\theta (\Omega ^R)}= \left( \lambda {\mathsf {S}}{\mathsf {L}}^k_\theta \eta , {\mathcal {V}}^+_{k,\lambda } \mu \right) _{L^2_\theta (\Omega ^R)}. \end{aligned}$$

Applying Lemma 2.33 leads to

$$\begin{aligned} \Phi ^{\sqrt{k^2-\lambda }}_{\Omega ^R} \left( {\mathsf {S}}{\mathsf {L}}^k_\theta \eta , {\mathcal {V}}^+_{k,\lambda } \mu \right) = \langle \gamma _1^i{\mathsf {S}}{\mathsf {L}}_\theta ^k \eta , \gamma _0^i {{\mathcal {V}}}^+_{k,\lambda } \mu \rangle _{\Gamma \cup \Gamma ^R}+ \left( \lambda {\mathsf {S}}{\mathsf {L}}^k_\theta \eta , {\mathcal {V}}^+_{k,\lambda } \mu \right) _{L^2_\theta (\Omega ^R)}. \end{aligned}$$

On the other hand, since \((-\Delta - k^2+\lambda ) {\mathcal {V}}^+_{k,\lambda } \mu = 0\) in \(\Omega ^R\), Green’s formula yields

$$\begin{aligned} \Phi ^{\sqrt{k^2-\lambda }}_{\Omega ^R}({\mathsf {S}}{\mathsf {L}}^k_\theta \eta ,{\mathcal {V}}^+_{k,\lambda }\mu ) = \langle \gamma _0^i{\mathsf {S}}{\mathsf {L}}_\theta ^k \eta , \gamma _1^i {\mathcal {V}}^+_{k,\lambda } \mu \rangle _{\Gamma \cup \Gamma ^R}. \end{aligned}$$

As the single layer potential satisfies the radiation condition in Definition 3.2 (cf. Proposition 4.2) and \({\mathcal {V}}^+_{k,\lambda } \mu \) the adjoint version (Definition 3.5), we get

$$\begin{aligned} \langle \gamma _1^i{\mathsf {S}}{\mathsf {L}}_\theta ^k \eta , \gamma _0^i {\mathcal {V}}^+_{k,\lambda } \mu \rangle _{\Gamma ^R} = \langle \gamma _0^i{\mathsf {S}}{\mathsf {L}}_\theta ^k \eta , \gamma _1^i {\mathcal {V}}^+_{k,\lambda } \mu \rangle _{\Gamma ^R} \end{aligned}$$

by the same arguments as in the proof of Lemma B.3. Then,

$$\begin{aligned} \langle \gamma _1^i{\mathsf {S}}{\mathsf {L}}_\theta ^k \eta , \gamma _0^i {\mathcal {V}}^+_{k,\lambda } \mu \rangle _{\Gamma } = \langle \gamma _0^i{\mathsf {S}}{\mathsf {L}}_\theta ^k \eta , \gamma _1^i {\mathcal {V}}^+_{k,\lambda } \mu \rangle _{\Gamma } -( \lambda {\mathsf {S}}{\mathsf {L}}^k_\theta \eta ,{\mathcal {V}}^+_{k,\lambda } \mu )_{L^2_\theta (\Omega ^R)}. \end{aligned}$$

The first term in the right-hand side can be bounded as

$$\begin{aligned} \left| \langle \gamma _0^i{\mathsf {S}}{\mathsf {L}}_\theta ^k \eta , \gamma _1^i {\mathcal {V}}^+_{k,\lambda } \mu \rangle _{\Gamma }\right|&\le \Vert \gamma _0^i {\mathsf {S}}{\mathsf {L}}^k_\theta \eta \Vert _{H^{s+\frac{1}{2}}_\theta (\Gamma )}\Vert \gamma _1^i {\mathcal {V}}^+_{k,\lambda } \mu \Vert _{H^{-s-\frac{1}{2}}_\theta (\Gamma )}\\&\lesssim \Vert \eta \Vert _{H^{s-\frac{1}{2}}_\theta (\Gamma )} \Vert \mu \Vert _{H^{-s+\frac{1}{2}}(\Gamma )}, \end{aligned}$$

where the last inequality follows from the continuity of \(\gamma _0^i {\mathsf {S}}{\mathsf {L}}_\theta ^k = {{\mathsf {V}}}_\theta ^k\) and Corollaries B.10 or B.8 depending on whether \(\Gamma \) is Lipschitz or smoother, respectively. For the second term, it holds

$$\begin{aligned} \left| ( \lambda {\mathsf {S}}{\mathsf {L}}^k_\theta \eta ,{\mathcal {V}}^+_{k,\lambda } \mu )_{L^2_\theta (\Omega ^R)}\right|&\lesssim \left\Vert {\mathsf {S}}{\mathsf {L}}^k_\theta \eta \right\Vert _{L^2_\theta (\Omega ^R)} \left\Vert {\mathcal {V}}^+_{k,\lambda } \mu \right\Vert _{L^2_\theta (\Omega ^R)}\\&\lesssim \left\Vert \eta \right\Vert _{H^{s-\frac{1}{2}}_\theta (\Gamma )} \left\Vert \mu \right\Vert _{H^{-s+\frac{1}{2}}_\theta (\Gamma )}, \end{aligned}$$

where the last inequality is due to the continuity of \({\mathsf {S}}{\mathsf {L}}^k_\theta \), and Lemma B.11. Mapping properties for \(\gamma _1^i{\mathsf {S}}{\mathsf {L}}_\theta ^k = {{{\mathsf {K}}}^k_\theta }'\) are obtained by density arguments.

For the double layer potential and its traces, pick \(g\in {\mathcal {D}}_\theta (\Gamma )\) and use the representation formula in Theorem 4.9—with \({\mathcal {U}}^+_{k,\lambda } g\) extended by zero to \(\Omega ^-\)—to obtain

$$\begin{aligned} \mathrm {D}{\mathsf {L}}^k_\theta g = {\mathcal {U}}^+_{k,\lambda } g+ {\mathsf {S}}{\mathsf {L}}^k_\theta (\gamma _1^i{\mathcal {U}}^+_{k,\lambda } g ) -{\mathcal {N}}^k_\theta ( -\lambda {\mathcal {U}}^+_{k,\lambda } g). \end{aligned}$$

Thus, we obtain the estimate

$$\begin{aligned}&\Vert \mathrm {D}{\mathsf {L}}^k_\theta g\Vert _{H^{s+1}(\Omega ^R)}\\&\lesssim \Vert {\mathcal {U}}^+_{k,\lambda } g \Vert _{H^{s+1}(\Omega ^R)}+ \Vert {\mathsf {S}}{\mathsf {L}}^k_\theta (\gamma _1^i{\mathcal {U}}^+_{k,\lambda } g ) \Vert _ {H^{s+1}(\Omega ^R)}+ \Vert {\mathcal {N}}^k_\theta ({\mathcal {U}}^+_{k,\lambda } g)\Vert _{H^{2}(\Omega ^R)}. \end{aligned}$$

By Lemma B.11, the mapping properties of \({\mathsf {S}}{\mathsf {L}}_\theta ^k\) and Theorem 4.4, we obtain

$$\begin{aligned}&\Vert \mathrm {D}{\mathsf {L}}^k_\theta g\Vert _{H^{s+1}(\Omega ^R)} \lesssim \Vert g\Vert _{H^{s+\frac{1}{2}}_\theta (\Gamma )} +\Vert \gamma _1^i{\mathcal {U}}^+_{k,\lambda } g \Vert _{H^{s-\frac{1}{2}}_\theta (\Gamma )}+ \Vert {\mathcal {U}}^+_{k,\lambda } g\Vert _{L^2_\theta (\Omega ^R)}. \end{aligned}$$

Finally, by using Corollary B.10, we have that

$$\begin{aligned} \Vert \mathrm {D}{\mathsf {L}}^k_\theta g\Vert _{H^{s+1}(\Omega ^R)} \lesssim \Vert g\Vert _{H^{s+\frac{1}{2}}_\theta (\Gamma )}. \end{aligned}$$

Bounds for the norms in \({\mathcal {G}}\setminus {\overline{\Omega }}^{\mathcal {G}}\) are derived by using \({\mathcal {U}}_{k,\lambda }^-\) and repeating the same procedure. The continuity of \(\gamma _0^i \mathrm {D}{\mathsf {L}}^\theta _k = {{\mathsf {K}}}^k_\theta \) is direct from the trace continuity in Theorem 2.29. The Neumann trace can be estimated as follows

$$\begin{aligned}&\left\Vert \gamma ^i_1 \mathrm {D}{\mathsf {L}}^k_\theta g\right\Vert _{H^{s-\frac{1}{2}}_\theta (\Gamma )} \\&\lesssim \left\Vert \gamma ^i_1 {\mathcal {U}}_{k,\lambda }^+g\right\Vert _{H^{s-\frac{1}{2}}_\theta (\Gamma )}+ \left\Vert \gamma ^i_1 {\mathsf {S}}{\mathsf {L}}^k_\theta (\gamma ^i_1 {\mathcal {U}}_{k,\lambda }^+g)\right\Vert _{H^{s-\frac{1}{2}}_\theta (\Gamma )}+ \left\Vert \gamma _1^i {\mathcal {N}}^k_\theta ({\mathcal {U}}_{k,\lambda }^+g)\right\Vert _{L^2_\theta (\Gamma )}. \end{aligned}$$

The first term on the right-hand side is bounded by Corollary B.10 whereas the second one is bounded by the continuity of \({{{\mathsf {K}}}_\theta ^k}^\prime \). The last term is bounded by the continuity of the Neumann trace, that of the Newton potential and Corollary B.10. \(\square \)

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Aylwin, R., Jerez-Hanckes, C. & Pinto, J. On the Properties of Quasi-periodic Boundary Integral Operators for the Helmholtz Equation. Integr. Equ. Oper. Theory 92, 17 (2020). https://doi.org/10.1007/s00020-020-2572-9

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