In 1997 Fokas [1] introduced a novel flexible approach to solve initial or boundary value problems for various two dimensional linear and integrable nonlinear PDE's and then to deal with multidimensional problems [2, 3]. In particular, this method was applied in [4] to treat the Laplace equation in a convex polygon and in [5] to other linear two-dimensional PDEs, including the modified Helmholtz and Helmholtz equations. The readers are also referred to see [6-8] for a systematic exposition of the method, its various applications and more references therein. The fundamental problems for the elliptic PDEs in nonpolygonal convex regions remain open [6], which is our main motivation of the present investigation.

We observe that the implementation of Fokas transform method has two fundamental steps. The first step is that Riemann-Hilbert technique is used to construct integral representation of the solution in terms of spectral functions, which is base on the Lax pair or the differential form of equation. The second one is known as a generalized Dirichlet to Neumann map, which determines unknown functions including in the spectral functions. This step is accomplished through so-called the global relation and some invariant properties. In some particular cases, these unknown functions can be reduced to seek a solution of a system of algebraic equations or one of a Riemann-Hilbert problem. However, it is difficult to determine these unknown functions in many cases.

The aim of this paper is to improve Fokas transform approach to study boundary value problems for the following basic elliptic PDEs in a semi-disk domain $\Omega$

$ \begin{equation}\label{eq1} \Delta q(z, \overline{z})+4\alpha q(z, \overline{z})=0, \ \ z \in \Omega, \end{equation} $

(1.1)

where

$ \Omega=\{(x, y): x^2+y^2\leq r^2, -r\leq x\leq r, y\geq 0\} $

and set

$ \begin{eqnarray*}& L_{1}=\{(x, y): y=0, -r<x<r\}, \\ & L_{2}=\{(x, y):x^2+y^2=r^2, -r\leq x\leq r, y\geq 0\}, \\ & \partial\Omega=L_{1}\cup L_{2} \end{eqnarray*} $

(see Figure 1).

Without loss of generality, we may assume $r=1$. When $\alpha=0$, equation (1.1) becomes a Laplace equation; when $\alpha=-\beta^2$, equation (1.1) is said to be a modified Helmholtz equation; when $\alpha=\beta^2$, equation (1.1) is called a Helmholtz equation.

The rest of the paper is arranged as follows. In Section 2 we derive the integral representations for the solutions of the elliptic PDEs above-metioned in terms of the spectral functions using the differential forms of equations and the Riemann-Hilbert technique. In Section 3 we discuss generalized Dirichlet to Neumann maps. As an illustration, the Dirichlet boundary value problem for Helmholtz equation in a semi-disk domain is investigated. Using the global relation and some symmetric properties, the unknown functions are determined by the solution of a Fredholm's integral equation of the first kind. A short summary of this work is given in Section 4.

In this section, based on the differential form of equation and Riemann-Hilbert technique we will derive the integral representations for the solutions of Laplace, modified Helmholtz and Helmholtz equation in terms of spectral functions. That is, we have

Theorem 1   Let $\Omega$ be a semi-disk domain in the complex plane $\mathbb{C}$ (see Figure 1). Assume that the modified Helmholtz equation has a solution $q(z, \overline{z})$ in $\Omega$ such that it is sufficiently smooth to $\partial\Omega$. Then $q(z, \overline{z})$ can be expressed by

$ \begin{equation}\label{eq2} q(x, y)={1\over 4\pi{i}}{\sum\limits_{j=1}^4}{\int_{l_{j}}}{e^{i\beta(kz-{\overline{z}\over k})}}\rho_{j}(x, y, k){dk\over k}, \end{equation} $

(2.1)

where

$ z=x+iy, \bar {z}=x-iy, k\in\mathbb{C}, $

$ \rho_{1}(x, y, k)={\int_{-1}^1}e^{-i\beta(k-{1\over k})x}[-iq_{y}(x, 0)+i\beta(k+{1\over k})q(x, 0)]dx, $

(2.2)

$ \rho_{2}(x, y, k)={\int_{\pi}^{\theta_{2}}}e^{-i\beta(ke^{i\theta}-{1\over k}e^{-i\theta})}[iq_{r}(\cos\theta, \sin\theta) -\beta(ke^{i\theta}-{1\over k}e^{-i\theta})q(\cos\theta, \sin\theta)]d\theta, $

(2.3)

$ \rho_{3}(x, y, k)={\int_{\theta_{1}}^{\theta_{2}}}e^{-i\beta(ke^{i\theta}-{1\over k}e^{-i\theta})}[iq_{r}(\cos\theta, \sin\theta) -\beta(ke^{i\theta}-{1\over k}e^{-i\theta})q(\cos\theta, \sin\theta)]d\theta, $

(2.4)

$ \rho_{4}(x, y, k)={\int_{\theta_{1}}^{0}}e^{-i\beta(ke^{i\theta}-{1\over k}e^{-i\theta})}[iq_{r}(\cos\theta, \sin\theta) -\beta(ke^{i\theta}-{1\over k}e^{-i\theta})q(\cos\theta, \sin\theta)]d\theta, $

(2.5)

$ \begin{array}{l} {\theta _1} = \arg [\frac{{{{(x + 1)}^2} - {y^2}}}{{{{(x + 1)}^2} + {y^2}}} + i\frac{{2y(x + 1)}}{{{{(x + 1)}^2} + {y^2}}}],{\theta _2} = \arg [\frac{{{y^2} - {{(x - 1)}^2}}}{{{{(x - 1)}^2} + {y^2}}} + i\frac{{2y(1 - x)}}{{{{(x - 1)}^2} + {y^2}}}],\\ 0 \le {\theta _1},{\theta _2} \le \pi \end{array} $

and the rays $ l_{j}=\{k\in \mathbb{C}:\arg k={{(j-1)\pi}\over 2}\}$ for $j=1, 2, 3, 4$, and $l_1$ and $l_3$ are oriented from zero to infinity, while $l_2$ and $l_4$ are oriented from infinity to zero; see Figure 2.

Furthermore, the following global relation is valid:

$ \begin{align}\label{eq3} &{\int_{-1}^1}e^{-i\beta(k-{1\over k})x}[-iq_{y}(x, 0)+i\beta(k+{1\over k})q(x, 0)]dx\nonumber\\=&{\int_{\pi}^{0}}e^{-i\beta(ke^{i\theta}-{1\over k}e^{-i\theta})}[iq_{r}(\cos\theta, \sin\theta) -\beta(ke^{i\theta}-{1\over k}e^{-i\theta})q(\cos\theta, \sin\theta)]d\theta. \end{align} $

(2.6)

Proof   First, it sees from [6] that function $q(z, \overline{z})$ satisfies the modified Helmholtz equation if and only if the following differential form is closed:

$ \begin{equation}\label{eq31} dW={e^{-i\beta(kz-{\overline{z}\over k})}}[(q_z+ik\beta q)dz-(q_{\overline{z}}+{\beta\over ik}q)d{\overline{z}}], \quad k\in \mathbb{C}.\end{equation} $

(2.7)

This implies the associated global relation

$ \begin{equation}\label{eq4} {\int_{L_{1}\cup L_{2}}}{e^{-i\beta(kz-{\overline{z}\over k})}}[(q_z+ik\beta q)dz-(q_{\overline{z}}+{\beta\over ik}q)d{\overline{z}}]=0, \end{equation} $

(2.8)

where $L_1$ and $L_{2}$ are shown as in Figure 1. Note that if $z=x+iy, {\overline z}=x-iy, $ then

$ \partial_ {z}={1\over 2}(\partial_ x-i\partial_ y), \partial_ {\overline z}={1\over 2}(\partial_ x+i\partial_ y); $

if $z=re^{i\theta}, {\overline z}=re^{-i\theta}$, then

$ \partial_ {z}={1\over 2}e^{-i\theta}(\partial_ r+{1\over {ir}}\partial_ {\theta}), \partial_ {\overline z}={1\over 2}e^{i\theta}(\partial_ r-{1\over {ir}}\partial_ {\theta}). $

Hence we deduce from (2.8) that (2.6) holds.

Next, we will derive the integral representation (2.1). We perform the spectral analysis of the differential form

$ \begin{equation}\label{eq5} d[{e^{-i\beta(kz-{\overline{z}\over k})}}\mu(z, \overline z, k)]=dW(z, \overline z, k). \end{equation} $

(2.9)

It can be derived directly from (2.7) and (2.9) that the modified Helmholtz equation has the following Lax pair equations

$ \mu_z-ik\beta \mu=q_z+ik\beta q, $

(2.10)

$ \mu_{\overline{z}}+{i\beta\over k}\mu=-(q_{\overline{z}}+{\beta\over ik}q). $

(2.11)

Integrating (2.9), we find that for all $ z \in \Omega$,

$ \mu_{j}(z, \overline z, k)={\int_{z_{j}}^z}{e^{i\beta[k(z-\zeta)-{1\over k}(\overline{z-\zeta})]}}[(q_\zeta+ik\beta q)d\zeta-(q_{\overline{\zeta}}+{\beta\over ik}q)d{\overline{\zeta}}], $

here $\mu_{j}(z, \overline z, k)$ depend only on point $z_{j}$ and are independent of the paths of integrations. Meanwhile, $\mu_{j}(z, \overline z, k)$ is also the particular solution of the Lax pair equations (2.10) and (2.11).

We now choose the point $z_{j}$ and a suitable path such that we can define a piecewise analytic function $\mu_{j}(z, \overline z, k)$ in the $k$-plane. Set

$ \begin{array}{l} {z_0} = 1 + i,{z_1} = - 1,{z_2} = 1,\\ {z_3} = \frac{{{{(x + 1)}^2} - {y^2}}}{{{{(x + 1)}^2} + {y^2}}} + i\frac{{2y(x + 1)}}{{{{(x + 1)}^2} + {y^2}}},{z_4} = \frac{{{y^2} - {{(x - 1)}^2}}}{{{{(x - 1)}^2} + {y^2}}} + i\frac{{2y(1 - x)}}{{{{(x - 1)}^2} + {y^2}}}, \end{array} $

where $z_{3}$ is a intersection point of straight line through $z_{1}, z$ with the semicircle $L_{2}$ and $z_{4}$ is one of straight line through the point $z_{2}, z$ with $L_{2}$, which are uniquely determined by $z\in\Omega$; see Figure 3. We define $\mu_{j}(z, \overline z, k)(j=1, 2, 3, 4)$ by

$ \begin{array}{l} {\mu _j}(z,\bar z,k) = \mathop \smallint \nolimits_{{z_j}}^z {e^{i\beta [k(z - \zeta ) - \frac{1}{k}(\overline {z - \zeta } )]}}[({q_\zeta } + ik\beta q)d\zeta - ({q_{\bar \zeta }} + \frac{\beta }{{ik}}q)d\bar \zeta ](j = 1,2),\\ {\mu _j}(z,\bar z,k) = \int_{\widehat {{z_0}{z_j}z}} {{e^{i\beta [k(z - \zeta ) - \frac{1}{k}(\overline {z - \zeta } )]}}} \left[ {({q_\zeta } + ik\beta q)d\zeta - ({q_{\bar \zeta }} + \frac{\beta }{{ik}}q)d\bar \zeta } \right]\left( {{\rm{j = 3, 4}}} \right), \end{array} $

where the paths of integrations $\mu_{j}(z, \overline z, k)(j=1, 2)$ are taken as the straight-line segment and the paths of integration $\mu_{j}(z, \overline z, k)(j=3, 4)$ is chosen as the sum of the circular arc $\widehat{z_{0}z_{j}} $ and the straight-line segment $\overline{z_{j}z}$, as shown in Figure 3. Let

$ \begin{array}{l} \mu _j^0(z,\bar z,k) = \int_{{z_0}{z_j}} {{e^{i\beta [k(z - \zeta ) - \frac{1}{k}(\overline {z - \zeta } )]}}} [({q_\zeta } + ik\beta q)d\zeta - ({q_{\bar \zeta }} + \frac{\beta }{{ik}}q)d\bar \zeta ],\\ \mu _j^*(z,\bar z,k) = \int_{\overline {{z_j}z} } {{e^{i\beta [k(z - \zeta ) - \frac{1}{k}(\overline {z - \zeta } )]}}[({q_\zeta } + ik\beta q)d\zeta - ({q_{\bar \zeta }} + \frac{\beta }{{ik}}q)d\bar \zeta ]} \end{array} $

for $j=3, 4$. Then we have

$ \mu_{j}(z, \overline z, k)=\mu_{j}^{0}(z, \overline z, k)+\mu_{j}^{*}(z, \overline z, k)(j=3, 4). $

It is easy to see the boundedness and analytic domains of the functions $\mu_{1}, \mu_{2}, \mu_{3}^{*}, \mu_{4}^{*}$ in the $k$-plane are $D_{j}(j=1, 2, 3, 4)$, respectively, which are defined by

$ \begin{align*} D_{1}=\{k\in \mathbb{C}: 0\leq \arg k\leq {{\pi}\over 2}\}, \quad D_{2}=\{k\in \mathbb{C}: {{3\pi}\over 2}\leq \arg k\leq {2\pi}\}, \\ D_{3}=\{k\in \mathbb{C}: \pi\leq \arg k\leq {{3\pi\over 2}}\}, \quad D_{4}=\{k\in \mathbb{C}: {{\pi}\over 2}\leq \arg k\leq \pi\}; \end{align*} $

see Figure 4. Indeed, put $k=|k|e^{i\varphi}, \zeta=\xi+i{\eta}$, then we find

$ \mbox{Re}[ik(z-\zeta)-{i\over k}{\overline{(z-\zeta)}}]=-(|k|+{1\over |k|})[(x-\xi)\sin\varphi+(y-\eta)\cos\varphi]. $

Consider first the function $\mu_{1}(z, \overline z, k)$. Notice that ${x-\xi}\ge0, {y-\eta}\ge0$, so the exponential associated with $\mu_{1}(z, \overline z, k)$ is bounded if and only if $\sin\varphi\ge0$ and $\cos\varphi\ge0$. This gives the domain of $\mu_{1}(z, \overline z, k)$ is equal to $D_{1}$. Similarly, we obtain that the domains of $\mu_{2}, \mu_{3}^{*}, \mu_{4}^{*}$ are equal to $D_{j} (j=2, 3, 4)$, respectively.

The Lax pair equation (2.10) gives

$ \begin{equation}\label{eq8} \mu=-q+O({1\over k}), \quad k\rightarrow \infty, \end{equation} $

(2.12)

which can be verified directly for each of the functions $\mu_{1}, \mu_{2}, \mu_{3}^{*}, \mu_{4}^{*}$. Thus we are able to formulate a Riemann-Hilbert problem for the sectionally analytic function $\mu$ which is defined by

$ \mu= \left\{ \begin{array}{ll} \mu_{1}(z, \overline z, k), \qquad &k\in{D_{1}}, \\ \mu_{2}(z, \overline z, k), \qquad&k\in{D_{2}}, \\ \mu_{3}^{*}(z, \overline z, k), \qquad &k\in{D_{3}}, \\ \mu_{4}^{*}(z, \overline z, k), \qquad &k\in{D_{4}} \end{array}\right. $

for $z\in {\Omega}$. The relevant jumps are given by

$ \begin{equation}\label{eq81} \mu^{+}-\mu^{-}=\rho(x, y, k)e^{i\beta(kz-{\overline{z}\over k})}, \quad k\in L, \end{equation} $

(2.13)

where $L={\sum\limits_{j=1}^4}l_{j}$ depicted in Figure 2, and $\rho(x, y, k)=\rho_{j}(x, y, k)(j=1, 2, 3, 4)$ will be determined below.

In the following we will derive $\rho(x, y, k)$. Since $\mu_{j}(x, y, k)$ satisfy Lax pair equations (2.10) and (2.11) for $j=1, 2, 3, 4$, the difference of any two solutions of (2.10) and (2.11) satisfies $p(k)e^{i\beta(kz-{\overline{z}\over k})}$. Hence, when $k\in {l_{1}}$, we have

$ \mu^{+}-\mu^{-}=\mu_{1}-\mu_{2}=p_{12}(k)e^{i\beta(kz-{\overline{z}\over k})}, $

which implies that

$ \begin{array}{l} {p_{12}}(k) = \mathop \smallint \nolimits_{{z_1}}^{{z_2}} {e^{ - i\beta (kz - \frac{{\bar z}}{k})}}[({q_z} + ik\beta q)dz - ({q_{\bar z}} + \frac{\beta }{{ik}}q)d\bar z]\\ \;\;\;\;\;\;\;\;\;\; = \mathop \smallint \nolimits_{ - 1}^1 {e^{ - i\beta (k - \frac{1}{k})x}}[ - i{q_y}(x,0) + i\beta (k + \frac{1}{k})q(x,0)]dx = {\rho _1}(x,y,k), \end{array} $

that is, the equality (2.2) holds; when $k\in {l_{2}}$, we get

$ \begin{align*} \mu^{+}-\mu^{-}=\mu_{1}-\mu_{4}^{*}=\mu_{1}-(\mu_{4}-\mu_{4}^{0}) =(\mu_{1}-\mu_{4})+\mu_{4}^{0}= p_{14}(k)e^{i\beta(kz-{\overline{z}\over k})}+\mu_{4}^{0}, \end{align*} $

which gives

$ \begin{align*} p_{14}(k)e^{i\beta(kz-{\overline{z}\over k})}+\mu_{4}^{0}=&{e^{i\beta(kz-{\overline{z}\over k})}}{\int_{\widehat{z_{1}z_{4}}}}{e^{-i\beta(k{\zeta}-{\overline{\zeta}\over k})}}[(q_{\zeta}+ik\beta q)d{\zeta}-(q_{\overline{\zeta}}+{\beta\over ik}q)d{\overline{\zeta}}]\\ =&{e^{i\beta(kz-{\overline{z}\over k})}}{\int_{\pi}^{\theta_{2}}}e^{-i\beta(ke^{i\theta}-{1\over k}e^{-i\theta})}[iq_{r}(\cos\theta, \sin\theta)\\ &-\beta(ke^{i\theta}-{1\over k}e^{-i\theta})q(\cos\theta, \sin\theta)]d\theta\\ =&\rho_{2}(x, y, k){e^{i\beta(kz-{\overline{z}\over k})}}, \end{align*} $

which yields that (2.3) holds; similarly, we can conclude that spectral functions $\rho_3$ and $\rho_4$ possess the expressions (2.4) and (2.5), respectively.

The solution of the Riemann-Hilbert problem with the estimate (2.12) and the jump (2.13) along $L$ can be expressed by

$ \begin{equation}\label{eq9} \mu=-q+{1\over{2\pi i}}{\int_L}\rho(x, y, k){e^{i\beta(sz-{\overline{z}\over s})}}{ds\over{s-k}}, \quad k\in {\mathbb{C}\setminus L}. \end{equation} $

(2.14)

Thus the Lax pair (2.11) where $\mu$ is replaced by (2.14) implies that (2.1) holds. This completes the proof of Theorem 1.

Similar to the proof of Theorem 1, we easily obtain

Corollary 1   Under the conditions of Theorem 1, the solution $q(z, \overline{z})$ of Laplace equation (see (1.1) with $\alpha=0$) can be expressed by

$ {\partial q\over\partial z}={1\over 2\pi}{\sum\limits_{j=1}^4}{\int_{l_{j}}}{e^{ikz}}\rho_{j}(x, y, k)dk, $

where

$ {\rho _1}(x,y,k) = \frac{1}{2}[q(1,0){e^{ - ik}} - q( - 1,0){e^{ik}}] + \frac{i}{2}\mathop \smallint \nolimits_{ - 1}^1 {e^{ - ikx}}[kq(x,0) - i{q_y}(x,0)]dx, $

$ \begin{array}{l} {\rho _2}(x,y,k) = \frac{1}{2}[q(\cos {\theta _2},\sin {\theta _2}){e^{ - ik{e^{i{\theta _2}}}}} - q( - 1,0){e^{ik}}]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \frac{1}{2}\mathop \smallint \nolimits_\pi ^{{\theta _2}} {e^{ - ik{e^{i\theta }}}}[i{q_r}(\cos \theta ,\sin \theta ) - k{e^{i\theta }}q(\cos \theta ,\sin \theta )]d\theta , \end{array} $

$ \begin{array}{l} {\rho _3}(x,y,k) = \frac{1}{2}[q(\cos {\theta _2},\sin {\theta _2}){e^{ - ik{e^{i{\theta _2}}}}} - q(\cos {\theta _1},\sin {\theta _1}){e^{ - ik{e^{i{\theta _1}}}}}]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \frac{1}{2}\mathop \smallint \nolimits_{{\theta _1}}^{{\theta _2}} {e^{ - ik{e^{i\theta }}}}[i{q_r}(\cos \theta ,\sin \theta ) - k{e^{i\theta }}q(\cos \theta ,\sin \theta )]d\theta , \end{array} $

$ \begin{array}{l} {\rho _4}(x,y,k) = \frac{1}{2}[q(1,0){e^{ - ik}} - q(\cos {\theta _1},\sin {\theta _1}){e^{ - ik{e^{i{\theta _1}}}}}]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \frac{1}{2}\mathop \smallint \nolimits_{{\theta _1}}^0 {e^{ - ik{e^{i\theta }}}}[i{q_r}(\cos \theta ,\sin \theta ) - k{e^{i\theta }}q(\cos \theta ,\sin \theta )]d\theta \end{array} $

and $\theta_{1}$, $\theta_{2}$, $l_{j}(j=1, 2, 3, 4)$ are defined as in Theorem 1. Moreover, the following global relation holds

$ \begin{align*} i{\int_{-1}^1}e^{-ikx}[kq(x, 0)-iq_{y}(x, 0)]dx ={\int_{\pi}^{0}}e^{-ike^{i\theta}}[iq_{r}(\cos\theta, \sin\theta) -ke^{i\theta}q(\cos\theta, \sin\theta)]d\theta. \end{align*} $

Corollary 2   Under the conditions of Theorem 1, the solution $q(z, \overline{z})$ of Helmholtz equation (see (1.1) with $\alpha=\beta^2$) can be expressed as

$ \begin{equation}\label{eq10} q(x, y)={1\over 4\pi{i}}{\sum\limits_{j=1}^{12}}{\int_{l_{j}}}{e^{i\beta(kz+{\overline{z}\over k})}}\rho_{j}(x, y, k){dk\over k}, \end{equation} $

(2.15)

where

$ \begin{array}{l} {\rho _1}(x,y,k) = \mathop \smallint \nolimits_{ - 1}^1 {e^{ - i\beta (k + \frac{1}{k})x}}[ - i{q_y}(x,0) + i\beta (k - \frac{1}{k})q(x,0)]dx,\\ {\rho _2}(x,y,k) = \mathop \smallint \nolimits_\pi ^{{\theta _2}} {e^{ - i\beta (k{e^{i\theta }} + \frac{1}{k}{e^{ - i\theta }})}}[i{q_r}(\cos \theta ,\sin \theta ) - \beta (k{e^{i\theta }} + \frac{1}{k}{e^{ - i\theta }})q(\cos \theta ,\sin \theta )]d\theta ,\\ {\rho _3}(x,y,k) = \mathop \smallint \nolimits_{{\theta _1}}^{{\theta _2}} {e^{ - i\beta (k{e^{i\theta }} + \frac{1}{k}{e^{ - i\theta }})}}[i{q_r}(\cos \theta ,\sin \theta ) - \beta (k{e^{i\theta }} + \frac{1}{k}{e^{ - i\theta }})q(\cos \theta ,\sin \theta )]d\theta ,\\ {\rho _4}(x,y,k) = \mathop \smallint \nolimits_{{\theta _1}}^0 {e^{ - i\beta (k{e^{i\theta }} + \frac{1}{k}{e^{ - i\theta }})}}[i{q_r}(\cos \theta ,\sin \theta ) - \beta (k{e^{i\theta }} + \frac{1}{k}{e^{ - i\theta }})q(\cos \theta ,\sin \theta )]d\theta ,\\ {\rho _9}(x,y,k) = \mathop \smallint \nolimits_\pi ^{{\theta _1}} {e^{ - i\beta (k{e^{i\theta }} + \frac{1}{k}{e^{ - i\theta }})}}[i{q_r}(\cos \theta ,\sin \theta ) - \beta (k{e^{i\theta }} + \frac{1}{k}{e^{ - i\theta }})q(\cos \theta ,\sin \theta )]d\theta ,\\ {\rho _{10}}(x,y,k) = \mathop \smallint \nolimits_0^{{\theta _2}} {e^{ - i\beta (k{e^{i\theta }} + \frac{1}{k}{e^{ - i\theta }})}}[i{q_r}(\cos \theta ,\sin \theta ) - \beta (k{e^{i\theta }} + \frac{1}{k}{e^{ - i\theta }})q(\cos \theta ,\sin \theta )]d\theta ,\\ {\rho _5}(x,y,k) = - {\rho _3}(x,y,k),\;\;\;{\rho _6}(x,y,k) = - {\rho _4}(x,y,k),\;\;\;{\rho _7}(x,y,k) = - {\rho _1}(x,y,k),\\ {\rho _8}(x,y,k) = - {\rho _2}(x,y,k),\;\;\;{\rho _{11}}(x,y,k) = - {\rho _9}(x,y,k),\;\;\;{\rho _{12}}(x,y,k) = - {\rho _{10}}(x,y,k), \end{array} $

and $\theta_{1}$, $\theta_{2}$ are defined as in Theorem 1, and $l_{j} (j=1, 2, \dots, 12)$ are shown as in Figure 5. Moreover, there exists the following global relation

$ \begin{array}{l} \;\;\;\;\mathop \smallint \nolimits_{ - 1}^1 {e^{ - i\beta (k + \frac{1}{k})x}}[ - i{q_y}(x,0) + i\beta (k - \frac{1}{k})q(x,0)]dx\\ = \mathop \smallint \nolimits_\pi ^0 {e^{ - i\beta (k{e^{i\theta }} + \frac{1}{k}{e^{ - i\theta }})}}[i{q_r}(\cos \theta ,\sin \theta ) - \beta (k{e^{i\theta }} + \frac{1}{k}{e^{ - i\theta }})q(\cos \theta ,\sin \theta )]d\theta . \end{array} $

(2.16)

In Section 2, we see that the integral representations of the solutions for basic linear elliptic PDEs can be expressed in terms of spectral functions $\rho_{j}(x, y, k)$. However, $\rho_{j}(x, y, k)$ are determined by boundary values $q(x, 0), q_{y}(x, 0)$ and $q(\cos\theta, \sin\theta), q_{r}(\cos\theta, \sin\theta)$. For some boundary conditions such as Dirichlet boundary conditions, boundary values $q_{y}(x, 0)$ and $q_{r}(\cos\theta, \sin\theta)$ are unknown. The goal of generalized Dirichlet to Neumann map is to determine these unknown boundary values, which can be accomplished through the global relation and some symmetric properties.

For illustration, we now discuss the Helmholtz equation in a semi-disk domain $\Omega$ with the following Dirichlet boundary condition

$ q(x, y)=f(x, y), (x, y)\in {\partial\Omega=L_{1}\cup L_{2}}, $

where function $f(x, y)$ has appropriate smoothness.

Theorem 2   Let $\Omega$ be a semi-disk domain in the complex plane $\mathbb{C}$ described in Figure 1. Assume that the boundary value $f(x, y)$ has appropriate smoothness and that the Dirichlet boundary problem of the Helmholtz equation (see (2.1) with $\alpha=\beta^2$) has the solution with form (2.15). Then all spectral function $\rho_{j}(x, y, k)(j=1, 2, 3, 4)$ can be determined by the boundary value $f(x, y)$.

Proof   It follows from the global relation (2.16) that

$ \begin{equation}\label{eq3-1} G_{1}(k)-G_{2}(k)=F(k), \qquad k\in \mathbb{C}, \end{equation} $

(3.1)

where

$ \begin{align*} G_{1}(k)=&{\int_{-1}^{1}}q_{y}(x, 0){e^{-i\beta(k+{1\over k})x}}dx, \quad G_{2}(k)={\int_{0}^{\pi}}q_{r}(\cos\theta, \sin\theta)e^{-i\beta(ke^{i\theta}+{1\over k}e^{-i\theta})}d{\theta}, \\ F(k)=&\beta(k-{1\over{k}}){\int_{-1}^{1}}f(x, 0){e^{-i\beta(k+{1\over k})x}}dx\\ &+i\beta{\int_{0}^{\pi}}f_{r}(\cos\theta, \sin\theta)(ke^{i\theta}+{1\over k}e^{-i\theta})e^{-i\beta(ke^{i\theta}+{1\over k}e^{-i\theta})}d{\theta}. \end{align*} $

It can be verified directly that $G_{1}(k), G_{2}(k)$ have following symmetric relation

$ \begin{equation}\label{eq3-2} {\overline{G_{1}({\overline{k}})}}=G_{1}(-k), \quad {\overline{G_{2}({\overline{k}})}}=G_{2}(-ke^{-2i\theta}), \qquad k\in \mathbb{C}. \end{equation} $

(3.2)

We can deduce from (3.1) and (3.2) that

$ G_{2}(k)-G_{2}(ke^{-2i\theta})={\overline{F({-\overline{k}})}}-F(k), $

that is,

$ \begin{equation}\label{eq3-3} {\int_{0}^{\pi}}q_{r}(\cos\theta, \sin\theta)\cdot K_{1}(\theta, k)d{\theta}={\overline{F({-\overline{k}})}}-F(k), \end{equation} $

(3.3)

where

$ K_{1}(\theta, k)=2\sinh[\beta(k-{1\over k})\sin\theta)]e^{-i\beta(k+{1\over k})\cos\theta}. $

This is the first kind of Fredholm integral equation. So by (3.1) and (3.3), we may determine the functions $G_{1}(k)$ and $q_{r}(\cos\theta, \sin\theta)$, which implies that all spectral functions $\rho_{j}(x, y, k)$ can be characterized by a given boundary value $f(x, y)$.

We remark that spectral functions $\rho_{j}(x, y, k)$ can be expressed by $G_{1}(k)$ and $q_{r}(\cos\theta, \sin\theta)$, so we only determine $q_{r}(\cos\theta, \sin\theta)$ from equation (3.3), which results in $G_{2}(k)$, and hence determine $G_{1}(k)$ from equation (3.1) and needn't calculate $q_{y}(x, 0)$.

We improve Fokas method to study the Laplace, modified Helmholtz and Helmholtz equations in a semi-disk region $\Omega$. The integral representations of the solutions for these basic linear elliptic PDEs are derived in terms of spectral functions $\rho_{j}(x, y, k)$, where the spectral functions $\rho_{j}(x, y, k)$ depend on the boundary values $q(x, 0), q(\cos\theta, \sin\theta)$ and its derivatives $q_{y}(x, 0), q_{r}(\cos\theta, \sin\theta)$. For some specific boundary value problems, using the global relation and symmetric properties, all spectral functions $\rho_{j}(x, y, k)$ can be determined through the solution to a Fredholm integral equation of the first kind. The results here can be further applied to discuss numerical solutions or asymptotic analysis.

The method here can also be used to solve boundary value problems for linear elliptic PDEs on sector domains or disk ones.