© Wuhan University 2023

0 Introduction

Boundary value problems for (λ, k) bi-analytic functions are discussed in Refs.[1-6]. In Ref.[4], the author used the Cauchy-type integral of (λ, 1) bi-analytic functions to study a class of inverse Riemann problems of (λ, 1) bi-analytic functions. In this article, we study the general case by using the theory of boundary value problems for analytic functions.

Let $\mathrm{\Gamma }$ be a simple and closed smooth contour, oriented counter-clockwisely. $D^{+}$($D^{-}$) denote respectively the interior (exterior) region bounded by $\mathrm{\Gamma }$. And let $O=(\mathrm{0},\mathrm{0})$ in $D^{+}$. Our inverse boundary value problem, simply called I-BVP, is to find functions $\omega (t)$, $\vartheta (t)$ and constants ${\lambda }^{+}$, ${\lambda }^{-}$, satisfying the following boundary conditions

$\{\begin{array}{c}\frac{\partial f(z)}{\partial \overline{z}}=\frac{{\lambda }^{\pm }-\mathrm{1}}{\mathrm{4}{\lambda }^{\pm }}{\varphi }^{\pm }(z)+\frac{{\lambda }^{\pm }+\mathrm{1}}{\mathrm{4}{\lambda }^{\pm }}\overline{{\varphi }^{\pm }(z)},\hspace{1em}\hspace{1em}\hspace{1em}z\in D^{\pm },\\ \frac{\partial {\varphi }^{+}(z)}{\partial \overline{z}}=\theta (z),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}z\in D^{+},\\ \frac{\partial {\varphi }^{-}(z)}{\partial \overline{z}}=\varpi (z)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}z\in D^{-},\\ f^{+}(t)=a_i(t)f^{-}(t)+b_i(t)\omega (t),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}t\in \mathrm{\Gamma },i=\mathrm{1,2},\\ {\varphi }^{+}(t)={\beta }_i(t){\varphi }^{-}(t)+{\gamma }_i(t)\vartheta (t),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} t\in \mathrm{\Gamma },i=\mathrm{1,2},\\ f^{-}(t_i)=c_i,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{t}_i\in \mathrm{\Gamma },i=\mathrm{1}, \mathrm{2},\end{array}$(1)

where $\theta (t)$, $\varpi (t)$, $a_i(t)$, $b_i(t)$, ${\beta }_i(t)$, ${\gamma }_i(t)$($i=\mathrm{1},\mathrm{2}$)$\in H(\mathrm{\Gamma })$ (Hölder continuous). $c_i$ ($i=\mathrm{1},\mathrm{2}$) are known constants.

1 Inverse Riemann Boundary Value Problems

In this section, we will consider a class of inverse Riemann boundary value problems, simply called I-RBVP. It is to find $\varphi (z)$ and $\vartheta (t)$ meeting the following conditions

$\{\begin{array}{c}\frac{\partial {\varphi }^{+}(z)}{\partial \overline{z}}=\theta (z),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}z\in D^{+}\\ \frac{\partial {\varphi }^{-}(z)}{\partial \overline{z}}=\varpi (z),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}z\in D^{-}\\ {\varphi }^{+}(t)={\beta }_{\mathrm{1}}(t){\varphi }^{-}(t)+{\gamma }_{\mathrm{1}}(t)\vartheta (t),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} t\in \mathrm{\Gamma }\\ {\varphi }^{+}(t)={\beta }_{\mathrm{2}}(t){\varphi }^{-}(t)+{\gamma }_{\mathrm{2}}(t)\vartheta (t),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} t\in \mathrm{\Gamma }\end{array}$(2)

Suppose that ${\gamma }_{\mathrm{1}}\ne {\gamma }_{\mathrm{2}}$ and ${\beta }_{\mathrm{1}}{\gamma }_{\mathrm{2}}\ne {\beta }_{\mathrm{2}}{\gamma }_{\mathrm{1}}$, the solvable conditions and the representation of the solutions are obtained.

Assume that $\Lambda \in H^{\nu }(\mathrm{\Gamma })$($\mathrm{0}<\nu \le \mathrm{1}$). Let the Cauchy integral operators

$\left(S_{\mathrm{\Gamma }}[\Lambda ]\right)(z)=\{\begin{array}{c}\frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }\mathrm{i}}{\int }_{\mathrm{\Gamma }}\frac{\Lambda (\tau )}{\tau -z}\mathrm{d}\tau ,\hspace{1em}\hspace{1em}z\notin \mathrm{\Gamma }\\ \frac{\mathrm{1}}{\mathrm{\pi }\mathrm{i}}{\int }_{\mathrm{\Gamma }}\frac{\Lambda (\tau )}{\tau -t}\mathrm{d}\tau ,\hspace{1em}z=t\in \mathrm{\Gamma }\end{array}$(3)

and its projection operator

$\left(S_{\mathrm{\Gamma }}^{\pm }[\Lambda ]\right)(z)=\{\begin{array}{c}\frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }\mathrm{i}}{\int }_{\mathrm{\Gamma }}\frac{\Lambda (\tau )}{\tau -z}\mathrm{d}\tau ,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}z\in D^{\pm }\\ \frac{\mathrm{1}}{\mathrm{2}}\left[\left(S_{\mathrm{\Gamma }}[\Lambda ]\right)(t)\pm \Lambda (t)\right],\hspace{1em}z=t\in \mathrm{\Gamma }\end{array}$(4)

Let

${\varphi }_{\mathrm{1}}(z)=\{\begin{array}{c}\varphi (z)-\overline{z} \theta (z),\hspace{1em}\hspace{1em}z\in D^{+}\\ \varphi (z)-\overline{z} \varpi (z),\hspace{1em}\hspace{1em}z\in D^{-}\end{array}$(5)

then we have

$\{\begin{array}{c}{\varphi }^{+}(t)={\varphi }_{\mathrm{1}}^{+}(t)+\overline{t}\theta (t)\\ {\varphi }^{-}(t)={\varphi }_{\mathrm{1}}^{-}(t)+\overline{t}\varpi (t)\end{array}$(6)

By (6) and (2), we obtain

$\{\begin{array}{c}{\varphi }_{\mathrm{1}}^{+}(t)=G(t){\varphi }_{\mathrm{1}}^{-}(t)+\overline{t} (G(t)\varpi (t)-\theta (t)),\hspace{1em}t\in \mathrm{\Gamma }\\ \frac{\partial {\varphi }_{\mathrm{1}}(z)}{\partial \overline{z}}=\mathrm{0},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}z\notin \mathrm{\Gamma }\end{array}$(7)

and

$\vartheta (t)=\frac{{\beta }_{\mathrm{2}}(t)-{\beta }_{\mathrm{1}}(t)}{{\gamma }_{\mathrm{1}}(t)-{\gamma }_{\mathrm{2}}(t)}\left({\varphi }_{\mathrm{1}}^{-}(t)+\overline{t}\varpi (t)\right)$(8)

where

$G(t)=\frac{{\beta }_{\mathrm{1}}(t){\gamma }_{\mathrm{2}}(t)-{\beta }_{\mathrm{2}}(t){\gamma }_{\mathrm{1}}(t)}{{\gamma }_{\mathrm{2}}(t)-{\gamma }_{\mathrm{1}}(t)},G(t)\in H(\mathrm{\Gamma })\mathrm{a}\mathrm{n}\mathrm{d}G(t)\ne \mathrm{0}$(9)

Now we consider the Riemann boundary value problem, simply called RBVP. It is to find ${\varphi }_{\mathrm{1}}(z)$ meeting the conditions given in (7). Denote $\kappa =\frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }}{\left[\mathrm{a}\mathrm{r}\mathrm{g}G(t)\right]}_{\mathrm{\Gamma }}$, then the index of RBVP (7) is $\kappa$. Suppose ${\varphi }_{\mathrm{1}}(\mathrm{\infty })=\mathrm{0}$, thus from Refs.[7, 8], we have the following results.

Lemma 1   If $\kappa \ge \mathrm{0}$, then the solutions of RBVP (7) are

${\varphi }_{\mathrm{1}}(z)=X(z)\left(S_{\mathrm{\Gamma }}\left[\frac{\overline{t} (G\varpi -\theta )}{X^{+}}\right]\right)(z)+P_{\kappa -\mathrm{1}}(z)X(z),\hspace{1em}\hspace{1em}z\notin \mathrm{\Gamma }$(10)

where

$X(z)=\{\begin{array}{c}{X}^{+}(z)=\mathrm{e}\mathrm{x}\mathrm{p}\left\{\left(S_{\mathrm{\Gamma }}\left[\mathrm{l}\mathrm{o}\mathrm{g}((t-z_{\mathrm{0}}{)}^{-\kappa }G)\right]\right)(z)\right\},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}z\in D^{+}\hspace{1em}\hspace{1em}\hspace{1em}\\ X^{-}(z)=(z-z_{\mathrm{0}}{)}^{-\kappa }\mathrm{e}\mathrm{x}\mathrm{p}\left\{\left(S_{\mathrm{\Gamma }}\left[\mathrm{l}\mathrm{o}\mathrm{g}((t-z_{\mathrm{0}}{)}^{-\kappa }G)\right]\right)(z)\right\},\hspace{1em}\hspace{1em}{z}_{\mathrm{0}}\in D^{+},z\in D^{-}\end{array}$(11)

$X(z)$ is the canonical, $P_{\kappa -\mathrm{1}}(z)$ is arbitrary polynomial whose degree does not exceed $\kappa -\mathrm{1}$.

If $\kappa <\mathrm{0}$, then the RBVP (7) has a unique solution

${\varphi }_{\mathrm{1}}(z)=X(z)\left(S_{\mathrm{\Gamma }}\left[\frac{\overline{t} (G\varpi -\theta )}{X^{+}}\right]\right)(z),\hspace{1em}\hspace{1em}z\notin \mathrm{\Gamma }$(12)

if and only if

${\int }_{\mathrm{\Gamma }}\frac{t^j\overline{t} (G(t)\varpi (t)-\theta (t))}{X^{+}(t)} \mathrm{d}t=\mathrm{0},\hspace{1em}j=\mathrm{0},\mathrm{1},\cdots ,-\kappa -\mathrm{1}$(13)

Hence, we obtain

Theorem 1   Under the requirement $\varphi (\mathrm{\infty })=\overline{\mathrm{\infty }}\varpi (\mathrm{\infty })$, when $\kappa \ge \mathrm{0}$, the solutions of I-RBVP (2) are

$\varphi (z)=\{\begin{array}{c}X(z)\left(S_{\mathrm{\Gamma }}\left[\frac{\overline{t} (G\varpi -\theta )}{X^{+}}\right]\right)(z)+P_{\kappa -\mathrm{1}}(z)X(z)+\overline{z} \theta (z),\hspace{1em}\hspace{1em}z\in D^{+}\\ X(z)\left(S_{\mathrm{\Gamma }}\left[\frac{\overline{t} (G\varpi -\theta )}{X^{+}}\right]\right)(z)+P_{\kappa -\mathrm{1}}(z)X(z)+\overline{z} \varpi (z),\hspace{1em}\hspace{1em}z\in D^{-}\end{array}$(14)

where $X(z)$ is given in (11), $P_{\kappa -\mathrm{1}}(z)$ is arbitrary polynomial whose degree does not exceed $\kappa -\mathrm{1}$, and

$\vartheta (t)=\frac{{\beta }_{\mathrm{2}}(t)-{\beta }_{\mathrm{1}}(t)}{{\gamma }_{\mathrm{1}}(t)-{\gamma }_{\mathrm{2}}(t)}\left\{X^{-}(t)\left(S_{\mathrm{\Gamma }}^{-}\left[\frac{\overline{\tau } (G\varpi -\theta )}{X^{+}}\right]\right)(t)+P_{\kappa -\mathrm{1}}(t)X^{-}(t)+\overline{t} \varpi (t)\right\}$(15)

When $\kappa <\mathrm{0}$, the solutions of I-RBVP (2) are

$\varphi (z)=\{\begin{array}{c}X(z)\left(S_{\mathrm{\Gamma }}\left[\frac{\overline{t} (G\varpi -\theta )}{X^{+}}\right]\right)(z)+\overline{z} \theta (z),\hspace{1em}\hspace{1em}z\in D^{+}\\ X(z)\left(S_{\mathrm{\Gamma }}\left[\frac{\overline{t} (G\varpi -\theta )}{X^{+}}\right]\right)(z)+\overline{z} \varpi (z),\hspace{1em}\hspace{1em}z\in D^{-}\end{array}$(16)

if and only if

${\int }_{\mathrm{\Gamma }}\frac{t^j\overline{t} (G(t)\varpi (t)-\theta (t))}{X^{+}(t)} \mathrm{d}t=\mathrm{0},\hspace{1em}j=\mathrm{0},\mathrm{1},\cdots ,-\kappa -\mathrm{1}$(17)

and

$\vartheta (t)=\frac{{\beta }_{\mathrm{2}}(t)-{\beta }_{\mathrm{1}}(t)}{{\gamma }_{\mathrm{1}}(t)-{\gamma }_{\mathrm{2}}(t)}\left\{X^{-}(t)\left(S_{\mathrm{\Gamma }}^{-}\left[\frac{\overline{\tau } (G\varpi -\theta )}{X^{+}}\right]\right)(t)+\overline{t} \varpi (t)\right\}$(18)

2 Inverse Boundary Value Problems for (λ, 1) Bi-Analytic Functions

In this section, we will consider the I-BVP for (λ, 1) bi-analytic functions. It is to find $\omega (t)$ and constants ${\lambda }^{\pm }$ meeting the following conditions

$\{\begin{array}{c}\frac{\partial f(z)}{\partial \overline{z}}=\frac{{\lambda }^{\pm }-\mathrm{1}}{\mathrm{4}{\lambda }^{\pm }}\varphi (z)+\frac{{\lambda }^{\pm }+\mathrm{1}}{\mathrm{4}{\lambda }^{\pm }}\overline{\varphi (z)},\hspace{1em}z\in D^{\pm }\\ f^{+}(t)=a_{\mathrm{1}}(t)f^{-}(t)+b_{\mathrm{1}}(t)\omega (t),\hspace{1em}\hspace{1em}\hspace{1em}t\in \mathrm{\Gamma }\\ f^{+}(t)=a_{\mathrm{2}}(t)f^{-}(t)+b_{\mathrm{2}}(t)\omega (t),\hspace{1em}\hspace{1em}\hspace{1em}t\in \mathrm{\Gamma }\\ f^{-}(t_i)=c_i,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{t}_i\in \mathrm{\Gamma },i=\mathrm{1}, \mathrm{2}\end{array}$(19)

where $a_i(t)$,$b_i(t)$($i=\mathrm{1},\mathrm{2}$)$\in H(\mathrm{\Gamma })$, constants $c_i$($i=\mathrm{1},\mathrm{2}$) are known.

In the following discussion, we assume that $b_{\mathrm{1}}(t)\ne b_{\mathrm{2}}(t)$ and $a_{\mathrm{1}}(t)b_{\mathrm{2}}(t)\ne a_{\mathrm{2}}(t)b_{\mathrm{1}}(t)$. By (5), we get

$\varphi (z)=\{\begin{array}{c}{\varphi }_{\mathrm{1}}(z)+\overline{z} \theta (z),\hspace{1em}\hspace{1em}z\in D^{+}\\ {\varphi }_{\mathrm{1}}(z)+\overline{z} \varpi (z),\hspace{1em}\hspace{1em}z\in D^{-}\end{array}$(20)

Substituting (20) into the first equation in (19) and letting

$f_{\mathrm{1}}(z)=\{\begin{array}{c}f(z)-\frac{{\lambda }^{+}-\mathrm{1}}{\mathrm{8}{\lambda }^{+}}{(\overline{z})}^{\mathrm{2}}\theta (z)-\frac{{\lambda }^{+}+\mathrm{1}}{\mathrm{4}{\lambda }^{+}}z\overline{\Theta (z)},\hspace{1em}z\in D^{+}\\ f(z)-\frac{{\lambda }^{-}-\mathrm{1}}{\mathrm{8}{\lambda }^{-}}{(\overline{z})}^{\mathrm{2}}\varpi (z)-\frac{{\lambda }^{-}+\mathrm{1}}{\mathrm{4}{\lambda }^{-}}z\overline{\Omega (z)},\hspace{1em}z\in D^{-}\end{array}$(21)

where ${\Theta }^{\text{'}}(z)=\theta (z)$, ${\Omega }^{\text{'}}(z)=\varpi (z)$, we obtain

$\frac{\partial f_{\mathrm{1}}(z)}{\partial \overline{z}}=\frac{{\lambda }^{\pm }-\mathrm{1}}{\mathrm{4}{\lambda }^{\pm }}{\varphi }_{\mathrm{1}}(z)+\frac{{\lambda }^{\pm }+\mathrm{1}}{\mathrm{4}{\lambda }^{\pm }}\overline{{\varphi }_{\mathrm{1}}(z)},\hspace{1em}z\in D^{\pm }$(22)

(22) and the second equation in (7) imply $f_{\mathrm{1}}(z)$ is a $(\lambda ,\mathrm{1})$ bi-analytic function with associate function ${\varphi }_{\mathrm{1}}(z)$. By (22), we obtain

$f_{\mathrm{1}}(z)=\frac{{\lambda }^{\pm }-\mathrm{1}}{\mathrm{4}{\lambda }^{\pm }}{\varphi }_{\mathrm{1}}(z)\overline{z}+\frac{{\lambda }^{\pm }+\mathrm{1}}{\mathrm{4}{\lambda }^{\pm }}\overline{{\Phi }_{\mathrm{1}}(z)}+\psi (z),\hspace{1em}\hspace{1em}z\in D^{\pm }$(23)

where ${\Phi }_{\mathrm{1}}^{\text{'}}(z)={\varphi }_{\mathrm{1}}(z)$. Let $z\to t$ from the inner domain $D^{+}$ and the outer domain $D^{-}$, respectively, one get

$f_{\mathrm{1}}^{+}(t)=f^{+}(t)-\frac{{\lambda }^{+}-\mathrm{1}}{\mathrm{8}{\lambda }^{+}}{(\overline{t})}^{\mathrm{2}}\theta (t)-\frac{{\lambda }^{+}+\mathrm{1}}{\mathrm{4}{\lambda }^{+}}t\overline{\Theta (t)}=\frac{{\lambda }^{+}-\mathrm{1}}{\mathrm{4}{\lambda }^{+}}{\varphi }_{\mathrm{1}}^{+}(t)\overline{t}+\frac{{\lambda }^{+}+\mathrm{1}}{\mathrm{4}{\lambda }^{+}}\overline{{\Phi }_{\mathrm{1}}^{+}(t)}+{\psi }^{+}(t),t\in \mathrm{\Gamma }$(24)

and

$f_{\mathrm{1}}^{-}(t)=f^{-}(t)-\frac{{\lambda }^{-}-\mathrm{1}}{\mathrm{8}{\lambda }^{-}}{(\overline{t})}^{\mathrm{2}}\varpi (t)-\frac{{\lambda }^{-}+\mathrm{1}}{\mathrm{4}{\lambda }^{-}}t\overline{\Omega (t)}=\frac{{\lambda }^{-}-\mathrm{1}}{\mathrm{4}{\lambda }^{-}}{\varphi }_{\mathrm{1}}^{-}(t)\overline{t}+\frac{{\lambda }^{-}+\mathrm{1}}{\mathrm{4}{\lambda }^{-}}\overline{{\Phi }_{\mathrm{1}}^{-}(t)}+{\psi }^{-}(t),t\in \mathrm{\Gamma }$(25)

respectively.

By the second equation and the third equation in (19), we get

$\left|\begin{array}{cc}\mathrm{1}& b_{\mathrm{1}}(t)\\ \mathrm{1}& b_{\mathrm{2}}(t)\end{array}\right|f^{+}(t)=\left|\begin{array}{cc}{a}_{\mathrm{1}}(t)& b_{\mathrm{1}}(t)\\ a_{\mathrm{2}}(t)& b_{\mathrm{2}}(t)\end{array}\right|f^{-}(t),\hspace{1em}t\in \mathrm{\Gamma }$(26)

and

$\omega (t)=\frac{a_{\mathrm{2}}(t)-a_{\mathrm{1}}(t)}{b_{\mathrm{1}}(t)-b_{\mathrm{2}}(t)}{f}^{-}(t)$(27)

Submitting $f^{+}(t)$ in (24) and $f^{-}(t)$ in (25) into (26), we obtain

${\psi }^{+}(t)=\frac{\left|\begin{array}{cc}{a}_{\mathrm{1}}(t)& b_{\mathrm{1}}(t)\\ a_{\mathrm{2}}(t)& b_{\mathrm{2}}(t)\end{array}\right|}{\left|\begin{array}{cc}\mathrm{1}& b_{\mathrm{1}}(t)\\ \mathrm{1}& b_{\mathrm{2}}(t)\end{array}\right|}{\psi }^{-}(t)+\Xi (t),\hspace{1em}t\in \mathrm{\Gamma }$(28)

where

$\begin{array}{c}\begin{array}{l}\Xi (t)=\frac{\left|\begin{array}{cc}{a}_{\mathrm{1}}(t)& b_{\mathrm{1}}(t)\\ a_{\mathrm{2}}(t)& b_{\mathrm{2}}(t)\end{array}\right|}{\left|\begin{array}{cc}\mathrm{1}& b_{\mathrm{1}}(t)\\ \mathrm{1}& b_{\mathrm{2}}(t)\end{array}\right|}\left\{\frac{{\lambda }^{-}-\mathrm{1}}{\mathrm{4}{\lambda }^{-}}\left[{\varphi }_{\mathrm{1}}^{-}(t)\overline{t}+\frac{\mathrm{1}}{\mathrm{2}}{(\overline{t})}^{\mathrm{2}}\varpi (t)\right]+\frac{{\lambda }^{-}+\mathrm{1}}{\mathrm{4}{\lambda }^{-}}\left[\overline{{\Phi }_{\mathrm{1}}^{-}(t)}+t\overline{\Omega (t)}\right]\right\}\\ -\left\{\frac{{\lambda }^{+}-\mathrm{1}}{\mathrm{4}{\lambda }^{+}}\left[{\varphi }_{\mathrm{1}}^{+}(t)\overline{t}+\frac{\mathrm{1}}{\mathrm{2}}{(\overline{t})}^{\mathrm{2}}\theta (t)\right]+\frac{{\lambda }^{+}+\mathrm{1}}{\mathrm{4}{\lambda }^{+}}\left[\overline{{\Phi }_{\mathrm{1}}^{+}(t)}+t\overline{\Theta (t)}\right]\right\}\end{array}\end{array}$(29)

From (23) and (22), obviously,

$\frac{\partial \psi (z)}{\partial \overline{z}}=\mathrm{0},\hspace{1em}z\notin \mathrm{\Gamma }$(30)

Now we consider the following RBVP. It is to find $\psi (z)$ meeting the conditions (28) and (30), namely,

$\{\begin{array}{c}{\psi }^{+}(t)=G_{\mathrm{*}}(t){\psi }^{-}(t)+\Xi (t),\hspace{1em}t\in \mathrm{\Gamma }\\ \frac{\partial \psi (z)}{\partial \overline{z} }=\mathrm{0},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}z\notin \mathrm{\Gamma }\end{array}$(31)

where

$G_{\mathrm{*}}(t)=\frac{a_{\mathrm{1}}(t)b_{\mathrm{2}}(t)-a_{\mathrm{2}}(t)b_{\mathrm{1}}(t)}{b_{\mathrm{2}}(t)-b_{\mathrm{1}}(t)}$(32)

$G_{\mathrm{*}}(t)\in H(\mathrm{\Gamma })$ and $G_{\mathrm{*}}(t)\ne \mathrm{0}$. ${\kappa }_{\mathrm{*}}=\frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }}{\left[\mathrm{a}\mathrm{r}\mathrm{g}{G}_{\mathrm{*}}(t)\right]}_{\mathrm{\Gamma }}$ is the index of RBVP (31). If $\psi (\mathrm{\infty })$ is to be finite, then we have the following results from Refs.[7, 8].

Lemma 2   1) If ${\kappa }_{\mathrm{*}}+\mathrm{1}\ge \mathrm{0}$, then the solutions of RBVP (31) are

$\psi (z)=X_{\mathrm{*}}(z)\left(S_{\mathrm{\Gamma }}\left[\frac{\Xi }{X_{\mathrm{*}}^{+}}\right]\right)(z)+P_{{\kappa }_{\mathrm{*}}}(z)X_{\mathrm{*}}(z),\hspace{1em}\hspace{1em}z\notin \mathrm{\Gamma }$(33)

where $P_{{\kappa }_{\mathrm{*}}}(z)$ is arbitrary polynomial whose degree does not exceed ${\kappa }_{\mathrm{*}}$,

$X_{\mathrm{*}}(z)=\{\begin{array}{c}{X}_{\mathrm{*}}^{+}(z)=\mathrm{e}\mathrm{x}\mathrm{p}\left\{\left(S_{\mathrm{\Gamma }}\left[\mathrm{l}\mathrm{o}\mathrm{g}((t-z_{\mathrm{*}}{)}^{-{\kappa }_{\mathrm{*}}}{G}_{\mathrm{*}})\right]\right)(z)\right\},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}z\in D^{+}\hspace{1em}\hspace{1em}\hspace{1em}\\ X_{\mathrm{*}}^{-}(z)=(z-z_{\mathrm{*}}{)}^{-{\kappa }_{\mathrm{*}}}\mathrm{e}\mathrm{x}\mathrm{p}\left\{\left(S_{\mathrm{\Gamma }}\left[\mathrm{l}\mathrm{o}\mathrm{g}((t-z_{\mathrm{*}}{)}^{-{\kappa }_{\mathrm{*}}}{G}_{\mathrm{*}})\right]\right)(z)\right\},\hspace{1em}\hspace{1em}{z}_{\mathrm{*}}\in D^{+},z\in D^{-}\end{array}$(34)

2) If ${\kappa }_{\mathrm{*}}+\mathrm{1}<\mathrm{0}$, then the RBVP (31) has a unique solution (33) ($P_{{\kappa }_{\mathrm{*}}}\equiv \mathrm{0}$) if and only if

${\int }_{\mathrm{\Gamma }}\frac{t^j\Xi (t)}{X_{\mathrm{*}}^{+}(t)} \mathrm{d}t=\mathrm{0},\hspace{1em}j=\mathrm{0},\mathrm{1},\cdots ,-{\kappa }_{\mathrm{*}}-\mathrm{2}$(35)

3) $\omega (t)$ is written as following

$\omega (t)=\frac{a_{\mathrm{2}}(t)-a_{\mathrm{1}}(t)}{b_{\mathrm{1}}(t)-b_{\mathrm{2}}(t)}\left\{\frac{{\lambda }^{-}-\mathrm{1}}{\mathrm{4}{\lambda }^{-}}\left[{\varphi }_{\mathrm{1}}^{-}(t)\overline{t}+\frac{\mathrm{1}}{\mathrm{2}}{(\overline{t})}^{\mathrm{2}}\varpi (t)\right]+\frac{{\lambda }^{-}+\mathrm{1}}{\mathrm{4}{\lambda }^{-}}\left[\overline{{\Phi }_{\mathrm{1}}^{-}(t)}+t\overline{\Omega (t)}\right]+{\psi }^{-}(t)\right\},\hspace{1em}t\in \mathrm{\Gamma }$(36)

(26) can be changed to

$f^{-}(t)=\frac{b_{\mathrm{2}}(t)-b_{\mathrm{1}}(t)}{a_{\mathrm{1}}(t)b_{\mathrm{2}}(t)-a_{\mathrm{2}}(t)b_{\mathrm{1}}(t)}{f}^{+}(t)$(37)

Submitting $f^{+}(t)$ in (24) into (37), one get

$f^{-}(t)=\frac{b_{\mathrm{2}}(t)-b_{\mathrm{1}}(t)}{a_{\mathrm{1}}(t)b_{\mathrm{2}}(t)-a_{\mathrm{2}}(t)b_{\mathrm{1}}(t)}\left\{\frac{{\lambda }^{+}-\mathrm{1}}{\mathrm{4}{\lambda }^{+}}\left[{\varphi }_{\mathrm{1}}^{+}(t)\overline{t}+\frac{\mathrm{1}}{\mathrm{2}}{(\overline{t})}^{\mathrm{2}}\theta (t)\right]+\frac{{\lambda }^{+}+\mathrm{1}}{\mathrm{4}{\lambda }^{+}}\left[\overline{{\Phi }_{\mathrm{1}}^{+}(t)}+t\overline{\Theta (t)}\right]+{\psi }^{+}(t)\right\}$(38)

where ${\psi }^{+}(t)$ is from $\psi (z)$ (let $z\to t$ from the inner domain $D^{+}$) given in Lemma 2, namely,

${\psi }^{+}(t)=X_{\mathrm{*}}^{+}(t)\left(S_{\mathrm{\Gamma }}^{+}\left[\frac{\Xi }{X_{\mathrm{*}}^{+}}\right]\right)(t)+\mathrm{\hslash }\cdot P_{{\kappa }_{\mathrm{*}}}(t)X_{\mathrm{*}}^{+}(t)$(39)

so, we get

$A(t)\frac{\mathrm{1}}{{\lambda }^{+}}+B(t)\frac{\mathrm{1}}{{\lambda }^{-}}=C(t)$(40)

where

$\begin{array}{l}A(t)=\frac{b_{\mathrm{2}}(t)-b_{\mathrm{1}}(t)}{a_{\mathrm{1}}(t)b_{\mathrm{2}}(t)-a_{\mathrm{2}}(t)b_{\mathrm{1}}(t)}\left\{{\varphi }_{\mathrm{1}}^{+}(t)\overline{t}+\frac{\mathrm{1}}{\mathrm{2}}{(\overline{t})}^{\mathrm{2}}\theta (t)-\overline{{\Phi }_{\mathrm{1}}^{+}(t)}-t\overline{\Theta (t)}\right\}\\ \hspace{1em}\hspace{1em}+\frac{b_{\mathrm{2}}(t)-b_{\mathrm{1}}(t)}{a_{\mathrm{1}}(t)b_{\mathrm{2}}(t)-a_{\mathrm{2}}(t)b_{\mathrm{1}}(t)}{X}_{\mathrm{*}}^{+}(t)\left(S_{\mathrm{\Gamma }}^{+}\left[\frac{-{\varphi }_{\mathrm{1}}^{+}\overline{\tau }-\frac{\mathrm{1}}{\mathrm{2}}{(\overline{\tau })}^{\mathrm{2}}\theta +\overline{{\Phi }_{\mathrm{1}}^{+}}+\tau \overline{\Theta }}{X_{\mathrm{*}}^{+}}\right]\right)(t)\end{array}$(41)

$B(t)=X_{\mathrm{*}}^{+}(t)\left(S_{\mathrm{\Gamma }}^{+}\left[\frac{{\varphi }_{\mathrm{1}}^{-}\overline{\tau }+\frac{\mathrm{1}}{\mathrm{2}}{(\overline{\tau })}^{\mathrm{2}}\varpi -\overline{{\Phi }_{\mathrm{1}}^{-}}-\tau \overline{\Omega }}{X_{\mathrm{*}}^{+}}\right]\right)(t)$(42)

$C(t)=\frac{b_{\mathrm{2}}(t)-b_{\mathrm{1}}(t)}{a_{\mathrm{1}}(t)b_{\mathrm{2}}(t)-a_{\mathrm{2}}(t)b_{\mathrm{1}}(t)}\left\{{\varphi }_{\mathrm{1}}^{+}(t)\overline{t}+\frac{\mathrm{1}}{\mathrm{2}}{(\overline{t})}^{\mathrm{2}}\theta (t)+\overline{{\Phi }_{\mathrm{1}}^{+}(t)}+t\overline{\Theta (t)}+\mathrm{\hslash }\cdot \mathrm{4}{P}_{{\kappa }_{\mathrm{*}}}(t)X_{\mathrm{*}}^{+}(t)\right\}$

$\hspace{1em}\hspace{1em}+\frac{b_{\mathrm{2}}(t)-b_{\mathrm{1}}(t)}{a_{\mathrm{1}}(t)b_{\mathrm{2}}(t)-a_{\mathrm{2}}(t)b_{\mathrm{1}}(t)}{X}_{\mathrm{*}}^{+}(t)\left(S_{\mathrm{\Gamma }}^{+}\left[\frac{-{\varphi }_{\mathrm{1}}^{+}\overline{\tau }-\frac{\mathrm{1}}{\mathrm{2}}{(\overline{\tau })}^{\mathrm{2}}\theta -\overline{{\Phi }_{\mathrm{1}}^{+}}-\tau \overline{\Theta }}{X_{\mathrm{*}}^{+}}\right]\right)(t)$

$+X_{\mathrm{*}}^{+}(t)\left(S_{\mathrm{\Gamma }}^{+}\left[\frac{{\varphi }_{\mathrm{1}}^{-}\overline{\tau }+\frac{\mathrm{1}}{\mathrm{2}}{(\overline{\tau })}^{\mathrm{2}}\varpi +\overline{{\Phi }_{\mathrm{1}}^{-}}+\tau \overline{\Omega }}{X_{\mathrm{*}}^{+}}\right]\right)(t)-\mathrm{4}{f}^{-}(t)$(43)

By the fourth equations in (19), we have the following system of equations

$\{\begin{array}{c}A(t_{\mathrm{1}})\frac{\mathrm{1}}{{\lambda }^{+}}+B(t_{\mathrm{1}})\frac{\mathrm{1}}{{\lambda }^{-}}=C(t_{\mathrm{1}})\\ A(t_{\mathrm{2}})\frac{\mathrm{1}}{{\lambda }^{+}}+B(t_{\mathrm{2}})\frac{\mathrm{1}}{{\lambda }^{-}}=C(t_{\mathrm{2}})\end{array}$(44)

Furthermore, we obtain

$\{\begin{array}{c}{\lambda }^{+}(t)=\frac{A(t_{\mathrm{1}})B(t_{\mathrm{2}})-B(t_{\mathrm{1}})A(t_{\mathrm{2}})}{C(t_{\mathrm{1}})B(t_{\mathrm{2}})-B(t_{\mathrm{1}})C(t_{\mathrm{2}})}\\ {\lambda }^{-}(t)=\frac{A(t_{\mathrm{1}})B(t_{\mathrm{2}})-B(t_{\mathrm{1}})A(t_{\mathrm{2}})}{A(t_{\mathrm{1}})C(t_{\mathrm{2}})-C(t_{\mathrm{1}})A(t_{\mathrm{2}})}\end{array}\hspace{1em}$(45)

Remark 1   Take $\mathrm{\hslash }=\mathrm{1}$ while ${\kappa }_{\mathrm{*}}+\mathrm{1}\ge \mathrm{0}$. Take $\mathrm{\hslash }=\mathrm{0}$ while ${\kappa }_{\mathrm{*}}+\mathrm{1}<\mathrm{0}$.

Summarizing the above discussion, we have the following result.

Theorem 2   For the I-BVP of (λ, 1) bi-analytic functions, the solutions are given in Theorem 1, (36) and (45), namely

$\vartheta (t)=\frac{{\beta }_{\mathrm{2}}(t)-{\beta }_{\mathrm{1}}(t)}{{\gamma }_{\mathrm{1}}(t)-{\gamma }_{\mathrm{2}}(t)}\left\{X^{+}(t)\left(S_{\mathrm{\Gamma }}^{+}\left[\frac{\overline{\tau } (G\varpi -\theta )}{X^{+}}\right]\right)(t)+\mathrm{\hslash }\cdot P_{\kappa -\mathrm{1}}(t)X^{+}(t)+\overline{t} \varpi (t)\right\},t\in \mathrm{\Gamma },$

($\mathrm{\hslash }=\mathrm{1}$ while $\kappa \ge \mathrm{0}$, $\mathrm{\hslash }=\mathrm{0}$ while $\kappa <\mathrm{0}$.)

$\omega (t)=\frac{a_{\mathrm{2}}(t)-a_{\mathrm{1}}(t)}{b_{\mathrm{1}}(t)-b_{\mathrm{2}}(t)}\left\{\frac{{\lambda }^{-}-\mathrm{1}}{\mathrm{4}{\lambda }^{-}}\left[{\varphi }_{\mathrm{1}}^{-}(t)\overline{t}+\frac{\mathrm{1}}{\mathrm{2}}{(\overline{t})}^{\mathrm{2}}\varpi (t)\right]+\frac{{\lambda }^{-}+\mathrm{1}}{\mathrm{4}{\lambda }^{-}}\left[\overline{{\Phi }_{\mathrm{1}}^{-}(t)}+t\overline{\Omega (t)}\right]+{\psi }^{-}(t)\right\},\hspace{1em}t\in \mathrm{\Gamma },$

${\lambda }^{+}(t)=\frac{A(t_{\mathrm{1}})B(t_{\mathrm{2}})-B(t_{\mathrm{1}})A(t_{\mathrm{2}})}{C(t_{\mathrm{1}})B(t_{\mathrm{2}})-B(t_{\mathrm{1}})C(t_{\mathrm{2}})}$

${\lambda }^{-}(t)=\frac{A(t_{\mathrm{1}})B(t_{\mathrm{2}})-B(t_{\mathrm{1}})A(t_{\mathrm{2}})}{A(t_{\mathrm{1}})C(t_{\mathrm{2}})-C(t_{\mathrm{1}})A(t_{\mathrm{2}})}$

where A(t), B(t), C(t) are given in (41), (42), (43), respectively. ${\varphi }_{\mathrm{1}}^{\pm }(t)$ is from ${\varphi }_{\mathrm{1}}(z)$ given in Lemma 1. ${\psi }^{-}(t)$ is from $\psi (z)$ given in Lemma 2. For ${\varphi }_{\mathrm{1}}(z)$ and $\psi (z)$, there exist four cases.

1) When $\kappa \ge \mathrm{0}$ and ${\kappa }_{\mathrm{*}}+\mathrm{1}\ge \mathrm{0}$, ${\varphi }_{\mathrm{1}}(z)$ and $\psi (z)$ are given in (10) and (33), respectively.

2) When $\kappa \ge \mathrm{0}$ and ${\kappa }_{\mathrm{*}}+\mathrm{1}<\mathrm{0}$, ${\varphi }_{\mathrm{1}}(z)$ is given in (10) and $\psi (z)$ is given in (33)($P_{{\kappa }_{\mathrm{*}}}\equiv \mathrm{0}$) if and only if the condition (35) is satisfied.

3) When $\kappa <\mathrm{0}$ and ${\kappa }_{\mathrm{*}}+\mathrm{1}\ge \mathrm{0}$, ${\varphi }_{\mathrm{1}}(z)$ is given in (12) if and only if the condition (13) is satisfied and $\psi (z)$ is given in (33).

4) When $\kappa <\mathrm{0}$ and ${\kappa }_{\mathrm{*}}+\mathrm{1}<\mathrm{0}$, ${\varphi }_{\mathrm{1}}(z)$ and $\psi (z)$ are given in (12) and (33)($P_{{\kappa }_{\mathrm{*}}}\equiv \mathrm{0}$) if and only if the conditions (13) and (35) are satisfied, respectively.

Remark 2   The upper conclusions may be applied to interface problems of the elastic system in plane, for instance, the welding problems and the quasi-static system of thermoelasticity, etc.Refs.[9-14].

References