\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 65, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/65\hfil Oblique derivative problems] {Oblique derivative problems for generalized Rassias equations of mixed type with several characteristic boundaries} \author[G. C. Wen\hfil EJDE-2009/65\hfilneg] {Guo Chun Wen} \address{Guo Chun Wen \newline LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China} \email{Wengc@math.pku.edu.cn} \thanks{Submitted December 16, 2008. Published May 14, 2009.} \subjclass[2000]{35M05, 35J70, 35L80} \keywords{Oblique derivative problems; generalized Rassias equations; \hfill\break\indent several characteristic boundaries} \begin{abstract} This article concerns the oblique derivative problems for second-order quasilinear degenerate equations of mixed type with several characteristic boundaries, which include the Tricomi problem as a special case. First we formulate the problem and obtain estimates of its solutions, then we show the existence of solutions by the successive iterations and the Leray-Schauder theorem. We use a complex analytic method: elliptic complex functions are used in the elliptic domain, and hyperbolic complex functions in the hyperbolic domain, such that second-order equations of mixed type with degenerate curve are reduced to the first order mixed complex equations with singular coefficients. An application of the complex analytic method, solves \eqref{e1.1} below with $m=n=1$, $a=b=0$, which was posed as an open problem by Rassias. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \section{Formulation of oblique derivative problems} Tricomi problems for second-order equations of mixed type with parabolic degenerate lines possess important applications to gas dynamics, and have been discussed in \cite{b1}-\cite{s4}, \cite{w4,w5}. In this article, we generalize those results to second-order equations of mixed type with parabolic degeneracy and several characteristic boundaries. Let $D$ be a simply connected bounded domain in the complex plane $\mathbb{C}$ with the boundary $\partial D =\Gamma\cup L$, where $\Gamma\subset\{\hat y=y - x^n>0\}$ and is an element in $C^2_\mu$ with $0<\mu<1$ and with end points $z_*=-R-iR^n,z^*=R+iR^n$; and $L=L_1\cup L_2\cup L_3\cup\dots \cup L_{2N}$, where $N$ is an odd positive integer, and for $l=1,\dots ,N$, \begin{gather*} L_{2l-1} = \big\{x + \int_0^{y-x^n} \sqrt{|K(t)|}dt = a_{l-1},\; x \in [a_{l-1},a_l]\big\},\\ L_{2l} = \big\{x-\int_0^{y-x^n}\sqrt{|K(t)|}dt=a_l,\; x\in[a_{l-1},a_l]\big\}\,. \end{gather*} Herein $-R=a_00\}$, $D^-=D\cap\{y-x^n<0\}$, and $G(y-x^n)=\int_0^{y-x^n} \sqrt{|K(t)|}dt$. Without loss of generality, we may assume that the boundary $\Gamma$ possesses the form $x=-R+\tilde G(\hat y)$ and $x=R-\tilde G(\hat y)$ near $z_*$ and $z^*$ with the condition $d\tilde G(\hat y)/d\hat y=\pm\tilde H(\hat y)=0$ at $z=z_*, z^*$ respectively. Otherwise through a conformal mapping as stated in \cite{w5}, this requirement can be realized. In this paper, we use the hyperbolic unit $j$ with the condition $j^2=1$ in $\overline{D^-}$, and $x+jy$, $W(z)=U(z)+jV(z)=[H(\hat y)u_x-ju_y]/2$ are called the hyperbolic number and hyperbolic complex function in $D^-$, and $x+iy$, $W(z)=U(z)+iV(z)=[H(\hat y)u_x-iu_y]/2$ are called the complex number and elliptic complex function in $\overline{D^+}$ respectively (see \cite{w1}). Consider generalized Rassias equation of mixed type with parabolic degeneracy $$K(y-x^n)u_{xx}+u_{yy}+au_x+bu_y+cu+d=0\quad\text{in } D,\label{e1.1}$$ where $\hat y=y-x^n$, $a,b,c,d$ are real functions of $z\in \overline D$, $u,u_x,u_y \in \mathbb{R}$, and suppose that \eqref{e1.1} satisfies the following conditions, \begin{itemize} \item[(C1)] For continuously differentiable functions $u(z)$ in $D^*=\overline D\backslash\{\tilde a_0,\tilde a_1,\dots ,\tilde a_N\}$, the coefficients $a,b,c,d$ satisfy $$\begin{gathered} \tilde L_\infty[\eta,D^+] = L_\infty[\eta,D^+] + L_\infty[\eta_x,D^+] \le k_0,\quad \eta = a,b,c,\\ \tilde L_\infty[d,D^+] \le k_1, \quad \tilde C[d,\overline{D^-}]=C[d,\overline{D^-}]+C[d_x,\overline{D-}] \le k_1,\\ \tilde C[\eta,\overline{D^-}]\le k_0,\quad \eta=a,b,c, \\ c\le0 \quad \text{in } D^+,\\ |a(x,y)|{|\hat y|^{1-m/2}}=\varepsilon_1(\hat y) \quad \text{as }\hat y\to 0, m\ge2, z\in\overline{D^-}, \end{gathered}\label{e1.2}$$ where $\tilde a_l=a_l+ia_l^n,l=0,1,\dots,N,\,\hat y=y-x^n$, and $\varepsilon_1(\hat y)$ is a non-negative function such that $\varepsilon_1(\hat y)\to0$ as $\hat y\to0$. \item[(C2)] For any continuously differentiable functions $u_1(z),u_2(z)$ in $D^*$, the function $F(z,u,u_z)=au_x+bu_y+cu+d$ satisfies \begin{aligned} &F(z,u_1,u_{1z}) - F(z,u_2,u_{2z}) \\ &= \tilde a(u_1 - u_2)_x + \tilde b(u_1 - u_{2})_y + \tilde c(u_1 - u_2)\quad\text{in }D, \end{aligned}\label{e1.3} in which $\tilde a,\tilde b,\tilde c$ satisfy the same conditions as those of $a,b,c$ in \eqref{e1.2}, and $k_0, k_1$ are positive constants such that $k_0\ge2$, $k_1\ge\max[1,6k_0]$. \end{itemize} To write the complex form of the above equation, denote \begin{gather*} \begin{aligned} W(z) &= U + iV = \frac12[H(y - x^n)u_x - iu_y] \\ &= u_{\tilde z} = \frac{H(y - x^n)}2[u_x - iu_Y] = H(y - x^n)u_Z, \end{aligned}\\ \begin{aligned} H(y - x^n)W_{\overline Z}& =\frac{H(y - x^n)}2[W_x + iW_Y]\\ &=\frac12[H(y - x^n)W_x + iW_y] =W_{\overline{\tilde z}}\quad\text{in }\overline{D^+}, \end{aligned}\end{gather*} where $Z(z)=x + iY = x + iG(\hat y)$, $\hat y=y - x^n$ in $\overline{D^+}$. We have \begin{align*} & K(y - x^n)u_{xx} + u_{yy} \\ &= H(y - x^n)[H(y - x^n) u_x - iu_y]_x + i[H(y - x^n)u_x - iu_y]_y -[iH_y+HH_x]u_x \\ &= 2\{H[U + iV]_x + i[U + iV]_y\} - [iH_y /H + H_x]Hu_x\\ &= 4H(y - x^n)W_{\overline Z} - [iH_y/ H + H_x]Hu_x = -[au_x + bu_y + cu + d]; \end{align*} i.e., \begin{aligned} &H(y - x^n)W_{\overline Z}\\ &= H[W_x + iW_Y]/2 \\ &= H[(U + iV)_x + i(U + iV)_Y]/2 \\ &= \{(iH_y/H + H_x - a/H)Hu_x -bu_y - cu - d\}/4 \\ &= \{(iH_y/H + H_x - a/H)(W + \overline W) + ib(\overline W - W) - cu - d\}/4 \\ &= A_1(z,u,W)W + A_2(z,u,W)\overline W + A_3(z,u,W)u + A_4(z,u,W)\\ &= g(Z)\quad\text{in }D^+_Z, \end{aligned}\label{e1.4} in which $D^+_Z=D_Z$ is the image domains of $D^+$ with respect to the mapping $Z=Z(z)$. Moreover denote \begin{gather*} W(z) = U + jV = \frac12[H(y - x^n)u_x - ju_y] \\ = \frac{H(y - x^n)} 2[u_x - ju_Y] = H(y - x^n)u_Z, \\ H(y - x^n)W_{\overline Z} = \frac{H(y - x^n)}2[W_x + jW_Y] = \frac12[H(y - x^n)W_x + jW_y] = W_{\overline{\tilde z}}\quad\text{in }\overline{D^-}, \end{gather*} in which $Z(z)=x + jY = x + jG(\hat y)$, $\hat y=y - x^n$ in $\overline{D^-}$. Then we obtain \begin{align*} &-K(y - x^n)u_{xx} - u_{yy} \\ &= H(y - x^n)[H(y - x^n)u_x - ju_y]_x + j[H(y - x^n)u_x - ju_y]_y -[jH_y + HH_x]u_x \\ & = 2\{H[U + jV]_x+j[U + jV]_y\} - [jH_y/H + H_x]Hu_x \\ & =4H(y - x^n)W_{\overline Z} - [jH_y/ H + H_x]Hu_x \\ & = au_x + bu_y + cu + d, H(y - x^n)W_{\overline Z} \\ & =H[(U + jV)_x + j(U + jV)_Y]/2 \\ &=\{(jH_y/H + H_x)Hu_x + au_x + bu_y + cu + d\}/4 \\ &=\{(jH_y/H+H_x+a/H)(W+\overline W)+jb(\overline W-W)+cu+d\}/4 \\ &= H\{e_1[U_x+V_Y+V_x+U_Y]/2+e_2[U_x+V_Y-V_x-U_Y]/2\} \\ &= H\{e_1[(U + V)_x + (U + V)_Y]/2 + e_2[(U - V)_x - (U - V)_Y]/2\} \\ &= H[e_1(U + V)_\mu +e_2(U - V)_\nu] \\ &= \frac14\{(e_1 - e_2)[H_y/H]Hu_x + (e_1 + e_2) [(H_x + a/H)Hu_x + bu_y + cu + d]\}, \end{align*} and in $D^-$, we have $$\begin{gathered} (U+V)_\mu=\frac1{4H}\{2[H_y/H+H_x+a/H]U-2bV+cu+d\}, \\ (U - V)_\nu=\frac1{4H}\{-2[H_y/H-H_x-a/H]U-2bV+cu+d\}, \end{gathered} \label{e1.5}$$ where $e_1 = (1 + j)/2$, $e_2 = (1 - j)/2$, $2x = \mu + \nu$, $2Y=\mu-\nu$, $\partial x/\partial\mu=1/2=\partial Y/\partial\mu$ $\partial x/\partial\nu=1/2=-\partial Y/\partial\nu$. Hence the complex form of \eqref{e1.1} can be written as $$W_{\overline{\tilde z}}=A_1W+A_2\overline W+A_3u+A_4\quad\text{in }\overline D,$$ $$u(z) = \begin{cases}\displaystyle 2\mathop{\rm Re}\int_{z_*}^z[\frac{U(z)}{H(y - x^n)} + iV(z)]dz + c_0 &\text{in }\overline{D^+},\vspace{1.2mm}\\ \displaystyle2\mathop{\rm Re}\int_{z_*}^z[\frac{U(z)}{H(y - x^n)} - jV(z)]dz + c_0 &\text{in }\overline{D^-}, \end{cases} \label{e1.6}$$ where $c_0=u(z_*)$, and the coefficients $A_l=A_l(z,u,W)$ are as follows $$\begin{gathered} A_1 = \begin{cases} \displaystyle\frac14[-\frac{a}{H}+\frac{iH_y}{H}+H_x-ib],\vspace{1.2mm} \\ \displaystyle \frac14[\frac a{H}+\frac{jH_y}{H}+H_x-jb],\end{cases} \quad A_2 = \begin{cases}\displaystyle \frac14[-\frac{a}{H}+\frac{iH_y}{H}+H_x+ib],\vspace{1.2mm} \\ \displaystyle\frac14[\frac a{H}+\frac{jH_y}{H}+H_x+jb],\end{cases} \\ \displaystyle A_3 = \begin{cases}\displaystyle-\frac c4,\vspace{1.2mm} \\ \displaystyle\frac c4,\end{cases} \quad A_4 = \begin{cases}\displaystyle -\frac d4&\text{in } \overline{D^+},\vspace{1.2mm} \\ \displaystyle\frac d4 &\text{in }\overline{D^-}. \end{cases} \end{gathered} \label{e1.7}$$ For convenience, sometimes $\tilde a_l=a_l+ia_l^{n}$ $(l=0,1,\dots ,N)$ in the $z=x+iy$-plane are replaced by $\hat t_1=a_0$, $\hat t_l=a_{l-2}$ $(l=3,\dots ,N+1)$, $\hat t_2=a_N$ in $\hat z=x+i\hat y$-plane, and the hyperbolic complex number $\hat z=x+j\hat y$, the function $F[z(Z)]$ are simply written as $z=x+j\hat y, F(z)$ respectively. The oblique derivative boundary-value problem for \eqref{e1.1} may be formulated as follows: \subsection*{Problem P} Find a continuous solution $u(z)$ of \eqref{e1.1} in $\overline D$, where $u_x,u_y$ are continuous in $D^*$, and satisfy the boundary conditions $$\begin{gathered} \frac12\frac{\partial u}{\partial\nu} = \frac1{H(y - x^n)}\mathop{\rm Re}[\overline{\lambda(z)}u_{\tilde z}] = \mathop{\rm Re}[\overline{\Lambda(z)}u_z] = r(z) \quad\text{on } \Gamma \cup \tilde L, u(\tilde a_0)=c_0, \\ \frac1{H(y - x^n)}\mathop{\rm Im}[\overline{\lambda(z)}u_{\tilde z}]|_{z=z_l} = \mathop{\rm Im}[\overline{\Lambda(z)}u_{\tilde z}]|_{z=z_l} = b_l, u(\tilde a_l) = c_l,\quad l=1,\dots ,N. \end{gathered} \label{e1.8}$$ Herein $\tilde L=L_1\cup L_3\cup\dots \cup L_{2N-1}$, $\nu$ is a given vector at every point $z\in\Gamma\cup\tilde L$, $u_{\tilde z}=[H(y - x^n)u_x-iu_y]/2$, $\Lambda(z)=\cos(\nu,x)-i\cos(\nu,y)$, $\cos(\nu,x)$ means the cosine of angle between $\nu$ and $x$, $\lambda(z)=\mathop{\rm Re}\lambda(z)+i \mathop{\rm Im}\lambda(z)$, if $z\in\Gamma$, and $u_{\tilde z}=[H(y - x^n)u_x-ju_y]/2$, $\lambda(z)=\mathop{\rm Re}\lambda(z)+j\mathop{\rm Im}\lambda(z)$, if $z\in\tilde L$, $b_l,c_l\,(l=1,\dots ,N),c_0$ are real constants, and $r(z),b_l,c_l\,(l=1\dots ,N),c_0$ satisfy the conditions $$\begin{gathered} C^1_\alpha[\lambda(z),\Gamma]\le k_0,\quad C^1_\alpha[\lambda(z),\tilde L]\le k_{0},\quad C^1_\alpha[r(z),\Gamma]\le k_2, \\ C^1_\alpha[r(z),\tilde L_1]\le k_2,\quad \cos(\nu,n)\ge0\quad \text{on } \Gamma,\\ \cos(\nu,n)<1\quad\text{on }\tilde L, \\ |b_l|,|c_l|,|c_0| \le k_2,\quad l = 1,\dots ,N,\\ \max_{z\in \tilde L}\frac1{|\mathop{\rm Re}\lambda(z)-\mathop{\rm Im}\lambda(z)|}\le k_0, \end{gathered} \label{e1.9}$$ in which $n$ is the outward normal vector at every point on $\Gamma$, $\alpha,k_0,k_2$ are positive constants with $0 < \alpha < 1$ and $k_2\ge k_0)$ . The number $$K=\frac12(K_1+K_2+\dots +K_{N+1})$$ is called the index of Problem P, where $K_l = \big[\frac{\phi_l}\pi\big] + J_l,\quad J_l = 0\text{ or }1$, $$e^{i\phi_l} = \frac{\lambda(\hat t_l-0)}{\lambda(\hat t_l+0)},\quad \gamma_l = \frac{\phi_l}\pi - K_l,\quad l = 1,2,\dots ,N+1,$$ in which $\hat t_1=a_0$, $\hat t_2=a_N$, $\hat t_3=a_1,\dots ,\hat t_{N+1}=a_{N-1}$, $\lambda(t)=e^{i\pi/2}$ on $L_0$, $L_0=D\cap\{y - x^n=0\}$ on $x$-axis, and $\lambda(\hat t_1+0)=\lambda(\hat t_3-0)=\lambda(\hat t_3+0)=\dots =\lambda(\hat t_N-0)=\lambda(\hat t_N+0)=\lambda(\hat t_2-0)=\exp(i\pi/2)$. Here $K=-1/2$ or $(N-1)/2$ on the boundary $\partial D^+$ of $D^+$ can be chosen, in the last case we can add $N$ point conditions $u(\tilde a_l)=c_l\,(l=1,\dots ,N)$. It is clear that we can require that $-1/2\le\gamma_l<1/2\,(l=0,1,\dots ,N)$. Moreover if $\cos(\nu,n)\equiv 0$ on $\Gamma$, the case is just the boundary condition of Tricomi problem, from \eqref{e1.8}, we can determine the value $u(z^*)$ by the value $u(z_*)$, namely $$u(z^*) = 2\mathop{\rm Re} \int_{z_*}^{z^*}u_zdz + u(z_*) = 2 \int_0^S \mathop{\rm Re}[z'(s)u_z]ds + c_0 = 2 \int^S_0 r(z)ds + c_0 = c_N,$$ and $u(z) = 2\mathop{\rm Re} \int^z_{\tilde a_0} u_zdz + u(\tilde a_0) = 2 \int_0^s \mathop{\rm Re}[z'(s)u_z]ds + c_0 = 2 \int^{s}_0 r(z)ds + c_0=\phi(z)$ on $\Gamma$, and for $l=0,1,\dots ,N-1$, $u(z) = 2\mathop{\rm Re} \int^{z}_{\tilde a_l} u_zd\bar z + u(\tilde a_l) = 2 \int_0^{s_l} \mathop{\rm Re}[\overline{z'(s)}u_z]ds + c_l = 2 \int^{s_l}_0 r(z)ds + c_l = \psi(z)$ on $L_{2l+1}$, in which $\overline{\Lambda(z)}=z'(s)$ on $\Gamma$, $z(s)$ is a parameter expression of arc length $s$ of $\Gamma$ with the condition $z(0)=z_*$, $S$ is the length of the boundary $\Gamma$, and $\overline{\Lambda(z)}=\overline{z'(s)}$ on $L_l$, $z(s)$ is a parameter expression of arc length $s$ of $L_l$ with the condition $z(0)=\tilde a_l, l=0,\dots ,N-1$. If we consider $$\mathop{\rm Re}[\overline{\lambda(z)}(U+jV)]=0\quad\text{on }L_0,$$ where $\lambda(z)=1=e^{i0\pi}$, then $\gamma_1 = \gamma_2 = -1/2$, $\gamma_l=0$ $(l = 2,\dots ,N+1)$ or $\gamma_1 = 1/2$, $\gamma_2 = -1/2$, $\gamma_l = 0$ $(l=2,\dots ,N+1)$, thus $K = 0$ or $-1/2$. For \eqref{e1.1} with $c = 0$, when $K=-1/2$ or $(N-1)/2$, the last point condition in \eqref{e1.8} can be replaced by $$Lu_{\tilde z}(z'_l) = \mathop{\rm Im}[\overline{\lambda(z)}u_{\tilde z}]|_{z=z'_l} = H(y'_l - x'^n_l)c_l=c'_l,\quad l=1,\dots ,N, \label{e1.10}$$ where $z'_l=x'_l+iy'_l=x'_l+ix'^n_l$ $(l=1,\dots ,N)$ are distinct points on $\Gamma\backslash\{\tilde a_0\cup\tilde a_N\}$, and $c_l\,(l=1,\dots ,N)$ are real constants, in this case the condition $\cos(\nu,n)\ge0$ on $\Gamma$ in \eqref{e1.9} can be cancelled. The boundary value problem is called Problem Q. Noting that $\lambda(z),r(z) \in C^1_\alpha(\Gamma)$, $\lambda(z),r(z) \in C^1_\alpha (\tilde L)\,(0<\alpha<1)$, we can find two twice continuously differentiable functions $u_0^\pm(z)$ in $\overline{D^\pm}$, for instance, which are the solutions of the oblique derivative problem with the boundary condition in \eqref{e1.8} for harmonic equations in $D^\pm$ (see \cite{w2}), thus the functions $v(z)=v^\pm(z)=u(z)-u_0^\pm(z)$ in $D^\pm$ is the solution of the following boundary value problem in the form \begin{gather} K(y - x^n)v_{xx} + v_{yy} + \hat av_x + \hat bv_y + \hat cv + \hat d=0\quad\text{in }D, \label{e1.11} \\ \begin{gathered} \mathop{\rm Re}[\overline{\lambda(z)}v_{\tilde z}(z)]=R(z)\quad\text{on } \Gamma\cup\tilde L,\\ v(\tilde a_0)=c_0, \\ \mathop{\rm Im}[\overline{\lambda(z_l)}v_{\tilde z}(z_l)] = b'_l,\quad v(\tilde a_l) = c_l \text{ or } \mathop{\rm Im}[\overline{\lambda(z'_l)}v_{\tilde z}(z'_l)] = c'_l,\quad l = 1,\dots ,N. \end{gathered} \label{e1.12} \end{gather} Herein $W(z)=U+iV=v^+_{\tilde z}$ in $D^+$, $W(z)=U+jV=v^-_{\tilde z}$ in $\overline{D^-}$, $R(z)=0$ on $\Gamma\cup\tilde L$, $b_l=0$, $c_0=c_l=0$, $l=1,\dots ,N$. Hence later on we only discuss the case of the homogeneous boundary condition and the index $K=(N-1)/2$, the other case can be similarly discussed. From $v(z)=v^\pm(z)= u(z)-u^\pm_0(z)$ in $\overline{D^\pm}$, we have $u(z)=v^+(z)+u^+_0(z)$ in $\overline{D^+}$, $u(z)=v^-(z)+u_0^-(z)$ in $\overline{D^-}$, $v^+(z)=v^-(z)-u^+_0(z)+u^-_0(z)$, $v^+_y=v^-_y-u^+_{0y}+u^-_{0y}=2\hat R_0(x)$, and $v^-_y=2\tilde R_0(x)$ on $L_0=D\cap\{y=0\}$, where $\hat R_0(x),\tilde R_0(x)$ are undetermined real functions. The boundary vale problem \eqref{e1.11}, \eqref{e1.12} is called Problem $\tilde P$ or $\tilde Q$. Here we mention that if the domain $D$ is general, then we can choose a univalent conformal mapping, such that $D$ is transformed onto a special domain with the partial boundary $\Gamma$ as stated before, then the $u_x$ in Conditions (C1),(C2) should be replaced by $u_z$. For the boundary condition \eqref{e1.8} on the boundary $\partial D$ of general domain $D$, we require that the boundary conditions about $u(z)$ and $u_x$ in \eqref{e1.8} satisfy the similar conditions. \section{Representation of solutions to oblique derivative problems} The representation of solutions of Problem P or Q for equation \eqref{e1.1} is as follows. \begin{theorem} \label{thm2.1} Under Conditions {\rm (C1), (C2)}, any solution $u(z)$ of Problem P or Q for equation \eqref{e1.1} in $D^-$ can be expressed as \begin{aligned} u(z) &= \int_0^{y-x^n}V(z)dy+ u(x) = 2\mathop{\rm Re} \int_{z_*}^z [\frac{\mathop{\rm Re} W}{H(\hat y)}+ \begin{pmatrix}i\\-j\end{pmatrix} \mathop{\rm Im} W]dz + c_0\\\ &\qquad \text{in }\begin{pmatrix}\overline{D^+}\\ \overline{D^-}\end{pmatrix}, \\ W(z) &= \Phi[Z(z)]+\Psi[(Z(z)]= \hat\Phi[Z(z)]+\hat\Psi[(Z(z)], \Psi(Z)=T(Z)-\overline{T(\overline Z)}, \\ \hat\Psi(Z) & = T(Z)+\overline{T(\overline Z)},\quad T(Z)=-\frac1{\pi}\int \int_{D^+_t} \frac{f(t)}{t-Z}d\sigma_t \quad\text{in }\overline{D^+_Z}, \\ W(z) &= \phi(z)+\psi(z)=\xi(z)e_1+\eta(z)e_2\;\quad\text{in }\,\overline{D^-}, \\ \xi(z) &= \zeta(z)+\int_{0}^{y-x^n}g_1(z)dt=\int_{S_1}g_1(z)dt + \int_0^{y-x^n}g_1(z)dt,\quad z\in s_1,\\ \eta(z) &= \theta(z)+\int_{0}^{y-x^n}g_2(z)dt =\int_{S_2}g_2(z)dt + \int_0^{y-x^n}g_2(z)dt,\quad z\in s_2, \\ g_l(z) & = \tilde A_l(U + V) + \tilde B_l(U - V) + 2\tilde C_lU + \tilde D_lu + \tilde E_l,\quad l=1,2.\end{aligned}\label{e2.1} Herein $Z=x+jG(y - x^n), f(Z)=g(Z)/H$, $U=Hu_x/2$, $V=-u_y/2$, $\begin{pmatrix}i\\ -j\end{pmatrix}$ is a $2\times 1$ matrix, $\xi(z) = \int_{S_1}g_1(z)dt$ in $D^-$, $\zeta(x)+\theta(x)=0$ on $L_0$, $s_1,s_2$ are two families of characteristics in $D^-$: $$s_1:\,\frac{dx}{dy}=H(y - x^n),\quad s_2:\, \frac{dx}{dy}=-H(y - x^n) \label{e2.2}$$ passing through $z=x+j(y - x^n)\in\overline{D^-}$, $S_1, S_2$ are characteristic curves from the points on $\tilde L=L_1\cup L_3\cup\dots \cup L_{2N-1}, \tilde L'=L_2\cup L_4\cup\dots \cup L_{2N}$ to two points on $L_0$ respectively, and $$\begin{gathered} W(z)=U(z)+jV(z)=\frac12Hu_x-\frac j2u_y, \\ \xi(z) = \mathop{\rm Re} W(z) + \mathop{\rm Im} W(z),\quad \eta(z) = \mathop{\rm Re} W(z) - \mathop{\rm Im} W(z), \\ \tilde A_{1} = \tilde B_2 = -\frac b2,\quad \tilde A_{2} = \tilde B_1 = \frac b2,\quad \tilde C_{1} = \frac a{2H} + \frac{m(1-nx^{n-1}H)}{4(y - x^n)}, \\ \tilde C_{2} = -\frac a{2H} + \frac{m(1 + nx^{n-1}H)}{4(y - x^n)},\quad \tilde D_{1} = -\tilde D_{2} = \frac c2,\quad \tilde E_1 = -\tilde E_2 = \frac d{2}, \end{gathered}\label{e2.3}$$ in which we choose $H(y - x^n)=|y - x^n|^{m/2}$, where $m$ is as stated before. \end{theorem} \begin{proof} From \eqref{e1.5}, \eqref{e1.6}, we see that equation \eqref{e1.1} in $\overline{D^-}$ can be reduced to the system of integral equations: \eqref{e2.1}. Moreover we can derive $H(0)u_x/2=U(x)=[\zeta(x)+\theta(x)]/2=0$, i.e. $\zeta(x) = -\theta(x)$ on $L_0$, and then $\zeta(z) = \int_{S_1}g_1(z)dt$, $\theta(z) = -\zeta(x+G(y-x^n))$ in $\overline{D^-}$. Here we mention that by using the way of symmetrical extension with respect to $L_l\,(l=1,2,\dots ,2N)$, we can extend the function $W(Z), u(z)$ from $\overline{D^-}$ onto the exterior of $D^-$. \end{proof} In the following, we prove the uniqueness of solutions of Problem P for \eqref{e1.1}. \begin{theorem} \label{thm2.2} Suppose that \eqref{e1.1} satisfies Condition {\rm (C1), (C2)}. Then Problem $P$ for \eqref{e1.1} in $D$ has a unique solution. \end{theorem} \begin{proof} Let $u_1(z), u_2(z)$ be two solutions of Problem P for \eqref{e1.1}. Then $u(z)=u_1(z)-u_2(z)$ is a solution of the generalized Rassias homogeneous equation $$K(y - x^n)u_{xx} + u_{yy} + \tilde au_x + \tilde bu_y + \tilde cu=0\quad\text{in }D, \label{e2.4}$$ satisfying the boundary conditions $$\begin{gathered} \frac12\frac{\partial u}{\partial\nu}=\frac1{H(\hat y)} \mathop{\rm Re}[\overline{\lambda(z)}u_{\tilde z}(z)]=0\quad\text{on } \Gamma\cup\tilde L, \\ u(\tilde a_0)=0,\quad \mathop{\rm Im}[\overline{\lambda(z_l)}u_{\tilde z}(z_l)]=0,\quad u(\tilde a_l)=0,\quad l=1,\dots ,N, \end{gathered} \label{e2.5}$$ where the function $W(z)=U(z)+jV(z)=[H(\hat y)u_x-ju_y]/2$ in the hyperbolic domain $D^-$ can be expressed in the form $$\begin{gathered} W(z)=\phi(x)+\psi(z)=\xi(z)e_1+\eta(z)e_2, \\ \xi(z) = \zeta(z)+ \int_{0}^{y-x^n} [\tilde A_1(U + V) + \tilde B_1(U - V) + 2\tilde C_1U + \tilde D_1u]dy,\quad z \in s_1, \\ \eta(z) = \theta(z) + \int_{0}^{y-x^n} [\tilde A_2(U + V) + \tilde B_2(U - V) + 2\tilde C_2U + \tilde D_2u]dy,\quad z \in s_2, \end{gathered}\label{e2.6}$$ where $\phi(z)=\zeta(z)e_1+\theta(z)e_2$ is a solution of equation $W_{\overline{\tilde z}}=0$ in $D^-$, and $$u(z)=2 \mathop{\rm Re}\int_{z_*}^z[\frac{\mathop{\rm Re} W(z)}{H(y - x^n)}+ \begin{pmatrix}i\\-j\end{pmatrix} \mathop{\rm Im} W]dz\quad \text{in } \begin{pmatrix}\overline{D^+}\\ \overline{D^-}\end{pmatrix} \label{e2.7}$$ By a similar way as in \cite[Section 2, Chapter V]{w5}, we can verify $u(z)=0$ in $\overline{D^-}$, especially $u_{\hat y}=0$ on $L_0$. Now we verify that the above solution $u(z)\equiv0$ in $D^+$. If the maximum $M=\max_{\overline{D^+}}u(z)>0$, it is clear that the maximum point $z'\not\in D^+$. If the maximum $M$ attains at a point $z'\in\Gamma$ and $\cos(\nu,n)>0$ at $z'$, we get $\partial u/\partial\nu>0$ at $z'$, which contradicts the first formula of \eqref{e2.5}. If $\cos(\nu,n)=0$ at $z'$, denote by $\Gamma'$ the longest curve of $\Gamma$ including the point $z'$, so that $\cos(\nu,n)=0$ and $u(z)=M$ on $\Gamma'$, then there exists a point $z_0\in\Gamma\backslash\Gamma'$, such that at $z_0$, $\cos(\nu,n)>0, {\partial u}/{\partial n}>0,\cos(\nu,s)>0\,(<0)$, ${\partial u}/{\partial s}\ge0\,(\le0)$, hence the inequality $$\frac{\partial u}{\partial\nu}=\cos(\nu,n)\frac{\partial u}{\partial n}+\cos(\nu,s)\frac{\partial u}{\partial s}>0\quad \text{at } z_0 \label{e2.8}$$ holds, in which $s$ is the tangent vector at $z_0\in\Gamma$, it is impossible. Thus $u(z)$ attains its positive maximum at a point $z=z'\in L_0$. By the Hopf Lemma, we can see that it is also impossible. Hence $u(z)=u_1(z)-u_2(z)=0$ in $\overline{D^+}$, thus we have $u_1(z)=u_2(z)$ in $\overline{D}$. This completes the proof. \end{proof} \section{Solvability of oblique derivative problems} In this section, we prove the existence of solutions of Problem P for equation \eqref{e1.1}. From the discussion in Section 1, we can only discuss the complex equation $$W_{\bar{\tilde z}}=A_1(z,u,W)W+A_2(z,u,W)\overline{W}+A_3(z,u,W)u+A_4(z,u,W) \quad\text{in } \label{e3.1}$$ with the relation $$u(z)=\begin{cases}\displaystyle 2\mathop{\rm Re}\int_{z_*}^z[\frac{\mathop{\rm Re} W(z)}{H(y-x^n)}+i\mathop{\rm Im} W(z)]dz+c_0 &\text{in }\overline{D^+},\vspace{1.8mm}\\ \displaystyle 2\mathop{\rm Re}\int_{z_*}^z[\frac{\mathop{\rm Re} W(z)}{H(y-x^n)}-j\mathop{\rm Im} W(z)]dz+c_0 &\text{in }\overline{D^-}, \end{cases}\label{e3.2}$$ and the homogeneous boundary conditions $$\begin{gathered} \mathop{\rm Re}[\overline{\lambda(z)}W(z)]=R(z)\quad\text{on }\Gamma\cup\tilde L,\\ u(\tilde a_0)=c_0, \\ \mathop{\rm Im}[\overline{\lambda(z_l)}u_{\tilde z}(z_l)] = b'_l,u(\tilde a_l) = c_l \text{ or } \mathop{\rm Im}[\overline{\lambda(z'_l)}u_{\tilde z}(z'_l)] = c'_l,\quad l = 1,\dots ,N, \end{gathered}\label{e3.3}$$ where $R(z)=0$ on $\Gamma\cup L_1$ and $c_0=b'_l=c_l=c'_l=0$, $l=1,\dots ,N$. The boundary value problem \eqref{e3.1}, \eqref{e3.2}, \eqref{e3.3} is called Problem $\tilde A$, which is corresponding to Problem $\tilde P$ or $\tilde Q$. It is clear that Problem $\tilde A$ can be divided into two problems, i.e. Problem $A_1$ of equation \eqref{e3.1}, \eqref{e3.2} in $D^+$ and Problem $A_2$ of equation \eqref{e3.1}, \eqref{e3.2} in $D^-$. The boundary conditions of Problems $A_1$ and $A_2$ as follows: $$\begin{gathered} \mathop{\rm Re}[\overline{\lambda(z)}W(z)] = R(z)\quad\text{on }\Gamma\cup L_0,\\ u(\tilde a_l) = c_l\text{ or } \mathop{\rm Im}[\overline{\lambda(z'_l)}W(z'_l)] = c'_l, \quad l=1,\dots ,N, \end{gathered} \label{e3.4}$$ where $\lambda(z)=-i, R(x)=\hat R_0(x)$ on $L_0$, and $$\begin{gathered} \mathop{\rm Re}[\overline{\lambda(z)}W(z)] = R(z)\quad\text{on }\tilde L\cup L_0,\\ \mathop{\rm Im}[\overline{\lambda(z_l)}W(z_l)] = b'_l,\quad l=1,\dots ,N, \end{gathered}\label{e3.5}$$ in which $\lambda(z) = a(z) + jb(z)$, $R(z) = 0$ on $\Gamma\cup\tilde L$ in \eqref{e1.12}, $\lambda(z)=1 + j$, $R(z) = -\tilde R_0(x)$ on $L_0,\hat R_0(x), \tilde R_0(x)$ on $L_0$ are as stated in \eqref{e1.12}, because $\mathop{\rm Re} W(x)=0$ on $L_0$, thus $1+j$ can be replaced by $j$. Introduce a function $$X(Z)=\prod^{N+1}_{l=1}(Z-\hat t_l)^{\eta_l}, \label{e3.6}$$ where $\hat t_1=-R$, $\hat t_2=R$, $\hat t_l=a_{l-2}$, $l=3,\dots ,N+1$, the numbers $\eta_l=1-2\gamma_l$ if $\gamma_l\ge0$, $\eta_l=\max(-2\gamma_l,0)$ if $\gamma_l<0$, $\gamma_l\,(l=1,2)$ are as stated in Section 1, $\eta_3=\dots =\eta_{N+1}=1$, where we choose a branch of multi-valued function $X(Z)$ such that $\arg X(x)=\eta_2\pi/2$ on $L_0\cap\{x>a_{N-1}\}$. Obviously that $X(Z)W[z(Z)]$ satisfies the complex equation $$\begin{gathered} {[X(Z)W]_{\overline{Z}}}=X(Z)[A_1W+A_2\overline W+A_3u+A_4]/H =X(Z)g(Z)/H\;\;\text{in}D_{Z}, \end{gathered}\label{e3.7}$$ and the boundary conditions \begin{gather*} \mathop{\rm Re}[\overline{\hat\lambda(z)}X(Z)W(z)] = R(z) = 0\quad\text{on }\Gamma,\\ \mathop{\rm Re}[\overline{\hat\lambda(z)}X(Z)W(z)] = 0\quad\text{on }\tilde L, \\ u(\tilde a_0)=0,\quad \mathop{\rm Im}[\overline{\lambda(z_l)}W(z_l)]=0,\quad u(\tilde a_l)=0,\quad l=1,\dots ,N, \end{gather*} where $D_Z=Z^+_Z$, $\hat\lambda(z)=\lambda(z)e^{i\arg X(Z)}$. Noting that \begin{gather*} e^{i\hat\phi_l}=\frac{\hat\lambda(\hat t_l-0)}{\hat\lambda(\hat t_l+0)} =\frac{\lambda(\hat t_l-0)}{\lambda(\hat t_l+0)} \frac{e^{i\arg X(\hat t_l-0)}}{e^{i\arg X(\hat t_l+0)}}=e^{i(\phi_l+\tilde\eta_l)}, \\ \tau_l=\frac{\hat\phi_l}\pi-\hat K_l=0,\quad l= 1,\dots ,N+1, \end{gather*} in which $\tilde\eta_l=\eta_l\pi/2$, $l=1,2$, $\tilde\eta_l=\eta_l\pi$, $l=3,\dots ,N+1$, which are corresponding to the numbers $\gamma_l$ $(1\le l\le N+1)$ in Section 1. If $\hat K_l=-1$, $\hat K_l=1,l=2,\dots ,N+1$, or $\hat K_l=-1$, $\hat K_l=0,l=2,\dots ,N+1$, then the index $\hat K=(\hat K_1+\dots +\hat K_{N+1})/2=(N-1)/2$ or $-1/2$ of $\hat\lambda(z)$ on $\Gamma\cup L_0$ is chosen. For the case $\hat K=(N-1)/2$, we need to add $N$ point conditions $u(\tilde a_l)=c_l$ $(l=1,\dots ,N)$ in \eqref{e1.8} and \eqref{e1.10}, such that Problem $\tilde P$ or $\tilde Q$ is well-posed. \begin{theorem} \label{thm3.1} Let \eqref{e1.1} satisfy Conditions {\rm (C1), (C2)}. Then any solution of Problem $A_1$ for \eqref{e1.1} in $D^+$ satisfies the estimate $$\begin{gathered} \hat C_\delta[W(z), \overline{D^+}] = C_\delta[X(Z)(\mathop{\rm Re} W(Z)/H + i\mathop{\rm Im} W(Z)), \overline{D^+}] + C_\delta[u(z), \overline{D^+}] \le M_1, \\ \hat C_\delta[W(z),\overline{D^+}]\le M_2(k_1+k_2), \end{gathered} \label{e3.8}$$ where $X(Z)$ is as stated in \eqref{e3.6}, $\delta <\min[2,m]/(m+2)$ is a sufficiently small positive constant, $M_1=M_1(\delta,k,H,D^+)$, $M_2=M_2(\delta,k_0,H,D^+)$ are positive constants, and $k=(k_0,k_1,k_2)$. \end{theorem} \begin{proof} We first assume that any solution $[W(z)$, $u(z)]$ of Problem $A_1$ satisfies the estimate $$\hat C[W(z),\overline{D^+}] = C[X(Z)(\mathop{\rm Re} W(Z)/H + i\mathop{\rm Im} W(Z),\overline{D_Z}] + C[u(z) ,\overline{D^+}] \le M_3, \label{e3.9}$$ where $M_3$ is a non-negative constant, and then give that $[W(z), u(z)]$ satisfy the H\"older continuous estimates in $\overline{D_Z}$. Firstly, we verify the H\"older continuity of solutions $[W(z), u(z)]$ in $\overline{D_Z}\cap\{\mathop{\rm dist}(Z,\{\hat t_1\cup\hat t_2\cup\dots \cup\hat t_{N+1}\})\ge\varepsilon\},$ in which $\varepsilon$ is a sufficiently small positive constant. Substituting the solution $[W(z),u(z)]$ into \eqref{e3.7} and noting $\mathop{\rm Re} W(Z)=R(x)=0$ on $L_0$, we can extend the function $X(Z)W[z(Z)]$ onto the symmetrical domain $\tilde D_Z$ of $D_Z$ with respect to the real axis $\mathop{\rm Im} Z=0$, namely set \tilde W(Z)=\begin{cases} X(Z)W[z(Z)] &\text{in}\; D_Z, \3pt] -\overline{X(\overline Z)W[z(\overline Z)]} &\text{in }\tilde D_Z, \end{cases} which satisfies the boundary conditions \begin{gather*} \mathop{\rm Re}[\overline{\tilde\lambda(Z)}\tilde W(Z)]=0\quad\text{on }\Gamma\cup\tilde\Gamma, \\ \tilde\lambda(Z) = \begin{cases} \lambda[z(Z)], \\[3pt] \overline{\lambda[z(\overline Z)]}, \\ 1, \end{cases} \quad \tilde R(Z) = \begin{cases} 0 &\text{on }\Gamma, \\ 0 &\text{on }\tilde\Gamma, \\ 0 &\text{on }L_0, \end{cases} \end{gather*} where \tilde\Gamma is the symmetrical curve of \Gamma about \mathop{\rm Im} Z=0. It is easy to see that the corresponding function u(z) in \eqref{e3.2} can be extended to the function \tilde u(Z), where \tilde u(Z)=u[z(Z)] in D_Z(=D^+_Z) and \tilde u(Z)=-u[z(\overline Z)] in \tilde D_Z. Noting (C1), (C2) and the condition \eqref{e3.9}, we see that the function \tilde f(Z)=X(Z)g(Z)/H in D_Z and \tilde f(Z)=-\overline{X(\overline Z)g(\overline Z)}/H in \tilde D_Z satisfies the condition L_\infty[y^{\tau}H\tilde f(Z),D_Z'] \le M_4, in which D_Z'=D_Z\cup\tilde D_Z\cup L_0, \tau= \max(1-m/2,0), M_4=M_4(\delta,k,H,D,M_3) is a positive constant. On the basis of \cite[Lemma 2.1, Chapter I]{w5}, we can verify that the function \tilde\Psi(Z)=T(Z)-\overline{T(\overline Z)}\,(T(Z)=-1/\pi \int \int_{D_t} [\tilde f(t)/(t - Z)]d\sigma_t\} over D_Z) satisfies the estimates $$C_{\beta}[\tilde\Psi(Z),\overline{D_Z}]\le M_5,\quad \tilde\Psi(Z)-\tilde\Psi(\hat t_l) =O(|Z-\hat t_l|^{\beta_l}),\quad 1\le l\le N+1, \label{e3.10}$$ in which \beta=\min(2,m)/(m+2)-2\delta=\beta_l\,(1\le l\le N+1), \delta is a constant as stated in \eqref{e3.8}, and M_5=M_5(\delta,k,H,D,M_3) is a positive constant. On the basis of Theorem \ref{thm2.1}, the solution X(Z)\tilde W(z) can be expressed as X(Z)\tilde W(Z)=\tilde\Phi(Z)+\tilde\Psi(Z), where \tilde\Phi(Z) is an analytic function in D_{Z} satisfying the boundary conditions \begin{gather*} \mathop{\rm Re}[\overline{\tilde\lambda(Z)}\tilde\Phi(Z)]=-\mathop{\rm Re}[\overline{\tilde \lambda(Z)}\tilde\Psi(Z)]=\hat R(Z)\quad\text{on }\Gamma\cup\tilde L, \\ u(\tilde a_0)=0, u(\tilde a_l)=0\,\text{ or }\mathop{\rm Im}[\overline{\lambda(z'_l)}\tilde W(z'_l)]=0,\quad l=1,\dots ,N. \end{gather*} There is no harm in assuming that \tilde\Psi(\hat t_l)=0, otherwise it suffices to replace \tilde\Psi(Z) by \tilde\Psi(Z)-\tilde\Psi(\hat t_l)\,(1\le l\le N+1). For giving the estimates of \tilde\Phi(Z) in D_Z\cap\{{\rm dist}(Z,\Gamma)\ge\varepsilon(>0)\}, from the integral expression of solutions of the discontinuous Riemann-Hilbert problem for analytic functions, we can write the representation of the solution \tilde\Phi(Z) of Problem A_1 for analytic functions, namely \begin{gather*} \tilde\Phi[Z(\zeta)]=\frac{X_0(\zeta)}{2\pi i} \Big[ \int_{\partial D_t} \frac{(t+\zeta)\tilde\lambda[Z(t)]\hat R[Z(t)]dt}{(t-\zeta)tX_0(t)}+ Q(\zeta)\Big] , \\ \begin{aligned} Q(\zeta) &=i\sum_{k=0}^{[\hat K]} (c_k\zeta^k +\overline{c_k}\zeta^{-k}) \\ &\;\;\;+ \begin{cases} 0,&\text{when 2\hat K = N - 1 is even}, \\ \displaystyle ic_*\frac{\zeta_1 + \zeta}{\zeta_1 - \zeta},\; c_* = i \int_{\partial D_t} \frac{\tilde\lambda[Z(t)]\hat R[Z(t)]dt}{X_0(t)t}, &\text{when 2\hat K=N-1 is odd}, \end{cases} \end{aligned} \end{gather*} (see \cite{w2,w3}), where X_0(\zeta) = \Pi_{l=1}^{N+1}(\zeta - \hat t_l)^{ \tau_l}, \tau_l (l=1,\dots ,N+1) are as before, Z=Z(\zeta) is the conformal mapping from the unit disk D_\zeta=\{|\zeta|<1\} onto the domain D_Z such that the three points \zeta=-1,i,1 are mapped onto Z=-1,Z'(\in\Gamma),1 respectively. Taking into account |X_0(\zeta)| = O(|\zeta - \hat t_l|^{\tau_l}),\quad |\hat\lambda[Z(\zeta)]\hat R[Z(\zeta)]/ X_0(\zeta)| = O(|\zeta - \hat t_l|^{\tilde\eta_l-\tau_l}), and according to the results in \cite{w2}, we see that the function \tilde\Phi(Z) determined by the above integral in D_Z\cap\{{\rm dist}(Z,\Gamma)\ge \varepsilon(>0)\} is H\"older continuous and \tilde\Phi(\hat t_l)=0\,(1\le l\le N+1). Thus, from \eqref{e3.10} and the above integral representation of \tilde\Phi(Z), we can give the following estimates $$C_\delta[\tilde\Phi(Z),D_\varepsilon]\le M_6,\quad C_\delta[X(Z)u_x,D_\varepsilon]\le M_6, \quad C_\delta[X(Z)u_y,D_\varepsilon]\le M_6, \label{e3.11}$$ where D_\varepsilon = \overline{D_Z}\cap\{{\rm dist}(Z,L_0)\ge\varepsilon\}, \varepsilon is arbitrary small positive constant, M_6 = M_6(\delta,k,H, D_\varepsilon,M_3) is a non-negative constant. Similarly we can get $$C_{\delta}[H(\hat y)u_x,D'_{\varepsilon}]\le M_7,\quad C_{\delta}[u_y,D'_{\varepsilon}]\le M_7, \label{e3.12}$$ in which D'_\varepsilon = \overline{D_Z} \cap\{{\rm dist}(Z,\Gamma\cup\tilde\Gamma)\ge\varepsilon\}, \varepsilon is arbitrary small positive constant, and M_7=M_7(\delta, k,H,D'_\varepsilon,M_3) is a non-negative constant. \end{proof} Next, for giving the estimates of X(Z)u_x, X(Z)u_x in \tilde D_l=D_l\cap\overline{D_Z} (D_l=\{|Z-\hat t_l|<\varepsilon (>0)\}, 1\le l\le2) separately, denote X(Z)=\tilde X+i\tilde Y as in \eqref{e3.6}, we first conformally map the domain D_Z'=D_Z\cup\tilde D_Z\cup L_0 onto a domain D_\zeta, such that L_0 is mapped onto himself, where D_\zeta is a domain with the partial boundary \Gamma\cup\tilde\Gamma, and \Gamma\cup\tilde\Gamma is a smooth curve including the line segment \mathop{\rm Re}\,\zeta=\hat t_l near \zeta=\hat t_l\,(1\le l\le2). Through the above mapping, the index \tilde K=(N-1)/2 is not changed, and the function \tilde\Psi[Z(\zeta)] in the neighborhood \zeta(D_l) of \hat t_l\,(1\le l\le2) is H\"older continuous. For convenience denote by D_Z, D_l, \tilde W(Z) the domains and function D_\zeta,\zeta(D_l),\tilde W[Z(\zeta)] again. Secondly reduce the the above boundary condition to this case, i.e. the corresponding function \tilde\lambda(Z)=1 on \Gamma\cup\tilde\Gamma near Z=\hat t_l\,(1\le l\le 2). In fact there exists an analytic function S(Z) in D'_Z= D_Z\cup\tilde D_Z\cup L_0 satisfying the boundary condition \mathop{\rm Re} S(Z) = -\arg\tilde\lambda(Z)\quad\text{on }\Gamma\cup\tilde\Gamma \quad\text{near }\hat t_l,\quad \mathop{\rm Im} S(\hat t_l) = 0, and the estimate C_\alpha[S(Z),D_l\cap D'_Z]\le M_8=M_8(\delta,k,H,D,M_3)<\infty, then the function e^{jS(Z)}X(Z)W(Z) is satisfied the boundary condition \mathop{\rm Re}[e^{iS(Z)}X(Z)W(Z)]=0\quad\text{on }\Gamma\cup\tilde\Gamma\quad \text{near }Z=\hat t_l\,(1\le l\le2). Next we symmetrically extend the function \Phi^*(Z) in D'_Z onto the symmetrical domain D^*_Z with respect to \mathop{\rm Re} Z=\hat t_l\,(1\le l\le2), namely let \hat W(Z)=\begin{cases} e^{iS(Z)}X(Z)W(Z)\quad\text{in }D'_Z,\vspace{1mm} \\ -\overline{e^{iS(Z')}X(Z')W(Z')}\quad\text{in }D^*_Z, \end{cases} where Z'=-\overline{(Z-\hat t_l)}+\hat t_l, later on we shall omit the secondary part e^{iS(Z)}. After the above discussion, as stated in \eqref{e2.1}, the solution X(Z)W(z) can be also expressed as X(Z)W(Z)=\Phi(Z)+\Psi(Z), where X(Z)=\tilde X+i\tilde Y, X(Z) is as stated in \eqref{e3.6}, \Psi(Z) in \hat D_Z=\{D^*_Z\cup D'_Z\}\cap\{Y=G(y-x^n)>0\} is H\"older continuous, and \Phi(Z) is an analytic function in \hat D_Z satisfying the boundary conditions in the form \begin{gather*} \mathop{\rm Re}[\overline{\tilde\lambda(Z)}\Phi(Z)] = \hat R(Z)\quad\text{on }\Gamma \cup L_0,\\ u(\hat t_l) = 0,\quad l=1,\dots , N+1, \end{gather*} because in the above case the index of \tilde\lambda(Z) on \partial D_Z is \tilde K=(N-1)/2. Hence by the similar way as in the proof of \eqref{e3.12}, we have C_\delta[X(Z)H(\hat y)u_x,\tilde D_l]\le M_9,\quad C_\delta[X(Z)u_y,\tilde D_l]\le M_{10}, \quad 1 \le l \le 2, where M_l = M_l (\delta,k,H,D,M_3)(l=9,10) is a non-negative constant. As for the solution of Problem P in the neighborhood of \hat t_l (3\le l\le N+1), we can use a similar way. Finally we use the reduction to absurdity, suppose that \eqref{e3.9} is not true, then there exist sequences of coefficients \{A^{(m)}_l\} (l=1,2,3,4), \{\lambda^{(m)}\}, \{r^{(m)}\} and \{c^{(m)}_l\} (l=0,1,\dots ,N), which satisfy the same conditions of coefficients as stated in \eqref{e1.8}, \eqref{e1.9}, such that \{A^{(m)}_l\} (l=1,2,3,4), \{\lambda^{(m)}\}, \{r^{(m)}\}, \{c^{(m)}_l\} in \overline{D^+}, \Gamma, L_0 weakly converge or uniformly converge to A^{(0)}_l (l=1,2,3,4), \lambda^{(0)}, r^{(0)}, \{c^{(0)}_l\} (l=0,1,\dots ,N) respectively, and the solutions of the corresponding boundary value problems \begin{gather*} \begin{aligned} W^{(m)}_{\overline Z} &= F^{(m)}(z,u^{(m)},W^{(m)}),F^{(m)}(z,u^{(m)}, W^{(m)}) \\ &=A^{(m)}_1W^{(m)}+A^{(m)}_2\overline{W^{(m)}}+A^{(m)}_3u^{(m)} +A^{(m)}_4\quad\text{in }\overline{D^+}, \end{aligned} \\ \mathop{\rm Re}[\overline{\lambda^{(m)}(z)}W^{(m)}(z)]=R^{(m)}(z)\quad\text{on }\Gamma\cup L_0, \\ u^{(m)}(\tilde a_0)=c^{(m)}_0,\quad u^{(m)}(\tilde a_l)=c^{(m)}_l\text{ or }LW^{(m)}(z'_l)=c'^{(m)}_l,\quad l=1,\dots ,N, \end{gather*} and \begin{align*} u^{(m)}(z)&=u^{(m)}(x)-2\int_0^y V^{(m)}(z)dy\\ &=2\mathop{\rm Re} \int_{z_*}^z [\frac{\mathop{\rm Re} W^{(m)}}{H(\hat y)}+i\mathop{\rm Im} W^{(m)}]dz+c_0^{(m)}\quad\text{in }\overline{D^+} \end{align*} have the solutions [W^{(m)}(z),u^{(m)}(z)], but \hat C[W^{(m)}(z),\overline{D^+}] (m=1,2,\dots ) are unbounded, hence we can choose a subsequence of [W^{(m)}(z),u^{(m)}(z)] denoted by [W^{(m)}(z),u^{(m)}(z)] again, such that h_m=\hat C[W^{(m)}(z),\overline{D^+}]\to\infty as m\to\infty, we can assume h_m\ge\max[k_1,k_2,1]. It is obvious that [\tilde W^{(m)}(z),\tilde u^{(m)}(z)_m]=[W^{(m)}(z)/h_m, u^{(m)}(z)_m/h_m] are solutions of the boundary value problems \begin{gather*} \tilde W^{(m)}_{\overline{Z}} =\tilde F^{(m)}(z,\tilde u^{(m)},\tilde W^{(m)}), \\ \tilde F^{(m)}(z,\tilde u^{(m)},\tilde W^{(m)}) = A^{(m)}_1\tilde W^{(m)} + A^{(m)}_2\overline{\tilde W^{(m)}} + A^{(m)}_3\tilde u^{(m)} + A^{(m)}_4/h_m\quad\text{in }\overline{D^+}, \\ \mathop{\rm Re}[\overline{\lambda^{(m)}(z)}\tilde W^{(m)}(z)]=R^{(m)}(z)/h_m\;\quad\text{on }\Gamma \cup L_0, \\ \tilde u^{(m)}(\tilde a_0)=c^{(m)}_0 /h_m,\\ \tilde u^{(m)}(\tilde a_l)=c^{(m)}_l /h_m\text{ or } L\tilde W^{(m)}(z'_l)=c^{(m)}_l /h_m,\quad l=1,\dots ,N, \end{gather*} and \begin{align*} \tilde u^{(m)}(z)&=\frac{u^{(m)}(x)}{h_m}-2\int_0^{\hat y} \tilde V^{(m)}(z)dy\\ &= 2\mathop{\rm Re}\int_{z_*}^z [\frac{\mathop{\rm Re}\tilde W^{(m)}}{H(\hat y)} + i\mathop{\rm Im}\tilde W^{(m)}]dz + \frac{c_0^{(m)}}{h_m}\quad\text{in }\overline{D^+}. \end{align*} We see that the functions in the above boundary value problems satisfy the same conditions. From the representation \eqref{e2.1}, the above solutions can be expressed as \begin{gather*} \begin{aligned} \tilde u^{(m)}(z)&=\frac{u^{(m)}(x)}{h_m}-2\int_0^y \tilde V^{(m)}(z)dy\\ &=2\mathop{\rm Re}\int_{z_*}^z[\frac{\mathop{\rm Re}\tilde W^{(m)}}{H(\hat y)} + i\mathop{\rm Im}\tilde W^{(m)}]dz + \frac{c_0^{(m)}}{h_m}\quad\text{in }\overline{D^+}, \end{aligned} \\ \tilde W^{(m)}(z)=\tilde\Phi^{(m)}[Z(z)]+\tilde\Psi^{(m)}[Z(z)], \\ \tilde\Psi^{(m)}(Z) = T(Z) - \overline{T(\overline Z)},\quad T(Z) = -\frac1\pi \iint_{D^+}\frac{\tilde f^{(m)}(t)}{t - Z} d\sigma_t,\quad\text{in }\overline{D^+}, \end{gather*} As in the proof of \eqref{e3.10}, and notice that y^{\tau}H(\hat y)\tilde f^{(m)}(Z)=y^{\tau}X(Z)g^{(m)}(Z)\in L_\infty(D_Z), \tau=\max(0,1-m/2), we can verify that C_{\beta}[\tilde\Psi(Z),\overline{D^+}]\le M_{11},\quad \tilde\Psi(Z)| _{Z=t_j} = O(|Z - t_j|^{\beta_j}),\quad j = 1,2, where M_{13}=M_{13}(\delta,k,H,D^+) is a non-negative constant. Noting that Conditions (C1), (C2) and the complex equation and boundary conditions about \tilde W^{(m)}_{x}, which satisfy the conditions similar to those about \tilde W^{(m)}(Z), we have C[X(Z)\tilde W^{(m)}_{x}(Z),\overline{D^+}]\le M_{12} =M_{12}(\delta,k,H,\overline{D^+}). Hence we can derive that sequence of functions: \[ \{X(Z)(\mathop{\rm Re}\tilde W^{(m)}(Z)/H(\hat y) + i\mathop{\rm Im} \tilde W^{(m)}(Z))\} satisfies the estimate \hat C_{\delta}[\tilde W^{(m)}(Z),\overline{D_Z}]\le M_{13} =M_{13}(\delta,k,H,D^+)<\infty.  Hence from $\{X(Z)[\mathop{\rm Re}\tilde W^{(m)}(z)/H+i\mathop{\rm Im}\tilde W^{(m)}(z)]\}$ and the sequence of corresponding functions $\{\tilde u^{(m)}(z)\}$, we can choose the subsequences denoted by $\{X(Z)[\mathop{\rm Re}\tilde W^{(m)}(z)/H+i\mathop{\rm Im}\tilde W^{(m)}(z)]\}, \quad \{\tilde u^{(m)}(z)\}$ again, which uniformly converge to $X(Z)[\mathop{\rm Re}\tilde W^{(0)}(z)/H+i\mathop{\rm Im}\tilde W^{(0)}(z)]$, $\tilde u^{(0)}(z)$ respectively, it is clear that $[\tilde W^{(0)}(z), \tilde u^{(0)}(z)]$ is a solution of the homogeneous problem of Problem $A_1$. On the basis of Theorem \ref{thm2.2}, the solution $\tilde W^{(0)}(z)=0$, $\tilde u^{(0)}(z)=0$ in $\overline{D^+}$, however, from $\hat C[\tilde W^{(m)}(z), \overline{D^+}]=1$, we can derive that there exists a point $z^*\in\overline{D^+}$, such that $\hat C[\tilde W^{(0)}(z^*),\overline{D^+}]=1$, it is impossible. This shows that \eqref{e3.9} is true, where the constant $M_3=M_3(\delta,k,H,D^+)$, and then the first estimate in \eqref{e3.8} can be derived. The second estimate in \eqref{e3.8} is easily verified from the first estimate in \eqref{e3.8}. \begin{theorem} \label{thm3.2} Under the same conditions as in Theorem $3.1$, Problem $A_1$ for \eqref{e3.1}, \eqref{e3.2} in $D^+$ is solvable, and then Problem Q for \eqref{e1.1} with $c=0$ in $D^+$ has a solution. Moreover, Problem P for \eqref{e1.1} in $D^+$ is solvable. \end{theorem} \begin{proof} Applying using the estimates in Theorem \ref{thm3.1} and the Leray-Schauder theorem, we can prove the existence of solutions of Problem $A_1$ for \eqref{e3.1} with $A_3=0$ in $D^+$. We consider the equation and boundary conditions with the parameter $t\in[0,1]$: $$W_{\overline{\tilde z}}-tF(z,u,W)=0,\quad F(z,u,W)=A_1W+A_2\overline W+A_4\,\quad\text{in }\overline{D_Z}, \label{e3.13}$$ and introduce a bounded open set $B_M$ of the Banach space $B=\hat C_\delta(\overline{D_Z})$, whose elements are functions $w(z)$ satisfying the condition w(Z)\in\hat C_\delta(\overline{D^+}),\;\;\hat C_\delta[w(Z),\overline{D_Z}]