\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 72, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/72\hfil Coefficients of degenerate elliptic equations] {An approach for constructing coefficients of degenerate elliptic complex equations} \author[G.-C. Wen \hfil EJDE-2013/72\hfilneg] {Guo Chun Wen} % in alphabetical order \address{Guo Chun Wen \newline LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China} \email{Wengc@math.pku.edu.cn} \thanks{Submitted November 30, 2012. Published March 17, 2013.} \subjclass[2000]{35J55, 35R30, 47G10} \keywords{Degenerate elliptic complex equations; coefficients of equations; \hfill\break\indent method of integral equations; H\"older continuity of a singular integral} \begin{abstract} This article deals with the inverse problem for degenerate elliptic systems of first order equations with Riemann-Hilbert type map in simply connected domains. Firstly the formulation and the complex form of the problem for the first-order elliptic systems with the degenerate rank 0 are given, and then the coefficients of the systems are constructed by a new complex analytic method. Here we verify and apply the H\"older continuity of a singular integral operator. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Formulation of the inverse problem for degenerate elliptic complex equations of first order} In \cite{b1,i1,k1,s1,t1,t2,w7,w8}, the authors discussed the inverse problem of second-order elliptic equations without degeneracy. In this article, by using the methods of integral equations and complex analysis, the existence of solutions of the inverse problem for degenerate elliptic complex equations of first order with Riemann-Hilbert type map is discussed. Let $D(\supset\{0\})$ be a simply connected bounded domain in the complex plane $\mathbb{C}$ with the boundary $\partial D=\Gamma\in C^1_\mu(0<\mu<1)$. There is no harm in assuming that the domain $D$ is $\{|z|<1\}$ with boundary $\Gamma=\{|z|=1\}$. Consider the linear elliptic systems of first-order equations with degenerate rank 0, $$\begin{gathered} H_1(y)u_{x}-H_2(y)v_{y}=au+bv \quad \text{in }D\\ H_1(y)v_{x}+H_2(y)u_{y}=cu+dv \quad \text{in }D, \end{gathered}\label{e1.1}$$ in which $H_j(y)=|y|^{m_j/2}h_j(y),h_j(y)$ $(j = 1,2)$ are positive continuous functions in $\overline D$, $m_j\,(j = 1,2,m_2 < \min(1,m_1))$ are positive constants, and $a,b,c,d\,(j=1,2)$ are functions of $x+iy\,(\in D)$ satisfying the conditions $a,b,c,d\in L_\infty(D)$, which is called Condition $C$. In this article, the notation is the same as in references \cite{v1,w1,w2,w3,w4,w5,w6,w7,w8}. The following degenerate elliptic system is a special case of system \eqref{e1.1} with $H_j(y)=|y|^{m_j/2}$ $(j=1,2)$: $$\begin{gathered} |y|^{m_1/2}u_{x}-|y|^{m_2/2}v_{y}=au+bv \quad \text{in }D,\\ |y|^{m_1/2}v_{x}+|y|^{m_2/2}u_{y}=cu+dv \quad \text{in }D, \end{gathered} \label{e1.2}$$ For convenience, we mainly discuss equation \eqref{e1.2}, and equation \eqref{e1.1} can be similarly discussed. From the elliptic condition in \eqref{e1.2} (see \cite[(1.3), Chpater II]{w5}), namely $$J=4K_1K_4-(K_2+K_3)^2=4H^2(y)=4[H_1(y)/H_2(y)]^2>0\quad\text{in }\overline D\backslash\gamma$$ and $J=0$ on $\gamma=\{-1 0,j=1,2,m_2 < \min(1,m_1))$ are as stated in Section 1, and $H_1(y)=H_1[\operatorname{Im} z(Z)]$, $z(Z)$ is as stated in \eqref{e1.3}. It is clear that the function $g(Z)/H_1(y)=g(Z)/H_1[\operatorname{Im} z(Z)]$ belongs to the space $L_1(D_0)$ and in general is not belonging to the space $L_p(D_0)\,(p>2)$, and the integral $\Psi(Z_0)$ is definite when $\operatorname{Im} Z_0\ne0$. If $Z_0\in D_0$ and $\operatorname{Im} Z_0=0$, we can define the integral $\Psi(Z_0)$ as the limit of the corresponding integral over $D_0\cap\{|\operatorname{Re} t -\operatorname{Re} Z_0|\ge\varepsilon\}\cap\{|\operatorname{Im} t-\operatorname{Im} Z_0|\ge\varepsilon\}$ as $\varepsilon\to0$, where $\varepsilon$ is a sufficiently small positive number. The H$\rm\ddot{o}$lder continuity of the singular integral will be proved by the following method. \begin{theorem} \label{thm3.1} If the function $g(Z)$ in $D_Z$ satisfies the condition in \eqref{e3.2}, and $H_1(y)=y^{m_1/2}h_1(y)$, where $m_1$ is a positive number, $h_1(y)$ is a continuous positive function, then the integral in \eqref{e3.2} satisfies the estimate $$C_{\beta}[\Psi(Z),\overline{D_Z}]\le M_1,\label{e3.3}$$ in which $\beta = (2 - m_2)/(m + 2) - \delta$, $m = m_1 - m_2$, $\delta$ is a sufficiently small positive constant, and $M_1=M_1(\beta,k_3,H_1,D_Z)$ is a positive constant. \end{theorem} \begin{proof} We first give the estimates for $\Psi(Z)$ of \eqref{e3.2} in $D\cap\{\operatorname{Im} Y\ge0\}$, and verify the boundedness of the function in \eqref{e3.2}. As stated Section 1, if $H_1(y) = y^{m_1/2}h_1(y)$, then $H_1(y) \ge sY^{m_1/(m+2)}$, where $s$ is a positive constant. For any two points $Z_0=x_0\in\gamma = (-1,1)$ on $x$-axis and $Z_1=x_1+iY_1(Y_1>0)\in D_0$ satisfying the condition $2\operatorname{Im} Z_1/\sqrt3\le|Z_1-Z_0|\le2\operatorname{Im} Z_1$, this means that the inner angle at $Z_0$ of the triangle $Z_0Z_1Z_2$ $(Z_2=x_0+iY_1\in D_0 )$ is not less than $\pi/6$ and not greater than $\pi/3$, choose a sufficiently large positive number $q$, from the H\"older inequality, we have $L_1[\Psi(Z),D_0] \le L_q[g(Z),D_0]L_p[1/H_1(\operatorname{Im} t)(t-Z),D_0]$, where $p = q/(q - 1)$ $(>1)$ is close to 1. In fact we can derive it as follows \begin{aligned} |\Psi(Z_0)| &\le \big|\frac1\pi\int \int_{D_0} \frac{g(t)/H_1(\operatorname{Im} t)}{t-Z_0}d\sigma_t\big| \\ &\le \frac1{s\pi}L_q[g(Z),D_0] \Big[\int \int_{D_0} \Big|\frac1{t^{m_1/(m+2)}(t - Z_0)}\Big|^pd\sigma_t \Big]^{1/p}\\ &= \frac1{s\pi}L_q[g(Z),D_0]\,J_1^{1/p}, \end{aligned}\label{e3.4} in which \begin{align*} J_1& = \int \int_{D_0} \Big|\frac1{t^{m_1/(m+2)} (t - Z_0)}\Big|^pd\sigma_t\\ & \le \int \int_{D_0}\frac1{|t|^{pm_1/(m+2)}|\operatorname{Im}(t-Z_0)|^{p\beta_0}| \operatorname{Re}(t-Z_0)|^{p(1-\beta_0)}}d\sigma_t\\ & \le \Big|\int_0^{d_0} \frac1{Y^{pm_1/(m+2)}|Y - Y_0|^{p\beta_0}}dY \int_{d_1}^{d_2} \frac1{|x - x_0|^{p(1-\beta_0)}}dx\Big|\le k_4, \end{align*} where $d_0 = \max_{Z\in\overline{D_0}}\operatorname{Im} Z$, $d_1 = \min_{Z\in\overline{D_0}}\operatorname{Re} Z$, $d_2 = \max_{Z\in\overline{D_0}}\operatorname{Re} Z$, $\beta_0 = (2 - m_2)/(m+2) - \varepsilon$, $\varepsilon\,(<1/p-m_1/(m+2))$ is a sufficiently small positive constant, we can choose $\varepsilon=2(p-1)/p\,(\le(2-m_2)/(m+2))$, such that $p(1-\beta_0)<1$ and $p[m_1/(m+2)+\beta_0]<1$, and $k_4=k_4(\beta,k_3,H_1,D_0)$ is a non-negative constant. Next we estimate the H\"older continuity of the integral $\Psi(Z)$ in $\overline{D_0}$; i.e., \begin{aligned} &|\Psi(Z_1) - \Psi(Z_0)|\\ &\le \frac{|Z_1 - Z_0|}\pi \Big|\int \int_{D_0} \frac{g(t)/H_1(\operatorname{Im} t)}{(t - Z_0)(t - Z_1)}d\sigma_t\Big|\\ &\le \frac{|Z_1 - Z_0|}{s\pi}L_q[g(Z),D_0]\Big[\int \int_{D_0} \Big|\frac1{t^{m_1/(m+2)}(t - Z_0)(t - Z_1)}\Big|^pd\sigma_t\Big]^{1/p}, \end{aligned} \label{e3.5} and \begin{align*} J_2& = \int \int_{D_0} \big|\frac1{t^{m_1/(m+2)}(t - Z_0)(t - Z_1)}\big|^pd\sigma_t\\ & \le \int \int_{D_0} \frac{|\operatorname{Re}(t - Z_0)|^{p(\beta_0/2-1)}|\operatorname{Re}(t - Z_1) |^{p(\beta_0/2-1)}}{|t|^{pm_1/(m+2)}|\operatorname{Im}(t - Z_0)|^{p\beta_0/2}|\operatorname{Im}(t - Z_1)|^{p\beta_0 /2}}d\sigma_t\\ & \le \int_0^{d_0} \frac1{Y^{pm_1/(m+2)}|\operatorname{Im}(Y - Z_0)|^{p\beta_0/2} |\operatorname{Im}(Y - Z_1)|^{p\beta_0/2}}dY\\ & \quad\times\int_{d_1}^{d_2}\frac1{|\operatorname{Re}(t - Z_0)|^{p(1-\beta_0/2)} |\operatorname{Re}(t - Z_1)|^{p(1-\beta_0/2)}}d\operatorname{Re} t\\ & \le k_5 \int_{d_1}^{d_2}\frac1{|x-x_0)|^{p(1-\beta_0/2)}|x-x_1|^{p(1 -\beta_0/2)}}dx, \end{align*} where $\beta_0=(2-m_2)/(m+2) - \varepsilon$ is chosen as before and $$k_5 = \max_{Z_0,Z_1\in D_0} \int_0^{d_0}[Y^{pm_1/(m+2)}|\operatorname{Im}(Y - Z_0) |^{p\beta_0/2}|\operatorname{Im}(Y - Z_1)|^{p\beta_0/2}]^{-1}dY.$$ Denote $\rho_0=|\operatorname{Re}(Z_1-Z_0)|=|x_1-x_0|$, $L_1=D_0\cap\{|x-x_0|\le2\rho_0,Y=Y_0 \}$ and $L_2=D_0\cap\{2\rho_0<|x-x_0|\le2\rho_1<\infty,Y=Y_0\}\supset[d_1,d_2] \backslash L_1$, where $\rho_1$ is a sufficiently large positive number, we can derive \begin{align*} J_2 & \le k_5\Big[\int_{L_1}\frac1{|x-x_0|^{p(1-\beta_0/2)}|x-x_1|^{p(1- \beta_0/2)}}dx\\ & \quad +\int_{L_2}\frac1{|x-x_0|^{p(1-\beta_0/2)}|x-x_1|^{p(1-\beta_0/2)}} dx\Big]\\ & \le k_5\Big[|x_1 - x_0|^{1-2p+p\beta_0}\int_{|\xi|\le2}\frac1{|\xi|^{p(1- \beta_0/2)}|\xi\pm1|^{p(1-\beta_0/2)}}d\xi\\ & \quad +k_6|\int_{2\rho_0}^{2\rho_1}\rho^{p\beta_0-2p}d\rho|\Big] \\ &\le k_7|x_1 - x_0|^{1-p(2-\beta_0)}\\ &= k_7|x_1-x_0|^{p((2-m_2)/(m+2)-\varepsilon+1/p-2)}, \end{align*} in which we use $|x-x_0|=\xi|x_1-x_0|$, $|x-x_1|=|x-x_0-(x_1-x_0)|=|\xi\pm1| |x_1-x_0|$ if $x\in L_1$, $|x-x_0|=\rho\le2|x-x_1|$ if $x\in L_2$, choose that $p$ $(>1)$ is close to 1 such that $1-p(2-\beta_0)<0$, and $k_j=k_j(\beta,k_3,H,D_0)\,(j=6,7)$ are non-negative constants. Thus we obtain $$|\Psi(Z_1) - \Psi(Z_0)| \le k_7|Z_1 - Z_0||x_1 - x_0 |^{(2-m_2)/(m+2)-\varepsilon+1/p-2} \le k_8|Z_1 - Z_0|^\beta, \label{e3.6}$$ in which we use that the inner angle at $Z_0$ of the triangle $Z_0Z_1Z_2$ $(Z_2=x_0+iY_1\in D_0 )$ is not less than $\pi/6$ and not greater than $\pi/3$, and choose $\varepsilon = 2(p - 1)/p$, $\beta = (2 - m_2)/(m + 2) - \delta$, $\delta = 3(p- 1)/p$, $k_8 = k_8 (\beta,k_3,H_1,D_0)$ is a non-negative constant. The above points $Z_0=x_0$, $Z_1 =x_1+iY_1$ can be replaced by $Z_0=x_0+iY_0$, $Z_1=x_1+iY_1\in\overline{D_0}$, \$0