\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Nonlinear perturbations \hfil EJDE--2000/05} {EJDE--2000/05\hfil C. J. Vanegas \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.~{\bf 2000}(2000), No.~05, pp.~1--10. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Nonlinear perturbations of systems of partial differential equations with constant coefficients \thanks{ {\em 1991 Mathematics Subject Classifications:} 35F30, 35G30, 35E20. \hfil\break\indent {\em Key words and phrases:} Right inverse, nonlinear differential equations, fixed point theorem. \hfil\break\indent \copyright 2000 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted April 6, 1999. Published Janaury 8, 2000.} } \date{} % \author{C. J. Vanegas} \maketitle \begin{abstract} In this article, we show the existence of solutions to boundary-value problems, consisting of nonlinear systems of partial differential equations with constant coefficients. For this purpose, we use the right inverse of an associated operator and a fix point argument. As illustrations, we apply this method to Helmholtz equations and to second order systems of elliptic equations. \end{abstract} \newcommand{\adj}{\mathop{\rm adj}} \section{Introduction} Let $G\subset{\mathbb R}^n$ be a bounded region with smooth boundary, and let $(B(G),\|.\|)$ be a Banach space of functions defined on $G$. For each natural number $n$, let $B^n(G)$ denote the space of functions $f$ satisfying $D^mf\in B(G)$ for all multi-index $m$ with $|m|\leq n$. Then under the norm $\|f\|_n= \max_{|m|\leq n}\|D^mf\|$, the space $B^n(G)$ becomes a Banach space. We consider the system $$\label{eqfirst} D_{0}\omega = f(\textbf{x},\omega,\frac{\partial \omega}{\partial x_{1}},\ldots,\frac{\partial \omega} {\partial x_{n}})\quad \mbox{in G}\,,$$ where $D_0$ is a linear differential operator of first order with respect to the real variables $x_1,\ldots,x_n$, the vector $\textbf{x}$ has components $(x_1,\ldots,x_n)$, and the unknown $\omega$ and the right-hand side $f$ are vectors of $m$ components, with $m\geq n$. To this system of differential equations, we add the boundary condition $$\label{eqsecond} Aw = g \quad\mbox{on }\partial G\,,$$ where $g$ is a given $m$-dimensional vector-valued function that belongs to the Banach space $B^1(\partial G)$. The operator $A$ is chosen so that (\ref{eqsecond}) leads to a well-posed problem on $B^1(G)\cap \ker D_0$. For finding a solution to this nonlinear problem, we use a right inverse of the operator $D_0$ and a fix point argument \cite{vekm,tut}. First, we construct the right inverse for a first order differential operator of constant coefficients. Then using that the operator $D_0$, in its matrix form, commutes with the elements of the formal adjoint matrix, we obtain the right inverse. In fact, we obtain a formal algebraic inversion through the associated operators determinant and adjoint matrix of $D_0$. In the last section of this article, we describe a natural generalization to high order systems, and show two applications of this method. \section{The Right Inverse of $D_0$.} The operator $D_0$ in (\ref{eqfirst}) is represented in a matrix form as $$D_0 = \left( \begin{array}{cccc} D_{11} & \ldots & D_{1m} \\ \vdots \\ D_{m1} & \cdots & D_{mm} \end{array} \right)\,,$$ where $D_{ij}$ is the differential operator of first order with respect to the real variables $x_{1} \ldots x_{n}$. The determinant of $D_0$ is computed formally, and is a scalar linear differential operator with constant coefficients. Note that $\det D_0$ maps the space $B^m(G)$ into the space $B(G)$. As a general hypothesis, we assume that the differential operator $\det D_0$ possesses a continuous right inverse: $$\label{eqinv} T_{\det D_0}: B(G)\rightarrow B^m(G)$$ which is an operator that improves the differentiability of functions in $B(G)$ by $m$ orders. The adjoint matrix associated with $D_0$, in algebraic sense, is computed formally, resulting a linear matrix differential operator, denotes by $\adj D_0$, with constant coefficients and of order $m - 1$ respect to the real variables $x_{1} \ldots x_{n}$, i.e., $m - 1$ is the order of the highest derivative that appears in the coefficients of the matrix. We observe that $\adj D_0$ maps the space $B^m(G)$ into the space $B^1(G)$. Under the assumptions above, we obtain the following result. \newtheorem{Theorem}{Theorem}[section] \begin{Theorem} The differential operator $$\adj D_0(T_{\det D_0}): B(G)\rightarrow B^1(G)$$ is a right inverse operator for $D_0$. \end{Theorem} \paragraph{Proof.} Note that $D_0\adj D_0 = \det D_0 I$, which is satisfied due to the fact that $D_0$ is a differential operator with constant coefficients. From this remark and (\ref{eqinv}) the proof follows. \quad $\square$ \section{First-Order Nonlinear Systems} We define the fitting operator $$\Omega: B^1(\partial G)\rightarrow B^1(G)\cap \ker D_0$$ by the relation $$\label{eqside-cond.} A(\Omega \phi) = A(\phi) \quad\mbox{for each } \phi \in B^1(\partial G).$$ i.e., to each $\phi \in B^1(\partial G)$ we associate the unique $B^1(G)$-solution to (\ref{eqside-cond.}) in $\ker D_0$. \begin{Theorem}\label{eq:theo.1} The boundary-value problem (\ref{eqfirst})-(\ref{eqsecond}) is equivalent to the fixed point problem for the operator $$\label{eqoper} T(\omega,h_{1}, \ldots, h_{n})= (W,H_{1}, \ldots, H_{n})\,,$$ where \begin{eqnarray} \label{eqmega} &W=\Omega g + (I-\Omega)\adj D_{0} (T_{\det D_{0}}I) f(\textbf{x},\omega,h_{1}, \ldots, h_{n})& \\ &\label{eqhache} H_{j}=\frac{\partial}{\partial x_{j}}(\Omega g+(I-\Omega)\adj D_{0} (T_{\det D_{0}}I)) f(\textbf{x},\omega,h_{1},\ldots,h_{n}),& \end{eqnarray} with $j = 1, \ldots, n$. \end{Theorem} \paragraph{Proof.} Let $\omega \in B^1(G)$ be a solution to (\ref{eqfirst})-(\ref{eqsecond}). To the function $$\label{eqsisi} \Psi = w-\adj D_{0}(T_{\det D_{0}}I)f(\textbf{x},\omega,\frac{\partial \omega}{\partial x_{1}},\ldots,\frac{\partial \omega}{\partial x_{n}})$$ we apply the operator $D_{0}$ to obtain $$D_{0}\Psi = D_{0}\omega-D_{0}\adj D_{0}(T_{\det D_{0}}I)f(\textbf{x},\omega, \frac{\partial \omega}{\partial x_{1}},\ldots, \frac{\partial \omega}{\partial x_{n}})= 0\,.$$ Thus, $\Psi \in \ker D_{0}$. To $\Psi$ we apply the operator $A$ and obtain \begin{eqnarray*} A\Psi &=& A\omega-A\adj D_{0}(T_{\det D_{0}}I) f(\textbf{x},\omega,\frac{\partial \omega} {\partial x_{1}},\ldots,\frac{\partial \omega}{\partial x_{n}}) \\ &=& g-A\adj D_{0}(T_{\det D_{0}}I) f(\textbf{x},\omega,\frac{\partial \omega} {\partial x_{1}},\ldots,\frac{\partial \omega}{\partial x_{n}})\,. \end{eqnarray*} According to the definition of the operator $\Omega$, we have $$\Psi = \Omega g-\Omega \adj D_{0}(T_{\det D_{0}}I) f(\textbf{x},\omega,\frac{\partial \omega} {\partial x_{1}},\ldots,\frac{\partial \omega}{\partial x_{n}})\,.$$ Substituting this expression in (\ref{eqsisi}) and differentiating with respect to $x_{j}$, we conclude that $(\omega,\frac{\partial \omega}{\partial x_{1}},\ldots,\frac{\partial \omega} {\partial x_{n}})$ is a fixed point of (\ref{eqoper}). On the other hand if $(\omega,h_{1}, \ldots, h_{n})$ is a fixed point of (\ref{eqoper}), we can carry out the differentiation of (\ref{eqmega}) with respect to $x_{j}$ for each $j=1, \ldots, n$. Because $\omega$ is in $B^1(G)$, we obtain $$\frac{\partial w}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} (\Omega g+(I-\Omega)\adj D_{0} (T_{\det D_{0}}I)) f(\textbf{x},\omega,h_{1},\ldots,h_{n})\,.$$ Comparing these equations with (\ref{eqhache}), it follows that $\frac{\partial \omega}{\partial x_{j}} = h_{j}$ for $j=1,\ldots,n$. Substituting these equations in (\ref{eqmega}) and then applying the operator $D_{0}$ we obtain $D_0 \omega = f(\textbf{x},\omega,\frac{\partial \omega} {\partial x_{1}}, \ldots, \frac{\partial \omega}{\partial x_{n}})$. Applying $\Omega$ to (\ref{eqmega}) we conclude that $$\Omega \omega=\Omega\Omega g + \Omega(I-\Omega)\adj D_{0} (T_{\det D_{0}}I) f(\textbf{x},\omega,h_{1}, \ldots, h_{n})= \Omega g\,.$$ By the definition of the operator $\Omega$, it follows that $A(\omega)= g$, and hence, $\omega$ is a solution of (\ref{eqfirst})-(\ref{eqsecond}). \quad $\square$\medskip Consider the polycylinder \begin{eqnarray*} M&=&\{(\omega,h_{1}, \ldots, h_{n})\in \prod_{i=1}^{n+1}B(G): \| \omega-\omega_{0}\|\leq a_{0},\\ &&\quad \| h_{j}-{h_{j}}_{0}\| \leq a_{j}, j=1, \ldots, n\} \end{eqnarray*} where $\omega_{0}\in B^{1}(G)$ and ${h_{j}}_{0}\in B(G)$ are taken as the coordinates of the polycylinder mid-point, and $a_{0},a_{1},\ldots,a_{n}$ are positive real numbers. From the definition of the operators $T_{\det D_{0}}, \adj D_{0}$, and $\Omega$, it follows that the operators \begin{eqnarray} \label{eqacoto 1} &(I-\Omega)\adj D_0(T_{\det D_0}I): B(G)\rightarrow B(G)\quad\mbox{and}& \\ \label{eqacoto 2} &\frac{\partial}{\partial x_{j}}(I-\Omega)\adj D_0(T_{\det D_{0}}I): B(G)\rightarrow B(G)& \end{eqnarray} are continuous and hence bounded. Therefore, for all $(\omega,h_{1},\ldots,h_{n})\in M$ we have \begin{eqnarray*} \| W-\omega_{0}\|&=& \|\Omega g+(I-\Omega) \adj D_0(T_{\det D_0}I)f(\textbf{x},\omega,h_{1}, \ldots, h_{n})- \omega_{0}\| \\ &=& \|(I-\Omega)\adj D_0(T_{\det D_0}I) [f(\textbf{x},\omega,h_{1}, \ldots, h_{n})-D_{0}\omega_{0}]\\ &&+(I-\Omega)\adj D_0(T_{\det D_0}I)D_{0}\omega_{0}+\Omega g -\omega_{0}\| \\ &\leq& \|(I-\Omega)\adj D_0(T_{\det D_0}I)\| \| f(\textbf{x},\omega,h_{1}, \ldots, h_{n})- D_{0}\omega_{0}\|+K_{0} \end{eqnarray*} and \begin{eqnarray*} \lefteqn{ \| H_{j}-{h_{j}}_{0}\| }\\ &=& \|\frac{\partial}{\partial x_{j}}[\Omega g+(I-\Omega) \adj D_0(T_{\det D_0}I)] f(\textbf{x},\omega,h_{1}, \ldots, h_{n})-{h_{j}}_{0}\| \\ &\leq& \|\frac{\partial}{\partial x_{j}}(I-\Omega) \adj D_0(T_{\det D_0}I)\| \| f(\textbf{x},\omega,h_{1}, \ldots, h_{n})- D_{0}\omega_{0}\|+K_{j}\,, \end{eqnarray*} where \begin{eqnarray*} &K_{0}=\|(I-\Omega)\adj D_0(T_{\det D_0}I)D_{0}\omega_{0}+\Omega g -\omega_{0}\|&\\ &K_{j}=\| \frac{\partial}{\partial x_{j}}(I-\Omega)\adj D_0(T_{\det D_0}I) D_{0}\omega_{0}+\frac{\partial}{\partial x_{j}}\Omega g-{h_{j}}_{0}\|,& \end{eqnarray*} for $j=1,\ldots,n$. For a positive real number $R$ and $j=1,2,\dots n$, we set \begin{eqnarray*} a_{0}&=&\|(I-\Omega)\adj D_0(T_{\det D_0}I)\| R + K_{0}\\ a_{j}&=&\|\frac{\partial}{\partial x_{j}}(I-\Omega)\adj D_0(T_{\det D_0}I) \| R + K_{j}\,. \end{eqnarray*} For the rest of this article, we will denote by $M_{R}$ the polycylinder $M$ with the parameters $a_{0},a_{1},\ldots,a_{n}$ as defined above. \begin{Theorem}\label{eq:Theo.2} Let $R$ be a positive real number such that $f$ maps the polycylinder $M_{R}$ into $B(G)$ and satisfies the growth condition $$\| f(\textbf{x},\omega,h_{1},\ldots,h_{n}) - D_{0}\omega_{0}\| \leq R, \quad \forall\, (\omega,h_{1},\ldots,h_{n})\in M_{R}\,.$$ Then the operator $T$ maps continuously the polycylinder $M_{R}$ into itself. \end{Theorem} \paragraph{Proof.} Let $(\omega,h_{1},\ldots,h_{n})$ be an element in $M_{R}$ and $(W,H_{1},\ldots,H_{n})$ its image under $T$. Since $(\omega,h_{1}, \ldots, h_{n}) \in M_R$, by the definitions of the operators $T_{\det D_{0}}, \adj D_{0}$ and $\Omega$, it follows that $W\in B^{1}(G)\subset B(G)$. Since $\frac{\partial}{\partial x_{j}}: B^1(G)\rightarrow B(G)$, it follows that $H_{j}\in B(G)$ for all $j=1, \ldots, n$. Therefore, $T: M_R \rightarrow \prod_{i=1}^{n+1}B(G)$. That $(W,H_{1},\ldots,H_{n})$ is in $M_{R}$ follows from the boundedness of the operators (\ref{eqacoto 1})-(\ref{eqacoto 2}), the hypotheses on $f$, and the definition of $M_R$. \quad $\square$ \begin{Theorem}\label{eq:Theo.3} Suppose $f$ maps the polycylinder $M_{R}$ into the space $B(G)$, and that $f$ is Lipschitz continuous with constant $L$ satisfying $$L<\min \{\|(I-\Omega)\adj D_0(T_{\det D_0}I)\|^{-1},\; \|\frac{\partial}{\partial x_{j}} (I-\Omega)\adj D_0(T_{\det D_0}I)\|^{-1}\},$$ for $j=1,\ldots,n$. Then $T$ is a contraction. \end{Theorem} \paragraph{Proof.} Let $(\omega,h_{1},\ldots,h_{n})$, $(\omega',h_{1}',\ldots,h_{n}')$ be elements of $M_{R}$, and \newline $(W,H_{1},\ldots,H_{n})$, $(W',H_{1}',\ldots,H_{n}')$ be their images under $T$. Since the operators (\ref{eqacoto 1}) and (\ref{eqacoto 2}) are bounded and $f$ is Lipschitz with constant $L$, it follows that \begin{eqnarray*} \| W-W'\| &\leq& \|(I-\Omega)\adj D_0(T_{\det D_0}I)\| L \|(\omega,h_{1},\ldots,h_{n})-(\omega',h_{1}',\ldots,h_{n}') \|\\ &\leq&\| (\omega,h_{1},\ldots,h_{n})-(\omega',h_{1}',\ldots,h_{n}')\|\,. \end{eqnarray*} Similarly, $$\| H_{j}-H_{j}' \| \leq \|(\omega,h_{1},\ldots,h_{n})-(\omega',h_{1}',\ldots,h_{n}')\|$$ for $j=1,\ldots,n$. Therefore, $T$ is a contraction. \quad $\square$\medskip With the aid of Theorems \ref{eq:theo.1}, \ref{eq:Theo.2} and \ref{eq:Theo.3}, we obtain existence and uniqueness of a solution for Problem (\ref{eqfirst})-(\ref{eqsecond}). \begin{Theorem} Suppose that $f$ satisfies the hypotheses of Theorems \ref{eq:Theo.2} and \ref{eq:Theo.3}. Then Problem (\ref{eqfirst})-(\ref{eqsecond}) possesses exactly one solution in the polycylinder $M_{R}$. \end{Theorem} \paragraph{Proof.} By definition $M_{R}$ is a closed subset in the space $B(G)$. Applying Theorems \ref{eq:Theo.2} and \ref{eq:Theo.3}, we realize that $T$ maps $M_{R}$ into itself, and it is a contraction; therefore, according to the Fixed Point Theorem there exists a unique fixed point in $M_{R}$. As a consequence of Theorem \ref{eq:theo.1} this fixed point is a solution to Problem (\ref{eqfirst})-(\ref{eqsecond}). \quad $\square$ \section{High-Order Systems} In this section we apply the method developed in the above section to high-order equations. Consider the system of differential equations $$\label{eqorder} D_{0}\omega = f(\textbf{x},D^{r}\omega)$$ where $D^{r}$ is a differential operator of order $r$, and $D_{0}$ is a linear differential operator of order $r$. The unknown $\omega$ and the right-hand side $f$ are vector-valued functions of $m$ components, with $m\geq n$. We will assume that the associated differential operator $\det D_0$ has a continuous right inverse, $T_{\det D_0}: B(G)\rightarrow B^{rm}(G)$. To system (\ref{eqorder}) we add the boundary condition $$\label{eqconorder} A\omega = g \quad\mbox{on } \partial G\,,$$ where $g$ is a vector-valued function with $m$ components in $B^r(\partial G)$. The operator $A$ is chosen so that (\ref{eqconorder}) becomes a well-posed problem on $B^r(G)\cap \ker D_0$. We define the fitting operator $\Omega: B^r(\partial G)\rightarrow B^r(G)\cap \ker D_0$ as follows: For each function $\phi \in B^r(\partial G)$, $\Omega (\phi)$ is the unique $B^r(G)$-solution in $\ker D_0$ to the equation $A(\Omega (\phi)) = A(\phi)$. The results established in section 3 are also valid for systems of order $r > 1$. However, (\ref{eqmega}) and (\ref{eqhache}) need to be increased to include equations corresponding to the higher-order derivatives. We will analyze the case when $D_0$ is a diagonal operator. Let $D_0$ be a linear differential operator of order $r$, which can be represented as $D_0 = PI$, where $P$ is a linear differential operator of order $r$ with a continuous right inverse $T_P: B(G)\rightarrow B^r(G)$. Let us assume that the operator $T_P$ satisfies homogeneous boundary condition $A(T_P\phi) = 0$ for all $\phi\in B(G)$; thus the identity $(I-\Omega)\adj D_{0} (T_{\det D_{0}}I) = T_PI$ holds. Under these conditions, the equivalent system (\ref{eqmega})-(\ref{eqhache}) can be simplified. Furthermore, we need only the continuity $T_P$ for homogeneous conditions, and an estimate on $\Omega$ for non-homogeneous conditions. As a consequence of this we have the following result \begin{Theorem}\label{eq:Theo.4} Suppose that \begin{eqnarray} \label{eqdiag} &D_0\omega = PI\omega = \tilde{f} & \\ \label{eqcodiag} &A(\omega) = 0& \end{eqnarray} is a well-posed problem in the sense of $$\label{eqTP} T_P: B(G)\rightarrow B^r(G),$$ where $\tilde{f}$ is a vector-valued function of dimension $m$, depending only on the coordinates $x_1,\ldots,x_n$. If the right-hand side in (\ref{eqorder}) satisfies a certain growth condition, and is Lipschitz with a constant sufficiently small, then Problem (\ref{eqorder})-(\ref{eqconorder}) is well-posed in the sense of (\ref{eqTP}). \end{Theorem} \section{Examples.} \subsection*{Example 1: Helmholtz type equations.} Let $G = G_1\times G_2$ be a bounded simply connected region in ${\mathbb R}^3$ with smooth boundary $\partial G$. Here $G_1$ is the region containing the component $x_1$, and $G_2$ is the region containing the components $x_2$ and $x_3$. On the domain $G$, we consider the system $$\label{eqHel} D_{0}\omega = f(\textbf{x},\omega,\frac{\partial \omega_1}{\partial x_2}, \frac{\partial \omega_1}{\partial x_3}, \frac{\partial \omega_2}{\partial x_1}, \frac{\partial \omega_2}{\partial x_3}, \frac{\partial \omega_3}{\partial x_1}, \frac{\partial \omega_3}{\partial x_2}),$$ where $\textbf{x}=(x_1,x_2,x_3)$ is a vector in ${\mathbb R}^3$, $\omega = (\omega_1,\omega_2,\omega_3)$ and $f = (f_1,f_2,f_3)$ are vector-valued functions, and the right-hand side $f$ does not dependent on $\frac{\partial \omega_i}{\partial x_i}$, $i=1,2,3$. For $\lambda>0$, let $$D_0 = \left( \begin{array}{cccc} \lambda & -\frac{\partial} {\partial x_3} & \frac{\partial}{\partial x_2} \\[3pt] \frac{\partial}{\partial x_3} & \lambda & -\frac{\partial}{\partial x_1} \\[3pt] -\frac{\partial}{\partial x_2} & \frac{\partial}{\partial x_1} & \lambda \end{array} \right)\,.$$ From (\ref{eqHel}) it follows that for $i\ne j$, $$\mathop{\rm curl} \omega + \lambda\omega = \left(\begin{array}{c} f_1(x,\omega,\frac{\partial \omega_1}{\partial x_2},\ldots,\frac{\partial \omega_i}{\partial x_j},\ldots)\\[3pt] f_2(x,\omega,\frac{\partial \omega_1}{\partial x_2},\ldots,\frac{\partial \omega_i}{\partial x_j},\ldots)\\[3pt] f_3(x,\omega,\frac{\partial \omega_1}{\partial x_2},\ldots,\frac{\partial \omega_i}{\partial x_j},\ldots)\end{array}\right)\,.$$ To the system (\ref{eqHel}) we add the Dirichlet boundary condition \begin{eqnarray} \label{eqboun} &\omega_1 = g_1 \quad\mbox{on } \partial G& \\ &\omega_2 = g_2 \quad\mbox{on } \partial G_1 \times \partial G_2 \,,& \nonumber \end{eqnarray} where $g_1$ and $g_2$ are given real-valued functions in the space of $\alpha$-H\"older continuous and differentiable functions $C^{1,\alpha}$. We look for solutions to Problem (\ref{eqHel})-(\ref{eqboun}) in the space of $\alpha$-H\"older continuous functions $C^\alpha(G)$. After some calculations, we obtain $\det D_0 = \lambda(\lambda^2 + \Delta)$, where $\Delta$ denotes the Laplace operator, and $\lambda^2$ is not an eigenvalue for the Helmholtz operator $\Delta + \lambda^2$. Therefore, this operator possesses a continuous right inverse $T_{\Delta + \lambda^2}: C^\alpha(G)\rightarrow C^{\alpha,2}(G)$. Similarly, we obtain the associated adjoint matrix $$\adj D_0 = \left( \begin{array}{cccc} \lambda^2 + \frac{\partial^2}{\partial x_1^2} & \frac{\partial^2}{\partial x_2\partial x_1} + \lambda \frac{\partial}{\partial x_3} & \frac{\partial^2}{\partial x_1\partial x_3} - \lambda \frac{\partial}{\partial x_2} \\[3pt] \frac{\partial^2}{\partial x_1 \partial x_2} - \lambda \frac{\partial}{\partial x_3} & \lambda^2 + \frac{\partial^2}{\partial x_2^2} & \frac{\partial^2}{\partial x_2\partial x_3} + \lambda \frac{\partial}{\partial x_1} \\[3pt] \frac{\partial^2}{\partial x_3 \partial x_1} + \lambda \frac{\partial}{\partial x_2} & \frac{\partial^2}{\partial x_3 \partial x_2} - \lambda \frac{\partial}{\partial x_1} & \lambda^2 + \frac{\partial^2}{\partial x_3^2} \end{array} \right)\,.$$ Note that the operator $T_{\Delta + \lambda^2}I$ improves the differentiability properties of a function by two, not by three orders. The operator $\adj D_0$ decreases the differentiability properties by two orders only in the $ii$ components with respect to $x_i$. However, it was assumed that the derivatives $\frac{\partial \omega_i}{\partial x_i}$, $i = 1,2,3$ do not appear in the right-hand side $f$ of (\ref{eqHel}). Therefore, $\adj D_0(T_{\Delta + \lambda^2}I)$ improves the properties of differentiability by one order, and we can consider all the equations except those associated with $\frac{\partial \omega_i}{\partial x_i}$, $i = 1,2,3$ in Problem (\ref{eqmega})-(\ref{eqhache}). Now, we study the kernel of $D_0$. Let $(\omega_1,\omega_2,\omega_3)$ be a solution of the homogeneous problem $$\label{eqkernel} D_0\omega = 0\,.$$ When we apply the operator $\adj D_0$ on the left in the above equation, it follows that $(\Delta + \lambda^2)\omega_i = 0$ for $i = 1,2,3$. Due to (\ref{eqkernel}), the three components are linearly dependent. Therefore, we will assume $w_1$ as an arbitrary given function which satisfies the equation $(\lambda^2 + \Delta)w_1 = 0$ and is also defined on $\partial G$. In view of (\ref{eqkernel}), we obtain \begin{eqnarray} \lambda w_1 - \frac{\partial \omega_2}{\partial x_3} + \frac{\partial \omega_3}{\partial x_2} & =& 0 \nonumber \\ \frac{\partial \omega_1}{\partial x_3} + \lambda w_2 - \frac{\partial \omega_3}{\partial x_1} & = & 0 \label{eqclkernel}\\ -\frac{\partial \omega_1}{\partial x_2} + \frac{\partial \omega_2}{\partial x_1} + \lambda w_3 & = & 0\,. \nonumber \end{eqnarray} When we differentiate the first equation respect to $x_1$, the second respect to $x_2$, and the third respect to $x_3$, after summing the results, we have $$\label {eqtodas} \frac{\partial \omega_1}{\partial x_1} + \frac{\partial \omega_2}{\partial x_2} + \frac{\partial \omega_3}{\partial x_3} = 0\,.$$ Using (\ref{eqclkernel}) and (\ref{eqtodas}) we have, in matrix form, $$\label {eqDuno} D_1\left( \begin{array}{c} w_2 \\ w_3 \end{array} \right)\; = \; \left( \begin{array}{c} -\frac{\partial \omega_1}{\partial x_1}\\ -\lambda w_1 \end{array} \right)$$ and $$\label {eqDdos} D_2\left( \begin{array}{c} w_2 \\ w_3 \end{array} \right)\; = \; \left( \begin{array}{c} -\frac{\partial \omega_1}{\partial x_3}\\ \;\;\frac{\partial \omega_1}{\partial x_2} \end{array} \right)$$ where $$D_1=\left( \begin{array}{cc} \frac{\partial}{\partial x_2} & \frac{\partial}{\partial x_3}\\[3pt] -\frac{\partial}{\partial x_3} & \frac{\partial}{\partial x_2} \end{array} \right)\quad \mbox{and}\quad D_2=\left( \begin{array}{cc} \lambda & -\frac{\partial}{\partial x_1}\\[3pt] \frac{\partial}{\partial x_1} & \lambda \end{array} \right)\,.$$ Since $\det D_1 = \frac{\partial^2}{\partial x_2^2} + \frac{\partial^2}{\partial x_3^2}$ and $\det D_2 = \lambda^2 + \frac{\partial^2}{\partial x_1^2}$, we can assume the existence of right inverse operators for $D_1$ and $D_2$. Since $(\lambda^2 + \Delta)w_1 = 0$, the integrability condition $$D_2 \left( \begin{array}{c} -\frac{\partial \omega_1}{\partial x_1}\\ -\lambda w_1 \end{array} \right)\; = \; D_1 \left( \begin{array}{c} -\frac{\partial \omega_1}{\partial x_3}\\ \;\;\frac{\partial \omega_1}{\partial x_2} \end{array} \right)$$ is fulfilled for the system (\ref{eqDuno})-(\ref{eqDdos}). Put $w = w_2 + iw_3$ and $z = x_2 - ix_3$. Then from (\ref{eqDuno}), we obtain the non-homogeneous Cauchy-Riemann System $$\label {eqcauchy} \frac{\partial \omega}{\partial \bar{z}} = F(\omega_1, \frac{\partial \omega_1}{\partial x_1}),$$ where $F$ is known. Thus $w$ can be uniquely determined up to a holomorphic function in $z$. Since $\omega$ satisfies $D_2 \omega = 0$, we apply the operator $\adj D_2$ on the left to this equation, and obtain $$\label{eqhh} (\lambda^2 + \frac{\partial^2}{\partial x_1^2})Iw = 0\,.$$ From (\ref{eqhh}) it follows that $(\lambda^2 + \frac{\partial^2}{\partial x_1^2})w_2 = 0$ and $(\lambda^2 + \frac{\partial^2}{\partial x_1^2})w_3 = 0$. When we prescribe the boundary values on $\partial G_1 \times \partial G_2$, $w_2$ becomes a uniquely determined function. Finally from the last equation in (\ref{eqclkernel}), we obtain $w_3 = \frac{1}{\lambda}(\frac{\partial \omega_1}{\partial x_2} - \frac{\partial \omega_2}{\partial x_1})$, and we cannot require additional values for $w_3$. Since this is a well-posed problem, it follows that (\ref{eqboun}) is well formulated. Therefore, applying the theory developed in section 3, we assure the existence of an unique solution for Problem (\ref{eqHel})-(\ref{eqboun}). \subsection*{Example 2: A second order elliptic operator.} Let $G$ be a bounded simply connected region in ${\mathbb R}^n$ with boundary sufficiently smooth. Consider the system $$\label {eqellip} D_0\omega = f(x,D^2\omega)\quad\mbox{in G}\,,$$ where $D^2$ is a second-order differential operator, not necessarily linear, and $D_0$ is a linear differential operator of second order. The unknown $\omega$ and the right-hand side $f$ are vectors of $m$ components. We assume that $D_0$ is a diagonal operator of the form $D_0 = PI$, where $P$ is an elliptic differential operator of second order with constant coefficients, $P = \sum^n_{i,j=1} a_{i,j}\frac{\partial^2}{\partial x_i\partial x_j}$. In addition to (\ref{eqellip}) we impose the Dirichlet boundary condition $$\label {eqbor} \omega = g \quad\mbox{on } \partial G,$$ where $g$ is a given vector-valued $m$-dimensional function belonging to $C^{2,\alpha}(\partial G)$. Then we look for a solution to (\ref{eqellip})-(\ref{eqbor}) in the space $C^{\alpha}(\bar{G})$. It is known that the operator $P$ possesses a continuous right inverse \cite{mir}, $T_P: C^\alpha(\bar{G})\rightarrow C^{2,\alpha}(\bar{G})$, which satisfies $A(T_P\phi) = 0$ for all $\phi\in C^\alpha(\bar{G})$. Since $\det D_0 = P^m$, there is a continuous right inverse operator $T_{\det D_0} = T_{P^m}: B(G)\rightarrow B^{2m}(G)$. We conclude by observing that now all the theory developed in sections 3 and 4 can be applied to this problem. \begin{thebibliography}{0} \bibitem{fich} Fichera C., {\it Linear Elliptic Differential Systems and Eigenvalue Problems}, Lecture Notes in Mathematics, Vol. 8, Springer-Verlag, Berlin-Heidelberg-New York (1965). \bibitem{foll} Folland G.B., {\it Introduction to Partial Differential Equations}, Princeton University Press, Princeton,New Jersey (1995). \bibitem{giltru}Gilbert D., Trudinger N.S., {\it Elliptic Partial Differential Equations of Second Order}, Grundlehren der mathematischen Wissenschaften, 224, Springer-Verlag, Berlin-Heidelberg-New York (1977). \bibitem{gilb}Gilbert R.P., {\it Constructive Methods for Elliptic Equations}, Lecture Notes in Mathematics, 365, Berlin (1974). \bibitem{gilbuch}Gilbert R.P., Buchanan J.L., {\it First Order Elliptic Systems: A Function Theoretic Approach}, Academy Press, New York, London (1983). \bibitem{horm}H\"ormander L., {\it Linear Partial Differential Operators}, third edition, Springer-Verlag (1969). \bibitem{mir} Miranda C., {\it Partial Differential Equations of Elliptic Type}, Springer-Verlag, Berlin-Heidelberg-New York (1970). \bibitem{tut} Tutschke W., {\it Partielle Differentialgleichungen. Klassische Funktionalanalytische und Komplexe Methoden}, BSB B.G. Teubner-Verlagsgesellschaft, Leipzig (1983). \bibitem{vekm} Vekua I.N., {\it New Methods for Solving Elliptic Equations}, Vol. 1, North-Holland Publ., Amsterdam (1968). \end{thebibliography} \bigskip \noindent{\sc Carmen J. Vanegas} \\ Department of Mathematics \\ Universidad Sim\'{o}n Bol\'{\i}var \\ Valle de Sartenejas-Edo Miranda \\ P O Box 89000, Venezuela \\ e-mail address: cvanegas@usb.ve \end{document}