V. S. Vinogradov

ON A BOUNDARY VALUE PROBLEM FOR LINEAR ELLIPTIC SYSTEMS OF FIRST-ORDER DIFFERENTIAL EQUATIONS IN THE PLANE

(Presented by Academician I. M. Vinogradov, 5 IX 1957)

Let us consider the system of differential equations

\[ \begin{aligned} a_{11}u_x+a_{12}u_y+b_{11}v_x+b_{12}v_y+c_{11}u+c_{12}v&=f_1;\\ a_{21}u_x+a_{22}u_y+b_{21}v_x+b_{22}v_y+c_{21}u+c_{22}v&=f_2. \end{aligned} \tag{1} \]

Here \(a_{ij}(x,y)\), \(b_{ij}(x,y)\) are bounded measurable functions satisfying the condition of uniform ellipticity

\[ 4 \begin{vmatrix} a_{11}&b_{11}\\ a_{21}&b_{21} \end{vmatrix} \cdot \begin{vmatrix} a_{12}&b_{12}\\ a_{22}&b_{22} \end{vmatrix} - \left\{ \begin{vmatrix} a_{11}&b_{12}\\ a_{21}&b_{22} \end{vmatrix} + \begin{vmatrix} a_{12}&b_{11}\\ a_{22}&b_{21} \end{vmatrix} \right\}^{2} \ge k>0, \]

\[ \begin{vmatrix} a_{11}&b_{11}\\ a_{21}&b_{21} \end{vmatrix} : \begin{vmatrix} b_{11}&b_{12}\\ b_{21}&b_{22} \end{vmatrix} >0, \qquad \begin{vmatrix} a_{12}&b_{12}\\ a_{22}&b_{22} \end{vmatrix} : \begin{vmatrix} b_{11}&b_{12}\\ b_{21}&b_{22} \end{vmatrix} >0 \tag{2} \]

in some simply connected, finite domain \(D\); \(c_{ij}(x,y)\), \(f_i(x,y)\) belong to \(L_p(D)\) for \(p>2\).

For the system (1) we shall solve the boundary value problem

\[ \alpha(t)u(t)+\beta(t)v(t)\big|_{\Gamma}=0, \tag{3} \]

where \(\Gamma\) is the boundary of the domain \(D\); \(\alpha(t)\), \(\beta(t)\in \operatorname{Lip}(\nu,\Gamma)\), \(0<\nu<1\), are functions prescribed on \(\Gamma\), Hölder continuous, and \(\alpha^2+\beta^2=1\).

Introducing the notation

\[ z=x+iy,\qquad w=u+iv,\qquad \frac{\partial}{\partial \bar z} = \frac12\left(\frac{\partial}{\partial x} +i\frac{\partial}{\partial y}\right), \]

\[ \frac{\partial}{\partial z} = \frac12\left(\frac{\partial}{\partial x} -i\frac{\partial}{\partial y}\right), \]

we can reduce the problem (1)—(3) to the form

\[ \frac{\partial w}{\partial \bar z} +\mu_1(z)\frac{\partial w}{\partial z} +\mu_2(z)\frac{\partial \overline{w}}{\partial \bar z} +a(z)w(z)+b(z)\overline{w(z)} =g(z), \tag{4} \]

\[ \operatorname{Re}\{(\alpha-i\beta)w\}\big|_{\Gamma}=0. \tag{5} \]

It follows from (2) that

\[ |\mu_1(z)|+|\mu_2(z)|\le q<1. \tag{6} \]

We shall seek the solution in the generalized sense of S. L. Sobolev, i.e. \(w(z)\in W_p^{(1)}(D)\), \(p>2\) \((^1)\).

For the system \(\partial w/\partial \bar z+aw+b\bar w=f\) this problem was solved by I. N. Vekua \((^2)\). The system (1) can be reduced to the indicated system in the case when \(\mu_1(z)\) and \(\mu_2(z)\) are differentiable \((^3)\). In the case when \(u(t)\) (or \(v(t)\)) is prescribed on \(\Gamma\), this problem was considered by Bers and Nirenberg \((^4)\).

We shall assume that \(\Gamma\) is a Jordan curve with continuous curvature. Then, with the aid of a conformal mapping \(\zeta=f(z)\), we can reduce the problem to the case when \(D\) is the disk \(|z|\le 1\), with \(0<c_1<|f'(z)|<c_2<\infty\) \((^{5,6})\) uniformly in \(D\).

Let us call the integer

\[ {}^{(2)}\ n=\frac{1}{2\pi}\{\arg(\alpha+i\beta)\}_{\Gamma} \]

the index of the problem; then the boundary condition can be rewritten in the form

\[ \operatorname{Re}\{t^{-n}e^{p(t)}w(t)\}_{\Gamma}=0, \tag{7} \]

where

\[ p(z)=q_1(z)+iq_2(z)=\frac{1}{2\pi}\int_{\Gamma} q_2(t)\frac{t+z}{t-z}\,ds, \tag{8} \]

\[ q_2(t)\big|_{\Gamma}=\arg[\alpha(t)+i\beta(t)]-n\arg t\in \operatorname{Lip}(\nu,\Gamma),\qquad 0<\nu<1. \]

If \(z\) approaches the boundary point \(t\), then \(p'(z)\) satisfies the condition \(|p'(z)|\le K|z-t|^{\nu-1}\) (7). Making the substitution \(w_1(z)=e^{p(z)}w(z)\), we bring the system and the boundary condition to the final form (for the new function and coefficients we keep the old notation)

\[ \frac{\partial w}{\partial \bar z} +\mu_1(z)\frac{\partial w}{\partial z} +\mu_2(z)\frac{\partial \bar w}{\partial z} +a(z)w+b(z)\bar w=g(z), \tag{9} \]

\[ \operatorname{Re}\{t^{-n}w(t)\}_{\Gamma}=0. \tag{10} \]

I. The case of nonnegative index, \(n\ge 0\).

From the results of (8,9) it follows that any function \(w(z)\in W_p^{(1)}(D)\), \(p>2\), satisfying condition (10) for \(n\ge 0\), can be represented in the form

\[ w(z)=-\frac{1}{\pi}\iint_D\left[\frac{\rho(\zeta)}{\zeta-z} +\frac{z^{2n+1}\overline{\rho(\zeta)}}{1-z\bar\zeta}\right]\,dT_{\zeta} + \]

\[ +a_0+\cdots+a^{\,n-1}z^{n-1}+icz^n-\bar a_{n-1}z^{n+1}\cdots-\bar a_0 z^{2n} =T_1\rho+\psi(z); \tag{11} \]

\(\rho(\zeta)\in L_p(D)\); \(a_0,\ldots,a_{n-1}\) are arbitrary complex numbers; \(c\) is an arbitrary real number. We shall regard \(L_p(D)\) and \(W_p^{(1)}(D)\) as spaces of complex functions over the field of real numbers.

Following the method proposed by I. N. Vekua (10), we substitute the expression for \(w(z)\) (11) into equation (9); then we obtain the singular integral equation

\[ \rho+\mu_1S_1\rho+\mu_2\bar S_1\rho+aT_1\rho+b\bar T_1\rho = g-\mu_1\psi'(z)-\mu_2\overline{\psi'(z)}-a\psi(z)-b\bar\psi(z), \tag{12} \]

where

\[ S_1\rho=\frac{\partial}{\partial z}T_1\rho = -\frac{1}{\pi}\iint_D\left[ \frac{\rho(\zeta)}{(\zeta-z)^2} +\frac{z^{2n+1}\bar\zeta}{(1-z\bar\zeta)^2}\overline{\rho(\zeta)} +(2n+1)z^{2n}\frac{\overline{\rho(\zeta)}}{1-z\bar\zeta} \right]\,dT_{\zeta}. \]

The singular operator

\[ S\rho=-\frac{1}{\pi}\iint_D\left[ \frac{\rho(\zeta)}{(\zeta-z)^2} +\frac{z^{2n+1}\bar\zeta}{(1-z\bar\zeta)^2}\overline{\rho(\zeta)} \right]\,dT_{\zeta} \]

is a bounded linear operator in \(L_p(D)\), \(1<p<\infty\). \(\|S\rho\|_{L_p}\le A_p\|\rho\|_{L_p}\), \(A_p\) depends continuously on \(p\). It may be calculated that \(A_2=1\). Therefore we can choose the number \(p\) from the interval \(2<p<2+\varepsilon\) so that \(qA_p<1\). Then the operator \(\rho+\mu_1S\rho+\mu_2\bar S\rho\) has an inverse; hence equation (12) is equivalent to a Fredholm equation.

We shall prove that the homogeneous equation (12) \((g(z)=\psi(z)=0)\) has no solutions different from zero. Let \(\rho(z)\) be such a solution. Then \(w(z)=T_1\rho\) satisfies the homogeneous equation (9) \((g(z)=0)\) and, by the known theorem on the representation of solutions (4, 13–15),

\[ w(z)=e^{S_0(z)}f[\chi(z)], \tag{13} \]

where \(\chi(z)\) is a homeomorphism of the disk \(|z|<1\) onto itself; \(f(\zeta)\) is a function analytic in the disk \(|\zeta|<1\); \(S_0(z)\), \(\chi(z)\), \(\psi(\zeta)=\chi^{-1}(\zeta)\in W_p^{(1)}(D)\), \(p>2\), and therefore satisfy the Hölder condition with exponent \(\frac{p-2}{p}\) (10), \(\operatorname{Im}S_0(z)\big|_{\Gamma}=0\).

It follows from (3) that the mapping \(\zeta=\chi(z)\) preserves orientation. Substituting (13) into (10), we obtain a boundary-value problem for the analytic function \(f(\zeta)\)

\[ \left.\operatorname{Re}\left\{\frac{e^{S_0(z)} f[\chi(z)]}{z^n}\right\}\right|_{\Gamma}=0 . \tag{14} \]

Solving this problem, we obtain for \(w(z)\) the expression

\[ \begin{aligned} w(z)=e^{S_0(z)+np[\chi(z)]}\bigl[a_0+a_1\chi(z)+\cdots+a_{n-1}\chi^{\,n-1}(z) +{}&\\ {}+ic\chi^n(z)-\cdots-\bar a_0\chi^{2n}(z)\bigr]; \end{aligned} \tag{15} \]

\(\rho(\zeta)\) is a function analytic in \(D\).

In view of (10), \(z^{-n}w(z)\) is a purely imaginary function; therefore on \(\Gamma\) it can be represented as the difference \(\Phi(z)-\overline{\Phi}(z)\), where \(\Phi(z)\) is analytic in the disk \(|z|\leqslant 1\). From the expression for \(T_1\rho\) it follows that \(\int_L z^{-k}w(z)\,ds=0,\ k=0,\ldots,2n\). Hence
\[ \left.w(z)\right|_{\Gamma}=\left.\operatorname{Im}\{z^{n+1}\Phi_1(z)\}\right|_{\Gamma}. \]
\(\Phi_1(z)\) is analytic in \(D\), but then \(w(z)\) has on \(\Gamma\) at least \(2(n+1)\) zeros. Therefore it follows from (15) that \(w(z)=0\) and \(\rho(z)=\partial w/\partial \bar z=0\).

Thus, we have proved the following theorem:

Theorem 1. In the case of a nonnegative index, the nonhomogeneous boundary-value problem (9), (10) is always solvable, and the corresponding homogeneous problem has \(2n+1\) linearly independent solutions.

II. The case of negative index, \(n<0\). Denote \(m=-n\). As in the case \(n\geqslant 0\), from \((8,9)\) there follows the representation for functions satisfying condition (10):

\[ w(z)=-\frac{1}{\pi}\iint_D\left[\frac{\rho(\zeta)}{\zeta-z}+ \frac{\bar\zeta^{\,2m-1}\rho(\bar\zeta)}{1-z\bar\zeta}\right]\,dT_\zeta =T_2\rho, \tag{16} \]

where \(\rho(\zeta)\in L_p(D)\) and satisfies \(2m-1\) conditions

\[ f_l(\rho)=\operatorname{Re}\left\{\iint_D \left[\zeta^{m-l-1}\rho(\zeta)+\bar\zeta^{\,m+l-1}\rho(\bar\zeta)\right]\,dT_\zeta\right\}=0, \]

\[ f_0(\rho)=\operatorname{Re}\left\{\iint_D \zeta^{m-1}\rho(\zeta)\,dT_\zeta\right\}=0, \tag{17} \]

\[ f_{-l}(\rho)=\operatorname{Im}\left\{\iint_D \left[\zeta^{m-l-1}\rho(\zeta)+\bar\zeta^{\,m+l-1}\rho(\bar\zeta)\right]\,dT_\zeta\right\}=0, \quad l=1,\ldots,m-1. \]

The functions \(\rho(\zeta)\) satisfying conditions (17) form a subspace \(L_{p,2m-1}(D)\) of the space \(L_p(D)\), which has index \(2m-1\). Therefore any element of \(L_p(D)\) can be represented in the form

\[ \rho(\zeta)=\rho_{2m-1}+\sum_{-(m-1)}^{m-1}\lambda_k\tau_k, \]

where \(\rho_{2m-1}\in L_{p,2m-1}(D)\); the \(\tau_k\) are linearly independent and \(\bar\tau_k\in L_{p,2m-1}(D)\). As \(\tau_k\) one may take
\[ \tau_l=\bar z^{\,m-l-1},\quad \tau_0=\bar z^{\,m-1},\quad \tau_{-l}=i\bar z^{\,m-l-1},\quad l=1,\ldots,m-1. \]

Substituting (16) into (9), we obtain the singular integral equation

\[ \rho+\mu_1S_2\rho+\mu_2\overline{S_2\rho}+aT_2\rho+b\overline{T_2\rho}=g, \]

\[ S_2\rho=\frac{\partial T_2\rho}{\partial z} =-\frac{1}{\pi}\iint_D\left[\frac{\rho(\zeta)}{(\zeta-z)^2}+ \frac{\bar\zeta^{\,2m}\rho(\bar\zeta)}{(1-z\bar\zeta)^2}\right]\,dT_\zeta . \tag{18} \]

A direct calculation shows that \(\|S_2\|_{L_2}=1\). Therefore equation (18) is equivalent to a Fredholm equation. Let us show that the corresponding homogeneous equation has no solutions different from zero. Let \(\rho(z)\) be such a solution,

\[ \rho(z)=\rho_{2m-1}+\sum_{-(m-1)}^{m-1}\lambda_k\tau_k, \]

\[ w(z)=T_2\rho=T_2\rho_{2m-1} +i\sum_{1}^{m-1}\lambda_{-k}\frac{z^{m-k}}{m-k} +\sum_{0}^{m-1}\lambda_k\frac{\bar z^{m-k}}{m-k}. \]

Using, as before, the theorem on the representation of solutions \((4,^{13-15})\), we shall have

\[ \operatorname{Re}\{t^m w(t)\}\big|_{\Gamma} = \operatorname{Re}\{t^m e^{S_0(z)}f[\chi(z)]\}\big|_{\Gamma} = \operatorname{Re}\left\{ \frac{\lambda_0}{m} + \sum_{k=1}^{m-1}\frac{\lambda_k+i\lambda_{-k}}{m-k}z^k \right\}. \tag{19} \]

Considering the inverse homeomorphism \(\psi(\zeta)=\chi^{-1}(\zeta)\) \((\psi(\zeta)|_{\Gamma}=\zeta e^{i\theta(\zeta)})\), we may write equality (19) in the form

\[ \left[ e^{S_0[\psi(\zeta)]-mq(\zeta)} \operatorname{Re}\{\zeta^m e^{mp(\zeta)}f(\zeta)\} \right]\big|_{\Gamma} = \operatorname{Re}\left\{ \frac{\lambda_0}{m} + \sum_{1}^{m-1} \frac{\lambda_k+i\lambda_{-k}}{m-k}z^k \right\}\bigg|_{\Gamma}; \]

\(p(\zeta)\) is a function analytic in \(D\), with \(\operatorname{Im}p(\zeta)|_{\Gamma}=\theta(\zeta)\), \(q(\zeta)=\operatorname{Re}p(\zeta)\). From the fact that \(\operatorname{Re}\{\zeta^m e^{mp(\zeta)}f(\zeta)\}\) has on \(\Gamma\) not fewer than \(2m\) zeros, it follows that \(\lambda_k=0\), \(k=0,\pm1,\ldots,\pm(m-1)\). Since \(m>0\), \(f(\zeta)=0\). Consequently, \(\rho=\partial w/\partial \bar z=0\). Thus the theorem has been proved:

Theorem 2. In the case of a negative index, the homogeneous problem has only the zero solution, and for the solvability of the nonhomogeneous problem it is necessary and sufficient that

\[ \rho(z)=[I+\mu_1S_2+\mu_2\overline{S}_2+aT_2+b\overline{T}_2]g \]

satisfy conditions (17).

In conclusion I express my gratitude to my adviser I. N. Vekua for valuable advice and guidance in carrying out this work.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
4 IX 1957

CITED LITERATURE

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