Full Text
UDC 517.9.46.2
MATHEMATICS
R. S. SAKS
ON A NONELLIPTIC DIRICHLET PROBLEM
(Presented by Academician S. L. Sobolev on 20 II 1967)
Consider an elliptic system of two differential equations with real coefficients, written in complex form as a single equation
\[ w_{\bar z \bar z}-A_1\bar w_z-A_2w_z-A_3\bar w_{\bar z}-A_4w_{\bar z}-A_5\bar w-A_6w=F, \tag{1} \]
where \(w=u_1+iu_2\); \(\bar w=u_1-iu_2\); \(\partial/\partial z=\frac12(\partial/\partial x-i\partial/\partial y)\), \(\partial/\partial \bar z=\frac12(\partial/\partial x+i\partial/\partial y)\); \(F(z,\bar z)\), \(A_j(z,\bar z)\) \((j=1,\ldots,6)\) are analytic functions of the variables \(z\) and \(\bar z\) in some cylindrical domain \((T,T^*)\), \(T\) being some domain of the plane \(z=x+iy\), and \(T^*\) the reflection of the domain \(T\) with respect to the real axis.
Every twice continuously differentiable solution of equation (1) in the domain \(T\) is an analytic function \((^{1,2})\).
For equation (1) the first boundary-value problem is studied in the following formulation:
Find a solution of equation (1) in a bounded simply connected domain \(D\subset T\) with boundary \(L\in A^{(2,\alpha)}\), belonging to the class \(C^2(D)\cap C^{(1,\alpha)}(D+L)\), under the condition
\[ w|_L=f, \tag{2} \]
where \(f\) is a given complex function of class \(C^{(k,\alpha)}(a)\), and \(k\ge 2\).
In the absence of lower-order terms \((A_j=0,\ j=1,\ldots,6)\), problem (2)—(1) is not Noetherian.* For example, in the disk \(|z-z_0|\le R\) the homogeneous problem has an infinite number of linearly independent solutions \((^3)\). However, problem (1)—(2) becomes Noetherian when terms containing \(\bar w\) are added.
Theorem 1. The Dirichlet problem for the equation
\[ \partial^2 w/\partial \bar z^2-A_1(z,\bar z)\,\partial\bar w/\partial z=0 \tag{3} \]
is Noetherian if
\[ A_1(z,\bar z)\ne 0\quad \text{on } L, \]
and the index of the problem is equal to
\[ \frac{1}{\pi}\,[\arg A_1(z,\bar z)]_L-2, \]
where \([\ ]_L\) denotes the increment of the argument of the function \(A_1(z,\bar z)\) when the domain \(D\) is traversed once in the positive direction.
Proof. Equation (3) is equivalent to the system
\[ \partial w/\partial \bar z-u=0, \]
\[ \partial u/\partial \bar z-A_1\bar u=0, \tag{4} \]
* Here problem (1)—(2) is called Noetherian if the homogeneous problem has a finite number of linearly independent solutions, and the nonhomogeneous problem is solvable subject to a finite number of orthogonality conditions on the function \(f\).
which is equivalent to the system of Volterra integral equations
\[ w-\int_{0}^{\bar z} u(z,\tau)\,d\tau=\psi(z),\qquad u-\int_{0}^{\bar z} A_{1}(z,\tau)\,\overline{u(\bar z,\bar\tau)}\,d\tau=\varphi''(z), \tag{5} \]
where \(\psi(z)\) and \(\varphi''(z)\) are arbitrary functions holomorphic in the domain \(D\), with \(\varphi(0)=\varphi'(0)=0\).
Solving the system (5), we obtain a general representation of an arbitrary solution, regular in the domain \(D\), of equation (3) in terms of two arbitrary holomorphic functions \(\varphi(z)\) and \(\psi(z)\):
\[ \begin{aligned} w={}&\psi+z\varphi''+A_{0}\bar\varphi+B\varphi'+C\varphi+{}\\ &+\int_{0}^{z} F_{1}(z,\bar z,t)\varphi(t)\,dt +\int_{0}^{\bar z} F_{2}(z,\bar z,\tau)\overline{\varphi(\bar\tau)}\,d\tau, \end{aligned} \tag{6'} \]
where \(B\), \(C(z,\bar z)\), \(F_{1}(z,\bar z,t)\), \(F_{2}(z,\bar z,\tau)\) are analytic functions of their arguments and are expressed through the coefficient \(A_{1}\); moreover, from (5) it follows that if \(w\in C^{(1,\alpha)}(D+L)\), then \(\varphi(z)\in C^{(2,\alpha)}(D+L)\), \(\psi(z)\in C^{(0,\alpha)}(D+L)\). Using the integral representations
\[ \varphi(z)=\frac{1}{\pi i}\int_{L}\frac{\mu_{1}(t)\,dt}{t-z}+iC_{1},\qquad \psi(z)=\frac{1}{\pi i}\int_{L}\frac{\mu_{2}(t)\,dt}{t-z}+iC_{2}, \tag{7} \]
where \(\mu_{1},\mu_{2}\) are real densities \(\mu_{1}(t)\in C^{(2,\alpha)}(L)\), \(\mu_{2}\in C^{(0,\alpha)}(L)\), and \(C_{1},C_{2}\) are real constants \({}^{(4)}\), the solution of equation (3) can be written in the form
\[ \begin{aligned} w(z,\bar z)={}& \frac{1}{\pi i}\int_{L}\frac{\mu_{2}(t)+B(t,\bar t)\mu_{1}'(t)+\bar t\mu_{1}''}{t-z}\,dt -\frac{1}{\pi i}\int_{L}\frac{\bar t-\bar z}{t-z}\mu_{1}'(t)\,dt \\ &-\frac{1}{\pi i}\int_{L}\frac{B(z,\bar t)-B(z,\bar z)}{t-z}\mu_{1}'(t)\,dt -\frac{A_{1}(z,\bar z)}{\pi i}\int_{L}\frac{\overline{\mu_{1}(t)}\,d\bar t}{\bar t-\bar z} +\frac{C}{\pi i}\int_{L}\frac{\mu_{1}\,dt}{t-z} \\ &+\frac{1}{\pi i}\int_{L}\mu_{1}(t)\int_{0}^{z}\frac{F_{1}(z,\bar z,\xi)\,d\xi}{t-\xi}\,dt +\frac{1}{\pi i}\int_{L}\mu_{1}(\bar t)\int_{0}^{\bar z}\frac{F_{2}(z,\bar z,\eta)\,d\eta}{\bar t-\eta}\,d\bar t +iC_{2}. \end{aligned} \tag{8} \]
The first integral in formula (8) is a holomorphic function of class \(C^{(0,\alpha)}(D+L)\) and can be represented by a Cauchy-type integral with real density \(\nu(t)\in C^{(0,\alpha)}(L)\) in the form
\[ \frac{1}{\pi i}\int_{L}\frac{\mu_{2}(t)+B(z,\bar t)\mu_{1}'+\bar t\mu_{1}''}{t-z}\,dt+iC_{2} = \frac{1}{\pi i}\int_{L}\frac{\nu(t)\,dt}{t-z}+iC_{3}, \tag{9} \]
where \(\nu(t)\) and \(C_{3}\) are uniquely determined through \(\mu_{1},\mu_{2},C_{2}\). Conversely, knowing \(\mu_{1},\nu,C_{3}\), one can uniquely determine \(\mu_{2},C_{2}\).
Substituting (9) into formula (8), we obtain a representation of an arbitrary solution \(w\) of problem (2)—(3) in terms of two arbitrary real functions \(\mu_{1}(t)\) and \(\nu(t)\):
\[ \begin{aligned} w(z,\bar z)={}& \frac{1}{\pi i}\int_{L}\frac{\nu(t)\,dt}{t-z}+iC_{3} +\frac{C(z,\bar z)}{\pi i}\int_{L}\frac{\mu_{1}(t)\,dt}{t-z} -\frac{A_{1}(z,\bar z)}{\pi i}\int_{L}\frac{\overline{\mu_{1}(t)\,dt}}{\bar t-\bar z} \\ &-\frac{1}{\pi i}\int_{L}\frac{\bar t-\bar z}{t-z}\mu_{1}'(t)\,dt -\frac{1}{\pi i}\int_{L}\frac{B(z,\bar t)-B(z,\bar z)}{t-z}\mu_{1}'(t)\,dt \\ &+\frac{1}{\pi i}\int_{L}\mu_{1}(t)\int_{0}^{z}\frac{F_{1}(z,\bar z,\xi)}{t-\xi}\,dt +\frac{1}{\pi i}\int_{L}\mu_{1}(\bar t)\int_{0}^{\bar z}\frac{F_{2}(z,\bar z,\eta)\,d\eta}{\bar t-\eta}\,d\bar t. \end{aligned} \tag{10} \]
Passing to the limit as \(z\to t_0\in L\), we obtain a singular integral equation of the form
\[ \nu(t_0)+\frac{1}{\pi i}\int_L \frac{\nu(t)\,dt}{t-t_0} +A_1(t_0,\bar t_0)\mu_1(t_0) -\frac{A_1(t_0,\bar t_0)}{\pi i}\int_L\frac{\mu_1\,d\bar t}{\bar t-\bar t_0} + \]
\[ +\,C(t_0,\bar t_0)\mu_1(t_0) +\frac{C(t_0,\bar t_0)}{\pi i}\int_L \frac{\mu_1\,dt}{t-t_0} +\int_L K_1(t_0,\bar t_0,t)\mu_1(t)\,dt+ \]
\[ +\int_L K_2(t_0,\bar t_0,\bar t)\mu_1(\bar t)\,d\bar t =f(t_0)-iC_3, \tag{11} \]
where the kernels \(K_1\) and \(K_2\) have at \(t=t_0\) a singularity of order less than 1. Equating the real and imaginary parts of expression (11), we obtain a system of two equations with real coefficients. It is easy to see that, for \(A_1(t_0,\bar t_0)\ne 0\), this system is of normal type \((4)\).
Equation (11) is equivalent to problem (3)—(2), i.e., if problem (3)—(2) has a solution, then equation (11) is solvable, and, conversely, if equation (11) is solvable, then problem (3)—(2) has a solution.
Since \(\varphi(0)=\varphi'(0)=0\), we obtain the conditions on \(\mu_1(t)\)
\[ \frac{1}{\pi i}\int_L \frac{\mu_1(t)\,dt}{t}+iC_1=0, \qquad \frac{1}{\pi i}\int_L \frac{\mu_1(t)}{t^2}\,dt=0. \tag{12} \]
Problem (3)—(2) is equivalent to equation (11) and to condition (12), which is used only in computing the index. We note that for \(f\in C^{(2,\alpha)}(L)\) all Hölder-continuous solutions \((\mu_1,\nu)\) of equation (11), for \(A_1\ne 0\) on \(L\), belong to the class \(C^{(2,\alpha)}(L)\), and, according to formula (10), the solution of problem (2)—(3) belongs to the class \(C^{(1,\alpha)}(D+L)\), but it may fail to belong not only to \(C^{(2,\alpha)}(D+L)\), but even to \(C^{(1,\beta)}(D+L)\) for \(\beta>\alpha\). Indeed, in formula (10) all integrals, except the second, belong to \(C^{(2,\alpha)}(D+L)\), while the second integral
\[ \eta(x,y)=-\frac{1}{\pi i}\int_L \frac{\bar t-\bar z}{t-z}\,\mu_1''(t)\,dt \]
is a function of class \(C^{(1,\alpha)}(D+L)\), when \(\mu_1(t)\in C^{(2,\alpha)}(L)\) \((4)\).
For the disk \(|z|\le 1\), when \(A_1=\mathrm{const}\ne 0\), a more concrete result is obtained. The general representation (6) in this case has the form:
\[ w(z,\bar z)=\psi(z)+\bar z\varphi(z)+A_1\int_0^{\bar z}\overline{(z-t)\varphi(t)}\,dt+ \]
\[ +|A_1|^2\int_0^z \varphi(t)\left\{\int_0^z d\tau_1\int_0^{\tau_1} I_0\bigl(2|A_1|(z-t)^{1/2}(\tau_1-\tau)^{1/2}\bigr)\,d\tau\right\}dt+ \]
\[ +A_1|A_1|^2\int_0^{\bar z}d\xi\int_0^\xi d\tau \left[\int_0^z I_0\bigl(2|A_1|(z-t)^{1/2}(\xi-\tau)^{1/2}\bigr)\,dt\cdot \int_0^{\bar\tau}\overline{\varphi(\eta)}\,d\eta\right], \tag{6″} \]
where \(I_0\) is the Bessel function of zero order of imaginary argument.
Expanding the functions \(\varphi(z)\) and \(\psi(z)\) in Taylor series,
\[ \varphi(z)=\sum_{n=0}^{\infty} a_n z^n,\qquad \psi(z)=\sum_{n=0}^{\infty} b_n z^n, \]
and \(f(\theta)\) in a Fourier series,
\[ f(\theta)=\sum_{n=-\infty}^{+\infty} f_n e^{in\theta}, \qquad \text{where }\quad f_n=\frac{1}{2\pi}\int_0^{2\pi} f(\theta)e^{in\theta}\,d\theta, \]
and sub-
substituting into formula (6), for \(|z|=1\) we obtain a system of linear equations for determining the coefficients \(a_n, b_n,\ n \geqslant 0\)
\[ I_1(2|A_1|-1)a_0=f_1,\qquad A_1^{-1}I_2(2|A_1|)a_0=\overline{f}_{-2}, \]
\[ |A_1|^{-n-2}\overline{A}_1 n! I_{n+2}(2|A_1|)a_n=\overline{f}_{-(n+2)}\qquad (n\geqslant 1), \]
\[ b_n+a_{n+1}\bigl[|A_1|^{-n-2}(n+1)!I_n(2|A_1|)-(n+1)!|A_1|^{-2}-1\bigr]=f_n\quad (n\geqslant 0), \]
where
\[ I_n(2|A_1|)=\sum_{k=0}^{\infty}\frac{|A_1|^{2k+n}}{k!(k+n)!}, \]
whence we obtain:
Theorem 2. The homogeneous problem \((3')\)—(2) for \(A_1=\mathrm{const}\ne 0\) in the disk \(|z|\leqslant 1\) has only the trivial solution, and the nonhomogeneous problem is solvable under the condition
\[ I_2(2|A_1|)\int_0^{2\pi}\overline{f}(\theta)e^{i\theta}\,d\theta = \overline{A}_1\bigl(I_1(2|A_1|)-1\bigr) \int_0^{2\pi} f(\theta)e^{-2i\theta}\,d\theta. \]
By analogous methods one obtains:
Theorem 3. The Dirichlet problem for the equation
\[ w_{z\overline z}-(A_3\overline w)_{\overline z}=0 \tag{13} \]
is Noetherian if
\[ A_3(z,\overline z)\ne 0 \quad \text{on } L, \]
and the index \(\varkappa\) of the problem is equal to
\[ \frac{1}{\pi}\,[\arg A_3(z,\overline z)]_L+2. \]
Theorem 4. The homogeneous problem \((13)\)—(2) for \(A_3=\mathrm{const}\ne 0\) in the disk \(|z|\leqslant 1\) has two nontrivial linearly independent solutions over the field of real numbers, and the nonhomogeneous problem is unconditionally solvable for \(f\in C^{(2,\alpha)}(L)\).
Theorem 5. The Dirichlet problem for the equation
\[ w_{z\overline z}-A_5\overline w=0 \tag{14} \]
is Noetherian if \(A_5(z,\overline z)\ne 0\) on \(L\) and \(f\in C^{(3,\alpha)}(L)\), and the index of the problem is equal to
\[ \frac{1}{\pi}\,[A_5(z,\overline z)]_L. \]
It is easy to see that the Dirichlet problem for the equation \(w_{z\overline z}-A_4w_z=0,\ A_4=\mathrm{const}\ne 0\), in the disk \(|z|\leqslant 1\), is not Noetherian, since the general representation in this case has the form \(w=\psi(z)+\varphi(z)e^{A_4z}\), where \(\varphi,\psi\) are arbitrary holomorphic functions in the disk.
In conclusion I express my deep gratitude to Corresponding Member of the Academy of Sciences of the USSR A. V. Bitsadze and to N. E. Tovmasyan for their constant attention to my work.
Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Received
20 XII 1965
References
- I. G. Petrovskii, Matem. sborn., 5 (47), 1, 3 (1939).
- I. N. Vekua, Matem. sborn., 31 (73), 2, 219 (1952).
- A. V. Bitsadze, Boundary value problems for elliptic equations of second order, “Nauka,” 1966.
- N. I. Muskhelishvili, Singular integral equations, Moscow, 1962.
- I. N. Vekua, New methods for solving elliptic equations, Moscow, 1948.