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UDC 517.919
MATHEMATICS
NGUYEN THYA HOP
ON THE NORMAL SOLVABILITY OF THE DIRICHLET PROBLEM FOR A SYSTEM OF ELLIPTIC TYPE OF A. V. BITSADZE
(Presented by Academician M. A. Lavrent’ev on 12 VII 1965)
Among elliptic boundary-value problems for which the Noether property is violated, the most typical is the Dirichlet problem for the system of A. V. Bitsadze
\[ \frac{\partial^2 u_1}{\partial x^2} - \frac{\partial^2 u_1}{\partial y^2} - 2\frac{\partial^2 u_2}{\partial x\,\partial y} = F_1(x,y), \]
\[ \frac{\partial^2 u_2}{\partial x^2} - \frac{\partial^2 u_2}{\partial y^2} + 2\frac{\partial^2 u_1}{\partial x\,\partial y} = F_2(x,y); \tag{1} \]
\[ u_1\big|_{\Gamma}=f_1(s),\qquad u_2\big|_{\Gamma}=f_2(s). \tag{2} \]
In the work \((^1)\) it was shown that the homogeneous problem \((1^0)—(2^0)\) \((F_1\equiv F_2\equiv 0,\ f_1\equiv f_2\equiv 0)\) has in a disk an infinite number of linearly independent solutions, while the nonhomogeneous problem for the same domain is normally solvable in the sense of Hausdorff \((^3)\).
The aim of the present work is to study the nature of the violation of the Noether property of problem (1)—(2) for a broad class of noncircular domains.
In complex form, problem (1)—(2) can be written as
\[ \partial^2 w/\partial \bar z^2 = F(z), \tag{1′} \]
\[ w\big|_{\Gamma}=f(s), \tag{2′} \]
where \(w=u_1+iu_2,\ F=F_1+iF_2,\ f=f_1+if_2\).
It is assumed that the derivative \(\partial^2/\partial\bar z^2\) is understood in the sense of S. L. Sobolev, \(F(z)\in L_p(\mathcal D)\), \(p>1\), \(\mathcal D\) is a simply connected domain with Lyapunov boundary \(\Gamma\), and the boundary condition \((2′)\) is satisfied almost everywhere, with \(f(s)\in L_2(\Gamma)\).
Without loss of generality, one may assume that \(F(z)\equiv 0\), i.e.
\[ \partial^2 u/\partial \bar z^2=0; \tag{I} \]
\[ u\big|_{\Gamma}=f(s). \tag{II} \]
This is easily verified by replacing the sought solution by
\[ w = u+ \frac{1}{\pi^2} \iint_{\mathcal D}\frac{d\xi_1\,d\eta_1}{t_1-z} \iint_{\mathcal D}\frac{F(\xi,\eta)}{t-t_1}\,d\xi\,d\eta . \]
The general solution of equation (I), as is known \((^2)\), has the form
\[ u(z)=\bar z\varphi(z)+\psi(z), \tag{3} \]
where \(\varphi(z)\), \(\psi(z)\) are arbitrary holomorphic functions in \(\mathcal D\).
Definition. A solution \(u(z)\) of equation (I) will be called regular if the functions \(\varphi(z)\) and \(\psi(z)\) in representation (3) belong to the class \(E_2(\mathcal D)\) \((^4)\).
We shall call problem D the problem of finding regular solutions in \(\mathscr D\) of equation (I), satisfying almost everywhere the boundary condition (II).
Below we study both the homogeneous problem \(D_0\) (\(f \equiv 0\)) and the nonhomogeneous problem D.
Problem (I)—(II) is easily reduced to a Fredholm integral equation of the first kind. Therefore the violation of Noetherianity for this problem and for a Fredholm integral equation of the first kind has the same character.
Let \(z(\omega)\) be a holomorphic function in the unit disk \(S\), continuous in the closed domain \(S+\gamma\); let \(\{\varphi_i(\omega)\}\) be the system of functions determined from the expansion of \(z(\omega)\) into a power series near zero
\[ z(\omega)=\sum_{k=0}^{i-1} a_k \omega^k+\omega^i\varphi_i(\omega). \tag{4} \]
Lemma 1. In the system \(\{\varphi_i(\omega)\}\), either all the functions are linearly independent, or among them there exists only a finite number \(m\) of linearly independent functions \(\varphi_1(\omega), \varphi_2(\omega), \ldots, \varphi_m(\omega)\). The latter case occurs if and only if \(z(\omega)\) is a rational function.
Let now \(z(\omega)\) be a conformal mapping of the unit disk \(|\omega|\le 1\) onto the domain \(\mathscr D\).
Theorem 1. Among simply connected domains bounded by analytic curves, the homogeneous problem \(D_0\) has nonzero regular solutions if and only if \(z(\omega)\) is a rational function.
This fact, under the assumption that \(\varphi(z), \psi(z) \in C^{(1,h)}(\mathscr D+\Gamma)\), was noted by N. E. Tovmasyan.
Along with problem (I)—(II), consider the homogeneous adjoint problem
\[ \partial^2 v/\partial z^2=0; \tag{I*} \]
\[ v|_{\Gamma}=0. \tag{II*} \]
Starting from Green’s formula
\[ \iint_{\mathscr D}\left[\bar v\,\frac{\partial^2 u}{\partial \bar z^2} -u\,\frac{\partial^2 \bar v}{\partial z^2}\right]\,dx\,dy = \frac{1}{2i}\int_{\Gamma}\left(\bar v\,\frac{\partial u}{\partial \bar z} -u\,\frac{\partial \bar v}{\partial z}\right)\,dz \]
for a solution \(u(z)\) of problem (I)—(II) and for any solution \(v(z)\) of the homogeneous adjoint problem (I)—(II), we have
\[ \int_{\Gamma} f\,\frac{\partial \bar v}{\partial z}\,dz=0. \tag{5} \]
Condition (5) is necessary for the solvability of problem (I)—(II). If condition (5) is also sufficient for the solvability of problem (I)—(II), then the latter problem is called Hausdorff. As a result of work [5] and Lemma 1 it follows
Theorem 2. For a simply connected domain bounded by a Lyapunov curve, problem (I)—(II) is Hausdorff if and only if \(z(\omega)\) is a rational function.
From Theorems 1 and 2 it follows
Theorem 3. For a simply connected domain bounded by a Lyapunov curve, problem (I)—(II) is not Noetherian.
Let \(\varphi_i(\omega)\) be the system of functions determined from (4), where \(z(\omega)\) is the indicated conformal mapping, and let \(\omega(z)\) be the function inverse to \(z(\omega)\). Denote by \(\Omega^{+}(s)\) the boundary value on \(\Gamma\) of any function of the class \(E_2(\mathscr D)\).
Theorem 4. In the case of the Hausdorff property of problem (I)—(II), for its solvability it is necessary and sufficient that the right-hand side \(f(s)\) in condition II have the form
\[ f(s)=\Omega^{+}(s)+\sum_{i=1}^{m} c_i \,\overline{\omega(s)}\, \overline{\varphi_i(\omega(s))}, \]
where \(c_i\) are arbitrary constants.
Novosibirsk State
University
Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Received
9 VII 1965
References
- A. V. Bitsadze, UMN, 3, no. 6 (28) (1948).
- A. V. Bitsadze, DAN, 59, no. 8 (1948).
- F. Hausdorff, On the Theory of Linear Equations in Metric Spaces, Theory of Sets, M.—L., 1937, p. 266.
- I. I. Privalov, Boundary Properties of Analytic Functions, 2nd ed., 1950.
- Nguyen Tkha Hop, DAN, 166, no. 4 (1966).