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MATHEMATICS
E. V. ZOLOTAREVA
A NECESSARY AND SUFFICIENT CONDITION FOR THE FREDHOLMNESS OF THE DIRICHLET PROBLEM FOR A CERTAIN CLASS OF ELLIPTIC SYSTEMS
(Presented by Academician S. L. Sobolev on 13 III 1962)
Consider a system of linear partial differential equations of second order of the form
\[ A u_{xx}+2B u_{xy}+C u_{yy}=0, \tag{1} \]
where \(A=\|A_{ik}\|\), \(B=\|B_{ik}\|\), \(C=\|C_{ik}\|\) are given real constant square matrices of order \(n\), and \(u(x,y)=(u_1(x,y),u_2(x,y),\ldots,u_n(x,y))\) is the unknown vector.
System (1) is called elliptic in the sense of Petrovskii if all \(2n\) roots of the characteristic equation
\[ \det |A+2B\lambda+C\lambda^2|=0 \tag{2} \]
are complex.
A general representation of solutions regular in a finite simply connected domain \(D\) of system (1) was obtained by A. V. Bitsadze \((^1)\).
In the present note we restrict ourselves to the case when (2) has one pair of complex-conjugate roots of multiplicity \(n\). Without loss of generality one may assume that the roots of (2) are \(\lambda=i\) and \(\bar\lambda=-i\). Then the general solution of system (1), following \((^1)\), is written in the form
\[ u(x,y)=\operatorname{Re}\sum_{l=1}^{n}\sum_{m=0}^{l-1} C_{lm} z^m \varphi_l^{(m)}(z), \tag{3} \]
where \(\varphi_l(z)\) are arbitrary holomorphic functions in the domain \(D\) of the variable \(z=x+iy\); the upper index \(m\) of the function \(\varphi_l(z)\) denotes the order of the derivative with respect to \(z\), and \(C_{lm}\) are constant vectors expressed exclusively in terms of the coefficients of system (1).
Following \((^2)\), we shall call system (1) weakly coupled if the vectors \(C_{l0}\) are linearly independent; otherwise system (1) is called strongly coupled. The \(C_{l0}\) are solutions of the following linear algebraic system of order \(n^2\):
\[ (A+2Bi-C)C_{10}=0, \]
\[ (A+2Bi-C)C_{20}+2(A+C)C_{10}=0, \tag{4} \]
\[ (A+2Bi-C)C_{k0}+2(A+C)C_{k-1,0}+(A-2Bi-C)C_{k-2,0}=0, \]
\[ k=3,\ldots,n. \]
All the remaining vectors \(C_{lk}\) are expressed linearly in terms of \(C_{l0}\).
The Dirichlet problem for system (1) consists in determining a solution of this system, regular in the domain \(D\), continuous up to the boundary \(\Gamma\), and satisfying the condition
\[ u=f(t) \quad \text{on } \Gamma, \]
where \(f(t)=(f_1(t),\ldots,f_n(t))\) is a given vector-function of a boundary point.
It is known that the Dirichlet problem for linear systems of elliptic type (in particular for system (1)), generally speaking, is not a problem of Fredholm type. The Fredholm alternative for the indicated problem always holds if the usual ellipticity requirement is strengthened by the condition of strong ellipticity (^3). However, the condition of strong ellipticity is only sufficient. There are examples of systems that are not strongly elliptic for which, nevertheless, the Dirichlet problem is always solvable and unique (^1). Therefore it is of interest to study the Dirichlet problem for elliptic systems that are not necessarily strongly elliptic.
In note (^4) the Dirichlet problem was considered in the case when \(n=2\) and the domain \(D\) is the unit disk.
In the present note we consider a bounded simply connected domain \(D\), whose boundary \(\Gamma\) satisfies the Lyapunov condition, and \(f(t)\) is a continuous vector-function satisfying a Hölder condition with exponent \(\leqslant 1\). The number of equations \(n\) is arbitrary.
Under the assumptions made above the following theorems are proved:
Theorem 1. If system (1) is weakly coupled, then: a) the homogeneous Dirichlet problem \((f(t)\equiv 0)\) has a finite number of linearly independent solutions; b) for solvability of the nonhomogeneous Dirichlet problem it is necessary to impose on \(f(t)\) as many additional conditions as there are solutions of the homogeneous Dirichlet problem.
Theorem 2. Weak coupling of system (1) is a necessary and sufficient condition for the Fredholm character of the Dirichlet problem *.
Since for system (1) the general solution (3) is known, the Dirichlet problem is nothing but a boundary value problem with respect to \(n\) unknown holomorphic functions \(\varphi_1(z),\ldots,\varphi_n(z)\). The most natural method for solving such a problem is to find a suitable integral representation of the unknown functions in the form of a Cauchy-type integral or analogous integrals. These expressions, substituted into the boundary conditions, give a system of integral equations; by investigating it one can draw a conclusion about the solvability of the Dirichlet problem.
In our case it is easy to show that in (3) \(\varphi_l(z)\) is \(H_{l-1}\)-holomorphic**. To prove Theorem 1 the following is used:
Lemma. Let \(\varphi_l(z)\) be \(H_{l-1}\)-holomorphic in \(D\). Then there exist a unique real-valued function \(\mu_l(t)\) and a unique real constant \(C_l\) such that the representation is valid
\[ \varphi_l(z)=\frac{1}{\pi i}\int_{\Gamma} \frac{ \mu_l(t)+\sum_{k=l+1}^{n}(-1)^{k-l}\dfrac{d^{\,k-l-1}}{dt^{\,k-l-1}} \left(\frac{\bar t^{\,k-l}}{(k-l)!}\,\mu'_k(t)\right) }{t-z}\,dt+iC_l, \tag{5} \]
where \(\mu_l(t)\) is \((l-1)\) times differentiable and \(\mu_l^{(l-1)}(t)\) satisfies a Hölder condition with exponent \(\leqslant 1\).
Starting from representation (5),
\[ \varphi_n(z)=\frac{1}{\pi i}\int_{\Gamma}\frac{\mu_n(t)}{t-z}\,dt+iC_n \]
and, regarding \(\mu_n(t)\) as found, we represent \(\varphi_{n-1}(z)\) as follows:
\[ \varphi_{n-1}(z)=\frac{1}{\pi i}\int_{\Gamma} \frac{\mu_{n-1}(t)-\bar t\,\mu'_n(t)}{t-z}\,dt+iC_{n-1} \]
and so on.
* Fredholmness of the Dirichlet problem is understood as formulated in Theorem 1.
** A function \(\varphi(z)\) is called \(H_m\)-holomorphic in \(D\) if it is continuous and satisfies a Hölder condition with exponent \(\leqslant 1\), together with its derivatives up to order \(m\) inclusive, in the closed domain \(\overline D\).
Thus, in the representation \(\varphi_l(z)\), in addition to the function \(\mu_l(t)\), there enter the functions \(\mu_{l+1}(t), \ldots, \mu_n(t)\), which are regarded as already known.
The Dirichlet problem is solved, with the aid of (5), by reduction to an equivalent system of singular integral equations with index \(\varkappa = 0\). This completes the proof of Theorem 1.
After this, in Theorem 2 it remains only to prove that, in the case of strong connectedness of system (1), the homogeneous Dirichlet problem has infinitely many solutions, while the nonhomogeneous Dirichlet problem is unsolvable.
In conclusion I express my deep gratitude to Corresponding Member of the Academy of Sciences of the USSR A. V. Bitsadze, whose attention I invariably enjoyed in carrying out this work.
Institute of Mathematics with Computing Center
of the Siberian Branch of the Academy of Sciences of the USSR
Received
6 III 1962
References
- A. V. Bitsadze, DAN, 112, No. 6, 983 (1957).
- A. V. Bitsadze, Equations of Mixed Type, 1959, p. 65.
- M. I. Vishik, Matem. sbornik, 29, No. 3, 615 (1951).
- E. V. Zolotareva, DAN, 132, No. 4, 751 (1960).