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UDC 517.949.2
MATHEMATICS
M. L. RASULOV
THE FUNDAMENTAL MATRIX OF AN ELLIPTIC SYSTEM OF SECOND ORDER WITH A COMPLEX PARAMETER
(Presented by Academician I. N. Vekua on 12 XII 1969)
The fundamental matrix for a general elliptic system without a parameter was constructed in the work \((^1)\), whose method was also applied in the case of an elliptic system with a real parameter \((^2)\).
In connection with the application of the contour-integral method \((^3)\) to the solution of parabolic problems, the present note is devoted to the construction and estimation of the fundamental matrix for the system
\[ A(x)\Delta u+\sum_{i=1}^{3} A_i(x)\frac{\partial u}{\partial x_i}+(A_0(x)-\lambda^2)u=0, \tag{1} \]
considered in a three-dimensional domain \(D\) of Euclidean space, where \(A(x), A_i(x)\) \((i=0,1,2,3)\) are square matrices of order \(m\).
It is assumed that in the closed domain \(D\) the matrices \(A(x), A_i(x)\) \((i=0,1,2,3)\) have continuous first-order derivatives with respect to all their arguments and that the roots \(\nu_i\) \((i=1,2,\ldots,p)\) of the characteristic equation
\[ \Delta(1,\nu)=\det(A(x)+\nu E)=0 \tag{2} \]
have constant multiplicity \(m_i\) and strictly negative real parts*
\[ \operatorname{Re}\nu_i(x)<0, \tag{3} \]
where \(\Delta(\beta,\nu)=\det(\beta A(x)+\nu E)\), and \(E\) is the identity matrix of order \(m\).
The fundamental matrix \(\widetilde P_0(x-\xi,\xi,\lambda)\), with singularity at the point \(x=\xi\), for the system
\[ \sum_{i=1}^{3} A(\xi)\frac{\partial^2 u}{\partial x_i^2}-\lambda^2 u=0 \]
is constructed in finite form
\[ P_0(x-\xi,\xi,\lambda)=\bigl(P_{0ks}(x-\xi,\xi,\lambda)\bigr)_{k,s=1}^{m}, \tag{4} \]
where the elements \(P_{0ks}(x-\xi,\xi,\lambda)\) admit the representations
\[ \begin{aligned} P_{0ks}(x-\xi,\xi,\lambda) &=\frac{1}{4\pi |x-\xi|} \sum_{i=1}^{p}\sum_{j=1}^{m_i} \frac{B_{sk}^{(i,j)}(\xi)(m_i-j)!}{(-\nu_i(\xi))^{m_i-j+1}} \\ &\quad\times \left\{ \frac{1}{(m_i-j)!} \exp\left[-\lambda\frac{|x-\xi|}{\sqrt{-\nu_i(\xi)}}\right] + \sum_{r=1}^{m_i-j} \frac{1}{(r!)^2(m_i-j-r)!} \left[ \left(-\frac{|x-\xi|}{2\sqrt{-\nu_i(\xi)}}\right)^r \lambda^r + \right.\right. \end{aligned} \]
* The fundamental matrices for such an equation in the case of a constant matrix \(A\) of second and third orders were constructed in \((^{4,5})\).
\[ + \sum_{q=1}^{r}(-1)^q\left(\frac{|x-\xi|}{2\sqrt{-v_i(\xi)}}\right)^{r-q} \sum_{j_q=1}^{r-1}\sum_{j_{q-1}=q-1}^{j_q-1}\cdots \sum_{j_1=1}^{j_2-1} \left(\frac{j_{\nu+1}+\nu}{2}\right)\lambda^{r-q} \right] \times \]
\[ \times \exp\left(-\lambda\frac{|x-\xi|}{\sqrt{-v_i(\xi)}}\right); \tag{5} \]
\[ |x-\xi| \text{ is the length of the vector } x-\xi, \]
\[ B_{sk}^{(i,j)}(\xi)= \frac{1}{(j-1)!}\, \frac{\partial^{j-1}}{\partial v^{j-1}} \left. \frac{\Delta_{sk}(1,v)} {\displaystyle\prod_{\substack{r=1\\ r\ne i}}^{p}(v-v_r(\xi))} \right|_{v=v_i(\xi)} \qquad (j=1,\ldots,m_i), \tag{6} \]
\(\Delta_{ks}(\beta,\gamma)\) is the cofactor of the element \((k,s)\) in the determinant \(\Delta(\beta,\gamma)\).
Formula (5) can be written in a more transparent form:
\[ P_{0ks}(x-\xi,\xi,\lambda)= \]
\[ = -\frac{1}{4\pi |x-\xi|} \sum_{i=1}^{p}\sum_{j=1}^{m_i} \frac{B_{sk}^{(i,j)}(\xi)}{(m_i-j+1)!} \frac{\partial^{m_i-j}}{\partial v^{m_i-j}} \left. \frac{ \exp\left[-\lambda\frac{|x-\xi|}{\sqrt{-v}}\right] }{v} \right|_{v=v_i(\xi)} . \tag{7} \]
In particular, if all roots \(v_i(x)\) of the characteristic equation (2) are simple \((m_i=1;\ i=1,2,\ldots,m=p)\), then from (7), taking (6) into account, we have
\[ P_{0ks}(x-\xi,\xi,\lambda)= -\frac{1}{4\pi |x-\xi|} \sum_{i=1}^{m} \frac{\Delta_{sk}(1,v_i(\xi))} {\displaystyle\prod_{\substack{r=1\\ r\ne i}}^{p}(v_i(\xi)-v_r(\xi))} \exp\left[-\lambda\frac{|x-\xi|}{\sqrt{-v_i(\xi)}}\right]. \tag{8} \]
According to the condition on the roots \(v_i(\xi)\) of the characteristic equation, the inequalities (3) hold, and, consequently, there exists a positive number \(\delta\) such that for all \(x,\xi\in \overline{D}\) the estimates
\[ \left| \frac{\partial^k P_0(x-\xi,\xi,\lambda)}{\partial x_i^k} \right| \le \frac{CB\exp(-\varepsilon|\lambda||x-\xi|)} {|x-\xi|^{k+1}} \qquad (k=0,1,2); \tag{9} \]
\[ \left| \frac{\partial P_0(x-\xi,\xi,\lambda)}{\partial \xi_i} \right| \le \frac{CB\exp(-2\varepsilon|\lambda||x-\xi|)} {|x-\xi|^2}; \tag{10} \]
\[ \frac{\partial^k}{\partial x_i^k} \left( \frac{\partial P_0(x-\xi,\xi,\lambda)}{\partial x_i} + \frac{\partial P_0(x-\xi,\xi,\lambda)}{\partial \xi_i} \right) \le \frac{CB\exp(-\varepsilon|\lambda||x-\xi|)} {|x-\xi|^{1+k}} \qquad (k=0,1), \tag{11} \]
where \(\lambda\) is any value from the domain \(R_\delta\):
\[ |\lambda|\ge R,\qquad |\arg \lambda|\le \pi/4+\delta; \tag{\(R_\delta\)} \]
\(C,R\) are sufficiently large positive constants; \(B\) is a square matrix of order \(m\), composed of ones; \(\varepsilon\) is some positive constant; inequalities (9)—(11) hold between the corresponding elements of the left- and right-hand sides.
The fundamental matrix \(P(x,\xi,\lambda)\) of system (1), with a singularity at the point \(x=\xi\), is constructed by the Levi—Carleman method \(({}^{6,7})\) in the form
\[ P(x,\xi,\lambda) = P_0(x-\xi,\xi,\lambda) + \int_D P_0(x-\eta,\eta,\lambda)\,h(\eta,\xi,\lambda)\,dD_\eta, \tag{12} \]
where \(h(\eta,\xi,\lambda)\) is the unknown density of the integral correction
\[ P_1(x,\xi,\lambda)= \int_D P_0(x-\eta,\eta,\lambda)\,h(\eta,\xi,\lambda)\,dD_\eta. \tag{13} \]
Substituting (12) into the left-hand side of equation (1) and (taking into account (9)—(11)) equating the resulting expression to zero, we arrive at the integral equation
\[ h(x,\xi,\lambda)=K(x,\xi,\lambda)+\int_D K(x,\eta,\lambda)h(\eta,\xi,\lambda)\,dD_\eta, \tag{14} \]
where
\[ K(x,\xi,\lambda)=\left\{(A(x)-A(\xi))\Delta_x+\sum_{i=1}^{3}A_i(x)\frac{\partial}{\partial x_i}+A_0(x)\right\}P_0(x-\xi,\xi,\lambda). \]
According to (9), under the restrictions imposed, the following inequality holds for the kernel:
\[ |K(x,\xi,\lambda)|\le CB\exp(-2\varepsilon|\lambda||x-\xi|)/|x-\xi|^2, \tag{15} \]
which is satisfied in the domain \(R_\delta\).
Estimate (15) makes it possible to construct a solution \(h(x,\xi,\lambda)\) of the integral equation (14) in the domain \(R_\delta\) by the method of successive approximations,
\[ h(x,\xi,\lambda)=K(x,\xi,\lambda)+\sum_{n=2}^{\infty}K_n(x,\xi,\lambda), \tag{16} \]
where \(K_n\) are the iterations of the kernel \(K\); moreover, for \(h\) the estimate
\[ |h(x,\xi,\lambda)|\le CB\exp(-\varepsilon|\lambda||x-\xi|)/|x-\xi|^2 \tag{17} \]
holds in the domain \(R_\delta\).
Thus one proves the
Theorem. Under the restrictions imposed, there exists a positive number \(\delta\) such that, for sufficiently large \(R\), for all complex values of \(\lambda\) satisfying the inequalities \((R_\delta)\), the system [1] has a fundamental matrix \(P(x,\xi,\lambda)\) of the form (12), analytic in \(\lambda\) in the domain \(R_\delta\), where for \(x,\xi\in\overline D\) the inequalities (9), (11) and the estimates
\[ |P_1(x,\xi,\lambda)|\le CB\exp(-\varepsilon|\lambda||x-\xi|)/\lambda|x-\xi|, \]
\[ |\partial^k P_1(x,\xi,\lambda)/\partial x_i^k| \le CB\exp(-\varepsilon|\lambda||x-\xi|)/|x-\xi|^k \quad (k=1,2), \]
\[ \left| \frac{\partial^k}{\partial x_i^k}\left[ \frac{\partial P(x,\xi,\lambda)}{\partial x_i} + \frac{\partial P_0(x-\xi,\xi,\lambda)}{\partial \xi_i} \right]\right| \le \frac{CB\exp(-\varepsilon|\lambda||x-\xi|)}{|x-\xi|^{1+k}}, \]
\[ |\partial^k P(x,\xi,\lambda)/\partial x_i^k| \le CB\exp(-\varepsilon|\lambda||x-\xi|)/|x-\xi|^{1+k} \quad (k=0,1,2). \]
If \(\Phi(x)\) is a vector-function having in \(D\) bounded continuous first-order derivatives with respect to all its arguments, then the vector-function
\[ u(x,\lambda,\Phi)=-\int_D P(x,\xi,\lambda)\Phi(\xi)\,dD_\xi \]
for all \(\lambda\) belonging to \(R_\delta\) is a solution of the nonhomogeneous equation
\[ A(x)\Delta u+\sum_{i=1}^{3}A_i(x)\frac{\partial u}{\partial x_i}+(A_0(x)-\lambda^2)u=\Phi(x). \]
An analogous theorem also holds for the case of a domain \(D\) of an arbitrary number of dimensions greater than one, with the corresponding complications of the formulas and estimates given, and \(P_{0ks}(x-\xi,\xi,\lambda)\) is expressed in terms of a Bessel function.
Azerbaijan State University
named after S. M. Kirov
Baku
Received
26 XI 1969
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