Abstract Generated abstract
The paper constructs and estimates the fundamental matrix for a second order elliptic system in a three-dimensional domain with a complex spectral parameter. Under smoothness assumptions on the matrix coefficients and a negativity condition on the real parts of the roots of the characteristic equation, an explicit fundamental matrix is first obtained for the frozen-coefficient principal system, including a simplified form when the characteristic roots are simple. The full variable-coefficient fundamental matrix is then built by the Levi-Carleman method through an integral correction whose density satisfies a resolvent-type integral equation solved by successive approximations. The resulting matrix is analytic in a sectorial domain for sufficiently large parameter values and satisfies exponential decay and derivative estimates, yielding an integral representation for solutions of the associated nonhomogeneous system.
Full Text
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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