MATHEMATICS

M. L. RASULOV

APPLICATION OF THE CONTOUR INTEGRAL METHOD TO THE SOLUTION OF MIXED PROBLEMS FOR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

(Presented by Academician S. L. Sobolev on 5 XI 1959)

In this note the contour integral method, given in \((^{1-3})\), is applied to the solution of the mixed problem (1)—(8). It is proved that, under conditions \(1^\circ\)—\(3^\circ\), this problem for sufficiently smooth \(\Phi^{(i)}(x)\), \(f^{(i)}(x,t)\) has a solution representable in the form of the sum (8) and (9).

Let \(D_1, D_2\) be bounded domains of points \(x=(x_1,x_2,x_3)\), lying inside the closed surface \(\Gamma_2\), and suppose that the boundary \(\Gamma_1\) of the domain \(D_1\) is also a closed surface contained inside the surface \(\Gamma_2\).

Consider the problem of finding solutions of the equations

\[ c^{(i)}(x) M\left(t,\frac{\partial}{\partial t}\right)v^{(i)} = \mathcal{L}^{(i)}\left(x,\frac{\partial}{\partial x}\right)v^{(i)} + f^{(i)}(x,t), \qquad x\in D_i, \tag{1} \]

satisfying the boundary conditions

\[ \lim_{x\to z} B^{(i)}\left(z,\frac{d}{dn_z},M\right)v(x,t)=\psi_i(z), \qquad z\in \Gamma_1,\quad i=1;\ 2; \]

\[ \lim_{x\to z} B^{(3)}\left(z,\frac{d}{dn_z},M\right)v(x,t)=\psi_3(z), \qquad z\in \Gamma_2, \tag{2} \]

and the initial conditions

\[ v^{(i)}(x,0)=\Phi_i(x), \qquad x\in D_i\quad (i=1,2), \tag{3} \]

where

\[ \mathcal{L}^{(i)}\left(x,\frac{\partial}{\partial x}\right) = \sum_{k=1}^{3} \left( \frac{\partial^2}{\partial x_k^2} + a_k^{(i)}(x)\frac{\partial}{\partial x_k} \right) + a(x); \qquad M=b_0(t)\frac{\partial}{\partial t}+b_1(t); \]

\[ B^{(i)}\left(z,\frac{d}{dn_z},M\right)v = \sigma_{i0}(z)\frac{dv^{(i)}(x,t)}{dn_z} + \sigma_{i1}(z)v^1(x,t) + \]

\[ + \sigma_{i2}(z)v^{(2)}(x,t) + \sigma_{i3}(z)M\left\{ \frac{dv^{(i)}(x,t)}{dn_z} + \sigma_{i4}(z)v^1(x,t) + \right. \]

\[ \left. + \sigma_{i5}(z)v^2(x,t) \right\} = \psi_i(z), \qquad z\in \Gamma_1,\quad i=1,\ 2; \]

\[ B^{(3)}\left(z,\frac{d}{dn_z},M\right)v = \sigma_{30}(z)\frac{dv^{(2)}(x,t)}{dn_z} + \sigma_{31}(z)v^{(2)}(x,t) + \]

\[ + \sigma_{32}(z)M\left\{ \frac{dv^{(2)}(x,t)}{dn_z} + \sigma_{33}(z)v^{(2)}(x,t) \right\} = \psi_3(z) \quad \text{for } z\in \Gamma_3; \]

\(n_z\) is the direction of the inward normal to \(\Gamma_i\) at the point \(z\in \Gamma_i\).

Suppose that the following conditions are satisfied:

\(1^\circ\). In the closed domain \(\overline{D_i}\), the functions \(a_k^{(i)}(x)\), \(a^{(i)}(x)\) are continuous, the functions \(c^{(i)}(x)\) are continuously differentiable once, and \(b_0(t)\), \(b_1(t)\) are continuous on \([0,T]\) and

\[ \int_0^t \frac{d\tau}{a_0(\tau)} > 0. \]

2°. The functions \(\sigma_{ij}(z)\) are continuous on the surface \(\Gamma_i\), and one of the functions \(\sigma_{i0}(z)\), \(\sigma_{i3}(z)\) does not vanish on \(\Gamma_i\).

3°. \(\Gamma_i\) are Lyapunov surfaces.

Consider the spectral problem:
\[ \mathcal L^{(i)}\left(x,\frac{\partial}{\partial x}\right)u^{(i)} -\lambda^2 c^{(i)}(x)u^{(i)}=\Phi^{(i)}(x) \quad \text{for } x\in D_i \ (i=1,2); \tag{4} \]
\[ \lim_{x\to z}B^{(i)}\left(z,\frac{d}{dn_z},\lambda^2\right)u(x,\lambda)=\psi_i(z) \quad \text{for } z\in\Gamma_1 \text{ and } i=1,2, \]
\[ \lim_{x\to z}B^{(3)}\left(z,\frac{d}{dn_z},\lambda^2\right)u(x,\lambda)=\psi_3(z) \quad \text{for } z\in\Gamma_2. \tag{5} \]

Denote by \(R_\delta\) the set of complex values of the parameter \(\lambda\) satisfying the conditions: \(|\lambda|\ge R\), \(\cos\arg \pi\ge\delta\), where \(R\) is sufficiently large and \(\delta\) is a sufficiently small positive number.

By the method of note (¹) the following lemmas are proved:

Lemma 1. Under condition \(1^\circ\) and for \(x\in D_i,\ \xi\in D_j\), for all \(\lambda\in R_\delta\):

1) There exist fundamental solutions \(P_{ij}(x,\xi,\lambda)\), analytic in \(\lambda\) in \(R_\delta\), with a singularity at the point \(x=\xi\), of the homogeneous equations corresponding to equations (4), admitting the representations
\[ P^{(i,j)}(x,\xi,\lambda) = P_0^{(j)}(x-\xi,\xi,\lambda) + \int_{D_i} P_0^{(i)}(x-y,y,\lambda)h^{(i,j)}(y,\xi,\lambda)\,d_yD_i, \tag{6} \]
where
\[ P_0^{(j)}(x-\xi,\xi,\lambda) = \exp\{-\lambda\sqrt{c^{(j)}(\xi)}\,|x-\xi|\}/4\pi|x-\xi|; \]
\(|x-\xi|\) denotes the length of the vector \(x-\xi\).

2) For the integral addition \(P_1^{(i,j)}(x,\xi,\lambda)\), its density \(h^{(i,j)}(x,\xi,\lambda)\), and the fundamental solution itself \(P^{(i,j)}(x,\xi,\lambda)\), the estimates
\[ \left|h^{(i,j)}(x,\xi,\lambda)\right| < c\exp\{-\varepsilon|\lambda||x-\xi|\}/|x-\xi|^{\beta_1}, \]
\[ \left|P_1^{(i,j)}(x,\xi,\lambda)\right| < c\exp\{-\varepsilon|\lambda||x-\xi|\}/|\lambda|^{\beta_2}|x-\xi|, \]
\[ \left| \frac{\partial^s P^{(i,j)}(x,\xi,\lambda)}{\partial x_k^s} \right| < c\exp\{-\varepsilon|\lambda||x-\xi|\}/|x-\xi|^{\beta_3+s}, \]
where \(\beta_1=3\) for \(i\ne j\); \(\beta_1=2\) for \(i=j\); \(\beta_2=0\) for \(i=j\), \(\beta_2=1\) for \(i\ne j\); \(\beta_3=1\) for \(i=j\); \(\beta_3=2\) for \(i\ne j\) \((s=0,1)\).

3) For \(x\in D_i\), the function
\[ u^{(i)}(x,\lambda) = -\sum_{j=1}^{3}\int_{D_j} P^{(i,j)}(x,\xi,\lambda)\Phi_j(\xi)\,d_\xi D_j \]
is a solution of equation (4).

Lemma 2. Under conditions \(1^\circ\)—\(3^\circ\), for all \(\lambda\in R_\delta\):

1) For \(x\in D_i\), there exist solutions \(u^{(i)}(x,\lambda)\) \((i=1,2)\), analytic in \(\lambda\in R_\delta\), of the corresponding homogeneous equations (4), satisfying the boundary conditions (5), representable in the form of simple-layer potentials:
\[ u_1^{(1)}(x,\lambda) = \int_{\Gamma_1} P^{(1,1)}(x,y,\lambda)\mu_1(y)\,d_y\Gamma_1, \quad x\in D_1; \]
\[ u_1^{(2)}(x,\lambda) = \int_{\Gamma_1} P^{(2,2)}(x,y,\lambda)\mu_2(y)\,d_y\Gamma_2 + \int_{\Gamma_2} P^{(2,2)}(x,y,\lambda)\mu_3(y)\,d_y\Gamma_2, \quad x\in D_2. \tag{7} \]

2) Substituting (7) into the boundary conditions (5), one obtains a system of three Fredholm integral equations of the second kind with respect to the densities \(\mu_i(y)\) \((i=1,2,3)\), which for \(\lambda\in R_\delta\) are solved by the iteration method indicated in (¹).

3) For the solution \(u_1^{(i)}(x,\lambda)\) \((i=1,2)\), for \(x\in D_i\) the estimates

\[ \left|\frac{d^k}{dn_z^k}u_1^{(i)}(x,\lambda)\right|<c,\qquad z\in \Gamma_i \quad\text{for } x\in \overline{D}_i\quad (i=1,2;\ k=0,1); \]

\[ \left|\partial^s u_1^{(i)}(x,\lambda)/\partial x_k^s\right| <c\exp\{-\varepsilon|\lambda|h\}/h^{s+1} \quad\text{for } x\in \overline{D}_{i1}\quad (i=1,2;\ s=0,1,2), \]

hold, where \(c\) is a constant; \(h\) is the minimal distance between the points of the boundaries of the domains \(D_{i1},D_i\) \((i=1,2)\); \(D_{i1}\) is an arbitrary domain lying, together with its boundary, in the domain \(D_i\).

From Lemmas 1 and 2 there follows the existence of the Green’s function

\[ G^{(i,j)}(x,\xi,\lambda) = P^{(i,j)}(x,\xi,\lambda)-Q^{(i,j)}(x,\xi,\lambda) \quad\text{for } x\in D_i,\ \xi\in D_j, \]

for whose regular part \(Q^{(i,j)}(x,\xi,\lambda)\) Lemma 2 is valid for every interior point \(\xi\) of the domain \(D_j\); moreover, the estimates of Lemma 2 are fulfilled for \(Q^{(i,j)}(x,\xi,\lambda)\) uniformly with respect to \(x\in \overline{D}_{i1}\), \(\xi\in \overline{D}_{j1}\) \((i,j=1,2)\).

Let \(S\) be a curve situated in the domain \(R_\delta\) of the complex \(\lambda\)-plane and asymptotically approaching the straight lines \(\cos\arg\lambda=\delta\).

With the aid of Lemmas 1 and 2 the following theorems are proved:

Theorem 1. Under conditions \(1^\circ\)—\(3^\circ\), if \(\Phi^{(i)}(x)\) is a continuously differentiable function in the domain \(D_i\), vanishing in some boundary strip of this domain, then the homogeneous equations corresponding to equations (1), for \(x\in D_i\), have solutions \(v_1^{(i)}(x,t)\), satisfying the boundary conditions (2) and the zero initial conditions \(\bigl(v_1^{(i)}(x,0)=0\bigr)\), representable by the formula

\[ v_1^{(i)}(x,t) = \frac{-1}{\pi\sqrt{-1}} \int_S \frac{u_1^{(i)}(x,\lambda)}{\lambda} \exp\left\{ \int_0^t \frac{\lambda^2-b_1(\tau)}{b_0(\tau)}\,d\tau \right\} d\lambda, \qquad x\in D_i. \tag{8} \]

Theorem 2. Under the conditions of Theorem 1, if \(f^{(i)}(x,t)\) is a function continuously differentiable once with respect to \(x\) and twice with respect to \(t\) for \(x\in \overline{D}_i\) \((i=1,2)\), \(t\in[0,T]\), vanishing in some boundary strip of the domain \(D_i\) for all \(t\) in the interval of variation of \(t\), then equations (1), for \(x\in D_i\), have solutions \(v_2^{(i)}(x,t)\) \((i=1,2)\), satisfying the homogeneous boundary conditions corresponding to conditions (2), and the initial conditions (3), representable by the formula

\[ v_2^{(i)}(x,t) = \frac{-1}{\pi\sqrt{-1}} \int_S \lambda\,d\lambda \sum_{j=1}^{3} \int_{D_j} G^{(i,j)}(x,\xi,\lambda) \left[ c^{(j)}(\xi)\Phi^{(j)}(\xi) +\right. \]

\[ \left. +\exp\left(\int_0^t \frac{\lambda^2-b_1(\tau)}{b_0(\tau)}\,dt\right) + \int_0^t \frac{f^{(j)}(\xi,\tau)}{b_0(\tau)} \exp\left( -\int_\tau^t \frac{\lambda^2-b_1(\tau_1)}{b_0(\tau_1)}\,d\tau_1 \right) d\tau \right] d_zD_j. \tag{9} \]

Remark 1. In the same way one can solve the analogous mixed problem when on the outer surface \(\Gamma_2\) a Dirichlet condition is prescribed for \(v^{(2)}\). In this case the second term in the second of formulas (7) is replaced by a double-layer potential.

Remark 2. As shown in note \((^3)\), for problem (1)—(3) under the condition \(\delta_{i2}(z)=0\) on \(\Gamma_i\), the solution can be constructed effectively.

Received
3 XI 1959

Cited Literature

\({}^1\) M. L. Rasulov, DAN, 125, No. 1 (1959).
\({}^2\) M. L. Rasulov, DAN, 125, No. 2 (1959).
\({}^3\) M. L. Rasulov, DAN, 128, No. 3 (1959).