UDC 517.946.91

APPLICATION OF THE CONTOUR INTEGRAL METHOD TO THE SOLUTION OF MIXED PROBLEMS WITH BOUNDARY CONDITIONS OF MIXED TYPE

M. L. RASULOV

§ 1. STATEMENT OF THE PROBLEM

In the present paper an application is given of the contour integral method [3] to the solution of a three-dimensional mixed problem for a single second-order equation of parabolic type, with boundary conditions of mixed type, also containing differentiation with respect to time.

For the solution of the corresponding spectral problem, the method of potential theory developed by V. D. Kupradze [2] is used.

Consider the equation

\[ c(x)\frac{\partial v}{\partial t} = \sum_{i=1}^{3} \left[ \frac{\partial}{\partial x_i} \left( k\frac{\partial v}{\partial x_i} \right) + a_i(x)\frac{\partial v}{\partial x_i} \right] + \]

\[ +\,a(x)v+f(x,t) \tag{1.1} \]

in a three-dimensional bounded domain \(D\) with boundary \(\Gamma\), which is a Lyapunov surface, under the boundary conditions

\[ v(y,t)=\Psi_1(y)\quad \text{for } y\in\Gamma_1, \tag{1.2} \]

\[ \lim_{x\to y} B\left(y,\frac{d}{dn_y},-\frac{\partial}{\partial t}\right)v(x,t) = \Psi_2(y) \quad \text{for } y\in\Gamma_2 \tag{1.3} \]

and the initial condition

\[ v(x,0)=\Phi(x),\quad x\in D, \tag{1.4} \]

where in the domain \(D+\Gamma\) the function \(a(x)\) is continuous, while \(c(x)\), \(k(x)\), \(a_i(x)\) are continuously differentiable; \(c(x), k(x)>0\); \(\Gamma_1\) is a connected part of the surface \(\Gamma\); \(\Gamma_2=\Gamma-\Gamma_1\);

\[ B\left(y,\frac{d}{dn_y},-\frac{\partial}{\partial t}\right)= \]

\[ = \left( \alpha_1(y)+\alpha_2(y)\frac{\partial}{\partial t} \right) \frac{d}{dn_y}; \tag{1.5} \]

\(\alpha_i(y)\) are continuously differentiable functions on \(\Gamma\), and one of them does not vanish; \(n_y\) is the direction of the normal at the point \(y\).

§ 2. CONNECTION BETWEEN THE GREEN’S FUNCTIONS OF AUXILIARY ADJOINT SPECTRAL PROBLEMS

By the spectral problem composed for (1.1)—(1.4) we shall mean the problem of finding a solution of the equation

\[ L\left(x,\frac{\partial}{\partial x},\lambda\right)u \equiv \sum_{i=1}^{3}\left[ \frac{\partial}{\partial x_i}\left(k\frac{\partial u}{\partial x_i}\right) +a_i(x)\frac{\partial u}{\partial x_i} \right] +\left(a(x)-\lambda^2 c(x)\right)u =\Phi(x) \tag{2.1} \]

under the boundary conditions

\[ u(y,\lambda)=\Psi_1(y)\quad \text{for } y\in\Gamma_1, \tag{2.2} \]

\[ \lim_{x\to y}B\left(y,-\frac{d}{dn_y},\lambda^2\right)u(x,\lambda) =\Psi_2(y),\quad y\in\Gamma_2, \tag{2.3} \]

where \(\lambda\) is a complex parameter.

Let \(\omega\) be a domain bounded by a closed surface \(\Gamma_1+\Gamma_3\) of Lyapunov type and containing inside it the domain \(D+\Gamma_2\). The coefficients of equation (2.1) and the derivatives of \(c(x), k(x), a_i(x)\) are extended continuously to the domain \(\omega+\Gamma_1+\Gamma_3\). In the domain \(\omega\) with boundary \(\Gamma_1+\Gamma_3\) consider the auxiliary spectral problem

\[ L\left(x,\frac{\partial}{\partial x},\lambda\right)u=h(x), \tag{2.4} \]

\[ u(y,\lambda)=0,\quad y\in\Gamma_1+\Gamma_3. \tag{2.5} \]

Let \(R,\delta\) be positive numbers, of which the first is sufficiently large and the second sufficiently small. Denote by \(R_\delta\) the set of values of \(\lambda\) satisfying the inequalities \(|\lambda|\ge R,\ \cos\arg\lambda\ge\delta\).

Under the assumptions of § 1, in works [3—6] the existence was proved of a solution, analytic in \(\lambda\), of the principal boundary-value problems for equation (2.4), represented by the formula

\[ u(x,\lambda)=-\iiint_{\omega} g(x,\xi,\lambda)h(\xi)\,d\omega_\xi, \tag{2.6} \]

where

\[ g(x,\xi,\lambda)=P(x,\xi,\lambda)-Q(x,\xi,\lambda). \tag{2.7} \]

This is the Green’s function; \(P(x,\xi,\lambda)\) is the fundamental solution in \(\omega\) with respect to the point \(x\) of the homogeneous equation (2.4), with a singularity at the point \(x=\xi\); \(Q(x,\xi,\lambda)\) is the regular part of the Green’s function. It is also proved that, for \(\lambda\in R_\delta\), the fundamental solution is representable in the form

\[ P(x,\xi,\lambda)=P_0(x-\xi,\xi,\lambda)+P_1(x,\xi,\lambda), \tag{2.8} \]

where

\[ P_0(x-\xi,\xi,\lambda)= \frac{1}{4\pi |x-\xi|} e^{-\lambda\sqrt{c(\xi)}\,|x-\xi|}, \tag{2.9} \]

\[ P_1(x,\xi,\lambda)= \iiint_{\omega}P_0(x-\eta,\eta,\lambda)h(\eta,\xi,\lambda)\,d\omega_\eta, \tag{2.10} \]

\(|x-\xi|\) is the length of the vector \(x-\xi\); for the density \(h\) of the integral correction \(P_1\) and for the fundamental solution \(P\), for \(\lambda\in R_\delta,\ x,\xi\in\omega+\Gamma_1+\Gamma_3\), the estimates hold

\[ \left|\frac{\partial^k h(x,\xi,\lambda)}{\partial x_i^k}\right| \leq \frac{c}{|x-\xi|^{2+k}} e^{-\varepsilon|\lambda||x-\xi|} \quad (k=0,1), \tag{2.11} \]

\[ \left|\frac{\partial^k P(x,\xi,\lambda)}{\partial x_i^k}\right| \leq \frac{c}{|x-\xi|^{1+k}} e^{-\varepsilon|\lambda||x-\xi|} \quad (k=0,1,2). \tag{2.12} \]

The regular part \(Q\) of the Green function \(g\) for \(\lambda \in R_\delta\) is represented in the form of a double-layer potential

\[ Q(x,\xi,\lambda)= \iint_{\Gamma} \frac{dP(x,z,\lambda)}{dn_z}\, \mu(z,\xi,\lambda)\,d\Gamma_z, \]

whose density \(\mu\) is determined by the formula

\[ \mu(y,\xi,\lambda)=2P(y,\xi,\lambda)+ \]

\[ +\iint_{\Gamma_1+\Gamma_3} R(y,z,\lambda)\,2P(z,\xi,\lambda)\,d\Gamma_z, \tag{2.13} \]

and for \(R\) the estimate

\[ |R(y,z,\lambda)| \leq \frac{c}{|y-z|^{2-\alpha}} e^{-\varepsilon|\lambda||y-z|}, \tag{2.14} \]

holds, where \(R\) is the resolvent of the integral equation satisfied by \(\mu(y,\xi,\lambda)\).

If \(\omega_1\) is a domain lying, together with its boundary, inside \(\omega\), then we have

\[ \left|\frac{\partial^k Q(x,\xi,\lambda)}{\partial x_i^k}\right| \leq \frac{c}{\sigma^{k+3}} e^{-\varepsilon|\lambda||x-\xi|} \quad (k=0,1,2) \quad \text{for } x,\xi\in\omega_1, \tag{2.15} \]

where \(\sigma\) is the smallest distance between the boundary points of the domains \(\omega\), \(\omega_1\). Further, for \(\lambda\in R_\delta\), \(x,\xi\in \omega+\Gamma_1+\Gamma_3\), the following inequalities hold for the Green function \(g\) represented by formula (2.7):

\[ \left|\frac{dg(y,z,\lambda)}{dn_z}\right| \leq \frac{c}{|y-z|^{2-\alpha}}, \tag{2.16} \]

\[ \left|\frac{dg(y,z,\lambda)}{dn_y}\right| \leq \frac{c}{|y-z|^{2-\alpha}} \quad \text{for } y,z\in\Gamma_1+\Gamma_3, \tag{2.17} \]

where \(\alpha\) is the Lyapunov exponent.

Obviously, all these assertions, under the assumptions of § 1, also remain valid for the adjoint problem

\[ L^*\left(x,\frac{\partial}{\partial x},\lambda\right)W\equiv \]

\[ \equiv \sum_{i=1}^{3} \left[ \frac{\partial}{\partial x_i} \left(k\frac{\partial W}{\partial x_i}\right) - \frac{\partial}{\partial x_i} \left(\overline{a_i(x)}\,W\right) \right] + \]

\[ + \left(\overline{a(x)}-\lambda^2 c(x)\right)W = g(x), \tag{2.18} \]

\[ W(y,\lambda)=0 \quad \text{for } y\in\Gamma_1+\Gamma_3. \tag{2.19} \]

Thus, for \(\lambda \in R_\delta\), the solution \(W(x,\lambda)\) of problem (2.18), (2.19) exists and is representable by the formula

\[ W(x,\lambda)=-\iiint_{\omega} g^*(x,\xi,\lambda)g(\xi)\,d\omega_\xi, \tag{2.20} \]

where \(g^*(x,\xi,\lambda)\) is the Green’s function of problem (2.18), (2.19).

It is known that for any pair of functions \(u(x)\), \(W(x)\) having continuous derivatives up to the second order in \(\omega\), and up to the first order in \(\omega+\Gamma_1+\Gamma_3\), Green’s formula holds:

\[ \begin{aligned} &\iiint_{\omega}\left\{ L\left(\xi,\frac{\partial}{\partial \xi},\lambda\right)u(\xi)\overline{W(\xi)} - u(\xi)L^*\left(\xi,\frac{\partial}{\partial \xi},\lambda\right)\overline{W(\xi)} \right\}\,d\omega_\xi \\ &= \iint_{\Gamma}\left\{ u(y)\frac{\overline{dW(y)}}{dn_y} - \frac{du(y)}{dn_y}\overline{W(y)} - \sum_{i=1}^{3} a_i(y)\cos(n,y_i)u(y)\overline{W(y)} \right\}\,d\Gamma_y . \end{aligned} \tag{2.21} \]

From this formula it is seen that if \(u(x,\lambda)\) is a solution of problem (2.4), (2.5), and \(W(x,\lambda)\) is a solution of the adjoint problem (2.18), (2.19), then

\[ \iiint_{\omega} L\left(x,\frac{\partial}{\partial x},\lambda\right)u(x,\lambda)\overline{W(x,\lambda)}\,d\omega_x = \]

\[ = \iiint_{\omega} u(x,\lambda)L^*\left(x,\frac{\partial}{\partial x},\lambda\right)\overline{W(x,\lambda)}\,d\omega_x . \tag{2.22} \]

Substituting in (2.22), in place of \(u(x,\lambda)\), \(W(x,\lambda)\), their expressions from (2.6) and (2.20), we obtain

\[ \iiint_{\omega} h(x)\,d\omega_x \iiint_{\omega} \overline{g^*(x,\xi,\lambda)}\,g(\xi)\,d\omega_\xi = \]

\[ = \iiint_{\omega}\overline{g(x)}\,d\omega_x \iiint_{\omega} g(x,\xi,\lambda)h(\xi)\,d\omega_\xi . \]

In the left-hand side of the last equality, interchanging \(x\) and \(\xi\), we shall have

\[ \iiint_{\omega}\iiint_{\omega} \left[ \overline{g^*(\xi,x,\lambda)}-g(x,\xi,\lambda) \right] h(\xi)\overline{g(x)}\,d\omega_\xi\,d\omega_x=0 . \tag{2.23} \]

In view of the arbitrariness of the functions \(h(\xi)\) and \(g(x)\), from (2.23) we obtain

\[ \overline{g^*(\xi,x,\lambda)}=g(x,\xi,\lambda) \tag{2.24} \]

or

\[ g^*(\xi,x,\lambda)=\overline{g(x,\xi,\lambda)} . \tag{2.25} \]

Thus, from (2.24), (2.25) it is seen that the complex conjugate of the Green’s function of problem (2.4), (2.5) with respect to the second point is the Green’s function of the adjoint problem (2.18), (2.19). Consequently, when one ...

from the points \(x,\xi\), with a boundary point of the domain \(\omega\) (the other must not belong to \(\Gamma_1+\Gamma_3\)), the function \(g(x,\xi,\lambda)\) must be equal to zero:

\[ L\left(x,\frac{\partial}{\partial x},\lambda\right)g(x,\xi,\lambda)=0 \quad \text{for } x\ne \xi, \tag{2.26} \]

\[ g(y,\xi,\lambda)=\overline{g^*(\xi,y,\lambda)}=0 \quad \text{for } y\in \Gamma_1+\Gamma_3,\ \xi\in \Gamma_1+\Gamma_3, \tag{2.27} \]

\[ L^*\left(\xi,\frac{\partial}{\partial \xi},\lambda\right)g^*(\xi,x,\lambda)= \]

\[ = L^*\left(\xi,\frac{\partial}{\partial \xi},\lambda\right)\overline{g(x,\xi,\lambda)}=0 \quad \text{for } \xi\ne x, \tag{2.28} \]

\[ g^*(y,x,\lambda)=\overline{g(x,y,\lambda)}=0 \]

\[ \text{for } y\in \Gamma_1+\Gamma_3,\quad x\in \Gamma_1+\Gamma_3. \tag{2.29} \]

§ 3. REPRESENTATION OF THE SOLUTION OF THE BASIC SPECTRAL PROBLEM FOR A HOMOGENEOUS EQUATION IN THE FORM OF A SUM OF POTENTIALS

Denote by \(D_1\) the domain \(\omega-D-\Gamma_2\). Let \(x\in D_1\). Then, applying in the domain \(D\) Green’s formula (2.21) to the solution \(u(\xi,\lambda)\) of the spectral problem (2.1)—(2.3) for the homogeneous equation (2.1), to the Green’s function \(g^*(\xi,x,\lambda)\) of the adjoint problem (2.18), (2.19), and taking (2.28) into account, we obtain

\[ \iint_{\Gamma}\left\{ u(y,\lambda)\frac{\overline{dg^*(y,x,\lambda)}}{dn_y} - \frac{du(y,\lambda)}{dn_y}\overline{g^*(y,x,\lambda)} - \sum_{i=1}^{3} a_i(y)\cos(n,y_i)U(y,\lambda)\overline{g^*(y,x,\lambda)} \right\}d\Gamma_y=0 \quad \text{for } x\in D_1. \tag{3.1} \]

In exactly the same way, if \(x\in D\), then, surrounding it by a ball \(Ш_\varepsilon(x)\) of radius \(\varepsilon\) with center at the point \(x\), lying inside the domain \(D\), applying Green’s formula (2.21) in the domain \(D-Ш_\varepsilon(x)\) to the same functions \(u(\xi,\lambda)\), \(g(\xi,x,\lambda)\) and letting \(\varepsilon\) tend to zero, in the limit we obtain

\[ u(x)=\iint_{\Gamma}\left\{ u(y,\lambda)\frac{\overline{dg^*(y,x,\lambda)}}{dn_y} - \frac{du(y,\lambda)}{dn_y}\overline{g^*(y,x,\lambda)} - \sum_{i=1}^{3} a_i(y)\cos(n,y_i)u(y,\lambda)\overline{g^*(y,x,\lambda)} \right\}d\Gamma_y \quad \text{for } x\in D. \tag{3.2} \]

Taking into account (1.5), (2.2), (2.3), (2.24), (2.29), from (3.1), (3.2) we have:

\[ \iint_{\Gamma_2}K(x,y,\lambda)u(y,\lambda)\,d\Gamma_y+f(x,\lambda)=0 \quad \text{for } x\in D_1, \tag{3.3} \]

\[ u(x,\lambda)=\iint_{\Gamma_2}K(x,y,\lambda)u(y,\lambda)\,d\Gamma_y+f(x,\lambda) \quad \text{for } x\in D, \tag{3.4} \]

where

\[ f(x,\lambda)=\iint_{\Gamma_1}\Psi_1(y)\frac{dg(x,y,\lambda)}{dn_y}\,d\Gamma_y -\iint_{\Gamma_2}\frac{\Psi_2(y)}{a_1(y)+\lambda^2 a_2(y)}\,g(x,y,\lambda)\,d\Gamma_y, \tag{3.5} \]

\[ K(x,y,\lambda)=\frac{dg(x,y,\lambda)}{dn_y} -\sum_{i=1}^{3} a_1(y)\cos(n,y_i)\,g(x,y,\lambda). \tag{3.6} \]

§ 4. Limiting Values of Potentials

We now proceed to derive the jump formulas for the double-layer potential

\[ W(x)=\iint_{\Gamma_2}K(x,y,\lambda)\mu(y)\,d\Gamma_y \tag{4.1} \]

and for the normal derivative of the single-layer potential

\[ V(x)=\iint_{\Gamma_2}g(x,y,\lambda)\mu(y)\,d\Gamma_y. \tag{4.2} \]

As is seen from expression (3.6) for the kernel \(K(x,y,\lambda)\), the estimate

\[ \left|K(x,y,\lambda)-\frac{d}{dn_y}\frac{1}{4\pi |x-y|}\right| \leq \frac{c(\lambda)}{|x-y|}, \tag{4.3} \]

holds, where \(c(\lambda)\) is a positive number depending only on \(\lambda\).

It is easy to see that, if \(z\in\Gamma_2\), and \(\Gamma_{2\delta}(z)\) is the part of \(\Gamma_2\) contained inside the sphere of radius \(\delta\) with center at the point \(z\), then \(W(x)\) can be represented in the form

\[ \begin{aligned} W(x)=& \iint_{\Gamma_2}\left[ K(x,y,\lambda)-\frac{d}{dn_y}\frac{1}{4\pi |x-y|} \right]\mu(y)\,d\Gamma_y \\ &+\iint_{\Gamma_2-\Gamma_{2\delta}(z)} \frac{d}{dn_y}\frac{1}{4\pi |x-y|}\mu(y)\,d\Gamma_y \\ &+\iint_{\Gamma_{2\delta}(z)} \frac{d}{dn_y}\frac{1}{4\pi |x-y|}\mu(y)\,d\Gamma_y . \end{aligned} \tag{4.4} \]

By virtue of (4.3), the first and second integrals on the right-hand side of (4.4) are continuous functions of the point \(x\). Consequently, their limiting values as \(x\to z\) are equal to their values at the point \(z\). As for the third integral, closing \(\Gamma_{2\delta}(z)\) by a smooth surface \(\pi_\delta(z)\) toward the domain \(D\), it can be represented in the form

\[ W_1(x)=\iint_{\Gamma_{2\delta}(z)} \frac{d}{dn_y}\frac{1}{4\pi |x-y|}\mu(y)\,d\Gamma_y= \]

\[ = \iint_{\Gamma_{2\delta}(z)+\bar{\Gamma}_{\delta}(z)} \frac{d}{dn_y}\frac{1}{4\pi |x-y|}\,\mu(y)\,d\Gamma_y - \iint_{\bar{\Gamma}_{\delta}(z)} \frac{d}{dn_y}\frac{1}{4\pi |x-y|}\,\mu(y)\,d\Gamma_y . \tag{4.5} \]

Let now the point \(x\in D_1\) tend to the point \(z\in \Gamma_2\). Then from (4.5) we obtain

\[ \lim_{\substack{x\to z\in\Gamma_2\\ x\in D_1}} W_1(x) = -\frac{\mu(z)}{2} + \]

\[ + \iint_{\Gamma_{2\delta}(z)+\bar{\Gamma}_{\delta}(z)} \frac{d}{dn_y}\frac{1}{4\pi |z-y|}\,\mu(y)\,d\Gamma_y - \]

\[ - \iint_{\bar{\Gamma}_{\delta}(z)} \frac{d}{dn_y}\frac{1}{4\pi |z-y|}\,\mu(y)\,d\Gamma_y = -\frac{\mu(z)}{2} + \]

\[ + \iint_{\Gamma_{2\delta}(z)} \frac{d}{dn_y}\frac{1}{4\pi |z-y|}\,\mu(y)\,d\Gamma_y . \tag{4.6} \]

Passing to the limit as \(x\to z\in\Gamma_2\), and taking into account (4.2) and the continuity of the first two integrals, from (4.4) we obtain

\[ \lim_{\substack{x\to z\in\Gamma_2\\ x\in D_1}} W = \iint_{\Gamma_2} \left[ K(z,y,\lambda) - \frac{d}{dn_y}\frac{1}{4\pi |z-y|} \right]\mu(y)\,d\Gamma_y + \]

\[ + \iint_{\Gamma_2-\Gamma_{2\delta}(z)} \frac{d}{dn_y}\frac{1}{4\pi |z-y|}\,\mu(y)\,d\Gamma_y + \]

\[ + \iint_{\Gamma_{2\delta}(z)} \frac{d}{dn_y}\frac{1}{4\pi |z-y|}\,\mu(y)\,d\Gamma_y - \frac{\mu(z)}{2}. \]

Thus, the validity of the formula has been proved:

\[ \lim_{\substack{x\to z\in\Gamma_2\\ x\in D_1}} W(x) = \lim_{\substack{x\to z\in\Gamma_2\\ x\in D_1}} \iint_{\Gamma_2} K(x,y,\lambda)\mu(y)\,d\Gamma_y = \]

\[ = -\frac{\mu(z)}{2} + \iint_{\Gamma_2} K(z,y,\lambda)\mu(y)\,d\Gamma_y . \tag{4.7} \]

Further, for \(x\in D\), by virtue of (2.25) we have

\[ V(x) = \iint_{\Gamma_2} g(x,y,\lambda)\mu(y)\,d\Gamma_y = \iint_{\Gamma_1+\Gamma_2} g(x,y,\lambda)\mu(y)\,d\Gamma_y . \tag{4.8} \]

If we take the derivative of both sides of (4.8) in the direction of the normal to \(\Gamma_2\) passing through the point \(x \in D\), and pass to the limit as \(x \to z\), from (4.8) we obtain

\[ \lim_{D \ni x \to z \in \Gamma_2} \iint_{\Gamma_2} \frac{d g(x,y,\lambda)}{d n_z}\,\mu(y)\,d\Gamma_y = -\frac{\mu(z)}{2} + \iint_{\Gamma_2} \frac{d g(z,y,\lambda)}{d n_z}\,\mu(y)\,d\Gamma_y, \tag{4.9} \]

\[ \lim_{D_1 \ni x \to z \in \Gamma_2} \iint_{\Gamma_2} \frac{d g(x,y,\lambda)}{d n_z}\,\mu(y)\,d\Gamma_y = \frac{\mu(z)}{2} + \iint_{\Gamma_2} \frac{d g(z,y,\lambda)}{d n_z}\,\mu(y)\,d\Gamma_y. \tag{4.10} \]

In what follows we shall also need the limiting values of the integral

\[ U_1(x)= \iint_{\Gamma_1} f(y)\, \frac{d g(x,y,\lambda)}{d n_y}\, d\Gamma_y \tag{4.11} \]

as \(D \ni x \to z \in \Gamma_1\).

To compute the limiting value of (4.11), assume that the interior Dirichlet problem

\[ L\left(x,\frac{\partial}{\partial x},\lambda\right)U=0, \tag{4.12} \]

\[ U(y)=f(y)\quad \text{for } y \in \Gamma_1+\Gamma_3 \tag{4.13} \]

has a unique solution for every continuous function on \(\Gamma_1+\Gamma_3\).

In particular, for example, when \(a_i(x)\equiv 0\), \(a(x)<0\) in \(\omega\), the fulfillment of the uniqueness condition for the Dirichlet problem (4.12), (4.13) follows from Green’s formula

\[ \iiint_{\omega} L\left(x,\frac{\partial}{\partial x},\lambda\right)u(x)\,\overline{W(x)}\,d\omega_x = \]

\[ = \iint_{\Gamma_1+\Gamma_3} \left[ k(y)\frac{d u(y)}{d n_y}\,\overline{W(y)} + \sum_{i=1}^{3} a_i(y)\cos(n,y_i)\,u(y)\,\overline{W(y)} \right]d\Gamma_y - \]

\[ - \iiint_{\omega} \left\{ k(x)\sum_{i=1}^{3} \frac{\partial u(x)}{\partial x_i} \frac{\partial \overline{W(x)}}{\partial x_i} - u(x)\sum_{i=1}^{3} \frac{\partial}{\partial x_i} \bigl(a_i(x)\overline{W(x)}\bigr) + \right. \]

\[ \left. + \bigl(\lambda^2 c(x)-a(x)\bigr)u(x)\overline{W(x)} \right\} d\omega_x. \tag{4.14} \]

If \(x \in \omega\), then, analogously to how formula (2.29) was obtained above, for the solution \(u(x)\) of problem (4.12), (4.13) it is easy to obtain the following representation:

\[ u(x)\iint_{\Gamma_1+\Gamma_3}\left\{ f(y)\frac{dg(x,y,\lambda)}{dn_y} -\frac{du(y)}{dn_y}g(x,y,\lambda) -\sum_{i=1}^{3}a_i(y)\cos(n,y_i)f(y)g(x,y,\lambda) \right\}\,d\Gamma_y . \tag{4.15} \]

By virtue of (2.29), from (4.15) for every \(x\in\omega\) we obtain

\[ u(x)=\iint_{\Gamma_1+\Gamma_3} f(y)\frac{dg(x,y,\lambda)}{dn_y}\,d\Gamma_y . \tag{4.16} \]

Obviously, in particular, if \(f(y)\) is a continuous function equal to zero on \(\Gamma_3\), then from (4.16), by virtue of the uniqueness of the solution of the interior Dirichlet problem (4.12), (4.13), we obtain

\[ \lim_{D\ni x\to z\in\Gamma_1} \iint_{\Gamma_1} f(y)\frac{dg(x,y,\lambda)}{dn_y}\,d\Gamma_y=f(z). \tag{4.17} \]

§ 5. SOLUTION OF THE BASIC SPECTRAL PROBLEM IN THE DOMAIN \(R_\delta\)

The solution \(u_1(x,\lambda)\) of the spectral problem (2.1), (2.3) for the homogeneous equation (2.1) will be sought in the form of a sum of potentials

\[ u_1(x,\lambda)=\iint_{\Gamma_2}K(x,y,\lambda)\mu(y)\,d\Gamma_y+f(x,\lambda) \quad\text{for } x\in D, \tag{5.1} \]

where \(\mu(y)\) is an unknown density subject to the additional condition

\[ \iint_{\Gamma_2}K(x,y,\lambda)\mu(y)\,d\Gamma_y+f(x,\lambda)=0 \quad\text{for } x\in D_1. \tag{5.2} \]

It is easy to see that for every continuous function \(\mu(y)\) the function \(u_1(x,\lambda)\), defined by formula (5.1), is a solution of the corresponding homogeneous equation (2.1).

Passing to the limit as \(D_1\ni x\to z\in\Gamma_2\) in (5.2), according to formula (4.7), we obtain the integral equation

\[ \mu(z,\lambda)=2f(z,\lambda)+\iint_{\Gamma_2}\mu(y,\lambda)\,2K(z,y,\lambda)\,d\Gamma_y \tag{5.3} \]

for the unknown density \(\mu(y)\), since the function \(f(x,\lambda)\), as is seen from its expression (3.5), is a continuous function in the domain \(\omega+\Gamma_1+\Gamma_3\). As is seen from (3.6), for the kernel \(2K(z,y,\lambda)\) the estimate

\[ |2K(z,y,\lambda)|\leq \frac{ce^{-\varepsilon|\lambda|\,|z-y|}}{|z-y|^{2-\alpha}}, \]

is valid, where \(\varepsilon>0\); \(c\) is a constant number; \(\alpha\) is a Lyapunov number.

Exactly as was done in [3], with the aid of estimate (5.4) by the method of successive approximations it is easy to prove that the integral equa-

tion (5.3) has a unique solution on \(\Gamma_2\), continuous in \(z\), analytic in \(\lambda\), and bounded in the domain \(R_\delta\). For the resolvent \(R(z,y,\lambda)\) of equation (5.3) the estimate

\[ |R(y,z,\lambda)| \leq \frac{c}{|y-z|^{2-\alpha}} e^{-\varepsilon|\lambda||y-z|}. \tag{5.4} \]

holds.

Next it is easy to see that if \(\mu(z,\lambda)\) is a solution of the integral equation (5.3), then condition (5.2) is satisfied under the uniqueness condition for the solution of the interior Dirichlet problem for equation (5.2) in the domain \(D_1\). Indeed, it is easy to see that the function

\[ W(x,\lambda)=\iint_{\Gamma_2} K(x,y,\lambda)\mu(y)\,d\Gamma_y+f(x,\lambda) \]

is a solution of the homogeneous equation (2.1) in the domain \(D_1\), bounded by the closed surface \(\Gamma_2+\Gamma_3\), and is equal to zero on the boundary of this domain. Then, by virtue of the uniqueness of the solution of the Dirichlet problem for equation (2.1) in the domain \(D_1\), the function \(W(x,\lambda)\equiv 0\) in \(D_1\), as was required to prove.

Now let us show that, when the listed conditions are fulfilled, if the function \(\Psi_1(y)\) is continuous on \(\Gamma_1\) and equal to zero on the boundary of the surface \(\Gamma_1\), while \(\Psi_2(y)\) is continuously differentiable on \(\Gamma_2\), then the function \(U_1(x,\lambda)\), defined by formula (5.1), is a solution of problem (2.1)—(2.3) for the homogeneous equation (2.1). For this it suffices to show that it satisfies conditions (2.2) and (2.3).

Indeed, as \(D\ni x\to z\in\Gamma_1\), from (5.1), taking into account (2.27), (3.5), (3.6), according to formula (4.17), for every \(z\in\Gamma_1\) not belonging to the boundary of the surface \(\Gamma_1\), we obtain

\[ \lim_{D\ni x\to z} u_1(x,\lambda)=\Psi_1(z). \]

Now let us show that \(u_1(x,\lambda)\) satisfies the boundary condition (2.3). By direct substitution of \(u_1(x,\lambda)\) into (2.3), taking into account the jump formula (4.9), we obtain

\[ \lim_{D\ni x\to z\in\Gamma_2} B\left(z,\frac{d}{dn_z},\lambda\right)u_1(x,\lambda)= \]

\[ = \lim_{D\ni x\to z\in\Gamma_2} B\left(z,\frac{d}{dn_z},\lambda\right) \iint_{\Gamma_2} K(x,y,\lambda)\mu(y)\,d\Gamma_y+ \]

\[ +\iint_{\Gamma_1}\Psi_1(y) B\left(z,\frac{d}{dn_z},\lambda\right) \frac{dg(z,y,\lambda)}{dn_y}\,d\Gamma_y +\frac{\Psi_2(z)}{2}- \]

\[ -\iint_{\Gamma_2} \frac{\Psi_2(y)}{a_1(y)+\lambda^2 a_2(y)} B\left(z,\frac{d}{dn_z},\lambda\right) g(z,y,\lambda)\,d\Gamma_y, \tag{5.5} \]

where it is assumed that \(z\) does not belong to the boundary of the surface \(\Gamma_2\). Further, taking into account the jump formula (4.10), from (5.2) we obtain

\[ \lim_{D_1\ni x\to z\in\Gamma_2} B\left(z,\frac{d}{dn_z},\lambda\right) \iint_{\Gamma_2} K(x,y,\lambda)\mu(y)\,d\Gamma_y+ \]

\[ +\iint_{\Gamma_1}\Psi_1(y)B\left(z,\frac{d}{dn_z},\lambda\right) \frac{dg(z,y,\lambda)}{dn_y}\,d\Gamma_y -\frac{\Psi_2(z)}{2}- \]

\[ -\iint_{\Gamma_2}\frac{\Psi_2(y)}{\alpha_1(y)+\lambda^2\alpha_2(y)} B\left(z,\frac{d}{dn_z},\lambda\right)g(z,y,\lambda)\,d\Gamma_y=0. \tag{5.6} \]

Let us note that \(\mu(z)\) is found from the integral equation (5.3), whose resolvent has the same singularity as the kernel \(K(z,y,\lambda)\). Consequently, the function \(f(z,\lambda)\) is the principal term of the function \(\mu(z,\lambda)\), determined by the method of successive approximations in the form of an infinite series. In view of the continuous differentiability of \(\Psi_2(y)\), for \(\alpha=1\) the function \(f(z,\lambda)\) is also continuously differentiable on \(\Gamma_2\), if \(z\) does not belong to the boundary of the surface \(\Gamma_2\). Therefore, if \(z\) does not belong to the boundary of the surface \(\Gamma_2\), then \(\mu(z,y,\lambda)\) is continuously differentiable. Then for such \(z\), as is known, the limiting values of the normal derivative of the double-layer potential

\[ \iint_{\Gamma_2}K(x,y,\lambda)\mu(y)\,d\Gamma_y \]

as \(x\) tends to \(z\in\Gamma_2\), remaining in the domains \(D\) and \(D_1\), exist and coincide [1].

Thus,

\[ \lim_{D\ni x\to z\in\Gamma_2} B\left(z,\frac{d}{dn_z},\lambda\right) \iint_{\Gamma_2}K(x,y,\lambda)\mu(y)\,d\Gamma_y = \]

\[ = \lim_{D_1\ni x\to z\in\Gamma_2} B\left(z,\frac{d}{dn_z},\lambda\right) \iint_{\Gamma_2}K(x,y,\lambda)\mu(y)\,d\Gamma_y . \]

Then, determining the right-hand side of the last equality from (5.6) and substituting into (5.5), we obtain

\[ \lim_{D\ni x\to z\in\Gamma_2} B\left(z,\frac{d}{dn_z},\lambda\right)U_1(x,\lambda)=\Psi_2(z), \]

which was required to be proved.

Let \(D_1\) be a domain lying, together with its boundary, inside the domain \(D\). From (5.1), with the aid of (5.3), it is easy to obtain the estimate

\[ \left|\frac{d^kU_1(x,\lambda)}{\partial x_i^k}\right| <\frac{c}{\sigma^{2+k}}e^{-\varepsilon|\lambda|\sigma} \quad (k=0,1,2)\quad \text{for } x\in\overline{D}_1, \tag{5.7} \]

where \(\sigma\) is the least distance between points of the boundaries of the domains \(D\), \(D_1\).

To construct the solution of problem (2.1)—(2.3), it is sufficient to construct the Green’s function of this problem. But the construction of the Green’s function of problem (2.1)—(2.3) is equivalent to solving an analogous problem for the corresponding homogeneous equation (2.1), since, in order to construct the regular part \(Q_1(x,\xi,\lambda)\) of the Green’s function \(G(x,\xi,\lambda)\) (problem (2.1)—(2.3)), determined by the formula

\[ G(x,\xi,\lambda)=P(x,\xi,\lambda)-Q_1(x,\xi,\lambda), \tag{5.8} \]

one must solve problem (2.1)—(2.3) for the corresponding homogeneous equation (2.1) with the following right-hand sides of the boundary conditions (2.2), (2.3):

\[ \Psi_1(y)=P(y,\xi,\lambda), \]

\[ \Psi_2(y)=B\left(y,\frac{d}{dn_y},\lambda^2\right)P(y,\xi,\lambda),\qquad \xi\in D, \tag{5.9} \]

where \(P(x,\xi,\lambda)\) is the fundamental solution of the homogeneous equation corresponding to (2.1). Consequently, \(Q_1(x,\xi,\lambda)\) is also determined by formula (5.1), but in the expressions (3.5) for \(f(x,\lambda)\) the functions \(\Psi_1(y)\), \(\Psi_2(y)\) are replaced by the functions (5.9):

\[ Q_1(x,\xi,\lambda)=\iint_{\Gamma_2} K(x,y,\lambda)\mu(y,\xi,\lambda)\,d\Gamma_y+f(x,\xi,\lambda), \tag{5.10} \]

where

\[ f(x,\xi,\lambda)=\iint_{\Gamma_1} P(y,\xi,\lambda)\, \frac{dg(x,y,\lambda)}{dn_y}\,d\Gamma_y - \]

\[ -\iint_{\Gamma_2} \frac{ B\left(y,\frac{d}{dn_y},\lambda^2\right)P(y,\xi,\lambda) }{ a_1(y)+\lambda^2 a_2(y) } \,g(x,y,\lambda)\,d\Gamma_y . \tag{5.11} \]

Then \(\mu(y,\xi,\lambda)\) is determined from the integral equation

\[ \mu(z,\xi,\lambda)=2f(z,\xi,\lambda)+ \iint_{\Gamma_2}2K(z,y,\lambda)\mu(y,\xi,\lambda)\,d\Gamma_y . \]

Thus, we have

\[ \mu(z,\xi,\lambda)=2f(z,\xi,\lambda)+ \iint_{\Gamma_2}R(z,y,\lambda)2f(y,\xi,\lambda)\,d\Gamma_y . \tag{5.12} \]

Obviously, in accordance with (2.12), (5.4), (5.11), it follows from (5.12) that the estimate

\[ |\mu(z,\xi,\lambda)|\leq \frac{c}{\sigma^2}e^{-\lambda |z-\xi|_\varepsilon} \quad \text{for } \xi\in D_1,\quad \lambda\in R_\delta , \tag{5.13} \]

holds, where \(\sigma\) is the distance from the nearest boundary point of the domain \(D_1\) to the point \(\xi\). In exactly the same way, from (5.10), in accordance with (5.4), (5.13), it follows that the estimate

\[ \left| \frac{\partial^k Q_1(x,\xi,\lambda)}{\partial x_i^k} \right| \leq \frac{c}{\sigma^{1+k}}e^{-\varepsilon |\lambda|\,|x-\xi|} \quad \text{for } x,\xi\in D_1,\quad \lambda\in R_\delta \tag{5.14} \]

holds \((k=0,1,2)\), where \(\sigma\) is the least distance between points of the boundaries of the domains \(D\), \(D_1\).

Thus, for every function \(\Phi(x)\) continuous in \(D\), equal to zero in some boundary strip of the domain \(D\), for all \(\lambda\in R_\delta\) the existence has been proved of a solution \(u(x,\lambda)\), analytic with respect to \(\lambda\), of problem (2.1)—(2.3), representable in the form

\[ u(x,\lambda)=u_1(x,\lambda)-\iiint_D G(x,\xi,\lambda)\Phi(\xi)\,dD_\xi , \]

where for \(u_1(x,\lambda)\) and \(G(x,\xi,\lambda)\) the estimates (2.12), (5.7), (5.14) are valid; moreover, for \(G(y,z,\lambda)\), when \(y,z\in\Gamma\), inequalities of the type (2.16), (2.17) also hold.

§ 6. Representation of the Solution of the Posed Mixed Problem in the Form of a Contour Integral

Let \(S\) be an unbounded contour in the \(\lambda\)-plane, lying in the domain \(R_\delta\), a sufficiently distant part of which coincides with the continuations of the rays
\(\cos \arg \lambda = \delta\).

Assume further that the following conditions are satisfied.

\(1^\circ\). In the domain \(D+\Gamma\) the functions \(k(x)\), \(c(x)\), \(a_i(x)\), \(\Phi(x)\), \(f(x,t)\) are continuously differentiable for \(0 \le t \le T\) with respect to all their arguments, while the functions \(\Phi(x)\), \(f(x,t)\) are equal to zero in some boundary strip of the domain \(D\); the function \(a(x)\) is continuous.

\(2^\circ\). The boundary \(\Gamma\) of the domain \(D\) is a Lyapunov surface with Lyapunov number \(\alpha=1\). The function \(\Psi_1(y)\) is continuous on \(\Gamma_1\) and is equal to zero on the boundary of the surface \(\Gamma_1\). On \(\Gamma_2\) the functions \(\alpha_1(y)\), \(\alpha_2(y)\), \(\Psi_2(y)\) are continuously differentiable, and one of the functions \(\alpha_1(y)\), \(\alpha_2(y)\) does not vanish.

\(3^\circ\). The interior Dirichlet problem for equation (2.1) in the domains \(\omega\) and \(D_1\) has a unique solution.

Entirely analogously to how this is done in [3], the following may be proved.

Theorem. Under conditions \(1^\circ\)—\(3^\circ\), the problem (1.1)—(1.4) has a solution \(v(x,t)\) representable by the formula

\[ v(x,t)=\frac{1}{\pi i}\int_S \left| \frac{u_1(x,\lambda)}{\lambda}+\right. \]

\[ \left. +\lambda \iiint_D G(x,\xi,\lambda)\left[\varphi(\xi)c(\xi)+ \right. \]

\[ \left. +\int_0^t e^{-\lambda^2\tau} f(\xi,\tau)\,d\tau \right] d\xi \right| e^{\lambda^2 t}\,d\lambda . \]

For the proof of this theorem it is sufficient to repeat the arguments carried out in the proofs of the corresponding Theorems 38 and 39 in [3].

Literature

1. Günter N. M. Theory of Potential and Its Application to the Basic Problems of Mathematical Physics. GITTL, 1953.
2. Kupradze V. D. Methods of Potential in the Theory of Elasticity. Fizmatgiz, 1963.
3. Rasulov M. L. The Method of the Contour Integral. Publishing House “Nauka,” 1964.
4. Rasulov M. L. The method of the contour integral and its application to the solution of multidimensional mixed problems for differential equations of parabolic type. Matem. sb., 60:4, 1963.
5. Rasulov M. L. DAN SSSR, 125, No. 1, 1959.
6. Rasulov M. L. DAN SSSR, 125, No. 2, 1959.

Received by the editors
April 15, 1966

Azerbaijan State University
named after S. M. Kirov