ON BOUNDARY VALUE PROBLEMS FOR EQUATIONS OF PARABOLIC TYPE WITH DISCONTINUOUS COEFFICIENTS

G. K. NAMAZOV

(Presented by Academician I. G. Petrovskii on 28 III 1962)

In the works of O. A. Oleinik \((^{1,2})\), the solution of boundary value problems for elliptic and parabolic equations of the second order with discontinuous coefficients was obtained as the limit of solutions of the corresponding boundary value problems for equations with smooth coefficients. Such an approach, based on a certain smoothing of the coefficients of the equation, the use of a priori estimates, and the embedding theorems of S. L. Sobolev \((^3)\) and S. M. Nikol’skii \((^4)\), makes it possible to obtain the solution of boundary value problems for a broad class of conjugation conditions on the surfaces of discontinuity of the coefficients and to investigate the smoothness of the solution obtained.

In the present note, by applying the methods developed in \((^{1,2})\), the solution of certain boundary value problems for parabolic equations of the second order with discontinuous coefficients is obtained.

1. Let a bounded domain \(D\) of the space \(x=(x_1,\ldots,x_n)\) be divided into a finite number of subdomains \(D_r\) \((r=1,2,\ldots,m)\) by \((n-1)\)-dimensional smooth nonintersecting surfaces. Denote by \(\Gamma\) the boundary of the domain \(D\); by \(\Gamma_{ik}\) the boundary separating the domains \(D_i\) and \(D_k\) \((i,k=1,2,\ldots,m)\); \(S=\Gamma \times [0,T]\), \(Q_r=D_r\times(0,T]\), \(r=1,2,\ldots,m\), \(S_{ik}=\Gamma_{ik}\times[0,T]\). In the cylinder \(Q=D\times(0,T]\) consider the parabolic equation

\[ A_0(x,t)\frac{\partial U}{\partial t} = \sum_{i,j=1}^{n} \frac{\partial}{\partial x_i} \left( A_{ij}(x,t)\frac{\partial U}{\partial x_j} \right) + \sum_{i=1}^{n} B_i(x,t)\frac{\partial U}{\partial x_i} + C(x,t)U+F(x,t), \tag{1} \]

whose coefficients are sufficiently smooth functions in \(\overline{Q}-\sum S_{ik}\) and have discontinuities of the first kind at the points of \(S_{ik}\). Let

\[ A_0 \geq \lambda_0>0,\qquad A_{ij}=A_{ji},\qquad \sum_{i,j=1}^{n} A_{ij}\xi_i\xi_j \geq \lambda \sum_{i=1}^{n}\xi_i^2,\qquad \lambda>0,\qquad C(x,t)\leq 0. \tag{2} \]

We shall assume that in a neighborhood of any point of the surface \(\Gamma_{ik}\) one can introduce local coordinates \(y_i=z_i(x_1,\ldots,x_n)\), \(i=1,2,\ldots,n\), under which \(\Gamma_{ik}\) passes into the plane \(y_n=0\), where the \(z_i(x_1,\ldots,x_n)\) are sufficiently smooth functions.

We seek a function \(U(x,t)\), continuous in \(Q\), which satisfies equation (1) at all points of \(Q-\sum S_{ik}\), the boundary conditions

\[ U\big|_{S}=0,\qquad U\big|_{t=0}=0 \quad \text{on } D-\sum \Gamma_{ik} \tag{3} \]

and conjugation conditions of the form

\[ l_{pq}U \equiv K_p\frac{dU}{dN_p} + K_q\frac{dU}{dN_q} + \delta_{pq}U = \varkappa_{pq} \quad \text{on } S_{pq}, \tag{4} \]

where \(K_i \geq K_0>0\), \(\delta_{pq}\geq 0\), and \(\varkappa_{pq}\) are known functions on \(S_{pq}\),

\[ \frac{d}{dN_p} = \sum_{i,j=1}^{n} A_{ij}^{p}(x,t)\cos(\nu_p,x_i)\frac{\partial}{\partial x_j}, \]

\(\nu_p\) is the normal to \(S_{pq}\), exterior with respect to the domain \(Q_p\), \(A_{ij}^{p}\) is the limiting value of \(A_{ij}\) on the boundary of the domain \(Q_p\).

We shall consider only those solutions \(U(x,t)\) of this problem which have derivatives \(\partial U/\partial x_i \in L_2(Q)\), \(i=1,2,\ldots,n\). In what follows we use the notation adopted in (5).

Theorem 1. Let \(\Gamma\) belong to the class \(A^{(2)}\), \(S_{pq}\) to the class \(A^{(l+2)}\); the functions
\[ A_{ij}\in C^{(l+1)}(\overline{Q}_r),\quad B_i,\ C,\ A_0,\ F\in C^{(l)}(\overline{Q}_r)\quad (r=1,2,\ldots,m), \]
\[ K_p\in C^{(l+1)}(\overline{S}_{pq}),\quad \delta_{pq},\varkappa_{pq}\in C^{(l)}(\overline{S}_{pq}). \]
Then, for \(l>n+k+1\) and \(k\ge 2\), there exists a unique solution \(U(x,t)\) of the problem (1), (3), (4). This solution belongs to the class \(C^{(k)}(\overline{Q}\cap \overline{Q}_r)\), \(r=1,2,\ldots,m\).

We shall obtain the solution of the problem (1), (3), (4) in the following way. Cover the domain \(D\) by a finite number of sufficiently small domains \(\omega_k\), \(k=1,2,\ldots,N\), such that in a domain \(\omega_k\) containing points of the surface \(\Gamma_{pq}\) it is possible to pass to local coordinates \(y_1^k,\ldots,y_n^k\) in such a way that \(y_n^k=y_n^l\) in the intersection of \(\omega_k\) and \(\omega_l\). Let \(e_k(x,t)\), \(k=1,2,\ldots,N\), be infinitely differentiable functions, equal to zero outside \(\omega_k\times[0,T]\) and in some neighborhood of its boundary, and such that everywhere in \(\overline{Q}\)
\[ 0\le e_k(x,t)\le 1,\qquad \sum_{k=1}^{n} e_k(x,t)\equiv 1. \]

By \(\varkappa_r\) and \(\delta_r\) denote smooth functions in \(\overline{Q}_r\) such that \(\varkappa_p-\varkappa_q=\varkappa_{pq}\), \(\delta_p-\delta_q=\delta_{pq}\) on the surface \(S_{pq}\). Let \(K_r(x,t)\) be a smooth function in \(\overline{Q}_r\), coinciding with \(K_r\) on the boundary of \(Q_r\). Denote by \(K(x,t)\) the function in \(Q\) equal to \(K_r(x,t)\) in the domain \(Q_r\). Define \(\varkappa\) and \(\delta\) analogously.

Construct the following functions:
\[ \varkappa_k^h=e_k(\varkappa)_k^h,\qquad \delta_k^h=e_k(\delta)_k^h,\qquad k=1,2,\ldots,N; \]
\[ A_0^h=\sum_{k=1}^{N} e_k(KA_0)_r^h,\qquad A_{ij}^h=\sum_{k=1}^{N} e_k(KA_{ij})_k^h, \]
\[ B_i^h=\sum_{k=1}^{N} e_k\left(KB_i-\sum_{j=1}^{N}\frac{\partial K}{\partial x_j}A_{ij}\right)_k^h,\qquad C^h=\sum_{k=1}^{N} e_k(KC)_k^h-\sum_{k=1}^{N}\left(\frac{\partial(\delta e_k)}{\partial y_n^k}\right)_k^h, \]
\[ F^h=\sum_{k=1}^{N} e_k(KF)_k^h-\sum_{k=1}^{N}\left(\frac{\partial(\varkappa e_k)}{\partial y_n^k}\right)_k^h, \]
where \((\theta)_k^h\) denotes the averaging (see (3)) of the function \(\theta\) in the domain \(\omega_k\times[0,T]\), with averaging radius \(h\) with respect to the variables \(y_1^k,\ldots,y_n^k,t\).

Consider the sequence of equations
\[ \sum_{i,j=1}^{n}\frac{\partial}{\partial x_i}\left(A_{ij}^h\frac{\partial U_h}{\partial x_j}\right) +\sum_{i=1}^{n}B_i^h\frac{\partial U_h}{\partial x_i} +C^hU_h+\sum_{k=1}^{N}\frac{\partial\delta_k^h}{\partial y_n^k}U_h \]
\[ = A_0^h\frac{\partial U_h}{\partial t}-F^h+\sum_{k=1}^{N}\frac{\partial\varkappa_k^h}{\partial y_n^k}, \qquad 0<h<1. \tag{5} \]

Transforming in (5) the terms containing derivatives with respect to \(y_n^1,\ldots,y_n^N\), we obtain
\[ \sum_{i,j=1}^{n}\frac{\partial}{\partial x_i}\left(A_{ij}^h\frac{\partial U_h}{\partial x_j}\right) +\sum_{i=1}^{n}B_i^h\frac{\partial U_h}{\partial x_i} +C^hU_h+\sum_{k=1}^{N}\sum_{i=1}^{n}\mu_{ik}\frac{\partial\delta_k^h}{\partial x_i}U_h \]
\[ = A_0^h\frac{\partial U_h}{\partial t}-F^h+\sum_{k=1}^{N}\sum_{i=1}^{n}\mu_{ik}\frac{\partial\varkappa_k^h}{\partial x_i}, \tag{6} \]
where \(\mu_{ik}\) are some smooth functions.

We shall obtain the solution of problem (1), (3), (4) as the limit, as \(h \to 0\), of the solutions of equations (6) under the boundary conditions

\[ U_h\big|_{t=0}=0,\qquad U_h\big|_{S}=0. \tag{7} \]

It is easy to show that \(U_h\) and \(\partial U_h/\partial x_i\) are uniformly bounded in the norm \(L_2(Q)\). Further, analogously to how this was done in \(\left(^{2}\right)\), we establish integral estimates for derivatives of higher order and, using the theorems of S. L. Sobolev \(\left(^{3}\right)\) and S. M. Nikol’skii \(\left(^{4}\right)\), obtain the existence of a limit \(U(x,t)\) for the sequence \(U_h\) as \(h\to 0\), possessing the differentiability properties indicated in Theorem 1.

The fulfillment of the conjugation condition is proved in the same way as in \(\left(^{2}\right)\). The continuity of \(U(x,t)\) at the points of \(S\) and \(D-\sum \Gamma_{ik}\) is established with the aid of barriers. The uniqueness of the solution of problem (1), (3), (4) is established by multiplying the equation satisfied by the difference of two solutions of the problem \(U_1\) and \(U_2\) by the function \((U_1-U_2)\psi_\varepsilon\), where \(\psi_\varepsilon\) denotes a function equal to zero in an \(\varepsilon\)-neighborhood of the boundary \(S\) and to one outside a \(2\varepsilon\)-neighborhood of this boundary, and integrating the resulting equality over the domain \(Q\). Integrating by parts and letting \(\varepsilon\) tend to zero, we obtain \(U_1 \equiv U_2\).

Let us note that, analogously, one can obtain the existence and uniqueness of the solution of boundary-value problems for general elliptic and hyperbolic equations of second order with discontinuous coefficients under conjugation conditions of the form (4) on the surfaces of discontinuity of the coefficients.

1. Consider the linear parabolic equation

\[ \frac{\partial U}{\partial t} = \sum_{i,j=1}^{n} A_{ij}(x,t)\frac{\partial^2 U}{\partial x_i\partial x_j} + \sum_{i=1}^{n} B_i(x,t)\frac{\partial U}{\partial x_i} + C(x,t)U + F(x,t) \tag{8} \]

in the cylinder \(Q=D\times(0,T]\). Let \(A_{ij}(x,t)\), \(B_i(x,t)\), \(C(x,t)\), and \(F(x,t)\) be smooth functions everywhere in the domain \(Q\), with the exception of a finite number of smooth surfaces \(S_r\) \((r=1,2,\ldots,m)\), where these functions have discontinuities of the first kind. Let the \(S_r\) divide \(Q\) into \(m+1\) domains \(Q_r\), and suppose that condition (2) is fulfilled in \(Q\).

Assume that in some neighborhood of any point there is a possible nonsingular change of coordinates

\[ \tau=t,\qquad y_i=y_i(x_1,\ldots,x_n,t),\qquad i=1,2,\ldots,n, \tag{9} \]

under which the surface of discontinuity of the coefficients is transformed into a piece of the plane \(y_n=0\).

We shall seek a continuous function \(U(x,t)\) in \(\overline{Q}\), having continuous first derivatives in \(Q\), satisfying equation (8) in \(Q-\sum_{i=1}^{N}S_i\) and the boundary conditions

\[ U\big|_{t=0}=\varphi,\qquad U\big|_{S}=\psi, \tag{10} \]

where \(\varphi\) and \(\psi\) are arbitrary continuous functions such that \(\psi|_{t=0}=\varphi|_{\Gamma}\). We note that this problem was considered by I. V. Girsanov in \(\left(^{6}\right)\) in the case when the coefficients \(A_{ij}\), \(B_i\) and the surfaces of discontinuity \(S_r\) do not depend on \(t\).

Theorem 2. Let the functions \(A_{ij}\), \(B_i\), \(C\), \(F\) belong to \(C^{(0,\mu)}(\overline{Q}_r)\), \(0<\mu<1\), \(r=1,2,\ldots,m+1\), and to the class \(C^{(l)}\) in some neighborhood of \(S_r\), \(l\ge n+1\), with \(S_r\in A^{(l+2)}\), and let the surface \(\Gamma\) be such that each of its points can be touched by some ball lying outside \(Q\). Then there exists in \(Q\) a unique solution of problem (8), (10).

The solution of problem (8), (10) can be obtained as the limit, as \(h \to 0\), of solutions of the equations

\[ \frac{\partial U_h}{\partial t} = \sum_{i,j=1}^{n} A_{ij}^{h}\frac{\partial^2 U_h}{\partial x_i\,\partial x_j} + \sum_{i=1}^{n} B_i^{h}\frac{\partial U_h}{\partial x_i} + C^{h}U_h+F^{h} \tag{11} \]

with boundary conditions

\[ U_h\big|_{t=0}=\varphi,\qquad U_h\big|_{S}=\psi . \tag{12} \]

Here

\[ A_{ij}^{h}=\sum_{k=1}^{N_0} e_k(A_{ij})_k^h;\quad B_i^{h}=\sum_{k=1}^{N_0} e_k(B_i)_k^h;\quad C^{h}=\sum_{k=1}^{N_0} e_k(C)_k^h \quad\text{and}\quad F^{h}=\sum_{k=1}^{N_0} e_k(F)_k^h, \]

\(e_k\) is an infinitely differentiable function equal to zero outside \(\Omega_k\) \((k=1,2,\ldots,N_0)\);

\[ \sum_{k=1}^{N_0}\Omega_k=Q;\qquad 0\le e_k(x,t)\le 1;\qquad \sum_{k=1}^{N_0} e_k=1, \]

and the domains \(\Omega_k\) are so small that in each of them a change of coordinates of the form (9) is possible, with \(y_n^k=y_n^l\) in the intersection of \(\Omega_k\) and \(\Omega_l\).

It follows from the maximum principle that the \(U_h\) are uniformly bounded with respect to \(h\) in \(Q\). Applying the a priori estimates of work (7), we obtain compactness, in the sense of uniform convergence of \(U_h\) and of all derivatives entering equation (11), in any closed domain \(Q'\subset Q\) not containing points of \(S_r\). Next, considering equation (11) in local coordinates \(y_1,y_2,\ldots,y_n,\tau\) in a neighborhood of a point \(P_0\in S_r\), we establish a priori estimates for derivatives of \(U_h\) up to order \(l\), involving no more than two differentiations with respect to \(y_n\). Applying the theorems of S. M. Nikol’skii (4), we obtain that the limiting function \(U(x,t)\) for \(U_h\) as \(h\to 0\) satisfies the required conditions in \(Q\).

The fulfillment of conditions (10) is proved with the aid of barriers. The uniqueness of the solution of problem (8), (10) follows from work (8).

In conclusion I express my sincere gratitude to Prof. O. A. Oleinik for guidance and assistance in writing this work.

Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR

Received
27 III 1962

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