THE FOURIER METHOD FOR NONSTATIONARY EQUATIONS WITH GENERAL BOUNDARY CONDITIONS

N. I. Brish, I. N. Valeshkevich

1. Let in a bounded domain \(\Omega\) with boundary \(S\) of the space \(x=(x_1,\ldots,x_n)\) there be given a differential operator of order \(2m\)

\[ Au=\sum_{|\alpha|\leq 2m} a_\alpha(x)D^\alpha u, \tag{1.1} \]

where \(\alpha=(\alpha_1,\ldots,\alpha_n)\) is a multi-index, \(|\alpha|=\alpha_1+\cdots+\alpha_n\), \(D=(D_1,\ldots,D_n)\), \(D_k=\frac{1}{i}\frac{\partial}{\partial x_k}\), \(D^\alpha=D_1^{\alpha_1}\cdots D_n^{\alpha_n}\). Suppose that the operator \(A\) is positively elliptic in \(\overline{\Omega}\). On the surface \(S\) we prescribe a normal system [1] of \(m\) boundary operators of orders \(k_j\leq 2m-1\)

\[ B_j u=\sum_{|\beta|\leq k_j} b_{\beta,j}(x)D^\beta u\quad (j=1,\ldots,m), \tag{1.2} \]

where \(0\leq k_1<k_2<\cdots<k_m\leq 2m-1\). We subject the operators \(B_j\) to the covering condition [1] (the complementing condition in [2]).

Consider in the cylinder \(Q=\Omega\times(0,T)\) the equation

\[ \frac{\partial^2 u}{\partial t^2}+Au=f(x,t). \tag{1.3} \]

For equation (1.3) we pose the following mixed problem: find in \(Q\) a solution \(u(x,t)\) of this equation satisfying the conditions

\[ \left.u\right|_{t=0}=\varphi(x),\qquad \left.\frac{\partial u}{\partial t}\right|_{t=0}=\psi(x), \tag{1.4} \]

\[ \left.B_j u\right|_{\Gamma}=0\qquad (j=1,\ldots,m), \tag{1.5} \]

where \(\Gamma=S\times[0,T]\) is the lateral surface of the cylinder \(Q\).

By a solution of problem (1.3)—(1.5) we mean a solution from the Sobolev functional space \(W_{x,t,2}^{2m,2}(Q)\). Recall that \(u\in W_{x,t,2}^{km,k}(Q)\) if the generalized derivatives \(D^\alpha \partial^s u/\partial t^s\in L_2(Q)\) for \(|\alpha|+sm\leq km\). The square of the norm \(\|u\|_{km,k}\) of a function \(u\in W_{x,t,2}^{km,k}(Q)\) is equal to the sum of the squares of the norms in \(L_2(Q)\) of such derivatives. We shall denote the norm in \(W_2^k(\Omega)\) by \(\|u\|_k\) (\(\|u\|_0=\|u\|\)), and the scalar product in \(L_2(\Omega)\) by \((u,v)\).

We shall prove the solvability of this problem by the Fourier method. The analogous problem for the case when \(B_j=\dfrac{\partial^{j-1}}{\partial \nu^{j-1}}\) was considered in [3].

1. In attempting to apply the method of separation of variables to the mixed problem (1.3)—(1.5), we arrive at the necessity of studying the following eigenvalue problem:

\[ Av=\lambda v, \tag{2.1} \]

\[ B_jv\big|_s=0\quad (j=1,\ldots,m). \tag{2.2} \]

We shall present some results pertaining to this problem. Suppose that \(S\in C^{2m}\), \(a_\alpha\in C^{|\alpha|}(\overline{\Omega})\), and \(b_{\beta,j}\in C^{2m-k_j}(S)\). Let \(A'\) be the operator formally adjoint to \(A\). It is defined by the formula

\[ A'u=\sum_{|\alpha|\le 2m} D^\alpha\bigl(\overline{a_\alpha}(x)u\bigr). \tag{2.3} \]

For \(l>k_j\ (j=1,\ldots,p)\), denote by \(W_2^l(\Omega,\{B_j\}_1^p)\) the subspace of functions \(u(x)\) from \(W_2^l(\Omega)\) that satisfy the boundary conditions \(B_ju|_s=0\ (j=1,\ldots,p)\). It is known [1] that for the system \(B_j\ (j=1,\ldots,m)\) one can find another normal system \(B'_j\ (j=1,\ldots,m)\) of boundary differential operators, formally adjoint to \(B_j\) with respect to \(A\), such that \(v\in W_2^{2m}(\Omega)\) satisfies the equation

\[ (Au,v)=(u,A'v) \tag{2.4} \]

for all \(u\in W_2^{2m}(\Omega,\{B_j\}_1^m)\) if and only if \(v\in W_2^{2m}(\Omega,\{B'_j\}_1^m)\). The system \(B'_j\) is not determined uniquely. However, any two systems \(B'_j\) and \(\widetilde{B}'_j\) formally adjoint to \(B_j\) are equivalent in the sense that the sets \(W_2^{2m}(\Omega,\{B'_j\}_1^m)\) and \(W_2^{2m}(\Omega,\{\widetilde{B}'_j\}_1^m)\) coincide.

Suppose that \(A\) is formally self-adjoint, i.e., \(A\) coincides with \(A'\). Let the system \(B_j\) be such that \(B'_j\) is equivalent to \(B_j\). In this case the system \(B_j\) and the problem (2.1), (2.2) are called formally self-adjoint. The existence of such systems \(B_j\) follows from [4].

Finally, suppose that there exists a Hermitian integro-differential bilinear form

\[ a(u,v)=\int_{\Omega}\sum_{|\alpha|,|\beta|\le m} a_{\alpha,\beta}(x)D^\alpha u\,\overline{D^\beta v}\,dx, \]

such that, for \(u,v\in W_2^{2m}(\Omega,\{B_j\}_1^m)\),

\[ (Au,v)=a(u,v), \tag{2.5} \]

and the corresponding quadratic form \(a(u,u)\) is coercive [5] on \(W_2^m(\Omega,\{B_j\}_1^r)\), where \(k_r\le m-1\), and \(k_{r+1}\ge m\). The latter means the existence of constants \(c\ge 0\) and \(c_1>0\) such that, for all \(u\in W_2^m(\Omega,\{B_j\}_1^r)\), the inequality

\[ a(u,u)\ge c_1\|u\|_m^2-c\|u\|^2 \tag{2.6} \]

holds.

Here the set \(\{B_j\}_1^r\) may be empty. In this case we assume that \(W_2^m(\Omega,\{B_j\}_1^r)=W_2^m(\Omega)\).

Consider \(A\) as an operator in \(L_2(\Omega)\) with domain of definition \(D(A)=W_2^{2m}(\Omega,\{B_j\}_1^m)\). It can be shown that the operator \(A\) is self-adjoint in \(L_2(\Omega)\). From a priori estimates for solutions of elliptic equations [2] it follows that, for \(\mu\) belonging to the resolvent

set \(A\), the operator \((A-\mu I)^{-1}\) is completely continuous in \(L_2(\Omega)\). Therefore problem (2.1), (2.2) has a countable set of real eigenvalues \(\{\lambda_k\}\), with \(|\lambda_k|\to\infty\) as \(k\to\infty\), if they are numbered in the appropriate way. To each \(\lambda_k\) there corresponds only a finite number of orthonormal eigenfunctions \(v_k \in W_2^{2m}(\Omega,\{B_j\}_1^m)\). The system \(\{v_k\}\) is complete in \(L_2(\Omega)\). From (2.5) and (2.6) it follows that among the \(\{\lambda_k\}\) there can be only a finite number of negative ones. In this case we number the system \(\{\lambda_k\}\) in increasing order.

It is easy to prove that the system \(\{v_k\}\) is complete in the spaces \(W_2^m(\Omega,\{B_j\}_1^l)\) and \(\overset{\circ}{W}{}_2^{2m}(\Omega,\{B_j\}_1^m)\). We denote by \(\varphi_k,\psi_k,f_k(t)\) the Fourier coefficients of the functions \(\varphi(x),\psi(x)\), and \(f(x,t)\) with respect to the orthonormal system \(\{v_k\}\). The following is true.

Lemma 1. If \(a_\alpha \in C^{|\alpha|+m(k-2)}(\overline{\Omega})\), \(\varphi \in W_2^{km}(\Omega)\), the functions \(\varphi,A\varphi,\ldots,A^{\left[\frac{k-2}{2}\right]}\varphi \in W_2^{2m}(\Omega,\{B_j\}_1^m)\) and, moreover, for odd \(k\),
\(A^{\frac{k-1}{2}}\varphi \in W_2^m(\Omega,\{B_j\}_1^l)\), then the equality holds

\[ \sum_{j=1}^{\infty}\lambda_j^k|\varphi_j|^2= \begin{cases} \left\|A^{\frac{k}{2}}\varphi\right\|^2, & k\ \text{even},\\[6pt] a\left(A^{\frac{k-1}{2}}\varphi,A^{\frac{k-1}{2}}\varphi\right), & k\ \text{odd}. \end{cases} \tag{2.7} \]

With the aid of Lemma 1 one proves

Lemma 2. If \(a_\alpha \in C^{|\alpha|+m(k-2)}(\overline{\Omega})\), \(f(x,t)\in W_{x,t,2}^{m(k-1),\,k-1}(Q)\), the functions
\(f,Af,\ldots,A^{\left[\frac{k-3}{2}\right]}f \in W_{x,t,2}^{2m,2}(Q,\{B_j\}_1^m)\) and, moreover, for even \(k\),
\(A^{\frac{k-2}{2}}f \in W_{x,t,2}^{m,1}(Q,\{B_j\}_1^l)\), then

\[ \sum_{j=1}^{\infty}\lambda_j^{k-s-1}\int_0^T |f_j^{(s)}(t)|^2\,dt \le c_2\|f\|_{m(k-1),\,k-1}^2, \tag{2.8} \]

where \(s=0,1,\ldots,k-1\), and

\[ \sum_{j=1}^{\infty}\lambda_j^{k-s-1}|f_j^{(s-1)}(t)|^2 \le c_3\|f\|_{m(k-1),\,k-1}^2, \tag{2.9} \]

where \(s=1,\ldots,k-1\). Moreover, the series (2.9) converge uniformly on \([0,T]\).

1. We pass to the solution of the mixed problem (1.3)—(1.5).

Theorem 1. The solution of the mixed problem (1.3)—(1.5) is unique.

Separating the variables, we obtain the formal solution of problem (1.3)—(1.5) in the following form:

\[ u(x,t)=\sum_{j=1}^{\infty}v_j(x)\left(\varphi_j\cos\sqrt{\lambda_j}\,t+ \frac{\psi_j}{\sqrt{\lambda_j}}\sin\sqrt{\lambda_j}\,t\right)+ \]

\[ +\sum_{j=1}^{\infty}\frac{v_j(x)}{\sqrt{\lambda_j}} \int_0^t f_j(\tau)\sin\sqrt{\lambda_j}\,(t-\tau)\,d\tau . \tag{3.1} \]

In this case those terms of the series (3.1) for which \(\lambda_j=0\) are defined by continuity (for example,
\[ \left.\frac{\sin\sqrt{\lambda_j}\,t}{\sqrt{\lambda_j}}\right|_{\lambda_j=0} =\lim_{\lambda\to 0}\frac{\sin\sqrt{\lambda}\,t}{\sqrt{\lambda}}=t \]
) and \(\sqrt{\lambda_j}=\sqrt{|\lambda_j|}\,e^{\frac{i}{2}\arg\lambda_j}\). The completeness of the system of eigenfunctions of the problem (2.1), (2.2) and the following theorem provide a justification of the formal Fourier method scheme for the solution of the mixed problem (1.3)—(1.5).

Theorem 2. Suppose that the following conditions are fulfilled: 1) \(\Omega\) is a bounded domain, bounded by a surface \(S\) of class \(C^{km}\); 2) the coefficients \(a_\alpha\in C^{|\alpha|+m(k-2)}(\overline{\Omega})\), \(b_{\beta,j}\in C^{km-k_j}(S)\); 3) \(\varphi\in W_2^{km}(\Omega)\), the functions \(\varphi,A\varphi,\ldots,A^{\left[\frac{k-2}{2}\right]}\varphi\in W_2^{2m}(\Omega,\{B_j\}_1^m)\) and, moreover, for odd \(k\), \(A^{\frac{k-1}{2}}\varphi\in W_2^m(\Omega,\{B_j\}_1^r)\); 4) \(\psi\in W_2^{m(k-1)}(\Omega)\), the functions \(\psi,A\psi,\ldots,A^{\left[\frac{k-3}{2}\right]}\psi\in W_2^{2m}(\Omega,\{B_j\}_1^m)\) and, moreover, for even \(k\), \(A^{\frac{k-2}{2}}\psi\in W_2^m(\Omega,\{B_j\}_1^r)\); 5) \(f(x,t)\in W_{x,t,2}^{m(k-1),\,k-1}(Q)\), the functions \(f,Af,\ldots,A^{\left[\frac{k-3}{2}\right]}f\in W_{x,t,2}^{2m,\,2}(Q,\{B_j\}_1^m)\) and, moreover, for even \(k\), \(A^{\frac{k-2}{2}}f\in W_{x,t,2}^{m,\,1}(Q,\{B_j\}_1^r)\). Then the series (3.1) converges in the norm \(W_2^{km}(\Omega)\) uniformly with respect to \(t\in[0,T]\), and the series obtained by termwise differentiating it \(l\) times with respect to \(t\) \((l\le k)\) converges in the norm \(W_2^{m(k-l)}(\Omega)\) uniformly in \(t\in[0,T]\). In this case the sum \(u(x,t)\) of the series (3.1) is a solution of the problem (1.3)—(1.5) in the space \(W_{x,t,2}^{km,\,k}(Q)\), and for all \(t\in[0,T]\) the inequality holds

\[ \sum_{l=0}^{k}\left\|\frac{\partial^l u}{\partial t^l}\right\|_{m(k-l)}^2 \le c_4\bigl(\|\varphi\|_{km}^2+\|\psi\|_{m(k-1)}^2+\|f\|_{m(k-1),\,k-1}^2\bigr), \tag{3.2} \]

where the constant \(c_4\) depends on \(T\), the domain \(\Omega\), \(a_\alpha\), and \(b_{\beta,j}\).

The proof of Theorem 2 is based on the following

Lemma 3. If conditions 1) and 2) of the theorem are fulfilled, then for any function \(\varphi\) satisfying condition 3) of this theorem, the inequality holds

\[ \|\varphi\|_{km}^2 \le \begin{cases} c_5\bigl(\|A^{\frac{k}{2}}\varphi\|^2+\|\varphi\|^2\bigr), & k\ \text{even},\\[6pt] c_5\bigl(|a(A^{\frac{k-1}{2}}\varphi,A^{\frac{k-1}{2}}\varphi)|+\|\varphi\|^2\bigr), & k\ \text{odd}. \end{cases} \tag{3.3} \]

Lemma 3 is proved with the aid of inequality (2.6) and a priori estimates in the norms \(W_2^l(\Omega)\) for solutions of boundary-value problems for elliptic equations, which were obtained by various authors (see, for example, [2]). Inequality (3.3) in the case of even \(k\) is also valid for a non-self-adjoint operator \(A\), and the restrictions on \(a_\alpha\) can be weakened (it is sufficient to require, for example, that \(a_\alpha\in C^{m(k-2)}(\overline{\Omega})\)). Its proof does not depend on inequality (2.6). We also note that in the case when zero is not an eigenvalue of the problem (2.1), (2.2), we may omit the term \(\|\varphi\|\) on the right-hand side of inequality (3.3). In particular, this term may be omitted for \(\varphi\) orthogonal to the null subspace of the operator \(A\).

Theorem 2 is proved by direct application of Lemmas 1—3. For the particular case of the mixed problem considered, detailed proofs of the propositions formulated here are given in [7].

The coercivity condition for the form \(a(u,u)\) is an essential restriction for the correct solvability of the mixed problem (1.3)—(1.5). This condition is equivalent to the semiboundedness from below of the operator \(A\) on the set \(W_2^{2m}(\Omega,\{B_j\}_1^m)\) [6]. Consequently, if it is not satisfied, then the problem (2.1), (2.2) has an infinite set of negative eigenvalues. In this case it is not difficult to show that the mixed problem (1.3)—(1.5) is posed incorrectly.

Indeed, let \(\lambda_p<0\), \(\lambda_p\to-\infty\) as \(p\to\infty\). Consider the mixed problem (1.3)—(1.5) for \(f=0\), \(\varphi=0\), and \(\psi=e^{-\sqrt[4]{|\lambda_p|}}v_p(x)\), where \(v_p\) is the eigenfunction corresponding to \(\lambda_p\). The solution of this problem has the form

\[ u=|\lambda_p|^{-\frac12} e^{-\sqrt[4]{|\lambda_p|}}\operatorname{sh}\sqrt{|\lambda_p|}\, t\cdot v_p(x). \]

Since, by Lemma 3 (for the case of even \(k\)),

\[ \|\psi\|_{km}^{2}\le c_6 e^{-2\sqrt[4]{|\lambda_p|}}(|\lambda_p|^k+1), \]

then, for sufficiently large \(p\), the function and its derivatives up to order \(km\) will be arbitrarily small in the mean. However, by direct calculation it is easy to verify that the integral of \(|u|^2\) over the cylinder \(Q\) of arbitrarily small height \(T\) will take arbitrarily large values if \(p\) is sufficiently large. This implies our assertion.

A concrete example of such an incorrectly posed problem is the mixed problem (1.3)—(1.5) in the case when \(A=\Delta^2\) (\(\Delta\) is the Laplace operator, \(n=2\)), and the boundary operators have the form

\[ B_1u=\sigma\Delta u-(1-\sigma)(\nu_1D_1+\nu_2D_2)^2u, \]

\[ B_2u=\frac{\partial\Delta u}{\partial\nu}+(1-\sigma)\frac{\partial}{\partial s}\bigl[(D_1^2u-D_2^2u)\nu_1\nu_2+(\nu_2^2-\nu_1^2)D_1D_2u\bigr], \]

if the constant \(\sigma<-3\) or \(\sigma>1\). Here \(\nu=(\nu_1,\nu_2)\) is the unit vector of the exterior normal, and \(s\) is the length of the arc of the contour. The corresponding quadratic form [7]

\[ a(u,u)=\iint_{\Omega}\{|\Delta u|^2+2(1-\sigma)[|D_1D_2u|^2-\operatorname{Re}(D_1^2u\overline{D_2^2u})]\}\,dx_1dx_2 \]

is coercive in the space \(W_2^m(\Omega)\) for \(-3<\sigma<1\), but is not coercive for \(\sigma\le -3\) or \(\sigma\ge 1\) [6]. All the remaining conditions here are satisfied for \(\sigma\ne -3\) and \(\sigma\ne 1\). Consequently, the mixed problem under consideration is posed correctly for \(-3<\sigma<1\), and incorrectly for \(\sigma<-3\) or \(\sigma>1\).

Let us note that for \(0\le\sigma<1\) this problem plays an important role in the theory of vibrations of plates. Problems of this kind for arbitrary \(n\ge2\) are considered in [7].

1. We have set forth the justification of the Fourier method for nonstationary equations containing the second derivative with respect to time. The case of nonstationary equations with the first derivative with respect to time, of the form

\[ \frac{\partial u}{\partial t}+Au=f(x,t), \tag{4.1} \]

\[ \frac{1}{i}\,\frac{\partial u}{\partial t}+Au=f(x,t) \tag{4.2} \]

under the boundary conditions (1.5) and the initial condition

\[ u\big|_{t=0}=\varphi(x). \tag{4.3} \]

This case is simpler; therefore we shall confine ourselves to the formulation of the final results. We consider the solutions of these problems in the spaces \(W_{x,t,2}^{2km,k}(Q)\).

Theorem 3. Suppose that the following conditions are satisfied:

1) \(\Omega\) is a bounded domain bounded by a surface \(S\) of class \(C^{2mk}\);

2) the coefficients \(a_\alpha \in C^{|\alpha|+2m(k-1)}(\overline{\Omega})\), \(b_{\beta,j}\in C^{2km-k_j}(S)\);

3) \(\varphi\in W_2^{(2k-1)m}(\Omega)\), and the functions \(\varphi, A\varphi,\ldots,A^{k-2}\varphi\in W_2^{2m}(\Omega,\{B_j\}_1^m)\) and \(A^{k-1}\varphi\in W_2^m(\Omega,\{B_j\}_1^r)\);

4) \(f\in W_{x,t,2}^{2m(k-1),k-1}(Q)\), and the functions \(f,Af,\ldots,A^{k-2}f\in W_{x,t,2}^{2m,1}(Q,\{B_j\}_1^m)\).

Then the series

\[ u(x,t)=\sum_{j=1}^{\infty} v_j(x)\left(\varphi_j e^{-\lambda_j t}+\int_0^t e^{-\lambda_j(t-\tau)}f_j(\tau)\,d\tau\right) \tag{4.4} \]

converges in the norm \(W_{x,t,2}^{2km,k}(Q)\), and its sum \(u(x,t)\) is a solution of the mixed problem (4.1), (4.3), (1.5). Moreover, the inequality

\[ \|u\|_{2km,k}\le c_7\bigl(\|\varphi\|_{(2k-1)m}+\|f\|_{2m(k-1),k-1}\bigr) \tag{4.5} \]

holds, where the constant \(c_7\) depends on \(T\), the domain \(\Omega\), \(a_\alpha\), and \(b_{\beta,j}\).

In the case of problem (4.1), (4.3), (1.5), an analogous remark holds concerning the correctness of its formulation, and the example given above of the operator \(A\) and the boundary operators \(B_j\) is, obviously, suitable in this case as well.

Passing to the consideration of the last mixed problem (4.2), (4.3), (1.5), we note that in this case the operator \(A\) need not be semibounded from below. The eigenvalue problem (2.1), (2.2) may have an infinite set of negative and positive eigenvalues.

Theorem 4. Suppose that conditions 1), 2) of Theorem 3 are satisfied, and that the following conditions are satisfied:

1) \(\varphi\in W_2^{2km}(\Omega)\), and the functions \(\varphi,A\varphi,\ldots,A^{k-1}\varphi\in W_2^{2m}(\Omega,\{B_j\}_1^m)\);

2) \(f\in W_{x,t,2}^{2km,k}(Q)\), and the functions \(f,Af,\ldots,A^{k-1}f\in W_{x,t,2}^{2m,1}(Q,\{B_j\}_1^m)\).

Then the series

\[ u(x,t)=\sum_{j=-\infty}^{+\infty} v_j(x)\left(\varphi_j e^{-i\lambda_j t}+i\int_0^t e^{-i\lambda_j(t-\tau)}f_j(\tau)\,d\tau\right) \tag{4.6} \]

converges in the norm \(W_2^{2km}(\Omega)\) uniformly with respect to \(t\in[0,T]\), and the series obtained by termwise differentiating it \(l\) times with respect to \(t\) \((l\le k)\) converges in the norm \(W_2^{2m(k-l)}(\Omega)\) uniformly in \(t\in[0,T]\). Moreover, the sum \(u(x,t)\) of the series (4.6) is a solution of problem (4.2), (4.3), (1.5) in the space \(W_{x,t,2}^{2km,k}(Q)\), and for all \(t\) the inequality

\[ \sum_{l=0}^{k}\left\|\frac{\partial^l u}{\partial t^l}\right\|_{2m(k-l)}^2 \le c_8\bigl(\|\varphi\|_{2km}^2+\|f\|_{2km,k}^2\bigr), \tag{4.7} \]

holds, where the constant \(c_8\) depends on \(T\), the domain \(\Omega\), \(a_\alpha\), and \(b_{\beta,j}\).

All the results carry over without essential changes to systems of the forms (1.3), (4.1), and (4.2), where \(Au=\sum_{|\alpha|\le p} a_\alpha(x)D^\alpha u\); \(u, f\) are column vectors of height \(N\); \(a_\alpha\) are \(N\times N\) matrices and \(Np\) is even, \(\frac{Np}{2}=m\), with boundary conditions of the form (1.5), where \(b_{\beta,j}\) are row vectors of length \(N\), \(j=1,\ldots,m\). In particular, such types of systems include the equations of elasticity theory and the Dirac equations. The system of Maxwell equations reduces to them.

References

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2. Agmon, S., Douglis, A., Nirenberg, L. Estimates of solutions of elliptic equations near the boundary. IL, Moscow, 1962.

3. Brish, N. I., Valeshkevich, I. N. DAN SSSR, 146, No. 6, 1962, pp. 1247–1250.

4. Aronszajn, N., Milgram, A. Differential operators on Riemannian manifolds. Rend. Circ. Mat. Palermo, ser. 2, 2, 1953, 1–61.

5. Agmon, S. The coerciveness problem for integro-differential forms. J. Analyse Math., 6, No. 2 (1958), 183–223.

6. Agmon, S. Remark on self-adjoint and semi-bounded elliptic boundary value problems. Proc. Internat. Sympos. Linear Spaces, Jerusalem, 1960; Jerusalem Acad. Press; Oxford-London-New York-Paris, Pergamon Press, 1961, 1–13.

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Received by the editors
December 21, 1964

Belorussian State University
named after V. I. Lenin