Full Text
Reports of the Academy of Sciences of the USSR
- Vol. 132, No. 5
MATHEMATICS
G. V. VIRABYAN
ON THE SPECTRUM OF AN OPERATOR AND ON THE DIRICHLET PROBLEM FOR THE EQUATION
\[ \square^{2}u+4\frac{\partial^{2}}{\partial t^{2}}\square u+2\frac{\partial^{4}u}{\partial t^{4}}=f(x,y,z,t) \]
(Presented by Academician S. L. Sobolev on 24 II 1960)
Let \(\Omega\) be a domain in the finite part of the four-dimensional space \(XYZT\), bounded by the surface
\(\Gamma(x,y,z,t)\equiv x^{2}+y^{2}+z^{2}+t^{2}-1=0\), the unit sphere with center at the origin.
\(1^\circ\). In the domain \(\Omega\) consider the Hilbert space \(H_B^*(\Omega)\), which is obtained by completing the linear manifold \(D_B\) of infinitely differentiable functions of finite support in \(\Omega\) (vanishing in some boundary strip of the domain \(\Omega\)) with scalar product
\[ \begin{aligned} (u,v)_B &=\idotsint_{\Omega}\biggl\{ \frac{\partial^{2}u}{\partial x^{2}}\frac{\partial^{2}v}{\partial x^{2}} +\frac{\partial^{2}u}{\partial y^{2}}\frac{\partial^{2}v}{\partial y^{2}} +\frac{\partial^{2}u}{\partial z^{2}}\frac{\partial^{2}v}{\partial z^{2}} +\frac{\partial^{2}u}{\partial t^{2}}\frac{\partial^{2}v}{\partial t^{2}} \\ &\quad +2\frac{\partial^{2}u}{\partial x\partial y}\frac{\partial^{2}v}{\partial x\partial y} +2\frac{\partial^{2}u}{\partial x\partial z}\frac{\partial^{2}v}{\partial x\partial z} +2\frac{\partial^{2}u}{\partial x\partial t}\frac{\partial^{2}v}{\partial x\partial t} +2\frac{\partial^{2}u}{\partial y\partial z}\frac{\partial^{2}v}{\partial y\partial z} \\ &\quad +2\frac{\partial^{2}u}{\partial y\partial t}\frac{\partial^{2}v}{\partial y\partial t} +2\frac{\partial^{2}u}{\partial z\partial t}\frac{\partial^{2}v}{\partial z\partial t} \biggr\}\,d\Omega +\iiint_{\Gamma}u\,d\sigma\cdot\iiint_{\Gamma}v\,d\sigma \\ &\quad +\iiint_{\Gamma}\frac{\partial u}{\partial n}\,d\sigma\cdot \iiint_{\Gamma}\frac{\partial v}{\partial n}\,d\sigma . \end{aligned} \tag{1} \]
For \(u,v\in D_B\), (1) takes the form
\[ (u,v)_B=\idotsint_{\Omega}\Delta^{2}u\cdot v\,d\Omega . \tag{1*} \]
In the space \(H_B^*(\Omega)\) consider the operator \(B^2\), defined by the formula
\(B^2=\Delta^{-2}\dfrac{\partial^{4}}{\partial t^{4}}\), where \(\Delta^{-2}\) is the operator inverse to the four-dimensional biharmonic Laplace operator under the boundary conditions
\[ u\big|_{\Gamma}=0,\qquad \frac{\partial}{\partial n}\bigg|_{\Gamma}=0; \]
\(n\) is the outward normal to the boundary surface \(\Gamma\).
Theorem 1. The operator \(B^2\) is a symmetric bounded and positive-definite operator on the dense manifold \(D_B\) of the space \(H_B^*(\Omega)\).
We shall denote the self-adjoint extension of the operator \(B^2\) in \(H_B^*(\Omega)\) by the same letter \(B^2\).
Theorem 2. The spectrum of the operator \(B^2\) in \(H_B^*(\Omega)\) is discrete.
Proof. The scheme of the proof of this theorem is essentially analogous to the method of P. Denchev \((^1)\). We shall briefly outline this scheme. Let \(R_n\) be the space of all polynomials of degree not exceeding \(n\); let \(N\) be its dimension. Introduce a scalar product in \(R_n\) by the formula: \((p,q)=\idotsint_{\Omega}pq\Gamma^2\,d\Omega\) for \(p,q\in R_n\). Consider in \(R_n\) the operators:
\[ L_1(p)=\frac{\partial^{4}}{\partial t^{4}}(\Gamma^{2}p),\qquad L_2(p)=\Delta^{2}(\Gamma^{2}p) \tag{2} \]
for all \(p\in R_n\). The operators \(L_1\) and \(L_2\) are symmetric and map \(R_n\) into itself; moreover, the operator \(L_2\) is positive. Then, by the well-known theorem from linear algebra [^1] on reducing two quadratic forms to a canonical basis, we conclude that there exist \(N\) numbers \(\lambda_1^2,\lambda_2^2,\ldots,\lambda_N^2\) and \(N\) linearly independent polynomials \(\gamma_1(x,y,z,t),\gamma_2(x,y,z,t),\ldots,\gamma_N(x,y,z,t)\) such that
\[ L_1(\gamma_k)-\lambda_k^2 L_2(\gamma_k)=0 \qquad (k=1,2,\ldots,N); \tag{3} \]
\[ (L_2\gamma_i,\gamma_j)=\delta_{ij} \qquad (i,j=1,2,\ldots,N). \tag{4} \]
The polynomials
\[ \chi_k(x,y,z,t)=\Gamma^2(x,y,z,t)\gamma_k(x,y,z,t) \qquad (k=1,2,\ldots,N) \tag{5} \]
will be eigenfunctions of the operator \(B^2\) in \(H_B^*(\Omega)\). Assigning the values \(n=1,2,\ldots\), we obtain an infinite system of polynomial eigenfunctions for the operator \(B^2\) in \(H_B^*(\Omega)\). This system will be complete in \(H_B^*(\Omega)\). Indeed, first note that every polynomial \(\mathcal P(x,y,z,t)\) that vanishes on the boundary surface \(\Gamma\) together with its normal derivative has the form
\[ \mathcal P(x,y,z,t)=\Gamma^2(x,y,z,t)\cdot P(x,y,z,t). \tag{6} \]
Next, every polynomial of the form (6) is a linear combination of the polynomial eigenfunctions \(\{\chi_k(x,y,z,t)\}\) of the operator \(B^2\) in \(H_B^*(\Omega)\). On the other hand, by polynomials of this form one can uniformly approximate, together with their derivatives, smooth finite functions in \(\Omega\) which, according to the definition of the Hilbert space \(H_B^*(\Omega)\), are in turn everywhere dense in \(H_B^*(\Omega)\) in the sense of the convergence of this space. This proves the completeness of the polynomial eigenfunctions of the operator \(B^2\) in \(H_B^*(\Omega)\), and the theorem is proved.
\(2^\circ\). In this paragraph we prove the possibility of applying the result obtained above to the study of the following boundary-value problem:
\[ L(u)\equiv \square^2 u+4\frac{\partial^2}{\partial t^2}\square u+2\frac{\partial^4 u}{\partial t^4} =f(x,y,z,t), \tag{7} \]
\[ u\big|_{\Gamma}=0, \tag{8} \]
\[ \frac{\partial u}{\partial n}\bigg|_{\Gamma}=0, \tag{8'} \]
where
\[ \square \equiv \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}-\frac{\partial^2}{\partial t^2} \]
is the four-dimensional wave operator, and \(n\) is the outward normal to the boundary surface \(\Gamma\).
The following theorem holds, which shows, in a certain sense, the connection of the boundary-value problem (7), (8), (8′) with the spectrum of the operator \(B^2\) in \(H_B^*(\Omega)\). Here \(\Omega\) is a finite domain bounded by a sufficiently smooth surface \(\Gamma\).
Theorem 3. If the operator \(B^2\) in \(H_B^*(\Omega)\) has a complete orthonormal system of eigenfunctions, then, for uniqueness of the solution of the boundary-value problem (7), (8), (8′), it is necessary and sufficient that the number \(\mu^*=1/2\) not be an eigenvalue of the operator \(B^2\) in \(H_B^*(\Omega)\).
Proof. Necessity is obvious: indeed, if \(\mu^*=1/2\) were an eigenvalue of the operator \(B^2\) in \(H_B^*(\Omega)\), then the corresponding eigenfunction \(u^*\) would be a nontrivial solution of the boundary-value problem; hence the boundary-value problem (7), (8), (8′) would have a nonunique solution.
Let now \(\mu^* = 1/2\) not be an eigenvalue for the operator \(B^2\) in \(H_B^*(\Omega)\), and suppose that \(u^*\) is a nontrivial solution of the boundary-value problem (7), (8), \((8')\). Then \(u^* \in H_B^*(\Omega)\) and, consequently, can be expanded in \(H_B^*(\Omega)\) in a series in the eigenfunctions \(\{\chi_k\}\) \((k=1,2,\ldots)\) of the operator \(B^2\), i.e.
\[ u^*=\sum_{k=1}^{\infty} a_k\chi_k . \tag{9} \]
Further, from
\[ a_k=(u^*,\chi_k)=\iiiint_{\Omega}\Delta^2u^*\chi_k\,d\Omega =\iiiint_{\Omega}u^*\Delta^2\chi_k\,d\Omega =\frac{1}{\lambda_k}\iiiint_{\Omega}\frac{\partial^4u^*}{\partial t^4}\chi_k\,d\Omega, \]
\[ a_k=\iiiint_{\Omega}\Delta^2u^*\chi_k\,d\Omega =2\iiiint_{\Omega}\frac{\partial^4u^*}{\partial t^4}\chi_k\,d\Omega \qquad (k=1,2,\ldots,N) \tag{10} \]
it follows that \(a_k=0\) \((k=1,2,\ldots)\), i.e. \(u^*(x,y,z,t)\equiv 0\). The theorem is proved.
Let now \(f(x,y,z,t)\in W_2^{(2)}(\Omega)\). Put
\[ F(x,y,z,t)=\frac{1}{6}\int_0^t (t-\tau)^3 f(x,y,z,\tau)\,d\tau . \tag{11} \]
Since \(F\in W_2^{(2)}(\Omega)\), we have
\[ F=F_0+\sum_{k=1}^{\infty}F_k\chi_k, \tag{12} \]
where \(F_0\) is a polynomial of order not exceeding 3.
Theorem 4. If the series
\[ \sum_{k=1}^{\infty}\frac{F_k^2}{(1/\lambda_k-2)^2} \tag{13} \]
converges, then the boundary-value problem has a solution in \(W_2^{(2)}(\Omega)\).
Proof. From condition (13) of the theorem it follows that the series
\[ \sum_{k=1}^{\infty} a_k\chi_k(x,y,z,t),\qquad a_k=\frac{F_k}{1/\lambda_k-2}, \tag{14} \]
converges in \(W_2^{(2)}(\Omega)\). Let
\[ \sum_{k=1}^{\infty} a_k\chi_k=u^*(x,y,z,t). \]
This means that
\[ S_n=\sum_{k=1}^{n}a_k\chi_k \to u^*\in W_2^{(2)}(\Omega). \tag{15} \]
We shall show that \(u^*\) is a solution of the boundary-value problem. To this end, first note that the function \(u^*\) satisfies the boundary conditions (8), \((8')\), since \(u^*\) is the limit, in the sense of \(W_2^{(2)}(\Omega)\), of functions from \(W_2^{(2)}(\Omega)\) satisfying these boundary conditions. Further, using the embedding theorems of S. L. Sobolev \((^2)\) and the relation
\[ \frac{\partial^4 F}{\partial t^4}=f(x,y,z,t), \tag{16} \]
one can prove that
\[ \iiint\!\!\int_{\Omega} u^* L\varphi\,d\Omega = \iiint\!\!\int_{\Omega} f\varphi\,d\Omega \tag{17} \]
for all \(\varphi\in\Phi_0\), where \(\Phi_0\) is a linear manifold of infinitely differentiable finite functions. This means precisely that \(u^*(x,y,z,t)\) is a solution of the boundary-value problem (7), (8), \((8')\). The theorem is proved.
\(3^\circ\). Let now \(H\) be the Hilbert space obtained by completing \(D_B\) in the sense of the scalar product
\[ (u,v)= \iiint\!\!\int_{\Omega} \left\{ \frac{\partial u}{\partial x}\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}\frac{\partial v}{\partial y} + \frac{\partial u}{\partial z}\frac{\partial v}{\partial z} + \frac{\partial u}{\partial t}\frac{\partial v}{\partial t} \right\}d\Omega . \tag{18} \]
In the space \(H\) consider the operator \(B=\Delta^{-1}\dfrac{\partial^2}{\partial t^2}\), where \(\Delta^{-1}\) is the operator inverse to the four-dimensional Laplace operator under zero boundary conditions. It can be proved\({}^{4}\) that the operator \(B\) on the linear manifold \(D_B\), dense in \(H\), is a symmetric, bounded, and positive-definite operator. We shall denote the hypermaximal extension of this operator by the same letter \(B\).
Theorem 5. The limiting spectrum of the operator \(B\) in \(H\) coincides with the interval \([0,1]\).
Proof. Consider the sequence of functions
\[ u_{k,n}(x,y,z,t)= \frac{ T_n\!\left(t\cos\frac{k}{n}\frac{\pi}{2} +\sqrt{x^2+y^2+z^2}\sin\frac{k}{n}\frac{\pi}{2}\right) }{ \sqrt{x^2+y^2+z^2} } + \frac{ (-1)^{k+1}T_n\!\left(t\cos\frac{k}{n}\frac{\pi}{2} -\sqrt{x^2+y^2+z^2}\sin\frac{k}{n}\frac{\pi}{2}\right) }{ \sqrt{x^2+y^2+z^2} }, \tag{19} \]
where \(n\ge 2;\ k=2,4,\ldots,n-1;\ T_n(x)=\cos(n\arccos x)\) is the Chebyshev polynomial of the first kind of degree \(n\). By direct verification\({}^{3,5}\) one can see that the function \(u_{k,n}(x,y,z,t)\) satisfies the boundary-value problem
\[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} - \nu_{k,n}^{\,2}\frac{\partial^2 u}{\partial t^2} =0, \tag{20} \]
\[ u|_{\Gamma}=0, \tag{21} \]
where \(\nu_{k,n}=\operatorname{tg}\dfrac{k}{n}\dfrac{\pi}{2}\), \(n\ge2\), and \(k\) takes even values up to \(n-1\). On the other hand, let us note that the solutions of the boundary-value problem (20), (21) are eigenfunctions for the operator \(B\) in \(H\) with eigenvalues
\[ \lambda_{k,n}^{2}=\frac{1}{1+\nu_{k,n}^{2}}, \]
and since the numbers
\[ \lambda_{k,n}^{2}= \frac{1}{1+\operatorname{tg}^2\frac{k}{n}\frac{\pi}{2}} \]
for \(n\ge2,\ k=2,4,\ldots,n-1\) are everywhere dense in the interval \([0,1]\), the limiting spectrum of the operator \(B\) in \(H\) coincides with \([0,1]\). The theorem is proved.
In conclusion I express my deep gratitude to Acad. S. L. Sobolev for discussion of this work.
Computing Center
Academy of Sciences of the Armenian SSR
Received
25 I 1960
CITED LITERATURE
- R. Denchev, DAN, 126, No. 2 (1959).
- S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
- R. A. Aleksandryan, DAN, 73, No. 5 (1950).
- G. V. Virabyan, DAN, 128, No. 1 (1959).
- R. Courant, D. Hilbert, Methods of Mathematical Physics, 2, 1951, p. 148.