Abstract
Full Text
MATHEMATICS
B. M. NAIMARK
COMPLETENESS OF THE SYSTEM OF EIGENFUNCTIONS AND ASSOCIATED FUNCTIONS OF STRONGLY ELLIPTIC SYSTEMS OF DIFFERENTIAL EQUATIONS
(Presented by Academician S. L. Sobolev on 12 XI 1956)
Let (x) be a point of a bounded domain (D) of (n)-dimensional Euclidean space, bounded by a smooth boundary (\Gamma); let (u(x)=(u_1(x),u_2(x),\ldots,u_N(x))) be a vector-valued function. The system of equations
[
\begin{aligned}
Lu={}&(-1)^m\sum_{(i),(j)}
\frac{\partial^m}{\partial x_{i_1}\cdots \partial x_{i_m}}
\left(
B^{(i_1\ldots i_m;\,j_1\ldots j_m)}(x)
\frac{\partial^m u}{\partial x_{j_1}\cdots \partial x_{j_m}}
\right)\
&+\mu\sum_{(i),(j)}
\frac{\partial^m}{\partial x_{i_1}\cdots \partial x_{i_m}}
\left(
K^{(i_1\ldots i_m;\,j_1\ldots j_m)}(x)
\frac{\partial^m u}{\partial x_{j_1}\cdots \partial x_{j_m}}
\right)
+T(x)u=f,\
&\qquad i_1,\ldots,i_m;\ j_1,\ldots,j_m=1,2,\ldots,n
\end{aligned}
\tag{1}
]
(where (B^{(i_1\ldots i_m;\,j_1\ldots j_m)}(x)), (K^{(j_1\ldots j_m;\,i_1\ldots i_m)}) are complex matrices of order (N), continuously differentiable (m) times; (T(x)) is a differential expression of order less than (2m)) is called strongly elliptic if:
-
For any (\xi_1,\ldots,\xi_n) such that (\sum_{i=1}^{n}\xi_i^2=1), the matrix
[
\sum_{(i),(j)}
B^{(i_1\ldots i_m;\,j_1\ldots j_m)}(x)
\xi_{i_1}\cdots \xi_{i_m}\xi_{j_1}\cdots \xi_{j_m},
\qquad
i_1,\ldots,i_m;\ j_1,\ldots,j_m=1,2,\ldots,n
]
is positive definite for every (x\in D). -
The matrices (B^{(i_1\ldots i_m;\,j_1\ldots j_m)}(x)) are Hermitian for every (x\in D).
-
The matrices (K^{(i_1\ldots i_m;\,j_1\ldots j_m)}(x)) are skew-symmetric for every (x\in D).
-
(B^{(i_1\ldots i_m;\,j_1\ldots j_m)}(x)=B^{(i_1\ldots i_m;\,i_2\ldots i_m)}(x)) for any set (i_1\ldots i_m;\ j_1\ldots j_m).
Any system of the form
[
Lu=(-1)^m\sum_{(i)}
A^{(i_1 i_2\ldots i_{2m})}(x)
\frac{\partial^{2m}u}{\partial x_{i_1}\cdots \partial x_{i_{2m}}}
+T_1(x)u=f,
]
for which the matrix
[
\sum_{(i)}
\left(
A^{(i_1\ldots i_{2m})}(x)+A^{(i_1\ldots i_{2m})*}(x)
\right)
\xi_{i_1}\cdots \xi_{i_{2m}}
]
is positive definite (1), can be reduced to system (1).
The boundary conditions
[
\left.
\frac{\partial^r u}{\partial x_{i_1}\cdots \partial x_{i_r}}
\right|{S}
=0,
\qquad
r=0,1,\ldots,m-\left[\frac{k}{2}\right]-1,
\tag{2}
]
where (S_{n-k}) is a manifold of dimension (n-k) bounding the domain (D), are called zero boundary conditions of Dirichlet type. In particular, when the boundary of the domain (D) is nondegenerate, they mean that (u(x)) vanishes on the boundary together with all derivatives up to order (m-1).
Definition. A chain of functions (u_0, u_1,\ldots,u_k,\ldots) is called a chain of eigenfunctions and associated functions of the Dirichlet problem for equations (1), (2), corresponding to the eigenvalue (\lambda), if (u_0 \ne 0) and the relations
[
\begin{aligned}
Lu_0-\lambda u_0&=0,\
Lu_1-\lambda u_1&=u_0,\
&\ldots\
Lu_k-\lambda u_k&=u_{k-1},\
&\ldots
\end{aligned}
\tag{3}
]
hold, and if all the functions (u_0,u_1,\ldots,u_k,\ldots) satisfy the boundary conditions (2).
Definition. The system of all eigenfunctions and associated functions of problem (1), (2) is called complete in the space (L^2(D)) of vector-functions (u(x)) with square summable over the domain (D), if the linear span of this system is everywhere dense in (L^2(D)).
Lemma 1. The operator (H), generated by problem (1), (2) and acting in (L^2(D)), and the operator (H^2) have the same linear spans of all eigenfunctions and associated functions.
Lemma 2. If (B) is a completely continuous operator of Hilbert–Schmidt type, then its resolvent is representable as the ratio of two entire functions, each of which has order of growth not exceeding the second ((^3)):
[
(E-\lambda B)^{-1}=\frac{D(\lambda)}{\Delta(\lambda)},\qquad
|D(\lambda)|\leqslant e^{M|\lambda|^2},\qquad
|\Delta(\lambda)|\leqslant e^{M_1|\lambda|^2}.
]
Lemma 3. If (B) is a completely continuous operator of Hilbert–Schmidt type, then the resolvent of the operator (B^2) is representable as the ratio of an entire operator-valued function (D(\lambda)) to an entire function (\Delta(\lambda)), each of which has order of growth not exceeding the first.
Raise the system (1), (2) successively (p) times to a square. Then, introducing operators of gradient and divergence type, one can write the system of equations obtained from (1), (2) in the form
[
\bigl[(G^*G)^n+T_n(x)\bigr]u=f,
]
where (T_n(x)) is an operator of order (\leqslant 2m\cdot 2^p), (n=2^p) ((^1)).
Now the problem for the eigenfunctions and associated functions of the operator (L^n) can be written in the following form:
[
(E+\mu A-\lambda B_p)g_n=B_pg_{n-1},
\tag{4}
]
where (A=(G^{-1}G^{-1}\ldots G^{-1}G^{-1})T_n(x)(G^{-1}G^{-1}\ldots G^{-1}G^{-1})) is a bounded operator; (B_p=(G^{-1}G^{-1})^n), (g_n=(G^G)^{n/2}u_n\in \mathfrak H) ((\mathfrak H) is the space of functions ((G^*G)^{n/2}u)).
Lemma 4. For sufficiently large (p), the operator (B_p) is an operator of Hilbert–Schmidt type.
Idea of the proof. Compare the system (1), (2) with the system (\Delta^m u=f) with the corresponding boundary conditions. The latter system generates the operators (G_\Delta, G_\Delta^), and the metric in the space of gradients is equivalent to the metric in the space of gradients (Gu). For large (n), the operator ((G_\Delta^{-1}G_\Delta^{-1})^n) is an operator of Hilbert–Schmidt type by virtue of the properties of the Green function of the Dirichlet problem. The lemma follows from this.
Find (p) from Lemma 4. The operator (B_p) satisfies the conditions of Lemma 2, and the operator (B_{p+1}) the conditions of Lemma 3. Hence the operator (B_{p+1}) has the resolvent (D(\lambda)/\Delta(\lambda)), where (|D(\lambda)|<e^{M|\lambda|}), (|\Delta(\lambda)|<e^{M_1|\lambda|}).
Lemma 5. The spectrum of equation (4) is discrete.
Theorem. Let the operator (B) have the following properties:
- From ((Bu,u)=0) it follows that (u=0).
- The operator (B) is self-adjoint, completely continuous, with a resolvent whose numerator and denominator may be taken to be of growth order not higher than one.
Suppose, moreover, that the operator (A) is bounded and ((E+\mu A)^{-1}) is bounded. Then the system of eigenvectors and associated vectors of the equation
[
(E+\mu A-\lambda B)f_n=Bf_{n-1},\qquad n=0,1,\ldots,
]
is complete in the space (\mathfrak H) for sufficiently small (\mu).
Proof. Since the operators (E+\mu A) and ((E+\mu A)^{-1}) are bounded, they map every dense set into a dense set. Making the substitution ((E+\mu A)f=g), it is enough to show that the system of eigenvectors and associated vectors of the equation
[
(E-\lambda B(E+\mu A)^{-1})g_n=B(E+\mu A)^{-1}g_{n-1},\qquad n=0,1,\ldots,
]
is complete in (\mathfrak H), or, denoting (B(E+\mu A)^{-1}=T), of the equation
[
(E-\lambda T)g_n=Tg_{n-1}.
]
Assume that the system of eigenvectors and associated vectors of the last equation is not complete. This means that there is a subspace (\Omega\subseteq\mathfrak H), orthogonal to all eigenvectors and associated vectors of equation (5). It is invariant with respect to (T^), and (E-\lambda T^) has a bounded inverse in (\Omega). Consider the part (T^_{\Omega}) of the operator (T^) lying in (\Omega). Since the numerator and denominator of the resolvent of the operator (B) are entire functions of growth order not higher than one, the resolvents of the operators (T,T^,T^{\Omega}) possess the same property. The operator (T^*) has no eigenfunctions, and therefore its resolvent has no poles and is an entire function of growth order not higher than one:
[
|(E-\lambda T^*_{\Omega})^{-1}|\leq e^{M|\lambda|}.
]
We shall now estimate the growth of the resolvent of the operator (T^*_{\Omega}) more precisely, using the fact that the operator (B) is self-adjoint. For this purpose write:
[
T^=(E+\mu A^)^{-1}B,\qquad B=(E+\mu A^)T^,
]
[
E-\lambda B
=E-\lambda(E+\mu A^)T^+\mu A^-\mu A^
=(E+\mu A^)(E-\lambda T^)-\mu A^*.
]
Assuming that (\lambda) is not an eigenvalue of the operators (B) and (T^), multiply the preceding equality on the left by ((E-\lambda B)^{-1}) and on the right by ((E-\lambda T^)^{-1}). We obtain:
[
(E-\lambda T^)^{-1}
=(E-\lambda B)^{-1}(E+A^)-\mu(E-\lambda B)^{-1}A^(E-\lambda T^)^{-1};
]
taking the norm of both sides of the equality, we obtain the inequality:
[
|(E-\lambda T^_{\Omega})^{-1}|
\leq
\frac{|\lambda|}{|\operatorname{Im}\lambda|}\,|E+A^|
+
\frac{|\lambda|}{|\operatorname{Im}\lambda|}\,|\mu A^|\cdot
|(E-\lambda T^_{\Omega})^{-1}|.
]
Denote by (\alpha) the angle in which
[
\frac{|\lambda|}{|\operatorname{Im}\lambda|}\,|\mu A^*|\leq \frac12 .
]
In it the inequality holds
[
|(E+\lambda T^_{\Omega})^{-1}|
\leq
\frac{|\lambda|}{|\operatorname{Im}\lambda|}\,|E+A^|
\leq
\frac{|E+A^|}{2\mu|A^|}.
]
In the angle supplementary to (\alpha),
[
\left|((E-\lambda T^*_{\Omega})^{-1}\varphi,\psi)\right|\leq e^{M|\lambda|}.
]
Since (((E-\lambda T^*_{\Omega})^{-1}\varphi,\psi)) is an entire function of (\lambda), we may apply to it the Phragmén–Lindelöf theorem on the growth of entire functions. From this theorem it follows that throughout the entire (\lambda)-plane the inequality
[
\left|((E-\lambda T^*_{\Omega})^{-1}\varphi,\psi)\right|<M_1(\varphi,\psi).
]
This means that ((E-\lambda T_\Omega^)^{-1}=C), where (C) is a bounded operator, or
(E=(E-\lambda T_\Omega^)C), whence (C=E,\ T_\Omega^=0), i.e., for (\varphi\in\Omega),
(T^\varphi=(E+\mu A^*)^{-1}B\varphi=0) for (\varphi\ne 0). Hence (B\varphi=0) for (\varphi\ne 0), and we arrive at a contradiction with the first condition of the theorem. Thus, (\Omega=0), and the system of eigenfunctions and associated functions of equation (5) is complete in (\mathfrak H).
Corollary. If in the preceding theorem the requirement that the operator ((E+\mu A)^{-1}) be bounded is replaced by the requirement that the spectrum of the operator (A) be discrete, then the theorem remains valid.
Indeed, taking then (\lambda_0) not belonging to the spectrum, we replace the operator (\mu A) by the operator (\mu A+\mu_0 A,\ \mu+\mu_0=\lambda_0). The rest of the proof remains unchanged.
Let us apply the lemmas stated above and the theorem to equation (4). Since the operator (B_{p+1}) is positive, equation (4) has a complete system of eigenvectors and associated vectors. Hence the (2^p)-th power of problem (1), (2) has a complete system of eigenfunctions and associated functions. Applying Lemma 1 and S. L. Sobolev’s embedding theorem ((^2)), we obtain the theorem:
Theorem. The system of eigenfunctions and associated functions of the Dirichlet problem for the strongly elliptic system (1), (2) has a system of eigenfunctions and associated functions complete in (L^2(D)) for sufficiently small (\mu).
Received
16 II 1955
REFERENCES CITED
(^1) M. I. Vishik, Matem. sborn., 29, no. 3 (1951).
(^2) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
(^3) T. Carleman, Math. Zs., 9 (1927).
(^4) M. V. Keldysh, DAN, 77, No. 1 (1951).
(^5) F. Browder, Proc. Nat. Acad. Sci. USA, 38, 39 (1952).