UDC 513.88:513.83+517.948.35+517.948.5

MATHEMATICS

I. V. GELMAN

STABILITY OF ABSTRACT BOUNDARY-VALUE PROBLEMS IN A BOUNDED DOMAIN UNDER WEAK PERTURBATIONS OF BOUNDARY OPERATORS*

(Presented by Academician V. I. Smirnov on 13 V 1967)

Let

\[ \mathcal P(x,D)=\sum_{|\alpha|\le m} a_\alpha(x)D^\alpha \]

be a differential expression with complex-valued coefficients, given in a bounded domain \(\Omega\) of Euclidean space \(E_n\), \(\Gamma=\partial\Omega\)**. Denote by \(P_0, P\) the minimal and maximal operators generated by \(\mathcal P(x,D)\) in \(L_2(\Omega)\), and let \(\hat P\) \((P_0\subset \hat P\subset P;\ D(\hat P)=\hat D)\) be a closed extension of \(P_0\). Put \(\hat\alpha=\dim\ker\hat P,\ \hat\beta=\dim\operatorname{coker}\hat P\).

In the present paper, under the assumption that \(\hat P\) is normally solvable and at least one of the dimensions \(\hat\alpha,\hat\beta\) is finite, sufficient conditions are obtained for the stability of the property of normal solvability and of the index of the operator \(\hat P\) under weak, in a certain sense, perturbations of its domain of definition. The article is adjacent to the works \((^5)\), where analogous questions were treated for small perturbations.

1. Consider the Hilbert space \(H_P\) of elements \(g(x)\in D(P)\) with scalar product \([g,h]=(g,h)+(Pg,Ph)\), where \((\,\cdot\,,\,\cdot\,)\) is the scalar product in \(L_2(\Omega)\). Analogously we construct the space \(H_{\bar P}\) for the differential expression \(\overline{\mathcal P}(x,D)\) formally adjoint to \(\mathcal P(x,D)\). The normal solvability of \(\hat P\) is equivalent to the representation \(((^5),\) Lemma 1)

\[ H_P=\hat D\dotplus\hat V, \]

\[ \hat V\{v:v\in H_P,\ (\hat P\varphi,Pv)=0,\ \forall\varphi\in\hat D;\ (\varphi_0,v)=0,\ \forall\varphi_0\in\ker\hat P\}. \tag{1} \]

Along with this, we shall consider the representation \(H_{\bar P}=\hat D^{*}\dotplus\hat V^{*}\) \((\hat D^{*}=D(\hat P^{*}),\ \hat P^{*}\) is the operator adjoint to \(\hat P\) in \(L_2(\Omega))\), which is constructed from the operator \(\hat P^{*}\) in the same way as the representation (1) is constructed from the operator \(\hat P\). Denote by \(\hat\chi\) \((\hat\chi^{*})\) the projection operator of \(H_P\) \((H_{\bar P})\) onto \(\hat V\) \((\hat V^{*})\) parallel to \(\hat D\) \((\hat D^{*})\).

Let \(\widetilde D\) \((D(P_0)\subset\widetilde D\subset H_P)\) be another subspace and let \(\widetilde P\) be the restriction of \(P\) to \(\widetilde D\); \(\widetilde D^{*}=D(\widetilde P^{*})\), \(\widetilde P^{*}\) is the operator adjoint in \(L_2(\Omega)\) to \(\widetilde P\); \(\widetilde\alpha=\dim\ker\widetilde P,\ \widetilde\beta=\dim\operatorname{coker}\widetilde P\).

Theorem 1. Let \(\hat P\) be normally solvable. 1) If \(\hat\alpha<\infty\) and the operator \(\hat\chi:\widetilde D\to L_2(\Omega)\) is completely continuous, then \(\widetilde P\) is normally solvable and \(\widetilde\alpha<\infty\). 2) If \(\hat\beta<\infty\) and the operator \(\hat\chi^{*}:\widetilde D^{*}\to L_2(\Omega)\) is completely continuous, then \(\widetilde P\) is normally solvable and \(\widetilde\beta<\infty\).

For arbitrary \(\Phi_{\pm}\)-operators in a Banach space, a similar result was obtained by G. Neubauer \(((^1),\) Theorem 6.7). From this result the conclusions of Theorem 1 follow only under the assumption of complete continuity of the operators \(\hat\chi:\widetilde D\to H_P,\ \hat\chi^{*}:\widetilde D^{*}\to H_{\bar P}\).

* Work on this article is connected with a question posed to the author by M. Sh. Birman.
** The coefficients \(a_\alpha(x)\) and the boundary \(\Gamma\) are assumed to be sufficiently smooth.

From Theorem 1 there follows the possibility of constructing a decomposition \(H_P=\hat D+\hat V\) of type (1) for the operator \(\hat P\). Let \(\varkappa\) be the projection operator of \(H_P\) onto \(\hat V\) parallel to \(\hat D\).

Definition 1. We shall call the subspaces \(\hat D\) and \(\tilde D\) completely continuous perturbations of one another if the operators \(\hat\varkappa:\tilde D\to H_P\) and \(\tilde\varkappa:\hat D\to H_P\) are completely continuous.

Apparently, for the first time (and in a more general situation) this class of perturbations was studied in \((^1)\). That, with respect to perturbations of this type, \(\operatorname{ind}\hat P\) is not stable is shown by the following

Example. Let \(N=\ker P\), \(\hat D\) be given, \(\hat N=\ker\hat P\), and \(\tilde D=\hat D\oplus\{\lambda\psi\}\), where \(\{\lambda\psi\}\) is a one-dimensional subspace generated by an arbitrary element \(\psi\in N\ominus\hat N\). Then \(\hat\varkappa\tilde D=\{\lambda\psi\}\), \(\tilde\varkappa\hat D=\{0\}\); nevertheless \(\operatorname{ind}\tilde P=\operatorname{ind}\hat P+1\).

Denote by \(\hat V^\perp,\tilde V^\perp\) the \([\cdot,\cdot]\)-orthogonal complements, respectively, of \(\hat D,\tilde D\) in \(H_P\).

Theorem 2. Let \(\hat P\) be a \(\Phi\) \((\Phi_\pm)\)-operator and let the subspaces \(\hat D,\tilde D\) be completely continuous perturbations of one another. In order that \(\operatorname{ind}\tilde P=\operatorname{ind}\hat P\), it is necessary and sufficient that
\[ \dim \tilde D\cap \hat V^\perp=\dim \hat D\cap \tilde V^\perp . \tag{2} \]

It is verified directly that
\[ \tilde D\cap \hat V^\perp=\ker(\tilde P^{*}\hat P+I),\qquad \tilde D\cap \tilde V^\perp=\ker(\hat P^{*}\tilde P+I). \]
We note that, for perturbations small in the sense of \((^5)\), both dimensions in formula (2) are equal to zero.

1. Denote by \(\hat\varkappa^\perp,\tilde\varkappa^\perp\) the \([\cdot,\cdot]\)-orthoprojectors, respectively, onto \(\hat V^\perp\) and \(\tilde V^\perp\). It can be shown that the subspaces \(\hat D,\tilde D\) are completely continuous perturbations of one another if and only if the operator \((\hat\varkappa^\perp-\tilde\varkappa^\perp):H_P\to H_P\) is completely continuous. The latter condition is equivalent to \((^2\), Lemma 3.14) the simultaneous complete continuity of the operators \(R[\hat P]-R[\tilde P]\), \(R[\hat P^{*}]-R[\tilde P^{*}]\), and \(\hat P R[\hat P]-\tilde P R[\tilde P]\) in \(L_2(\Omega)\), where, by definition,
   \[ R[K]=(K^{*}K+I)^{-1} \]
   for any densely defined and closed operator \(K\) in \(L_2(\Omega)\) (below \(K\) will denote an operator with these properties).

Consider in \(L_2(\Omega)\) the positive self-adjoint operator
\[ S[K]=(R[K])^{1/2}. \]

Lemma 1 (Cordes and Labrousse \((^2)\)). 1) The operators \(K\) and \(KS[K]\) are simultaneously normally solvable or not normally solvable. 2) \(\ker K=\ker KS[K]\); \(\operatorname{coker}K=\operatorname{coker}KS[K]\).

Definition 2. We shall call the subspaces \(D\) and \(\tilde D\) weak perturbations of one another if the operator
\[ (\hat P S[\hat P]-\tilde P S[\tilde P]):L_2(\Omega)\to L_2(\Omega) \]
is completely continuous.

Theorem 3. Let \(\hat P\) be a \(\Phi\) \((\Phi_\pm)\)-operator and let the subspaces \(\hat D\) and \(\tilde D\) be weak perturbations of one another. Then \(\tilde P\) is also a \(\Phi\) \((\Phi_\pm)\)-operator and \(\operatorname{ind}\tilde P=\operatorname{ind}\hat P\).

This theorem follows directly from Lemma 1 and the known assertions of perturbation theory for \(\Phi\) \((\Phi_\pm)\)-operators by completely continuous operators \((^{3,4})\).

Corollary. The conclusions of Theorem 3 are valid if the operator
\[ (S[\hat P]-S[\tilde P]):L_2(\Omega)\to H_P \]
is completely continuous.

1. As in the papers \((^5)\), introduce boundary operators \(\hat A,\hat B,\ldots,\hat A',\) proceeding from the representations
   \[ (Pu,w)-(u,\tilde P w)=\langle \hat A u|_\Gamma,\hat B w|_\Gamma\rangle+\langle \hat B u|_\Gamma,\hat A'w|_\Gamma\rangle = \tag{3} \]
   \[ =\langle \tilde A u|_\Gamma,\tilde B w|_\Gamma\rangle+\langle \tilde B'u|_\Gamma,\tilde A'w|_\Gamma\rangle, \]

where \(\langle\cdot,\cdot\rangle\) is the scalar product in some Hilbert space \(H(\Gamma)\) of vector functions defined on \(\Gamma\), such that the operators \(\hat P,\tilde P\) (\(\hat P^{*},\tilde P^{*}\)) are the restrictions of \(P\) (\(\bar P\)), respectively, to the subspaces
\[ \hat D=\{\varphi:\hat A\varphi|_{\Gamma}=0\},\quad \tilde D=\{\psi:\tilde A\psi|_{\Gamma}=0\} \]
\[ (\hat D^{*}=\{\varphi^{*}:\hat A'\varphi^{*}|_{\Gamma}=0\},\quad \tilde D^{*}=\{\psi^{*}:\tilde A'\psi|_{\Gamma}=0\}). \]
Put
\[ \hat Q=(S[\hat P])^{-1},\quad \tilde Q=(S[\tilde P])^{-1},\quad T=\hat Q-\tilde Q,\quad Q=\hat Q+\tilde Q. \]
Then from the representations (3) we obtain
\[ \bigl|(S[\hat P]-S[\tilde P])f\bigr|^{2} =(Tf,(S[\hat P]-S[\tilde P])f)+\gamma^{*}; \tag{4} \]
\[ \begin{aligned} \gamma={}& \langle \tilde B'P(S[\hat P]-S[\tilde P])f|_{\Gamma}, \tilde A(S[\hat P]-S[\tilde P])f|_{\Gamma}\rangle\\ &+\langle \tilde B'(S[\hat P]-S[\tilde P])f|_{\Gamma}, \tilde A'P(S[\hat P]-S[\tilde P])f|_{\Gamma}\rangle . \end{aligned} \tag{5} \]

We shall assume that the operators occurring in formulas (4), (5) have the following properties: 1) the restrictions of the operators \(T\) and \(Q\) to \(C_{0}^{\infty}(\Omega)\) are pseudodifferential operators**; 2) the symbols of these operators \(t(x,\xi)\) and \(q(x,\xi)\) admit asymptotic expansions with homogeneous in \(\xi\) terms \(t_j(x,\xi)\) and \(q_j(x,\xi)\) of orders, respectively, \(\tau_j\searrow-\infty\) and \(\chi_j\searrow-\infty\), such that the function \(h(j,k)=\chi_j+\tau_k\) satisfies the condition: if \(h(j_0,k_0)\ge 0\) and \(j_1+k_1>j_0+k_0\), then \(h(j_1,k_1)<h(j_0,k_0)\); 3) if the sequences \(\{\varphi_n\}\subset \hat D\) and \(\{\psi_n\}\subset \tilde D\) are such that \(\{\varphi_n-\psi_n\}\) is bounded in \(H_P\), then \(\{\tilde B'(\varphi_n-\psi_n)|_{\Gamma}\}\) is a bounded sequence and \(\{\tilde A\varphi_n|_{\Gamma}\}\) is a compact sequence in \(H(\Gamma)\); 4) if the sequences \(\{\varphi_n^{*}\}\subset \hat D^{*}\) and \(\{\psi_n^{*}\}\subset \tilde D^{*}\) are such that \(\{\varphi_n^{*}-\psi_n^{*}\}\) is bounded in \(H_P\), then \(\{\tilde B'(\varphi_n^{*}-\psi_n^{*})|_{\Gamma}\}\) is a bounded sequence and \(\{\tilde A'\varphi_n^{*}|_{\Gamma}\}\) is a compact sequence in \(H(\Gamma)\).

Lemma 2. Let conditions 1), 2) be fulfilled. Then the operator
\[ T:C_{0}^{\infty}(\Omega)\cap L_2(\Omega)\to L_2(\Omega) \]
is completely continuous.

Remark. If \(j_1+k_1>j_0+k_0\) implies \(h(j_1,k_1)<h(j_0,k_0)\) for all integers \(j_0,k_0\), then \(T\in \mathcal L_{-\infty}\) (i.e. \(t_j(x,\xi)\equiv0,\ j=0,1,2,\ldots\)).

Lemma 2 and the remark to it are derived from the obvious relation \(QT=-TQ\) by constructing an asymptotic series for the symbol of the operator \(QT\).

Theorem 4. Let the conditions 1)—4) formulated above be fulfilled. Then the subspaces \(\hat D\) and \(\tilde D\) are weak perturbations of one another, the operators \(\hat P\) and \(\tilde P\) either both are not, or both are, \(\Phi\)-(\(\Phi_{\pm}\))-operators, and in the latter case
\[ \operatorname{ind}\tilde P=\operatorname{ind}\hat P. \]

The proof of this theorem is based on the corollary to Theorem 3, Lemma 2, and an analysis of formulas (4), (5).

Let us note that the conclusions of Lemma 2 and Theorem 4 remain valid if conditions 1), 2) are replaced by the following: on \(C_{0}^{\infty}(\Omega)\) the operators \(\hat Q,\tilde Q\) and \(Q\) are elliptic pseudodifferential operators of order \(m\), while \(S[\hat P]\), \(S[\tilde P]\), and \(Q^{-1}\) are of order \(-m\).

In the case when the coefficients \(a_{\alpha}(x)\) are constant, condition 1) is fulfilled, and \(T\in\mathcal L_{-\infty}\). Thus, in this case the conclusions of Theorem 4 follow from only the conditions 3), 4) on the boundary operators.

The author expresses his gratitude to the participants of the seminar of Academician V. I. Smirnov, who made critical remarks during the discussion of this work.

Leningrad Forestry Academy
named after S. M. Kirov

Received
11 V 1967

References

1. G. Neubauer, Math. Ann., 162, No. 1 (1965).
2. H. O. Cordes, J.-P. Labrousse, J. Math. Mech., 12, No. 5 (1963).
3. B. Yood, Duke Math. J., 18, No. 3 (1951).
4. И. Ц. Гохберг, М. Г. Крейн, УМН, 12, No. 2 (1957).
5. И. В. Гельман, ДАН, 163, No. 2 (1965); 170, No. 6 (1966).
6. J. J. Kohn, L. Nirenberg, Comm. Pure Appl. Math., 18, No. 1/2 (1965).
7. L. Hörmander, Comm. Pure Appl. Math., 18, No. 3 (1965).

* \(|\cdot|\) is the norm of \(H_P\).
** In the sense of (6, 7).