Reports of the Academy of Sciences of the USSR

1. Volume 146, No. 3

MATHEMATICS

M. S. AGRANOVICH, A. S. DYNIN

GENERAL BOUNDARY-VALUE PROBLEMS FOR ELLIPTIC SYSTEMS IN A MULTIDIMENSIONAL DOMAIN

(Presented by Academician I. G. Petrovskii on 16 IV 1962)

In this note, for systems elliptic in the sense of I. G. Petrovskii, results are established analogous to those obtained in \((^1)\), where the case of a single elliptic equation was considered.

Let \(G\) be a bounded domain in the Euclidean space \(R^n\) of points \(x=(x^1,\ldots,x^n)\). We assume that its boundary \(\Gamma\) is an \((n-1)\)-dimensional infinitely smooth* surface admitting local “straightening” by means of coordinate transformations (see, for example, \((^{2-4})\)). Consider in the domain \(G\) the operator

\[ Au=A(x,D)u(x). \tag{1} \]

Here \(u(x)\) is a column of \(p\) functions; \(A(x,D)\) is a square matrix of order \(p\), consisting of linear partial differential operators of order \(s\) with complex coefficients infinitely smooth in \(\overline{G}\); \(D=(D_1,\ldots,D_n)\), where \(D_j=-i\,\partial/\partial x^j\). We assume the operator \(A\) to be elliptic:

\[ \det A_0(x,\xi)\ne 0 \quad \text{for } x\in \overline{G} \text{ and } \xi=(\xi_1,\ldots,\xi_n)\ne 0, \tag{2} \]

where \(A_0(x,D)\) is the principal part of \(A\) (homogeneous in \(D\) of degree \(s\)); the \(\xi_j\) are real. For \(n>2\), it follows from (2) that \(ps\) is even \((^5)\); for \(n=2\) this is assumed additionally.

On \(\Gamma\) let us prescribe the boundary operator

\[ Bu=B(x,D)u(x)\big|_{\Gamma}. \tag{3} \]

Here \(B(x,D)\) is a matrix with \(r=ps/2\) rows and \(p\) columns, consisting of linear partial differential operators; the coefficients in these operators will be, as in \((^1)\) (see also \((^6)\)), singular integral operators on \(\Gamma\) with infinitely smooth complex kernels and free terms (see \((^{7,8})\)). In particular, if all kernels are equal to 0, then \(B\) will be an ordinary differential operator. Let \(m_k\) be the highest order of differentiation in the \(k\)-th row of the matrix \(B\). Denote by \(B_0(x,D)\) the principal part of \(B\), obtained by discarding, in the \(k\)-th row, terms of order \(<m_k\) \((k=1,\ldots,r)\).

We shall subject the operators \(A\) and \(B\) on \(\Gamma\) to the condition of Ya. B. Lopatinskii (L), which we now formulate (cf. \((^{5,9,10})\)). Take an arbitrary point \(M\) on \(\Gamma\). In order not to introduce complicated notation, suppose that the origin of the coordinate system \(x\) is at \(M\) and that the axis \(x^n\) is directed along the inward normal to \(\Gamma\) at \(M\). On functions defined in the half-space \(R^n_+:\ x^n>0\), consider the operators

\[ A_0(0,D)u(x),\qquad B_0(0,D)u(x)\big|_{x^n=0}. \tag{4} \]

The first of them is obtained from \(A_0(x,D)\) by freezing the coefficients at \(x=0\). In the second, the coefficients are singular integral operators on the hyperplane \(x^n=0\). Their symbols (see \((^{7,8})\)) are obtained

* All assumptions on smoothness may be weakened.

by freezing at \(x=0\) the symbols of the corresponding integral operators entering into \(B_0(x,D)\).

In (4) perform the Fourier transform \(\dot F\) with respect to the variables \(x^1,\ldots,x^{n-1}\), and consider, for arbitrarily fixed \(\dot\xi=(\xi_1,\ldots,\xi_{n-1},0)\), the boundary-value problem on the half-line:

\[ A_0(0,\dot\xi,D_n)v(x^n)=0\qquad (x^n>0); \tag{5} \]

\[ B_0(0,\dot\xi,D_n)v(x^n)=h\qquad (x^n=0). \tag{6} \]

Here \(h\) is a numerical column of height \(r\). Following \({}^{(9)}\), we shall call a solution of system (5) stable if it tends to 0 as \(x^n\to+\infty\). For \(n>2\) it is proved that the dimension of the space \(\mathfrak M\) of stable solutions is equal to \(r\) \({}^{(5)}\); for \(n=2\) we shall assume this additionally. An operator \(A\) with this property is called properly elliptic.

The condition (L) at the point \(M\) of the boundary \(\Gamma\) consists in the fact that for every \(\dot\xi\ne0\) problem (5), (6) is uniquely solvable in \(\mathfrak M\).

For what follows it is necessary to write condition (L) in explicit form. To this end take some stable basis \(\omega_1(x^n),\ldots,\omega_r(x^n)\), i.e. a basis in the space \(\mathfrak M\), and put

\[ v(x^n)=a_1\omega_1(x^n)+\cdots+a_r\omega_r(x^n), \tag{7} \]

where \(a_j\) are numerical coefficients. Denote by \(\omega\) the matrix composed of the columns \(\omega_1,\ldots,\omega_r\). Substituting (7) into (6), we obtain a system of \(r\) linear algebraic equations with respect to \(a_1,\ldots,a_r\). The condition for its unique solvability is that the matrix

\[ B_0\omega(\dot\xi)=B_0(0,\dot\xi,D_n)\,\omega(x^n)\big|_{x^n=0}\qquad (\dot\xi\ne0) \tag{8} \]

must be nonsingular. This is precisely condition (L). Obviously, it does not depend on the choice of the stable basis.

Put \(\mathfrak A=(A,B)\). We shall call the operator \(\mathfrak A\) elliptic if \(A\) is properly elliptic and if \(A\) and \(B\) are connected on \(\Gamma\) by condition (L).

Let \(k\) be an integer \(\ge0\). Denote by \(H_k(G)\) the direct product of \(p\) (scalar) spaces of S. L. Sobolev \(W_2^{(k)}(G)\). If \(k\ge \max m_j+1\), then denote by \(H_{k-m-1/2}(\Gamma)\) the direct product of the spaces of L. N. Slobodetskii \({}^{(11)}\)

\[ W_2^{(k-m_j-1/2)}(\Gamma)\qquad (j=1,\ldots,r). \]

We agree to denote the norms in these vector spaces by \(\|\ \|\) with the corresponding indices. In this case the square of the norm of a vector is equal to the sum of the squares of the norms of its components.

In what follows \(l\) will be an integer \(\ge l_0=\max(s,m_j+1)\). By \(H_l(G,\Gamma)\) denote the direct product of the spaces \(H_{l-s}(G)\) and \(H_{l-m-1/2}(\Gamma)\). We introduce analogous notation for \(R_+^n,\dot R_+^n\) instead of \(G,\Gamma\).

The operator \(\mathfrak A\) acts from \(H_l(G)\) into \(H_l(G,\Gamma)\) and is bounded.

Theorem 1. Ellipticity of the operator \(\mathfrak A\) is equivalent to each of the following two conditions:

I. If \(u\in H_{l_0}(G)\), \(Au\in H_{l-s}(G)\), and \(Bu\in H_{l-m-1/2}(\Gamma)\), then \(u\in H_l(G)\) and the a priori estimate holds

\[ \|u\|_l\le C\bigl(\|Au\|_{l-s}+\|Bu\|_{l-m-1/2}+\|u\|_0\bigr), \tag{9} \]

where \(C\) is a constant independent of \(u(x)\).

II. \(\mathfrak A\) is a \(\Phi\)-operator from \(H_l(G)\) into \(H_l(G,\Gamma)\).

The latter means that the equation \(\mathfrak A u=0\) has a finite number \(\alpha\) of linearly independent solutions in \(H_l(G)\), that the range of the operator \(\mathfrak A\) is closed in \(H_l(G,\Gamma)\), and that the quotient space \(H_l(G,\Gamma)/\mathfrak A H_l(G)\) has finite dimension \(\beta\) \({}^{(12)}\). If the operator \(\mathfrak A\) is elliptic, then from I it follows that the numbers \(\alpha\) and \(\beta\) do not depend on \(l\). The difference \(\varkappa(\mathfrak A)=\alpha-\beta\) is called the index of the operator \(\mathfrak A\).

For lack of space there is no possibility here to dwell in detail on the prehistory of Theorem 1 (see \((^{2-5,9,10,13-16})\), where further references may be found). In the cited works differential boundary conditions were considered, and Theorem 1 had not yet been established for them in full generality. In \((^1)\) it is proved for integro-differential boundary conditions for \(p=1\).

Let us outline the derivation of assertion II from the ellipticity of the operator \(\mathfrak A\). First one constructs a stable basis \(\Omega_1,\ldots,\Omega_r\) for (5), where \(\Omega_j\) satisfies condition (6) with a column \(h\) all of whose elements are equal to 0 except the \(j\)-th, which is equal to 1. Denote by \(\Omega\) the matrix whose columns are the \(\Omega_j\). The inverse Fourier transforms of its elements are the “Poisson kernels” (cf. \((^3)\)). For \(p=1\) there are explicit formulas \((^3)\) for \(\Omega_j\). For \(p \geqslant 1\) one can show (cf. \((^{3,5})\)) that the following is true.

Lemma 1. The matrix \(\Omega\) admits the representation
\[ \Omega(\xi,x^n)=\int_{\gamma(\xi)} e^{ix^n\lambda} A_0^{-1}(0,\xi,\lambda)N(\xi,\lambda)\,d\lambda . \tag{10} \]

Here \(\gamma(\xi)\) is a contour in the half-plane \(\operatorname{Im}\lambda>0\), encircling the zeros of \(\det A_0(0,\xi,\lambda)\) lying there, and \(N\) is a matrix with \(p\) rows and \(r\) columns, the elements of whose \(j\)-th column, infinitely differentiable for \(\xi\neq 0\), are polynomials in \(\lambda\) and are positively homogeneous in \((\xi,\lambda)\) of degree \(m_j-s-1\) \((j=1,\ldots,r)\).

We shall call a regularizer for the operator \(\mathfrak A\) in the domain \(G\) such a bounded operator \(\mathfrak R\) from \(H_l(G,\Gamma)\) into \(H_l(G)\) that \(\mathfrak A\mathfrak R=I_1+T_1\) and \(\mathfrak R\mathfrak A=I_2+T_2\), where \(I_1\) and \(I_2\) are the identity operators in \(H_l(G,\Gamma)\) and \(H_l(G)\), while \(T_1\) and \(T_2\) are bounded operators from these spaces respectively into \(H_{l+1}(G,\Gamma)\) and \(H_{l}(G)\).

Lemma 2. The elliptic operator \(\mathfrak A\) possesses a regularizer.

For \(p=1\) the regularizer is constructed in \((^4)\) and \((^{21})\). The latter construction is generalized here to the case \(p \geqslant 1\) with the aid of Lemma 1.

First regularizers are constructed for (4) in \(R_+^n\) and for \(A_0(0,D)\) in \(R^n\) (their definitions are analogous to that just given). The first of them is constructed by the formulas
\[ \mathfrak R(f,g)=\mathfrak R_0 f+\mathfrak R_1(g-B\mathfrak R_0 f); \tag{11} \]
\[ \mathfrak R_0 f=MF^{-1}A_0^{-1}(0,\xi)|\xi|^{l_0}(1+|\xi|^{l_0})^{-1}FLf; \tag{12} \]
\[ \mathfrak R_1 g=F^{-1}\Omega(\xi,x^n)|\xi|^{l_0}(1+|\xi|^{l_0})^{-1}Fg, \tag{13} \]
where \(F\) is the Fourier transform with respect to \(x\); \(L\) is an operator extending functions from \(R_+^n\) to \(R^n\), bounded in the norm \(\|\ \|_l\) \((^{17})\), and \(M\) is the operator restricting functions from \(R^n\) to \(R_+^n\). The second is defined by formula (12), in which \(L\) and \(M\) are omitted. Next regularizers are constructed in \(R_+^n\) and in \(R^n\) for operators with coefficients close to constant ones. Finally, by means of a partition of unity, a regularizer for \(\mathfrak A\) in \(G\) is defined.

Theorem 2. Let \(\mathfrak A_1=(A,B_1)\) and \(\mathfrak A_2=(A,B_2)\) be elliptic operators with the same elliptic system \(A\) in \(\bar G\) and different boundary operators \(B_1\) and \(B_2\) on \(\Gamma\). Then
\[ \varkappa(\mathfrak A_1)-\varkappa(\mathfrak A_2)=\varkappa(S). \]

Here \(S\) is an operator in the direct product of \(r\) spaces \(L^2(\Gamma)\), defined by a square matrix of order \(r\) of singular integral operators. The symbol \(\sigma_S\) of this matrix is constructed explicitly from the matrices \(A\), \(B_1\), and \(B_2\) on \(\Gamma\).

* The orders of rows in \(B_1\) and \(B_2\) with identical numbers may be different.

This symbol will be a nondegenerate square matrix of order \(r\), defined and infinitely smooth on the bundle \(\Xi(\Gamma)\) of tangent vectors to \(\Gamma\) of length 1. To define \(\sigma_s\), let us return to the notation used in formulas (4)—(8). Let \(B_{1,0}\) and \(B_{2,0}\) be the principal parts of the operators \(B_1\) and \(B_2\); \(\dot{\xi}=(\xi_1,\ldots,\xi_{n-1})\) is the vector with components \((\xi_1,\ldots,\xi_{n-1},0)\) in the coordinate system \(x\), having length 1. Then (see (8))

\[ \sigma_s(\dot{\xi})=B_{1,0}\omega(\dot{\xi})\cdot [B_{2,0}\omega(\dot{\xi})]^{-1}. \tag{14} \]

This definition does not depend on the choice of a stable basis.

Theorem 2 clarifies the dependence of the index of an elliptic operator on the boundary condition. In the case \(p=1\), a similar result was obtained in \((^1)\). Earlier, in \((^9)\), explicit formulas were found for the index of an elliptic operator in the two-dimensional case \((n=2)\) when \(\max m_j < s\). We also note that in \((^{18})\) the dependence of the index of the elliptic operator \((A,B)\) on the coefficients of the system \(A\) inside the domain \(G\) is clarified.

For the proof of Theorem 2, we equalize the orders of the rows of the matrices \(B_1\) and \(B_2\) by multiplying them on the left the required number of times by the integro-differential operator \(\Lambda\) on \(\Gamma\)—the square root of \(-\Delta\), where \(\Delta\) is the Beltrami–Laplace operator (see \((^6,^8)\)). After this we consider a linear deformation of \(B_1\) into \(SB_2\), applying Theorem 1.

Corollary. If the symbol (14), considered as a continuous mapping of the bundle \(\Xi(\Gamma)\) into the space of nondegenerate matrices of order \(r\), is homotopic to a symbol depending only on the point of \(\Gamma\) (i.e., constant on tangent vectors issuing from one point on \(\Gamma\)), then \(\chi(\mathfrak{A}_1)=\chi(\mathfrak{A}_2)\).

It follows from this that the problem of homotopic classification of elliptic operators \((A,B)\) with fixed \(A\) (this is one of the variants of the general problem posed in \((^{10})\)) is naturally solved by allowing integro-differential operators as \(B\) (cf. \((^1,^6)\)).

Theorem 2 and the corollary from it allow one, for comparison of the indices of elliptic operators, to make use of the results of \((^{19},^{20})\); in particular, one obtains

Theorem 3. The indices of the elliptic operators \((A,B_1)\) and \((A,B_2)\) coincide if 1) \(ps<n-1\) or 2) \(\Gamma\) is homeomorphic to the \((n-1)\)-dimensional sphere and \(ps<2(n-1)\).

We take this opportunity to express our sincere gratitude to M. I. Vishik for his constant attention and valuable discussions.

All-Union Correspondence Machine-Building Institute
Petrozavodsk State University

Received
9 IV 1962

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