A. S. DYNIN

MULTIDIMENSIONAL ELLIPTIC BOUNDARY-VALUE PROBLEMS WITH ONE UNKNOWN FUNCTION

(Presented by Academician P. S. Aleksandrov, 2 VI 1961)

1. The solvability of the general boundary-value problem for an elliptic equation in a bounded domain of Euclidean space is investigated. A method is indicated for reducing such a problem to a system of integro-differential equations on the boundary of the domain, which makes it possible to apply the results of paper \((^1)\). The case of a second-order equation is analyzed most fully.

2. Notation. \(G\) is a bounded domain of Euclidean space \(R^n\) \((n > 1)\) with infinitely smooth boundary \(\dot G\);

\(x = (x_1,\ldots,x_n) \in R^n\); \(D = i^{-1}\left(\dfrac{\partial}{\partial x_1},\ldots,\dfrac{\partial}{\partial x_n}\right)\); \(\alpha = (\alpha_1,\ldots,\alpha_n)\) is a set of natural numbers, \(|\alpha| = \alpha_1 + \cdots + \alpha_n\); \(D^\alpha = i^{-|\alpha|}\partial^{|\alpha|}/\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}\); \(\xi_x\) is a tangent vector to the manifold \(\dot G\) at the point \(x \in \dot G\); \(\tau_x\) is the unit vector of the inward normal at the point \(x\); if \(\eta = (\eta_1,\ldots,\eta_n)\in R^{n*}\), then \(\eta^\alpha = \eta_1^{\alpha_1}\cdots \eta_n^{\alpha_n}\);

\(A=\sum_{|\alpha|\le 2k} a_\alpha(x)D^\alpha\) is an elliptic differential polynomial with infinitely differentiable complex coefficients on the set \(\bar G\), the closure of the set \(G\);

\(\sigma_A(\xi_x,z)=\sum_{|\alpha|=2k} a_\alpha(x)\times(\xi_x+z\tau_x)^\alpha\) is the symbol of the operator \(A\) \((z\) is a complex number);

\(B_i=\sum_{\beta\le m_i} B_i^{(\beta)}\dfrac{\partial^\beta}{\partial t^\beta}\) \((i=1,\ldots,k)\); \(B_i^{(\beta)}\) is a singular operator of order \(m_{i\beta}\le m_i-\beta\) on the manifold \(\dot G\) \((^1)\);

\(\sigma_{B_i}(\xi_x,z)=\sum_{m_{i\beta}+\beta=m_i}\sigma_{B_i^{(\beta)}}(\xi_x)z^\beta\) is the symbol of the operator \(B_i\) \(\bigl(\sigma_{B_i^{(\beta)}}(\xi_x)\) is defined in \((^1)\bigr)\);

\(E(\bar G)\) (respectively \(E(\dot G)\)) is the Schwartz space of infinitely differentiable functions on the set \(\bar G\) (respectively \(\dot G\)); \(W_2^{(l)}(G)\) (respectively \(W_2^{(l-1/2)}(\dot G)\)) \((l\) is any natural number) is the Sobolev space of generalized functions on \(G\) (respectively the Slobodetskii space on \(\dot G\) \((^3)\)).

The system \(\mathfrak A=\{A,B_1,\ldots,B_k\}\) defines the operators

\[ \mathfrak A:E(\bar G)\to E(\bar G)\times (E(\dot G))^k; \tag{1} \]

\[ \mathfrak A:W_2^{(l)}(G)\to W_2^{(l-2k)}(G)\times W_2^{(l-m_1-1/2)}(\dot G)\times\cdots\times W_2^{(l-m_k-1/2)}(\dot G) \tag{2} \]

\[ (l\ge \max\{2k,m_1+1,\ldots,m_k+1\}). \]

We shall call the operator \(\mathfrak A\) elliptic (cf. \((^4)\)) if for each-

for fixed \(\xi_x \ne 0\): a) the roots of the \(z\)-polynomial \(\sigma_A(\xi_x,z)\) are distributed equally between the upper and lower \(z\)-half-planes, and b) the \(z\)-polynomials \(\sigma_{B_i}(\xi_x,z)\) \((i=1,\ldots,k)\) are linearly independent modulo the \(z\)-polynomial
\[ \sigma_A^+(\xi_x,z)=\prod_{j\le k}(z-z_j(\xi_x)), \]
where \(z_j(\xi_x)\) \((j=1,\ldots,k)\) are the roots of the \(z\)-polynomial \(\sigma_A(\xi_x,z)\) lying in the upper \(z\)-half-plane. This definition is due to Lopatinskii, who also noted that for \(n>2\) condition a) is always satisfied \((^5)\).

The following proposition is a modification of the results of \((^6)\).

Theorem 1. In order that the operator \(\mathfrak A\) be elliptic, it is necessary and sufficient that the a priori estimate
\[ \|u\|_l \le C\left(\|Au\|_{l-2k}+\sum_{i\le k}\|B_i u\|_{l-m_i-\frac12}+\|u\|_0\right),\qquad u\in E(\overline G), \]
hold, where \(\|\ \|_s\) is the norm in \(W_2^{(s)}(G)\); \(\|\ \|_{s-\frac12}\) is the norm in \(W_2^{(s-\frac12)}(\dot G)\); \(C\) is a constant independent of \(u\).

The following theorem is analogous to Theorem 3 of \((^1)\) (cf. \((^4)\)).

Theorem 2. For the ellipticity of the operator \(\mathfrak A\) it is necessary and sufficient that the following set of conditions hold: a) the generalized solutions of the equation \(\mathfrak A u=0\) are infinitely differentiable; b) these solutions form a finite-dimensional subspace; c) the operators (1), (2) are normally solvable; d) the defects of the ranges of these operators are finite and equal.

Let \(\nu_{\mathfrak A}\) be the dimension of the space \(\mathfrak A^{-1}(0)\); let \(\rho_{\mathfrak A}\) be the defect of the ranges of the operators \(\mathfrak A\); let \(\chi_{\mathfrak A}=\nu_{\mathfrak A}-\rho_{\mathfrak A}\) be the index of the operator \(\mathfrak A\).

The following proposition is analogous to Theorem 4 of \((^1)\).

Theorem 3. 1) The index \(\chi_{\mathfrak A}\) of an elliptic operator is determined by its symbol
\[ \sigma_{\mathfrak A}(\xi_x,z)=\{\sigma_A(\xi_x,z),\ \sigma_{B_1}(\xi_x,z),\ldots,\sigma_{B_k}(\xi_x,z)\}. \]
2) The index \(\chi_{\mathfrak A}\) is constant under uniformly small changes of the first
\[ 2\max\{n,k,m_1,\ldots,m_k\} \]
derivatives of the symbol \(\sigma_{\mathfrak A}(\xi_x,z)\).

1. In this section we indicate a method for transforming the operator \(\mathfrak A\) into a system \(\mathfrak B\) of singular operators on the manifold \(\dot G\).

Consider the remainder \(\sigma'_i(\xi_x,z)\) \((i=1,\ldots,k)\) from the division, for fixed \(\xi_x\ne0\), of the \(z\)-polynomial \(\sigma_{B_i}(\xi_x,z)\) by the \(z\)-polynomial \(\sigma_A^+(\xi_x,z)\). Let \(B'_i\) \((i=1,\ldots,k)\) be a boundary operator with symbol \(\sigma'_i(\xi_x,z)\).

Lemma. The indices of the operators \(\mathfrak A\) and \(\mathfrak A'=\{A,B'_1,\ldots,B'_k\}\) are equal.

Indeed,
\[ \sigma_{B_i}(\xi_x,z)=\sigma_{B'_i}(\xi_x,z)+\sigma_A^+(\xi_x,z)R_i(\xi_x,z). \]
Substituting here, instead of \(R_i(\xi_x,z)\), the factor \((1-t)R_i(\xi_x,z)\) \((t\in[0,1])\), we find that \(\mathfrak A\) and \(\mathfrak A'\) are connected in the class of elliptic operators, so that it remains to apply Theorem 3.

Now put \(v_\beta=\partial^\beta u/\partial\tau^\beta\) \((\beta=0,1,\ldots,k-1)\). Then the system of operators \(B'_i\) \((i=1,\ldots,k)\) is transformed into a system \(\mathfrak B\) of singular operators acting in the space of vector-functions \((v_0,\ldots,v_{k-1})\).

Let
\[ \mathfrak D=\left\{A,\ 1,\ \frac{\partial}{\partial\tau},\ldots,\frac{\partial^{k-1}}{\partial\tau^{k-1}}\right\} \]
be the operator corresponding to the first boundary-value problem.

Theorem 4. \(\chi_{\mathfrak A}=\chi_{\mathfrak D}+\chi_{\mathfrak B}\).

Let us note that the index of the operator \(\mathfrak D\) is equal to zero if \(A\) is a strongly elliptic operator, and also if \(A\) is an operator of second order (for the latter see \((^7)\), where an outline of the proof is given; however, this follows from the fact that the first boundary-value problem satisfies the ellipticity condition with respect to any elliptic operator \(A\), while the set of elliptic operators of second order is linearly connected \((^7)\), after which one must use Theorem 3).

The symbol \(\sigma_{\mathfrak A}(\xi)\) decomposes into the product of an elliptic matrix \((^{1})\) and a nondegenerate diagonal matrix.

Theorem 5. The elliptic operator \(\mathfrak A=\{A,B\}\), where \(A\) is an operator of second order and the order \(B\) is arbitrary, has zero index.

Moscow State University
named after M. V. Lomonosov

Received
2 VI 1961

CITED LITERATURE

\(^{1}\) A. S. Dynin, DAN, 141, No. 1 (1961).
\(^{2}\) P. D. Lax, Comm. Pure and Appl. Math., 8, No. 4, 615 (1955); collected volume Matematika, 1, 1, 43 (1957).
\(^{3}\) L. N. Slobodetskii, Uch. zap. Leningrad. ped. inst., 197, 54 (1958).
\(^{4}\) M. Schechter, Comm. Pure and Appl. Math., 12, No. 4, 561 (1959); collected volume Matematika, 4, 6 (1960).
\(^{5}\) Ya. B. Lopatinskii, Ukr. matem. zhurn., 5, 123 (1953).
\(^{6}\) S. Agmon, A. Douglis, L. Nirenberg, Comm. Pure and Appl. Math., 12, No. 4, 623 (1959); L. N. Slobodetskii, Vestn. Leningrad. univ., 7, 28 (1960).
\(^{7}\) B. V. Boyarskii, Bull. Acad. Polon. Sci., Sér. Math., 8, No. 1, 19 (1960).