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||B_i u|}+\sum_{i\le k{l-m_i-\frac12}+|u|_0\right),\qquad
u\in E(\overline G),
]
hold, where (|\ |_s) is the norm in (W_2^{(s)}(G)); (|\ |) 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(\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)).

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).