MATHEMATICS

Ya. A. ROITBERG, Z. G. SHEFTEL

ENERGY INEQUALITIES FOR ELLIPTIC OPERATORS WITH DISCONTINUOUS COEFFICIENTS AND GENERAL BOUNDARY CONDITIONS AND CONJUGATION CONDITIONS

(Presented by Academician S. L. Sobolev on 20 VII 1962)

1°. It is known that energy inequalities of the form \(\|u\|_l^2 \le K(\|Au\|_0^2 + \|u\|_0^2)\) play an important role in proofs of existence, uniqueness, and smoothness of solutions of partial differential equations. Recently, in papers \((^{1-7})\), energy inequalities with a boundary norm were proved for a number of boundary-value problems; their use gives a number of advantages.

In the present note the authors prove energy inequalities with a boundary norm for elliptic operators with discontinuous coefficients and general boundary conditions and conjugation conditions on surfaces of discontinuity. The method of proof is a development of the method applied by M. Schechter in \((^{7})\). The inequalities obtained make it possible to investigate the solvability of general boundary-value problems for elliptic equations with discontinuous coefficients.

2°. Let \(G\) be a bounded domain of the \(n\)-dimensional Euclidean space \(E_n\) with boundary \(\Gamma\); let \(G_1\) be a subdomain of \(G\) with boundary \(\gamma\), having no common points with \(\Gamma\); \(G_2 = G/\overline{G}_1\)*. Consider the direct sum of Sobolev spaces \(W_2^l(G_1) + W_2^l(G_2) = W_2^l(G) = W_2^l\); obviously, \(W_2^l\) is obtained by taking the closure in the norm

\[ \|u\|_1^2 = \int_G \sum_{|\alpha|\le l} |D^\alpha u|^2 dx \]

of the set of functions defined in \(G\) and infinitely differentiable in each of the \(\overline{G_i}\); in particular, \(W_2^0(G)=L_2(G)\). Every function \(u \in W_2^l\) can be represented in the form \(u(x)=u_1(x)+u_2(x)\), where \(u_i(x)=u(x)\), \(x\in G_i\); \(u_i(x)=0\), \(x\in G/\overline{G}_i\) \((i=1,2)\). If \(l>0\), then for \(x\in\gamma\) by \(u_i(x)\) we mean the limiting value of \(u(x)\) from the side of \(G_i\).

Consider an elliptic differential operator \(A\) with discontinuous complex coefficients

\[ (Au)(x)= \begin{cases} (A^1u)(x), & x\in G_1,\\ (A^2u)(x), & x\in G_2, \end{cases} \tag{1} \]

where

\[ A^i=\sum_{|\mu|\le 2m} a_\mu^i(x)D^\mu,\quad x\in G_i\quad (i=1,2); \tag{2} \]

\[ \mu=(\mu_1,\ldots,\mu_n),\quad |\mu|=\mu_1+\cdots+\mu_n;\quad D^\mu=D_1^{\mu_1}\cdots D_n^{\mu_n},\quad D_k=\frac{1}{i}\frac{\partial}{\partial x_k}. \]

The ellipticity condition means that, for every real vector \(\xi=(\xi_1,\ldots,\xi_n)\ne 0\), the characteristic polynomials

\[ D^i(x,\xi)=\sum_{|\mu|=2m} a_\mu^i(x)\xi^\mu \]

are nonzero in \(G_i\) \((i=1,2)\).

* The case of two domains is considered only to simplify the notation. All results are also valid for a partition into a finite number of domains.

Let \(x\) be any fixed point on \(\Gamma(\gamma)\); consider the polynomial
\(P^i(\eta)=P^i(x,\tau+\eta\nu)\), where \(\tau\ne0\) is any real vector tangent to \(\Gamma(\gamma)\) at the point \(x\), and \(\nu\ne0\) is any real vector normal to \(\Gamma(\gamma)\) at \(x\). From ellipticity it follows that \(P^i(\eta)\) have no real roots; we shall require that the complex roots of these polynomials be distributed equally in the upper and lower half-planes (in the case \(n\ge3\) this condition is always fulfilled \((^8)\)); moreover,
\(P^i(\eta)=P^i_+(\eta)P^i_-(\eta)\), where \(P^i_+\) (\(P^i_-\)) are polynomials of degree \(m\), all of whose roots lie in the upper (lower) half-plane.

We shall also introduce the boundary operators

\[ B^i_k=\sum_{|\mu|\le m^i_k} b^i_{k\mu}(x)D^\mu \tag{3} \]

\[ (i=1,2,3;\quad k=1,\ldots,r_i;\quad r_1=r_2=r\ge2m;\quad r_3\ge m; \]

\[ m^1_k=m^2_k=m_k;\; m^i_k\le 2m-1), \]

where the complex functions \(b^i_{k\mu}(x)\) \((i=1,2)\) are defined on \(\gamma\), and \(b^3_{k\mu}(x)\) on \(\Gamma\); denote their characteristic polynomials by
\[ Q^i_k(x,\xi)=\sum_{|\mu|=m^i_k} b^i_{k\mu}(x)\xi^\mu; \]
if \(x\) is a fixed point on \(\Gamma(\gamma)\), then, as above for \(P^i(x,\xi)\), set \(Q^i_k(\eta)=Q^i_k(x,\tau+\eta\nu)\).

Definition 1. Let at any point \(x\in\gamma\), for any \(\tau\ne0\), \(\nu\ne0\), the following condition be satisfied: there exist natural numbers \(k_1,\ldots,k_{2m}\) such that the simultaneous fulfillment of the identities
\[ c_1Q^i_{k_1}(\eta)+\cdots+c_{2m}Q^i_{k_{2m}}(\eta) =C^i(\eta)P^i_{\pm}(\eta)\quad (i=1,2), \]
where \(c_1,\ldots,c_{2m}\) are complex constants and \(C^1(\eta),C^2(\eta)\) are polynomials, is possible only when \(c_1=\cdots=c_{2m}=0\); in other words,
\[ \binom{Q^1_{k_t}}{Q^2_{k_t}}\quad (t=1,\ldots,2m) \]
are linearly independent modulo
\[ \binom{P^1_+}{P^2_-}. \]
Then we shall say that the operators \(B^i_k\) \((i=1,2)\) jointly cover the operators \(A^1,A^2\).**

This definition is analogous to the definition of covering systems of operators \((^{6,7,9})\), according to which the boundary operators \(B_1,\ldots,B_r\) \((r\ge m)\) cover an elliptic operator \(A\) of order \(2m\) if among the characteristic polynomials \(Q_1(\eta),\ldots,Q_r(\eta)\) there are \(m\) linearly independent modulo \(P_+(\eta)\).

Definition 2. We shall say that the boundary operators (3) cover the operator \(A\) with discontinuous coefficients if the operators \(B^3_k\) cover \(A^2\), and the operators \(B^1_k,B^2_k\) jointly cover \(A^1,A^2\).

3°. Theorem. Let \(a^i_\mu(x)\in C^s(\overline{G_i})\), \(b^i_{k\mu}(x)\in C^{2m-m_k+s}(\gamma)\) \((i=1,2)\);
\(b^3_{k\mu}(x)\in C^{2m-m^3_k+s}(\Gamma)\); let the surfaces \(\Gamma\) and \(\gamma\) be of class \(C^{2m+s}\), where \(s\ge0\) is an integer. Then, and only then, when the operator \(A\) is elliptic and the boundary operators (3) cover it (in the sense of Definition 2), there exists a constant \(K>0\) such that

\[ \|u\|^2_{2m+s}\le K\left(\|Au\|^2_s+\sum_{k=1}^{r}\langle |B_ku|\rangle^2_{2m-m_k+s-1} +\sum_{k=1}^{r_3}\langle B^3_ku\rangle^2_{2m-m^3_k+s-1} +\|u\|^{(2)}_0\right), \]

\[ u\in W^{2m+s}_2(G). \tag{4} \]

* For \(i=1\) the plus subscript is taken here, and for \(i=2\) the minus subscript.

** It is easy to show that in this definition one may replace \(P^1_+,P^2_-\) by \(P^1_-,P^2_+\).

Here \([B_k u](x) = (B_k^1 u_1)(x) - (B_k^2 u_2)(x)\), \(x \in \gamma\); \(\langle v\rangle_l\) is the boundary norm \((^7)\), defined for functions given on \(\Gamma(\gamma)\) (in \((^9)\) this same norm is denoted by \(\langle v\rangle_{l+1}\)). Such norms were introduced earlier by L. N. Slobodetskii in \((^1,\,^2)\). One may assume, for example, that in the notation of \((^1,\,^2)\)
\[ \langle v\rangle_l = \|v\|_{W_2^{\,l+1/2}(\Gamma)} \]
(respectively, for \(\gamma\)).

The proof of sufficiency is first carried out locally; in the general case the inequality is obtained by means of a partition of unity. Suppose, for example, that \(x \in \gamma\); it is enough to prove that there exists a neighborhood \(U\) of the point \(x\) such that inequality (4) holds for all functions \(u \in W_2^{2m+s}(U)\) that vanish outside \(U\). In doing so it is sufficient to consider the case where \(U\) is an \(n\)-dimensional ball, \(U \cap \gamma\) is a piece of the hyperplane \(x_n=0\), and the operators
\[ A^i = \sum_{|\mu|=2m} a_\mu^i D^\mu,\qquad B_k^i = \sum_{|\mu|=2m-1} b_{k\mu}^i D^\mu \]
have constant coefficients (cf. (7)). The proof of this for \(s=0\) is carried out with the aid of the Fourier transform; here the condition of joint covering is used essentially. This condition, by means of an algebraic apparatus that is a development of the algebraic methods used in \((^7)\), makes it possible to apply the lemma of Aronszajn \((^{10},\,^{11},\,^7)\). To obtain the inequality for \(s=1\), we write the obtained inequality for difference quotients in the tangential direction and, after passing to the limit, obtain estimates of the derivatives in this direction; estimates for the normal derivatives are obtained by differentiating \(Au\) and using the inequalities already found. The argument is analogous for \(s=2,3,\ldots\).

In conclusion the authors express their deep gratitude to Yu. M. Berezanskii for posing the question and for discussing the results.

Stanislav State Pedagogical Institute

Drohobych State Pedagogical Institute
named after Iv. Ya. Franko

Received
20 VII 1962

CITED LITERATURE

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