UDC 517.514

MATHEMATICS

EMBEDDING THEOREMS FOR FUNCTIONS DEFINED IN UNBOUNDED DOMAINS, AND THEIR APPLICATION TO THE SPECTRAL THEORY OF ELLIPTIC SELF-ADJOINT OPERATORS

Yu. V. Rybalov

(Presented by Academician I. M. Vinogradov on 26 VI 1968)

As K. Clark showed \((^1)\), the embedding \(W_2^1(\Omega)\) into \(L_2(\Omega)\) is not completely continuous if the unbounded domain \(\Omega\) has infinite measure. A consequence of this is the non-discreteness of the spectrum of the Laplace operator in a domain containing a countable number of identical nonintersecting squares (see \((^2)\)).

In this paper, for a sufficiently general unbounded domain \(\Omega\), we consider the class \(W_{p,\alpha}^{l}(\Omega,g)\), which coincides with the class \(W_p^l(\Omega)\) in the case of a bounded domain (for \(\Omega=E_n\), \(W_{p,\alpha}^{l}(\Omega,g)\) is the class of L. D. Kudryavtsev \(W_{p,\alpha}^{l}(E)\) \((^{3-5})\)), and it is shown that the embedding \(W_{p,\alpha}^{l}(\Omega,g)\) into \(W_{p,\alpha_k}^{k}(\Omega,g)\) is completely continuous \((0\le k<l)\) for a certain exactly determined exponent of the weight degree \(\alpha_k\). With the help of this result an operator \(a(X,D)\) is constructed, differing from the elliptic self-adjoint operator \(L(X,D)\) by a positive factor of power growth at infinity, and it is proved that the operator \(a(X,D)\) has a discrete spectrum in an unbounded domain not containing a sphere of positive radius.

Let the unbounded domain \(\Omega\subset E_n\), \(n\ge 1\), let \(g\) be a bounded interior subdomain of \(\Omega\), let \(\alpha\) be a real number, and \(1\le p\le\infty\).

We introduce classes of functions as linear normed spaces of functions for which the following norms are meaningful and finite:

\[ |f;L_{p,\alpha}(\Omega)|=|(1+|X|)^{-\alpha}f;\ L_p(\Omega)|, \]

\[ |f;L_{p,c(X)}(\Omega)|=|c^{1/p}(X)f;\ L_p(\Omega)|, \]

\[ |f;W_{p,\alpha}^{l}(\Omega,g)|=|f;L_p(g)|+\sum_{|r|=l}|D^r f;\ L_{p,\alpha}(\Omega)|= \]

\[ =|f;L_p(g)|+|f;L_{p,\alpha}^{l}(\Omega)| \quad \bigl(|f;W_{p,\alpha}^{0}(\Omega,g)|=|f;L_{p,\alpha}(\Omega)|\bigr), \]

\[ |f;W_{p,\alpha,\alpha_0}^{l}(\Omega)|=|f;L_{p,\alpha_0}(\Omega)|+|f;L_{p,\alpha}^{l}(\Omega)|, \]

where \(D^r f\) is a generalized derivative of order

\[ |r|=\sum_{i=1}^{n} r_i . \]

Denote by \(\overset{\circ}{B}\) the closure in the norm of the class \(B\) of functions from \(C_0^\infty(\Omega)\). By definition, a domain \(\Omega\) with locally Lipschitz boundary belongs to the class \(Z_m\) if

\[ \Omega=G+\sum_{j=1}^{N}\Omega_j, \]

where \(G\) is a finite domain, \(N<\infty\), and the domain \(\Omega_j\) \((j=1,\ldots,N)\) has the following properties after a nonsingular linear transformation, its own for each \(j\):

1) for \(X\in\Omega_j\), the coordinates \(x_i\) have \(|x_i|\le c<\infty\) \((i=m+1,\ldots,n)\), while the remaining coordinates \(x_1,\ldots,x_m\) are unbounded;

2) if one passes from the orthogonal coordinates \(X\) to cylindrical coordinates
\[ Y:\quad y_1=|X_m|=\left(\sum_{i=1}^m x_i^2\right)^{1/2},\quad y_2=\varphi,\quad y_3=\theta_1,\ldots,y_m=\theta_{m-2},\quad y_{m+1}=x_{m+1},\ldots \]
\[ \ldots,y_n=x_n\quad (y_1=x_1\ \text{for }m=1), \]
then for some \(i>1\), depending on \(j\), the domain \(\Omega_j\) is written in the form
\[ \Omega_j=\{Y:\ a_k^{(j)}<y_k<b_k^{(j)}\ (k\ne i),\quad a_i^{(j)}<y_i<\psi_i^{(j)}(P_i)\ (\psi_i^{(j)}(P_i)\ge b_i^{(j)})\}, \]
where \(a_k^{(j)}, b_k^{(j)}\) are constants, \(|a_k^{(j)}|<\infty\) \((k=1,\ldots,n)\), \(b_1^{(j)}=\infty\), \(|b_k^{(j)}|<\infty\) \((k=2,\ldots,n)\), \(P_i=(y_1,\ldots,y_{i-1},y_{i+1},\ldots,y_n)\), and \(\psi_i^{(j)}(P_i)\) is a single-valued continuous bounded function of \(P_i\).

Theorem 1. Let \(f\in W_{p,\alpha}^{\,l}(\Omega,g)\), \(\Omega\in Z_m\), \(1\le m\le n\); let \(\alpha\) be a real number; \(1\le p\le\infty\); \(l,k\) natural numbers, \(0\le k<l\); \(\beta=\max(\alpha,m/p-1+\varepsilon)\), \(\varepsilon>0\); \(\alpha_k=\beta+l-k\); \(|r|=k\).

Then
\[ |D^r f;\,L_{p,\alpha_k}(\Omega)|\ll |f;\,W_{p,\alpha}^{\,l}(\Omega,g)|^* . \]

Corollary 1. Let \(\Omega\in Z_m\), \(1\le m\le n\); let \(\alpha\) be a real number; \(1\le p\le\infty\); \(l\) a natural number;
\[ \beta=\max(\alpha,m/p-1+\varepsilon),\quad \varepsilon>0;\quad \alpha_0=\beta+l. \]
Then the classes \(W_{p,\alpha}^{\,l}(\Omega,g)\), \(W_{p,\alpha,\alpha_0}^{\,l}(\Omega)\) coincide up to equivalence of norms.

Theorem 1 for \(\Omega=E_n\) is contained in the works \((^{3-5})\).

Theorem 2. Let \(\Omega\in Z_m\), \(1\le m\le n\); let \(\alpha\) be a real number; \(1\le p\le\infty\); \(l,k\) natural numbers, \(0\le k<l\); \(\beta=\max(\alpha,m/p-1+\varepsilon)\), \(\varepsilon>0\); \(\alpha_k=\beta+l-k\).

Then \(W_{p,\alpha}^{\,l}(\Omega,g)\) is embedded in \(W_{p,\alpha_k+\varepsilon}^{\,k}(\Omega,g)\) completely continuously.

Theorem 3. Let \(\alpha\ge0\); \(1\le p\le\infty\); \(l,k\) be natural numbers, \(0\le k<l\); \(\varepsilon>0\); \(\alpha_k=\alpha+l-k\).

Then the embedding \(\mathring W_{p,\alpha}^{\,l}(\Omega,g)\), \(\mathring W_{p,\alpha_k+\varepsilon}^{\,k}(\Omega,g)\) is completely continuous.

Corollary. Let the unbounded domain \(\Omega\) contain no sphere of positive radius; \(\alpha\ge0\); \(1\le p<\infty\); \(l,k\) be natural numbers, \(0\le k<l\); \(\varepsilon>0\); \(\alpha_k=\alpha+l-k\).

Then the embedding \(\mathring L_{p,\alpha}^{\,l}(\Omega)\) in \(\mathring L_{p,\alpha_k+\varepsilon}^{\,k}(\Omega)\) is completely continuous.

The stated theorems are sharp in the sense that the exponents of the weight cannot be decreased by \(\varepsilon>0\).

Theorem 4. Let the unbounded domain \(\Omega\) contain no sphere of positive radius. Suppose, further, that \(\alpha\ge0\); \(\varepsilon>0\); \(\alpha_0=\alpha+l+\varepsilon\); \(\beta=\alpha\) for \(\alpha>n/2-1\), \(\beta=\alpha+\varepsilon\) for \(\alpha\le n/2-1\); \(L(X,D)\) is a linear elliptic self-adjoint operator of order \(2l\) (\(D\) is the differentiation operator in the sense of distributions)
\[ L(X,D)=(-1)^l \sum_{\sum\alpha_i=l} \frac{l!}{\alpha_1!\cdots\alpha_n!} \frac{\partial^l}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}} \left( a_{\alpha_1\ldots\alpha_n}(X) \frac{\partial^l}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}} \right)+b(X), \]
\[ \sum_{\sum\alpha_i=l} \frac{l!}{\alpha_1!\cdots\alpha_n!} a_{\alpha_1\ldots\alpha_n}(X) (\xi_1^{\alpha_1}\cdots\xi_n^{\alpha_n})^2\ge \]
\[ \ge \frac{B}{(1+|X|^2)^\alpha} \sum_{\sum\alpha_i=l} \frac{l!}{\alpha_1!\cdots\alpha_n!} (\xi_1^{\alpha_1}\cdots\xi_n^{\alpha_n})^2 \quad \text{for } X\in\Omega,\ \vec{\xi}\in E_n, \]
\[ a(X,D)=c^{-1}(X)L(X,D); \]
\(a_{\alpha_1\ldots\alpha_n}(X)\), \(b(X)\), \(c(X)\) are infinitely differentiable coefficients in \(\Omega\);
\[ |a_{\alpha_1\ldots\alpha_n}(X)|\le A/(1+|X|^2)^\alpha;\quad 0\le \]

\[ \text{* By the expression } A\ll B \text{ here and below is meant } A\le cB,\ \text{where the positive constant } c \text{ does not depend on } f. \]

\[ \leq b(X)\leq A/(1+|X|^2)^{\beta+l};\qquad 0<c(X)\leq A/(1+|X|^2)^{\alpha_0}; \]
\(A,\ B\) are positive constants.

Define the operator \(T\) in \(L_{2,c(X)}(\Omega)\) as follows:
\[ D(T)=\overset{\circ}{L}{}^{\,l}_{2,\alpha}(\Omega)\cap \{f\in L_{2,c(X)}(\Omega): a(X,D)f\in L_{2,c(X)}(\Omega)\}, \]
\[ Tf=a(X,D)f,\qquad f\in D(T). \]

Then \(T\) is a closed linear operator; the spectrum \(\sigma(T)\) is discrete, has no finite limit points, and is situated in the half-plane \(\operatorname{Re}\lambda>0\); for \(\lambda\notin\sigma(T)\) the resolvent
\[ R_\lambda(T)=(\lambda I-T)^{-1} \]
is completely continuous in \(L_{2,c(X)}(\Omega)\).

We note that the operator \(L(X,D)\) is equivalent to the operator \(a(X,D)\) in the sense that the equations \(L(X,D)u=f\) for \(f\in L_{2,c^{-1}(X)}(\Omega)\) and
\[ a(X,D)u=c^{-1}(X)f=g \]
for \(g\in L_{2,c(X)}(\Omega)\) are equivalent (a generalized solution of one is a generalized solution of the other).

If, in the definition of the spectrum of the operator \(A\), one replaces the resolvent
\[ R_\lambda(A)=(\lambda I-A)^{-1} \]
by the resolvent
\[ R_{\lambda,c}(A)=(\lambda c(X)I-A)^{-1}, \]
then we obtain the definition of the \(c\)-spectrum of the operator \(A\).

Corollary. Let the hypotheses of Theorem 4 be satisfied. Define the operator \(T_1\) as follows:
\[ D(T_1)=\overset{\circ}{L}{}^{\,l}_{2,\alpha}\cap \{f\in L_{2,c(X)}(\Omega): L(X,D)f\in L_{2,c^{-1}(X)}(\Omega)\}, \]
\[ T_1f=L(X,D)f,\qquad f\in D(T). \]

Then \(T_1\) is a closed linear operator, the \(c\)-spectrum \(\sigma_c(T_1)\) is discrete, has no finite limit points, and is situated in the half-plane \(\operatorname{Re}\lambda>0\); for \(\lambda\notin\sigma_c(T_1)\) the resolvent
\[ R_{\lambda,c}(T_1)=(\lambda c(X)I-T_1)^{-1} \]
is completely continuous from \(L_{2,c^{-1}(X)}(\Omega)\) into \(L_{2,c(X)}(\Omega)\).

For \(n=1\) and \(l=1\), the \(c\)-spectrum of the operator \(L(X,D)\), under other restrictions on the coefficients, was first considered by K. Friedrichs \((^6)\).

Theorem 5. Let the hypotheses of Theorem 4 be satisfied, with the sole exception that
\[ \alpha_0=\beta+l. \]
Then the equation
\[ L(X,D)u=f \]
for a function \(f\) such that
\[ (1+|X|)^{\alpha_0}f\in L_2(\Omega), \]
has, moreover, a unique generalized solution
\[ u\in \overset{\circ}{L}{}^{\,l}_{2,\alpha}(\Omega), \]
and the inequality
\[ |u;\ \overset{\circ}{L}{}^{\,l}_{2,\alpha}(\Omega)|\leq |(1+|X|)^{\alpha_0}f;\ L_2(\Omega)| \]
holds.

The operator \(L(X,D)\) for \(l=1\) was considered earlier by L. D. Kudryavtsev \((^{7-9})\) in connection with the variational method for solving elliptic equations posed in a half-space.

In conclusion, the author expresses his gratitude to his scientific supervisor Prof. L. D. Kudryavtsev for posing the problem and for his attention to the work.

Moscow
Institute of Physics and Technology

Received
20 VI 1968

REFERENCES

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6. K. Friedrichs, Stud. and Essays present. to R. Courant, N. Y., 1948, p. 145.
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