UDC 517.946

MATHEMATICS

A. I. ACHIL’DIEV

ON THE EIGENFUNCTIONS OF THE FIRST BOUNDARY-VALUE PROBLEM FOR ELLIPTIC EQUATIONS DEGENERATING AT POINTS

(Presented by Academician A. N. Tikhonov on 18 III 1969)

Works \((^{1-6})\), etc., are devoted to equations of elliptic type admitting degeneration, or, in other terminology \((^1)\), to equations with singular coefficients. A detailed survey of the principal results on degenerate elliptic equations is contained in the monograph of M. M. Smirnov \((^5)\). In the present article a method is indicated for finding the eigenfunctions of elliptic equations degenerating at interior points, and some cases of completeness of the system of functions so obtained are given.

Let \(E_n\) be Euclidean space of \(n\) dimensions. Denote by \(r(x)\) the distance from the point \(x=(x_1,x_2,\ldots,x_n)\) to the origin \(O\). The boundary of an arbitrary set \(A\) of points of the space \(E_n\) will be denoted by \(\Gamma A\). By \(K_\rho\) we shall denote the \(n\)-dimensional open ball of radius \(\rho\) with center at the point \(O\).

Let a bounded closed domain \(G\) of class \(C_{(2,\alpha)}\) (see \((^9)\)) be given, and let the origin \(O\) be an interior point of \(G\). Denote by \(G_0\) the domain \(G-\Gamma G-O\). Consider in \(G\) the elliptic differential equation

\[ Lu \equiv -\frac{\partial}{\partial x_i}\left(A_{ij}\frac{\partial u}{\partial x_j}\right)+C(x)u(x)=\lambda\sigma(x)u(x). \tag{1} \]

Here and throughout what follows, summation from 1 to \(n\) over pairs of equal indices is understood. We assume that \(A_{ij}(x)=A_{ji}(x)\) and that the quadratic form \(A_{ij}\xi_i\xi_j\) is positive definite in any closed domain \(G_\rho=G-K_\rho\), where \(\rho\) is an arbitrary small positive number. The coefficients \(A_{ij}(x)\in C_{(1,\alpha)}(G-O)\), while \(C(x)>0\), \(\sigma(x)>0\) belong to the class \(C_{(0,\alpha)}\) in \(G-O\). The function \(\sigma(x)\) is summable in \(G\), and the relation

\[ \lim_{r(x)\to 0}\frac{\sigma(x)}{C(x)}=0 \tag{2} \]

is satisfied.

Denote by \(\mathscr{L}_2(A,\sigma)\) the Hilbert space of measurable functions whose squares are integrable over the domain \(A\) with weight \(\sigma(x)\).

Suppose there exists a twice continuously differentiable and positive function \(w(x)\) in \(K_d-O\), tending to infinity as \(r(x)\to 0\), and such that

\[ Lw>0 \quad \text{in } K_d, \tag{3} \]

where \(d\) is some positive number. Then the differential expression \(Lu\) will be called \(w\)-normal (see \((^6)\)).

Let us note some particular cases in which condition (3) is fulfilled for \(w(x)=r^{-\beta}(x)\), \(\beta>0\). Let the coefficients \(A_{ij}(x)\) have the form \(A_{ij}=r^\mu(x)a_{ij}(x)\), where \(a_{ij}(x)\in C_{(1,0)}(G)\), and

\[ a_{ij}(0)= \begin{cases} 1, & \text{for } i=j=1,2,\ldots,m;\quad 0\le m\le n,\\ 0, & \text{for the remaining } i,j=1,2,\ldots,n. \end{cases} \tag{4} \]

Then condition (3) will be satisfied if there exists a positive \(\beta\) such that in some ball \(K_d\) the inequality

\[ C(x)+\beta(\mu+m-\beta-2)r^{\mu-2}(x)\geq \tau r^\beta(x), \tag{5} \]

holds, where \(\tau\) is some positive number.

Let, for any positive number \(\lambda\), the differential expression
\[ L_\lambda u \equiv Lu-\lambda\sigma u \]
be \(w_\lambda\)-normal in the ball \(K_{d(\lambda)}\), where, on the basis of condition (2),
\[ C(x)-\lambda\sigma(x)\geq 0. \]
Consider the problem of eigenvalues and eigenfunctions for the differential expression \(Lu\).

Find those values of the parameter \(\lambda\) for which there exist functions \(u(x)\not\equiv 0\), bounded in \(G\), \(u(x)\in C_{(2,0)}(G_0)\), satisfying equation (1) in \(G_0\), continuous in \(G-O\), and subject to the condition

\[ u\big|_{\Gamma_G}=0. \tag{6} \]

We shall call these values \(\lambda\) eigenvalues, and the corresponding nontrivial solutions \(u(x)\) eigenfunctions.

The solution of problem (1), (6) is carried out by a certain limiting transition from the solutions of the following problem.

Find those values \(\lambda_\varepsilon\) for which there exist functions \(u_\varepsilon(x)\not\equiv 0\), continuous in \(G_\varepsilon\), \(0<\varepsilon<d\), belonging to \(C_{(2,0)}\) in the domain \(G_\varepsilon-\Gamma G_\varepsilon\), satisfying there the equation

\[ Lu_\varepsilon(x)=\lambda_\varepsilon\sigma(x)u_\varepsilon(x), \tag{7_\varepsilon} \]

and subject to conditions (6) and

\[ u_\varepsilon(x)\big|_{\Gamma K_\varepsilon}=0. \tag{8_\varepsilon} \]

Problem \((7_\varepsilon)\), (6) and \((8_\varepsilon)\) is, under our assumptions, regular and has been studied in detail (see \(({}^{7-9})\)).

For any fixed \(\varepsilon\) there exists a countable sequence, nondecreasing with increasing index, of eigenvalues \(\lambda_\varepsilon^{(k)}>0\), with the only limiting point at infinity, and the corresponding system of eigenfunctions \(\{u_\varepsilon^{(k)}(x)\}\), \(u_\varepsilon^{(k)}(x)\in C_{(2,\alpha)}(G_\varepsilon)\) (see \(({}^{7-9})\)). This system of functions is complete and orthonormal in \(\mathscr L_2(G_\varepsilon,\sigma)\).

From the minimax principle (see \(({}^8)\)) it follows that the \(k\)-th eigenvalues \(\lambda_\varepsilon^{(k)}\) do not increase under a monotone decrease of \(\varepsilon\) to zero. Therefore the limit exists

\[ \lim_{\varepsilon\to 0}\lambda_\varepsilon^{(k)}=\lambda^{(k)}. \tag{9_k} \]

It is proved that these \(\lambda^{(k)}\) are eigenvalues of problem (1), (6), corresponding to eigenfunctions \(u^{(k)}(x)\). This eigenfunction \(u^{(k)}(x)\) can be obtained as the limit, converging in \(G-O\) together with its first and second derivatives, of the sequence \(u_{\varepsilon_k}^{(k)}(x)\), \(\{\varepsilon_k\}\subset \{\varepsilon_{k-1}\}\), of the \(k\)-th eigenfunctions of problem \((7_\varepsilon)\), (6), \((8_\varepsilon)\), \(\varepsilon=\varepsilon_k\).

Theorem 1. Let the closed domain \(G\) belong to the class \(C_{(2,\alpha)}\), the coefficients \(A_{ij}(x)\in C_{(1,\alpha)}(G-O)\), the functions \(C(x)\) and \(\sigma(x)\) be positive and belong to the class \(C_{(0,\alpha)}\) in \(G-O\), \(\sigma(x)\) be summable in \(G\), and let relation (2) hold. Let, for any positive number \(\lambda\), the differential expression \(L_\lambda u\) be \(w_\lambda\)-normal in the ball \(K_{d(\lambda)}\).

Then there exists a countable sequence, nondecreasing with increasing index, of positive eigenvalues \(\lambda^{(k)}\), defined by equality \((9_k)\), with the only limiting point at infinity, and the corresponding system of eigenfunctions \(u^{(k)}(x)\) of problem (1), (6) is complete and orthonormal in the Hilbert space \(\mathscr L_2(G,\sigma)\).

Let \(L\) denote the symmetric operator, extended in the sense of Friedrichs (see \(({}^{3,5,7})\)), determined by the differential expression \(Lu\) on the set of functions twice continuously differentiable in \(G\), finite-

finite at the origin and on the boundary of the domain \(G\). Theorem 1 makes it possible to formulate the following result.

Theorem 2. If all the conditions of Theorem 1 are satisfied, then the operator \(L\) will be self-adjoint, positive definite, and have a discrete spectrum in the space \(\mathcal L_2(G,\sigma)\).

1. Consider the elliptic differential equation (1) in the whole space \(E_n\). Denote by \(K(a,b)\) the domain of points \(x\) satisfying the inequalities \(a \le r(x) \le b\). Suppose that \(A_{ij}=A_{ji}\), and that the quadratic form \(A_{ij}\xi_i\xi_j\) is positive definite in any closed domain \(K(\rho,\rho^{-1})\), where \(\rho\) is an arbitrary small positive number. Analogously to the preceding case, we consider the problem of eigenfunctions \(u(x)\), bounded in the whole space \(E_n\), belonging to \(C_{(2,0)}\) and satisfying equation (1) in any domain \(K(\delta,\delta^{-1})\), where \(\delta\) is an arbitrary small positive number. In this case the following results hold.

Theorem 3. Let the coefficients of the differential expression \(L_\lambda u\) in any closed domain \(K(\rho,\rho^{-1})\) satisfy the conditions: \(A_{ij}(x)\in C_{(1,\alpha)}\), \(C(x)\) and \(\sigma(x)\) are positive and belong to the class \(C_{(0,\alpha)}\). Let \(\sigma(x)\) be summable in \(E_n\), let condition (2) be satisfied, and let the analogous condition at infinity be satisfied

\[ \lim_{r(x)\to\infty}\frac{\sigma(x)}{C(x)}=0. \tag{2′} \]

Let, for any positive number \(\lambda\), the expression \(L_\lambda u\) be \(w_\lambda\)-normal in \(K_{d(\lambda)}\) and, analogously, \(W_\lambda'\)-normal outside the ball \(K_{d'(\lambda)}\), where \(d'(\lambda)\) is some positive number.

Then there exists a countable, nondecreasing with increasing index, sequence of positive eigenvalues \(\lambda^{(k)}\), with the only limit point at infinity, and the corresponding system of eigenfunctions \(w^{(k)}(x)\) of equation (1) in \(E_n\) is complete and orthonormal in the Hilbert space \(\mathcal L_2(E_n,\sigma)\).

If, as in the preceding paragraph, we denote by \(L'\) the Friedrichs extension of the symmetric operator defined by the differential expression \(Lu\) on the set of functions twice continuously differentiable in \(E_n\), finite at the origin and at infinity, then the following is valid.

Theorem 4. If all the conditions of Theorem 3 are satisfied, then the operator \(L'\) will be self-adjoint, positive definite, and have a discrete spectrum in the space \(\mathcal L_2(G,\sigma)\).

I take this opportunity to express my gratitude to L. G. Mikhailov and to the members of the seminar he leads for the discussion of this work and for valuable comments.

Physical-Technical Institute named after S. U. Umarov
Academy of Sciences of the Tajik SSR
Dushanbe

Received
22 I 1969

CITED LITERATURE

1. L. G. Mikhailov, A new class of singular integral equations and its applications to differential equations with singular coefficients, Dushanbe, 1963.
2. M. V. Keldysh, DAN, 77, No. 2 (1951).
3. S. G. Mikhlin, Vestn. Leningrad. Univ., 3, No. 8, 19 (1954).
4. O. A. Oleinik, Mat. sborn., 69 (111), 1, 111 (1966).
5. M. M. Smirnov, Degenerate Elliptic and Hyperbolic Equations, Moscow, 1966.
6. A. I. Achil’diev, DAN, 152, No. 1, 13 (1963).
7. S. G. Mikhlin, The Minimum Problem of a Quadratic Functional, 1952.
8. S. G. Mikhlin, Variational Methods in Mathematical Physics, 1957.
9. O. A. Ladyzhenskaya, N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type, 1964.