Mathematics

I. A. Solomeshch

On the Asymptotics of the Eigenvalues of Bilinear Forms Associated with Certain Elliptic Equations Degenerating on the Boundary

(Presented by Academician V. I. Smirnov on 23 I 1962)

Let in Euclidean space \(E_n\), with points \(x=(x_1,\ldots,x_n)\), there be given: an \(n\)-dimensional domain \(\omega\), a \(p\)-dimensional domain \(s\), and a manifold of points \(k\).

We shall denote: \(\rho(k,x)\) is the distance of the point \(x\) from \(k\); \(i\) is the vector index \((i_1,i_2,\ldots,i_n)\), \(|i|=i_1+i_2+\cdots+i_n\); \(D^i=\partial^{|i|}/\partial x_1^{i_1}\cdots \partial x_n^{i_n}\); \(|s|\) is the \(p\)-dimensional volume of \(s\).

The set of complex functions \(f\in L_2(\omega)\) having a finite integral

\[ \int_\omega \rho^\alpha(k,x)\sum_{|i|=m}|D^i f|^2\,dx, \tag{1} \]

will be denoted by \(W^m_{k,\alpha}(\omega)\); \(\alpha\) is a constant from \([0,2m)\). The closure of the set of infinitely differentiable functions finite in \(\omega\) in the norm determined by the integral (1) will be denoted by \(\dot W^m_{k,\alpha}(\omega)\).

Let an operator be given

\[ a\equiv \sum_{|i|,|j|=m} D^j(a_{ij}D^i)\qquad (2m>n) \tag{2} \]

with coefficients defined on all of \(E_n\).

The form
\[ \int_\omega \rho^\alpha(k,x)\sum_{|i|,|j|=m} a_{ij}D^i f\,D^j\bar g\,dx, \]
considered on functions from \(W^m_{k,\alpha}(\omega)\), will be denoted by \(a_{k,\alpha}(f,g,\omega)\). For \(\alpha=0\) we shall write
\[ a_{k,0}(f,g,\omega)\equiv a(f,g,\omega). \]

The problem of finding the eigenfunctions of the form \(a_{k,\alpha}(f,g,\omega)\), considered on some subspace \(V^m(\omega)\) \((\dot W^m_{k,\alpha}(\omega)\subseteq V^m(\omega)\subseteq W^m_{k,\alpha}(\omega))\), will be called the problem \(\{a_{k,\alpha},V^m(\omega)\}\).

For the form \(a_{k,\alpha}(f,g,\omega)\), by \(A_\omega(\lambda)\), \(B_\omega(\lambda)\) we shall denote the number of eigenvalues of the problem \(\{a_{k,\alpha},\dot W^m_{k,\alpha}(\omega)\}\), respectively \(\{a_{k,\alpha},W^m_{k,\alpha}(\omega)\}\), not exceeding \(\lambda\).

In what follows we assume that the coefficients of the operator (2) are continuous on the closure \(\overline{\Omega}\) of some bounded \(n\)-dimensional domain \(\Omega\) with sufficiently smooth boundary \(S\), and that the matrix \(\|a_{ij}\|\) is uniformly positive definite on \(\overline{\Omega}\), i.e.

\[ \sum_{|i|,|j|=m} a_{ij}(x)\eta_i\overline{\eta_j} \ge d \sum_{|i|=m}\frac{m!}{i_1!\cdots i_n!}|\eta_i|^2 \qquad (d>0) \]

for all complex \(\eta_i\) and \(x\in\overline{\Omega}\).

In the present paper the question of the asymptotics of the eigenvalues of the problem \(\{a_{S,\alpha},V^m(\Omega)\}\) is considered. In the case \(\alpha=0\), the formulas obtained are contained in the results of paper \((^5)\).

It follows from the papers \((^{2,3})\) that for any \(V^m(\Omega)\) there exists a sequence \(\{u_i\}\subset V^m(\Omega)\) of functions and a corresponding sequence of numbers \(0\leqslant \lambda_1\leqslant \lambda_2\leqslant\cdots\to\infty\) such that \(a_{k,\alpha}(u_i,f,\Omega)=\lambda_i(u_i,f)\) for all \(f\in V^m(\Omega)\), where \((u_i,f)\) is the ordinary scalar product, and the \(u_i\) form a complete system in \(V^m(\Omega)\).

We first carry out an estimate of the growth of the eigenvalues for domains of a special form.

Let \(\sigma\) be an \((n-1)\)-dimensional plane domain bounded by a finite number of sufficiently smooth \((n-2)\)-dimensional manifolds and having no reentrant angles. Moreover, the boundary of \(\sigma\) has no points at which more than \(n-1\) of the manifolds bounding it intersect.

Denote
\[ b \equiv \sum_{|i|,\,|j|=m} D^i(b_{ij}D^j), \]
where \(b_{ij}\) are constants and the matrix \(\|b_{ij}\|\) is positive definite; \(2m>n\);
\[ \omega(b)=(2\pi)^{-n}\int_{b(\xi)\leqslant 1} d\xi, \]
where
\[ b(\xi)=\sum_{|i|,\,|j|=m} b_{ij}\xi^{i+j}, \]
\(\xi\) is a point of \(E_n(\xi_1,\xi_2,\ldots,\xi_n)\), \(\xi^i=\xi_1^{i_1}\xi_2^{i_2}\cdots\xi_n^{i_n}\). Below, \(\varepsilon(\lambda)\), with different subscripts, denotes a function tending to \(0\) as \(\lambda\to\infty\).

By the methods of \((^1)\), using asymptotic estimates of eigenvalues for nondegenerate elliptic operators \((^5)\), one obtains the following assertions.

Lemma 1. Let \(\Pi\) be a cylinder with points \(x=(x',x_n)\); \(x'=(x_1,\ldots,x_{n-1})\in\sigma\), \(0\leqslant x_n\leqslant l\). For the form \(b(f,g,\Pi)\) the relations are valid:
\[ A_{\Pi}(\lambda)\geqslant |\Pi|\,\omega(b)\lambda^{n/2m}\bigl(1+\varepsilon_1(l^{-1})\bigr)\bigl(1+\varepsilon_{1,0}(\lambda l^{2m})\bigr), \]
\[ B_{\Pi}(\lambda)\leqslant |\Pi|\,\omega(b)\lambda^{n/2m}\bigl(1+\varepsilon_2(l^{-1})\bigr)\bigl(1+\varepsilon_{2,0}(\lambda l^{2m})\bigr). \]

Lemma 2. Let \(\Pi\) be the cylinder \(x'\in\sigma,\ 0\leqslant x_n\leqslant l=(\gamma/\lambda)^{\frac{1}{2m-\alpha}}\). For the problem \(\{b_{\sigma,\alpha}, W^m_{\sigma,\alpha}(\Pi)\}\), for every \(\gamma>0\) the estimate holds
\[ B_{\Pi}(\lambda)\leqslant |\sigma|K(\gamma)\lambda^{\frac n{2m}+\frac{\frac{\alpha}{2m}-1}{2m-\alpha}}\bigl(1+\varepsilon_3(l^{-1})\bigr), \]
where
\[ K(\gamma)=K_0\gamma^{-\frac{n-1}{2m-2}}, \]
and \(K_0\) is a constant.

These assertions make it possible, by the methods of R. Courant \((^1)\), with the use of the techniques of \((^4)\), to obtain successively the following theorems.

Theorem 1. Let \(\Pi\) be the cylinder \(x'\in\sigma,\ 0\leqslant x_n\leqslant l\). For the form \(b_{\sigma,\alpha}(f,g,\Pi)\) the relations are valid:
\[ A_{\Pi}(\lambda),\ B_{\Pi}(\lambda) =M(\lambda)\omega(b)|\sigma|\lambda^{\frac n{2m}+\frac{\beta-1}{2m-\alpha}} \bigl(1+\varepsilon_4(l^{-1})\bigr)+\cdots \quad \text{for } \beta>1, \]
\[ A_{\Pi}(\lambda),\ B_{\Pi}(\lambda) =\frac{\omega(b)}{2m-\alpha}\sigma|\lambda|^{\frac n{2m}}\ln\lambda \bigl(1+\varepsilon_5(l^{-1})\bigr)+\cdots \quad \text{for } \beta=1, \]
\[ A_{\Pi}(\lambda),\ B_{\Pi}(\lambda) =\omega(b)|\sigma|\lambda^{\frac n{2m}} \bigl(1+\varepsilon_6(l^{-1})\bigr)\int_0^l \frac{dx}{x^\beta}+\cdots \quad \text{for } \delta<1. \]

Here and below, \(\beta=\alpha n/2m\); the dots replace terms of lower order with respect to \(\lambda\); \(M(\lambda)\) is an unknown function of \(\lambda\), bounded below and above by positive constants.

Theorem 2. For the eigenvalues of the form \(a_{S,\alpha}(f,g,\Omega)\) the following asymptotic formulas hold:

\[ A_{\Omega}(\lambda),\ B_{\Omega}(\lambda) = M(\lambda)\lambda^{\frac{n}{2m}+\frac{\beta-1}{2m-\alpha}} \int_S w(a)\,dS+\ldots \qquad \text{for } \beta>1, \]

\[ A_{\Omega}(\lambda),\ B_{\Omega}(\lambda) = \frac{1}{2m-\alpha}\lambda^{\frac{n}{2m}}\ln\lambda \int_S w(a)\,dS+\ldots \qquad \text{for } \beta=1, \]

\[ A_{\Omega}(\lambda),\ B_{\Omega}(\lambda) = \lambda^{\frac{n}{2m}} \int_{\Omega}\frac{w(a)}{\rho^{\beta}(S,x)}\,dx+\ldots \qquad \text{for } \beta<1. \]

Remark 1. The formulas of Theorem 2 remain valid without change also for the more general form, taking real values,

\[ a_{S,\alpha}(f,g,\Omega) + \int_{\Omega} \sum_{|i|,\,|j|\le m;\ |i|+|j|<2m} r_{ij}\rho^{\alpha_{ij}}(S,x)D^i f\,\overline{D^j g}\,dx, \]

where \(r_{ij}\) are bounded functions measurable in \(\Omega\); \(\alpha_{ij}\) are constants satisfying the condition
\(\alpha_{ij}>\alpha-(2m-|i|-|j|)\).

Remark 2. The asymptotic estimates obtained are valid for the eigenvalues of the problems \(\{a_{S,\alpha}, V^m(\Omega)\}\).

Received
6 I 1962

References

1. R. Courant, D. Hilbert, Methods of Mathematical Physics, 1, Ch. 6; 2, Ch. 7, Moscow–Leningrad, 1951.
2. M. I. Vishik, Matem. sbornik, 35 (77), 3, 513 (1954).
3. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Leningrad, 1950.
4. I. A. Solomeshch, Matem. sbornik, 54 (96), 3, 295 (1961).
5. G. Ehrling, Math. Scand., 2, fasc. 2, 267 (1954).