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Reports of the Academy of Sciences of the USSR
- Volume 157, No. 4
MATHEMATICS
LIN TSZUN-CHI
PERTURBATION OF SOLUTIONS AND PERTURBATION OF EIGENVALUES AND EIGENFUNCTIONS OF SECOND-ORDER ELLIPTIC EQUATIONS UNDER PERTURBATION OF THE BOUNDARY
(Presented by Academician I. G. Petrovskii on 27 II 1964)
In the papers \((^2,\,^3)\), using the methods of the paper \((^1)\), we considered questions concerning the perturbation of solutions and the perturbation of eigenvalues and eigenfunctions of ordinary second-order differential equations under perturbation of the boundary. In the present note analogous problems are transferred to equations in partial derivatives. In problems on the perturbation of solutions under perturbation of the boundary, the first variations of the solutions were investigated in the work \((^4)\) of V. A. Kronberg.
A problem with perturbed boundaries can be reduced to a problem with perturbed coefficients; for this it is sufficient to transform the perturbed domain into the unperturbed one. The coefficients of the transformed equation will depend on the transformation—this makes the calculations in the investigation of asymptotics inconvenient, and it is expedient to study directly the perturbation of the solution under perturbation of the boundary. At the same time such a replacement makes it possible to use the results of the investigation of a problem with perturbed coefficients for solving a problem with a perturbed boundary, which is used in some cases in the work.
Let \(\Gamma_\varepsilon\) be a family of closed curves depending on the parameter \(\varepsilon\), whose equation in polar coordinates is given in the form \(\rho = 1 + \varepsilon \alpha(\varphi)\), where \(\alpha(\varphi)\) is some analytic function of \(\varphi\) with period \(2\pi\). Each curve \(\Gamma_\varepsilon\) bounds a domain \(\Omega_\varepsilon\); \(\Gamma_0\) (\(\Gamma_\varepsilon\) for \(\varepsilon = 0\)) is the circle \(\rho = 1\), bounding the disk \(\Omega_0\).
In Sec. 1 we consider the perturbation of solutions of a boundary-value problem; in Sec. 2, eigenvalue problems.
- Consider in the domain \(\Omega_\varepsilon\) the following perturbed problem \(A_\varepsilon\):
\[ L_\varepsilon u_\varepsilon \equiv \Delta u_\varepsilon + a(x,y)\frac{\partial u_\varepsilon}{\partial x} + b(x,y)\frac{\partial u_\varepsilon}{\partial y} + c(x,y)u_\varepsilon = f(x,y) \tag{1} \]
under perturbation of the boundary
\[ u_\varepsilon\big|_{\Gamma_\varepsilon} = u_\varepsilon(1+\varepsilon \alpha(\varphi),\varphi)=0, \tag{2} \]
where the coefficients \(a(x,y)\), \(b(x,y)\), \(c(x,y)\) and \(f(x,y)\) are analytic functions.
The unperturbed problem \(A_0\) consists in solving the problem
\[ L_0u_0 \equiv \Delta u_0 + a\frac{\partial u_0}{\partial x} + b\frac{\partial u_0}{\partial y} + cu_0 = f(x,y),\qquad u_0\big|_{\Gamma_0}=u_0(1,\varphi)=0. \]
Case 1, when the limiting problem is not on the spectrum. We shall seek the solution \(u_\varepsilon\) of the problem \(A_\varepsilon\) in the form
\[ u_\varepsilon = u_0+\varepsilon u_1+\cdots+\varepsilon^n u_n+\cdots \tag{3} \]
Substituting (3) into (1), (2) and comparing the coefficients of like powers of \(\varepsilon\), we obtain the following problems:
\[ L_0u_0 \equiv \Delta u_0 + a\frac{\partial u_0}{\partial x} + b\frac{\partial u_0}{\partial y} + cu_0 = f(x,y),\qquad u_0\big|_{\Gamma_0}=u_0(1,\varphi)=0; \]
\[ L_0u_n \equiv \Delta u_n+a\frac{\partial u_n}{\partial x}+b\frac{\partial u_n}{\partial y}+cu_n=0, \qquad u_n\big|_{\Gamma_0}=-\sum_{l=1}^{n}\frac{\alpha^l(\varphi)}{l!}\,D^{(l)}u_{n-l}, \]
where \(D^{(l)}\) is the normal derivative of order \(l\). Solving these problems successively, we obtain the functions \(u_0,u_1,\ldots,u_n,\ldots\).
Passing to an arbitrary bounded domain with analytic boundary and to an arbitrary elliptic operator with analytic coefficients would lead only to a certain complication of the calculations. We note that any elliptic operator is transformed into an operator of the type considered, and any finite simply connected domain with analytic boundary can be carried by a conformal transformation into a disk; moreover, the Laplace operator does not change its form. The case of three or more variables is, in principle, no different from the case considered.
We make the change of variables
\[ \bar{x}=x(1+\varepsilon v(x,y)),\qquad \bar{y}=y(1+\varepsilon v(x,y)), \]
\[ v(x,y)=\tilde v(\rho,\varphi),\qquad \text{on the boundary }\ \tilde v(1,\varphi)=\alpha(\varphi), \]
where \(v(x,y)\) is some fixed analytic function of \(x\) and \(y\) (for example, harmonic). This transformation carries the disk \(\Omega_0\) into the domain \(\Omega_\varepsilon\). By direct calculations it is not difficult to show that the Jacobian is nonzero for sufficiently small \(\varepsilon\). Consequently, there exists a one-to-one transformation of the disk \(\Omega_0\) onto the domain \(\Omega_\varepsilon\): \((x,y)\leftrightarrow(\bar{x},\bar{y})\), \(u(x,y)=\bar u(\bar{x},\bar{y})\). Under this transformation, equation (1) and boundary condition (2) are rewritten in the form
\[ L_\varepsilon \bar u \equiv A_\varepsilon(\bar{x},\bar{y})\frac{\partial^2\bar u}{\partial \bar{x}^2} +B_\varepsilon(\bar{x},\bar{y})\frac{\partial^2\bar u}{\partial \bar{x}\partial \bar{y}} +C_\varepsilon(\bar{x},\bar{y})\frac{\partial^2\bar u}{\partial \bar{y}^2} +D_\varepsilon(\bar{x},\bar{y})\frac{\partial \bar u}{\partial \bar{x}} + \]
\[ +E_\varepsilon(\bar{x},\bar{y})\frac{\partial \bar u}{\partial \bar{y}} +g_\varepsilon(\bar{x},\bar{y})\bar u =F_\varepsilon(\bar{x},\bar{y}), \tag{4} \]
\[ \bar u\big|_{\Gamma_0}=\bar u(1,\rho)=0, \tag{5} \]
where the coefficients are analytic functions of \(\varepsilon\); \(A_\varepsilon=1+O(\varepsilon)\), \(B_\varepsilon=O(\varepsilon)\), \(C_\varepsilon=1+O(\varepsilon)\).
As is known, the solution of problem (4), (5) for a differential equation of elliptic type with analytic coefficients depending analytically on \(\varepsilon\) is itself a function analytic in \(\varepsilon\):
\[ \bar u=u_0+\varepsilon u_1+\ldots+\varepsilon^n u_n+\ldots, \]
where \(u_0,u_1,\ldots,u_n\) are those defined above, since the expansion of an analytic function into a power series is unique. By analyticity this series converges for sufficiently small \(\varepsilon\), i.e., expansion (3) converges for sufficiently small \(\varepsilon\).
Case 2, when the limiting problem lies on the spectrum. In this case we shall seek the solution \(u_\varepsilon\) of problem \(\Lambda_\varepsilon\) in the form
\[ u_\varepsilon=\frac{c_0u_{-1}}{\varepsilon} +(u_0+c_1u_{-1}) +\varepsilon(u_1+c_2u_{-1}) +\ldots+\varepsilon^n(u_n+c_{n+1}u_{-1})+\ldots \tag{6} \]
Substituting (6) into (1), (2) and equating the coefficients of like powers of \(\varepsilon\), we obtain:
\[ L_0u_{-1}\equiv \Delta u_{-1}+a\frac{\partial u_{-1}}{\partial x} +b\frac{\partial u_{-1}}{\partial y} +cu_{-1}=0, \qquad u_{-1}\big|_{\Gamma_0}=u_{-1}(1,\varphi)=0; \tag{7} \]
\[ L_0u_0=f(x,y), \qquad u_0\big|_{\Gamma_0}=-c_0\alpha(\varphi)Du_{-1}; \tag{8} \]
\[ L_0u_n=0,\qquad u_n\big|_{\Gamma_0} = -\sum_{l=0}^{n}c_{n-l}\frac{D^{(l+1)}u_{-1}}{(l+1)!}\alpha^{l+1}(\varphi) -\sum_{l=1}^{n}\frac{\alpha^l(\varphi)}{l!}D^{(l)}u_{n-l}, \tag{9} \]
where for \(u_n\) we introduce the normalization
\[ \int_{(\Omega_0)} u_{-1} z_0\, d\Omega_0=(u_{-1},z_0)=1,\qquad (u_n,z_0)=0,\quad n=0,1,\ldots, \tag{10} \]
where \(z_0\) is the solution of the adjoint homogeneous problem
\[ L_0^*z_0\equiv \Delta z_0-\frac{\partial}{\partial x}(az_0)-\frac{\partial}{\partial y}(bz_0)+cz_0=0,\qquad z_0|_{\Gamma_0}=0, \]
\(u_{-1}(x,y)\) is the eigenfunction corresponding to the simple eigenvalue \(\lambda_0=0\) of problem (7), and the function \(u_0(x,y)\) is a solution of problem (8). The solvability condition for it is as follows:
\[ (f,z_0)=c_0\int_{(\Gamma_0)} \alpha(\varphi)D u_{-1}\frac{\partial z_0}{\partial N}\,d\Gamma_0. \tag{11} \]
It is obtained from Green’s formula
\[ (L_0u_0,z_0)-(u_0,L_0^*z_0)= \int_{(\Gamma_0)} \left( \frac{\partial u_0}{\partial N}z_0-u_0\frac{\partial z_0}{\partial N} \right)d\Gamma_0+ \]
\[ +\int_{(\Gamma_0)}[a\cos(N,x)+b\cos(N,y)]u_0z_0\,d\Gamma_0. \]
Condition (11) can be satisfied for any \(f(x,y)\), by choosing \(c_0\), if and only if
\[ \int_{(\Gamma_0)} \alpha(\varphi)D u_{-1}\frac{\partial z_0}{\partial N}\,d\Gamma_0\ne 0. \tag{12} \]
If condition (12) is fulfilled, then we determine \(c_0\); knowing \(c_0\), we find \(u_0\) as a particular solution of problem (8) under the normalization condition.
Suppose that all \(c_0,c_1,\ldots,c_{n-1}, u_0,u_1,\ldots,u_{n-1}\) have already been determined. From the solvability condition for problem (9)
\[ 0=-\int_{(\Gamma_0)}u_n\frac{\partial z_0}{\partial N}\,d\Gamma_0 =\int_{(\Gamma_0)} \left[ \sum_{l=0}^{n} c_{n-l}\frac{\alpha^{l+1}(\varphi)}{(l+1)!}D^{(l+1)}u_{-1} + \sum_{l=1}^{n}\frac{\alpha^l(\varphi)}{l!}D^{(l)}u_{n-l} \right]\frac{\partial z_0}{\partial N}\,d\Gamma_0 \]
we find \(c_n\), and then from problem (9) determine a particular solution \(u_n\) under condition (10).
Let us note that the boundary-value problem with a perturbed boundary can be reduced to a problem with perturbed (analytic) coefficients. By virtue of paper (1), after the transformation there is an expansion (6), which is analytic for \(\varepsilon\ne 0\).
In the case where the problem \(A_\varepsilon\) has associated functions, we shall seek the solution \(u_\varepsilon\) in the form
\[ u_\varepsilon= \frac{c_0\bar u_0}{\varepsilon^k} + \frac{c_0\bar u_1+c_1\bar u_0}{\varepsilon^{k-1}} +\ldots+ \frac{c_0\bar u_{k-1}+\ldots+c_{k-1}\bar u_0}{\varepsilon} + \]
\[ +\left[u_0+(c_1\bar u_{k-1}+\ldots+c_k\bar u_0)\right]+\ldots \]
\[ \ldots+\varepsilon^s\left[u_s+(c_{s+1}\bar u_{k-1}+\ldots+c_{k+s}\bar u_0)\right]+\ldots, \]
where \(c_0,c_1,\ldots,c_k,c_{k+1},\ldots,c_{k+s}\), \(\bar u_0,\bar u_1,\ldots,\bar u_{k-1}\), \(u_0,u_1,\ldots,u_s\) are determined in the same way as above; the functions \(\bar u_1,\bar u_2,\ldots,\bar u_{k-1}\) are called associated functions of the eigenfunction \(\bar u_0\) (see (1)).
- We shall consider in the domain \(\Omega_\varepsilon\) the perturbed eigenvalue problem
\[ L_\varepsilon u_\varepsilon-\lambda_\varepsilon u_\varepsilon \equiv \Delta u_\varepsilon +a(x,y)\frac{\partial u_\varepsilon}{\partial x} +b(x,y)\frac{\partial u_\varepsilon}{\partial y} +c(x,y)u_\varepsilon-\lambda_\varepsilon u_\varepsilon=0 \tag{13} \]
under perturbation of the boundary
\[ u_\varepsilon\big|_{\Gamma_\varepsilon} = u_\varepsilon(1+\varepsilon\alpha(\varphi),\varphi)=0, \tag{14} \]
where the coefficients \(a(x,y)\), \(b(x,y)\), and \(c(x,y)\) are analytic functions.
For \(\varepsilon=0\) one obtains the unperturbed problem:
\[ L_0u_0-\lambda_0u_0 \equiv \Delta u_0+a\frac{\partial u_0}{\partial x} +b\frac{\partial u_0}{\partial y} +cu_0-\lambda_0u_0=0, \qquad u_0\big|_{\Gamma_0}=u_0(1,\varphi)=0. \]
Here \(u_0\) is the normalized eigenfunction; for the time being we assume the eigenvalue \(\lambda_0\) to be a simple eigenvalue. We seek the eigenvalue \(\lambda_\varepsilon\) close to \(\lambda_0\) and the corresponding eigenfunction \(u_\varepsilon\) for the perturbed operator in the form
\[ \lambda_\varepsilon=\sum_{n=0}^{\infty}\varepsilon^n\lambda_n, \qquad u_\varepsilon=\sum_{n=0}^{\infty}u_n\varepsilon^n, \qquad u_0\ne0. \tag{15} \]
Substituting (15) into (13), (14) and comparing coefficients of like powers of \(\varepsilon\), we obtain
\[ L_0u_0-\lambda_0u_0=0, \qquad u_0\big|_{\Gamma_0}=u_0(1,\varphi)=0; \tag{16} \]
\[ L_0u_n-\lambda_0u_n = \sum_{k=1}^{n}\lambda_k u_{n-k}, \qquad u_n\big|_{\Gamma_0} = -\sum_{l=1}^{n}\frac{\alpha^l(\varphi)}{l!}D^{(l)}u_{n-l}, \tag{17} \]
where \(D^{(l)}\) is the derivative in the normal direction of order \(l\); moreover, for \(u_n\) we introduce the normalization
\[ \int_{(\Omega_0)}u_0 z_0\,d\Omega_0 = (u_0,z_0)=1, \qquad (u_n,z_0)=0, \qquad n\ne0. \tag{18} \]
To determine \(u_1\) and \(\lambda_1\) we consider the problem
\[ L_0u_1-\lambda_0u_1=\lambda_1u_0, \qquad u_1\big|_{\Gamma_0}=-\alpha(\varphi)Du_0. \tag{19} \]
The condition for its solvability is
\[ \lambda_1= \int_{(\Gamma_0)} \alpha(\varphi)Du_0\cdot\frac{\partial z_0}{\partial N}\,d\Gamma_0. \]
It can be obtained from Green’s formula, using the equations, the boundary conditions, and (18).
From the solvability condition we find \(\lambda_1\) and then determine \(u_1\) as a particular solution of problem (19) subject to the normalization condition.
Suppose that all \(\lambda_0,\lambda_1,\ldots,\lambda_{n-1}\) and \(u_0,u_1,\ldots,u_{n-1}\) have already been determined; to determine \(\lambda_n\) and \(u_n\) we consider the problem
\[ L_0u_n-\lambda_0u_n = \sum_{k=1}^{n}\lambda_k u_{n-k}, \qquad u_n\big|_{\Gamma_0} = -\sum_{l=1}^{n}\frac{\alpha^l(\varphi)}{l!}D^{(l)}u_{n-l}. \tag{20} \]
From the solvability condition for problem (20) we find the eigenvalue \(\lambda_n\), and then determine \(u_n\) as a particular solution of problem (20).
In a manner analogous to that indicated above, the perturbed problem with perturbed boundary can be reduced to a problem with perturbed (analytic) coefficients. In doing so we obtain that the expansions of the eigenvalue \(\lambda_\varepsilon\) and the eigenfunction \(u_\varepsilon\) are analytic for sufficiently small \(\varepsilon\ne0\).
In conclusion I express my deep gratitude to my scientific adviser, Corresponding Member of the Academy of Sciences of the USSR L. A. Lyusternik, for advice and systematic assistance in carrying out the present work.
Moscow State University
named after M. V. Lomonosov
Received
22 II 1964
References
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- Lin Tsun-chi, Vestn. MGU (in press).
- V. A. Kronberg, Prikl. matem. i mekh., 19, no. 4, 463 (1955).