Reports of the Academy of Sciences of the USSR
1958. Volume 119, No. 4

MATHEMATICS

M. I. VISHIK and Corresponding Member of the Academy of Sciences of the USSR L. A. LYUSTERNIK

ON THE ASYMPTOTICS OF SOLUTIONS OF PROBLEMS WITH RAPIDLY OSCILLATING BOUNDARY CONDITIONS FOR PARTIAL DIFFERENTIAL EQUATIONS

As is known, the solution (u(x,y)), bounded in the half-plane (y>0), of the Laplace equation (\Delta u=0) under the boundary condition (u|_{y=0}=\sin kx) has the form (u=e^{-ky}\sin kx) and, for large (k), differs noticeably from zero only near the boundary.

Phenomena of this kind of boundary effect occur in various problems of mathematical physics with rapidly oscillating boundary conditions. In this article we investigate the asymptotics of solutions of the first boundary-value problem for arbitrary elliptic equations with rapidly oscillating boundary conditions. Some of the possible definitions of rapid oscillation are given in §§ 2 and 3. Since a uniformly bounded sequence of functions oscillating ever more rapidly on the boundary (\Gamma) of the domain converges weakly to zero on (\Gamma), and the solution operator for an elliptic equation is continuous, it is easy to derive convergence of the corresponding solutions to zero in every interior subdomain. It is of interest, however, to construct the asymptotics of the solutions near the boundary. We shall construct it by the methods described in the works ((^{1,2})).

1. Let us first consider, in a plane domain (Q) with boundary (\Gamma), an elliptic equation of order (2k)

[
L_{2k}u=0,
\tag{1}
]

solvable under arbitrary boundary conditions of the first boundary-value problem. First consider the case when these boundary conditions have the form:

[
\left.\frac{\partial^l u}{\partial n^l}\right|_{\Gamma}
=
A_l(\varphi)e^{i\omega\varphi}
\quad
(l=0,1,\ldots,k-1),
\tag{2}
]

where (A_l(\varphi)) are sufficiently smooth functions that vanish outside some part (\varphi_0\leq \varphi\leq \varphi_1) of the boundary ((\varphi) is a parameter on the boundary). In some (\rho_0)-neighborhood (\Omega_{\varphi_0}) of this part of the boundary introduce a coordinate system ((\rho,\varphi)) (cf. ((^2))), where (\rho), for example, is the distance along the normal to (\Gamma). Passing to these variables, write equation (1) in the form

[
L_{2k}u \equiv H_{2k}\left(\frac{\partial}{\partial \rho},\frac{\partial}{\partial \varphi}\right)u+L_{2k-1}u=0,
\tag{3}
]

[
H_{2k}(\xi,\eta)=\sum_{j=0}^{2k}a_j(\rho,\varphi)\xi^j\eta^{2k-j},
\quad
H_{2k}(i\xi,i\eta)\geq C\left(\xi^{2k}+\eta^{2k}\right)
\tag{4}
]

for real (\xi,\eta). Expand (a_l(\rho,\varphi)) in powers of (\rho):

[
a_l(\rho,\varphi)=a_l(\varphi)+\sum_{s=1}^{p} b_{ls}(\varphi)\rho^s + b_{l,p+1}(\rho,\varphi)\rho^{p+1}.
]

Then

[
H_{2k}(\xi,\eta)=M'_0(\xi,\eta)+\ldots,\qquad
M_0(\xi,\eta)=\sum a_j(\varphi)\xi^j\eta^{2k-j}.
]

Accordingly, carrying out analogous expansions of all coefficients of the operator (L_{2k}) and making the substitution (t=\omega\rho) and (u=v(\rho,\varphi)e^{i\omega\varphi}), we represent equation (3) in the form

[
e^{i\omega\varphi}\omega^{2k}\left(M_0\left(\frac{\partial}{\partial t},i\right)+\frac1\omega M_1+\ldots\right)v=0.
\tag{5}
]

Thus, we see that a small parameter (1/\omega) appears in the equation, which makes it possible to carry out the second iterative process of work (²). Namely, as a first approximation (v_0) we take a solution of the ordinary differential equation with respect to (t)

[
M_0\left(\frac{\partial}{\partial t},i\right)v\equiv
\sum_{j=} ^{2k} a_j(\varphi)i^{2k-j}\frac{\partial^j v}{\partial t^j}=0
\tag{6}
]

with coefficients constant with respect to (t) and characteristic equation

[
M_0(\lambda,i)=\sum a_j(\varphi)i^{2k-j}\lambda^j=0.
\tag{7}
]

From (4) there follows the inequality (M_0(i,i\lambda)>0) for all real (\lambda). Therefore, by virtue of Lemma 4 of work (²), equation (7) has exactly (k) roots (-\lambda_1,-\lambda_2,\ldots,-\lambda_k) with negative real parts. The general solution of equation (6) of boundary-layer type has the form

[
v_0=\sum_{l=1}^{k} c_l(\varphi)\exp(-\lambda_l t)
=\sum_{l=1}^{k} c_l(\varphi)\exp(-\lambda_l\omega\rho);
\tag{8}
]

the coefficients (c_l(\varphi)) are determined uniquely from the condition that (v_0) satisfy the boundary conditions (2). The next approximation (v_1) (\left(v=v_0+\frac1\omega v_1+\ldots\right)) is found just as elementarily (²) from the equation (M_0(v_1,i)=-M_1v_0) under homogeneous boundary conditions of type (2); analogously, constructing further the second iterative process (²), we obtain successive approximations (v_2,\ldots,v_m). The functions (v_0,v_1,\ldots,v_m) are defined only in a neighborhood of (\Omega_{\rho_0}). They can be extended to the entire domain (¹,²) by multiplying them by a smooth function (\psi(\rho)); (\psi(\rho)\equiv1) for (0\le\rho\le\tfrac13\rho_0), (\psi(\rho)\equiv0) for (\tfrac23\rho_0<\rho), and by setting (v_i\equiv0) in (Q-\Omega_{\rho_0}).

Denote

[
z_{m+1}=u-\left(v_0+\frac1\omega v_1+\ldots+\frac1{\omega^m}v_m\right)e^{i\omega\varphi},
\tag{9}
]

where (u) is the exact solution of problem (1), (2). We have:

[
|L_{2k}z_{m+1}|<\frac{C}{\omega^{m-2k}},
\tag{10}
]

and (z_{m+1}) satisfies homogeneous boundary conditions of type (2). Hence, using estimates in the metrics (W_p^{(2k)}) (³), we obtain

[
|z_{m+1}|_{W_p^{(2k)}}\leqslant \frac{C}{\omega^{m-2k}}.
\tag{11}
]

Thus, we obtain the asymptotics of the solution and an estimate of the remainder term, valid for the function (u) itself and for its derivatives up to order (2k). With the aid of energy inequalities under smaller restrictions

one can obtain analogous asymptotics and estimates up to derivatives of order (\leqslant k).

1. Let a family of functions ({f_\varepsilon}), depending on the parameter (\varepsilon), be given on (\Gamma). We shall say that this family is (\dfrac{1}{\varepsilon})-oscillating on the interval (\mu(\varphi_0 \leqslant \varphi \leqslant \varphi_1)), (\mu \subset \Gamma), if for any (\varphi) from this interval

[
\left|\int_{\varphi_0}^{\varphi} f_\varepsilon(\varphi)\,d\varphi\right| < K\varepsilon
\qquad (K\text{ does not depend on }\varepsilon).
\tag{12}
]

The family ({f_\varepsilon}) (\dfrac{1}{\varepsilon})-oscillates on the whole boundary (\Gamma), if (\Gamma) can be covered by a finite number of intervals (\mu_i) ((i=1,\ldots,p)), in each of which it (\dfrac{1}{\varepsilon})-oscillates. With the aid of a partition of unity (1 \equiv \sum \psi_i), where (\psi_i \equiv 0) everywhere outside (\mu_i), the function (v_\varepsilon) is represented in the form (v_\varepsilon \equiv \sum \psi_i v_\varepsilon), where each term is different from 0 only in (\mu_i).

We shall seek the asymptotics of the solutions (u_\varepsilon) of equation (1) under the boundary conditions

[
\left.\frac{\partial^l u_\varepsilon}{\partial \rho^l}\right|{\Gamma}
= \alpha_l(\varphi) g(\varphi),
\tag{13}
]

where (\alpha_l(\varphi) \equiv 0) outside one of the intervals (\mu_i), and (g_{l\varepsilon}(\varphi)) are (\dfrac{1}{\varepsilon})-oscillating functions on (\mu_i). The general case of oscillating boundary conditions is obtained by summing such solutions over all intervals (\mu_i). We shall regard (\mu_i) as corresponding to the interval ((0,2\pi)) of the variable (\varphi).

Expand the function into a Fourier series

[
g_{l\varepsilon}
=
\sum_{k=-\infty}^{+\infty} a_{lk}(\varepsilon)e^{ik\varphi}
]

and solve, for each term, the boundary-value problem (1), (2) with (A_l(\varphi)=\alpha_l(\varphi)a_{lk}(\varepsilon)), (\omega=k). In this way we obtain successive approximations (v_{0k}e^{ik\varphi},\ldots,\dfrac{1}{k^m}v_{mk}e^{ik\varphi}), and, summing them, obtain the asymptotics of (u_\varepsilon) in the form

[
w_0+w_1+\cdots+w_m,\qquad
\text{where }\
w_s=\sum_{k=-\infty}^{+\infty}\frac{1}{k^s}v_{sk}e^{ik\varphi}.
\tag{14}
]

From condition (12) of rapid oscillation of the function (g_{l\varepsilon}(\varphi)) it follows that

[
|a_{lk}(\varepsilon)| \leqslant C|k|\varepsilon.
]

Therefore the terms (v_{sk}) corresponding to small values of (k) are small, while the terms with large values of (k) decay rapidly as (\rho) increases, by virtue of (8).

The following estimates hold:

[
|w_0| \leqslant C\frac{\varepsilon}{\rho}\quad \text{for }\rho>\varepsilon,
\qquad
|w_1| \leqslant C\min(\varepsilon^{1/2},\,\varepsilon|\ln\rho|),
]

[
|w_k| \leqslant C\varepsilon \quad \text{for } k \geqslant 2,
\qquad
|D^s w_{s+1}| \leqslant C\min(\varepsilon^{1/3},\,\varepsilon|\ln\rho|),
\tag{15}
]

[
|D^s w_p| \leqslant C\varepsilon \quad \text{for } p>s+1.
]

It can be verified that the remainder (z_\varepsilon=u_\varepsilon-(w_0+\cdots+w_{2k})) satisfies an equation of the form

[
L_{2k}z_\varepsilon=O(\varepsilon)
]

and homogeneous boundary conditions. Therefore, by (3),

[
|z_\varepsilon|_{W_p^{(2k)}} \le C\varepsilon .
]

Thus, the iterations given above yield a remainder that is small throughout the whole domain together with its derivatives up to order (2k).

If we want to obtain a remainder that is small in the metric (L_p) ((p>1)), then, as is clear from (15), it is enough to restrict ourselves to the first approximation (w_0):
(u_\varepsilon = w_0 + z_{\varepsilon 0}) ((|z_{\varepsilon 0}|{L_p}\le C\varepsilon)). If we want to ensure that (z\varepsilon) is small of order (\varepsilon) in the metric (C), then, as is clear from (14), it is enough to restrict ourselves to the first two terms (w_0+w_1).

1. One can give a more refined definition of oscillation on the interval (a\le \varphi \le b). A family of functions ({f_\varepsilon}) on the interval ([a,b]) is called (\dfrac{1}{\varepsilon})-oscillating of order (s) if:

1)
[
\left|\int_{a_\varepsilon}^{b_\varepsilon} f_\varepsilon\,d\varphi\right|\le C\varepsilon^s,
\qquad
\text{where } |a-a_\varepsilon|\le C\varepsilon,\quad |b-b_\varepsilon|\le C\varepsilon;
]

2) for the (s)-th successive principal primitive* the condition

[
\left|\int_{a_\varepsilon}^{\varphi} f_\varepsilon^{(-s)}\,d\varphi\right|\le C\varepsilon^s
]

is satisfied.

If the family of boundary conditions (g_{1\varepsilon}(\varphi)) is (\dfrac{1}{\varepsilon})-oscillating of order (s), then, applying the preceding constructions, we obtain the estimates: for (l\varepsilon)), (|w_s|\le C\min(\varepsilon^s,\ \varepsilon|\ln\rho|)), (|w_m|\le C_m\varepsilon^s) for (m>s), and the asymptotic representation (14) can be carried to remainder terms of order (\varepsilon^s).

1. The multidimensional case does not differ in any way from the two-dimensional one. Let us note that it is enough to have rapid oscillations in at least one direction on the boundary for the constructions and estimates given above to be valid. For example, the definition of (\dfrac{1}{\varepsilon})-oscillation given in item 2 is generalized as follows: a family of functions ({f_\varepsilon}) is called (\dfrac{1}{\varepsilon})-oscillating on a given part of the boundary if, in some local coordinate system, the integrals over any parallelepipedal domain are of order (\varepsilon). As in the two-dimensional case, we pass from a local definition to a definition of oscillation over the entire boundary.

The authors’ attention was drawn to the subject of the present note by talks of A. L. Gol’denveizer at their seminar. The authors express their gratitude to him.

References Cited

1. M. I. Vishik, L. A. Lyusternik, DAN, 113, No. 4 (1957).
2. M. I. Vishik, L. A. Lyusternik, Uspekhi Mat. Nauk, 12, issue 15 (77) (1957).
3. A. I. Koshelev, DAN, 110, No. 3 (1956).

* We shall call the primitive (f_\varepsilon^{(-1)}) of the function (f_\varepsilon) the principal primitive for which

[
\int_{a_\varepsilon}^{b_\varepsilon} f_\varepsilon^{(-1)}\,d\varphi = 0 .
]