MATHEMATICS

N. M. LEONTOVICH

ASYMPTOTIC EXPANSION OF BOUNDARY-VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS

(Presented by Academician S. L. Sobolev, 12 XII 1959)

Consider, in a bounded domain (Q) of (n)-dimensional space with boundary (\Gamma), the equation

[
L_\varepsilon u=\varepsilon^2\Delta\Delta u+L_2u=h
\tag{1}
]

under the boundary conditions

[
u|_\Gamma=0;
\tag{2}
]

[
\Delta u|_\Gamma=0,
\tag{3}
]

where (L_2) is an elliptic operator of the 2nd order,

[
L_2=-\sum_{i,j=1}^{n}\frac{\partial}{\partial x_i}
\left(a_{ij}(x)\frac{\partial}{\partial x_j}\right)
+\sum_{i=1}^{n} b_i(x)\frac{\partial}{\partial x_i}+c(x),
\quad
x=(x_1,\ldots,x_n),
]

[
c-\frac12\sum_{i=1}^{n}\frac{\partial b_i}{\partial x_i}\ge \alpha^2>0.
\tag{4}
]

The coefficients of the equation and the boundary of the domain have (N) derivatives, (h(x)\in W_2^{2N+2}).

We seek an approximate solution of the problem (A_\varepsilon) (equation (1) under boundary conditions (2), (3)) for small (\varepsilon) in the form of a segment of a series in powers of (\varepsilon).

We shall use the method developed by M. I. Vishik and L. A. Lyusternik ((^1)).

For the boundary-value problem under consideration it is necessary, along with the expansion described below of the operator (L_\varepsilon), to carry out the same expansion of the boundary operators (2), (3). Near the boundary we introduce a local coordinate system ((\rho,\varphi_1,\ldots,\varphi_{n-1})), where ((\varphi_1,\ldots,\varphi_{n-1})) are coordinates of a point of the boundary, and (\rho) is the distance along the normal. In equation (1) and in the boundary conditions (2), (3) we pass to the variables (t=\rho/\varepsilon), (\varphi_1,\ldots,\varphi_{n-1}), and expand the coefficients by Taylor’s formula with respect to the variable (t). Then

[
L_\varepsilon u\equiv \frac1{\varepsilon^2}
\left[R_0+\varepsilon R_1+\cdots+\varepsilon^N R_N+\varepsilon^{N+1}R\right]u,
\tag{5}
]

where

[
R_0\equiv \frac{\partial^4}{\partial t^4}
-a^0(\varphi)\frac{\partial^2}{\partial t^2},
\quad
\varphi=(\varphi_1,\ldots,\varphi_{n-1}),
\quad
a^0(\varphi)>0.
]

It is obvious that among the characteristic roots of the equation (R_0u=0) there is a negative one, (-\lambda(\varphi)), i.e. the degeneration of the problem (A_\varepsilon) into the problem (A_0) is regular ((^1)). The (R_k) are differential operators whose coefficients depend on (t) only polynomially.

The boundary operators (in the variables ((t,\varphi))) are written as follows:
[
u\big|{\Gamma}=u\big|,
]
[
\Delta u\big|{\Gamma}
=
\frac{1}{\varepsilon^{2}}
\left(
\frac{\partial^{2}}{\partial t^{2}}+\varepsilon l}+\varepsilon^{2}l_{2
\right)u\big|{t=0},
]
where (l) are differential operators of order 2.}) and (l_{2

Problem (A_{0}), i.e. the equation
[
L_{2}w=h
\tag{6}
]
with the boundary condition
[
w\big|_{\Gamma}=C,
\tag{7}
]
has a unique solution by virtue of (4).

We construct an approximate solution (\widetilde u_{\varepsilon}) of problem (A_{\varepsilon}) in the form
(\widetilde u_{\varepsilon}=\widetilde u+\varepsilon^{2}\widetilde v), where (\widetilde u) satisfies (1), and (\varepsilon^{2}\widetilde v) satisfies the corresponding homogeneous equation, i.e.
[
L_{\varepsilon}\widetilde u
\equiv
\varepsilon^{2}\Delta\Delta \widetilde u+L_{2}\widetilde u=h;
\tag{8}
]
[
L_{\varepsilon}\varepsilon^{2}\widetilde v
\equiv
\bigl[R_{0}+\varepsilon R_{1}+\cdots+\varepsilon^{N+1}R\bigr]\widetilde v=0.
\tag{9}
]
Fulfillment of the boundary conditions requires for (\widetilde u+\varepsilon^{2}\widetilde v):
[
\widetilde u\big|{\Gamma}+\varepsilon^{2}\widetilde v\big|=0;
\tag{10}
]
[
\Delta \widetilde u\big|{\Gamma}
+
\left(
\frac{\partial^{2}}{\partial t^{2}}+\varepsilon l}+\varepsilon^{2}l_{2
\right)
\widetilde v\big|_{t=0}=0.
\tag{11}
]

We seek (\widetilde u) and (\widetilde v) in the form
[
\widetilde u
=
u_{0}+\varepsilon u_{1}+\cdots+\varepsilon^{N}u_{N}+\cdots;
\qquad
\widetilde u_{N}
=
u_{0}+\varepsilon u_{1}+\cdots+\varepsilon^{N}u_{N};
\tag{12}
]
[
\widetilde v
=
v_{0}+\varepsilon v_{1}+\cdots+\varepsilon^{N}v_{N}+\cdots;
\qquad
\widetilde v_{N}
=
v_{0}+\varepsilon v_{1}+\cdots+\varepsilon^{N}v_{N}.
\tag{13}
]

Substituting (12) and (13) into (8) and (9) and, respectively, into (10) and (11), and collecting terms with equal powers of (\varepsilon), we obtain equations and boundary conditions for the successive determination of (u_i) and (v_i).

Terms for (\varepsilon^{0}):
[
L_{2}u_{0}=h,\qquad u_{0}\big|{\Gamma}=0.
]
(u\equiv w),}) is determined uniquely: (u_{0
[
R_{0}v_{0}
\equiv
\frac{\partial^{4}v_{0}}{\partial t^{4}}
-
a^{0}(\varphi)\frac{\partial^{2}v_{0}}{\partial t^{2}}
=0;
\tag{14^0}
]
[
\frac{\partial^{2}v_{0}}{\partial t^{2}}\bigg|{t=0}
=
-\Delta u\big|{\Gamma}.
\tag{15^0}
]
We take that solution of this equation which has the character of a boundary layer (1):
[
v,}=c_{0}(\varphi)e^{-\lambda(\varphi)t
\qquad
c_{0}(\varphi)=-\frac{\Delta u_{0}}{(\lambda(\varphi))^{2}}.
]

Terms for (\varepsilon^{i}):
[
L_{2}u_{i}=-\Delta\Delta u_{i-2},
\qquad
u_{i}\big|{\Gamma}=-v\big|{t=0}.
]
Since (v\big|{t=0}) does not depend on (\varepsilon), (u_i) also does not depend on (\varepsilon),
[
R}v_{i
=
-R_{1}v_{i-1}-R_{2}v_{i-2}-\cdots-R_{i}v_{0};
\tag{14^i}
]
[
\frac{\partial^{2}v_{i}}{\partial t^{2}}\bigg|{t=0}
=
-\Delta u\big|{\Gamma}
-
l\big|}v_{i-1{t=0}
-
l.}v_{i-2}\big|_{t=0
\tag{15^i}
]

Since
(-R_{1}v_{i-1}-R_{2}v_{i-2}-\cdots-R_{i}v_{0})
is a polynomial of degree (2i-1) in (t), multiplied by (e^{-\lambda(\varphi)t}), equation ((14^{i})) has a solution of the form
[
v_{i}=\bigl[tS_{2i-1}(t,\varphi)+c_{i}(\varphi)\bigr]e^{-\lambda(\varphi)t},
]

where the coefficients of the polynomial (S_{2i-1}) are found by the method of selection, (c_i(\varphi)) from the conditions of satisfaction of the boundary condition ((15^i)), and (v_i) is defined in the boundary strip along (\Gamma).

We now multiply (\widetilde v_N) by a smoothing function (\psi(t)), having derivatives of all orders, equal to 1 in some strip along (\Gamma) and to 0 outside some other, wider strip. We obtain (v_N), defined in the whole domain.

From the method of constructing the functions (\widetilde u_N) and (\widetilde v_N) it follows that (\widetilde u_N+\varepsilon^2 v_N) satisfies the following equation and boundary conditions:

[
L_\varepsilon(\widetilde u_N+\varepsilon^2 v_N)=h+\varepsilon^{N+1}G_1;
\tag{16}
]

[
(\widetilde u_N+\varepsilon^2 v_N)\big|_{\Gamma}=\varepsilon^{N+1}g_1;
\tag{17}
]

[
\Delta(\widetilde u_N+\varepsilon^2 v_N)\big|_{\Gamma}=\varepsilon^{N+1}g_2,
\tag{18}
]

where the functions (G_1,g_1,g_2) are of order (O(1)) with respect to (\varepsilon).

Let (\alpha) be a function of order (\varepsilon^{N+1}) satisfying (17) and (18). Then the function (z)

[
z=u_\varepsilon-(\widetilde u_N+\varepsilon^2 v_N-\alpha)
]

satisfies the equation

[
L_\varepsilon z=\varepsilon^{N+1}G;\qquad G=O(1)
\tag{19}
]

and the homogeneous boundary conditions (2), (3).

Consider the quadratic form:

[
(L_\varepsilon z,z)=\varepsilon^2(\Delta\Delta z,z)+(L_2z,z)=
]

[
=\varepsilon^2\iint_Q\left[\sum_{i=1}^n\frac{\partial^2 z}{\partial x_i^2}\right]^2\,d\Omega
+\iint_Q\sum_{i,j=1}^n a_{ij}\frac{\partial z}{\partial x_i}\frac{\partial z}{\partial x_j}\,d\Omega
+\iint_Q\left[c-\frac12\sum_{i=1}^n\frac{\partial b_i}{\partial x_i}\right]z^2\,d\Omega.
\tag{20}
]

The integrals over the boundary of the domain (\Gamma) are equal to 0, since (z) satisfies the homogeneous boundary conditions (2), (3).

From the last equality (20) we obtain:

[
\varepsilon^2\iint_Q\sum_{i,j=1}^n
\left(\frac{\partial^2 z}{\partial x_i\partial x_j}\right)^2\,d\Omega
+\iint_Q\sum_{i=1}^n
\left(\frac{\partial z}{\partial x_i}\right)^2\,d\Omega
+\iint_Q z^2\,d\Omega
\le
]

[
\le M\left|(\varepsilon^{N+1}G,z)\right|
\le M\varepsilon^{N+1}
\left{\iint_Q G^2\,d\Omega\right}^{1/2}
\left{\iint_Q z^2\,d\Omega\right}^{1/2}.
]

In deriving this inequality we have used the ellipticity of the operator (L_2), inequality (4), and the inequality

[
\iint_Q\sum_{i,j=1}^n
\left(\frac{\partial^2 z}{\partial x_i\partial x_j}\right)^2\,d\Omega
\le C_1
\iint_Q\left(\sum_{i=1}^n\frac{\partial^2 z}{\partial x_i^2}\right)^2\,d\Omega
]

(see (2), Ch. II).

Thus we obtain estimates in (\mathscr L^2) for (z), its first and second derivatives:

[
\left{\iint_Q z^2\,d\Omega\right}^{1/2}
\le C\varepsilon^{N+1};
\tag{21}
]

[
\left{\iint_Q\sum_{i=1}^n
\left(\frac{\partial z}{\partial x_i}\right)^2\,d\Omega\right}^{1/2}
\le C\varepsilon^{N+1};
\tag{22}
]

[
\left{\iint_Q \sum_{i,j=1}^{n}\left(\frac{\partial^2 z}{\partial x_i\partial x_j}\right)^2\,d\Omega\right}^{1/2}\leq C\varepsilon^N.
\tag{23}
]

We obtain estimates of the third and fourth derivatives from Lemma 1, ((^2)) Ch. II:

[
\sum_{s=0}^{4}\iint_Q
\left(
\sum_{\alpha_1,\ldots,\alpha_j}
\frac{\partial^s z}{\partial x_{\alpha_1}\cdots \partial x_{\alpha_j}}
\right)^2 d\Omega
\leq C'\left[
\iint_Q z^2\,d\Omega+\iint_Q(\Delta z)^2\,d\Omega+
\iint_Q(\Delta \Delta z)^2\,d\Omega+
\iint_Q\sum_{i=1}^{n}\left(\frac{\partial z}{\partial x_i}\right)^2\,d\Omega+
\iint_Q\left(\sum_{i=1}^{n}\frac{\partial \Delta z}{\partial x_i}\right)^2\,d\Omega
\right].
\tag{24}
]

Since

[
\varepsilon^2\Delta\Delta z=-L_2z+\varepsilon^{N+1}G,
\tag{25}
]

it follows that

[
\varepsilon^2\iint_Q(\Delta\Delta z)^2\,d\Omega
\leq
C\iint_Q(L_2,z)^2\,d\Omega+
\varepsilon^{2N+2}\iint_Q G^2\,d\Omega,
]

[
\iint_Q(\Delta\Delta z)^2\,d\Omega\leq C\varepsilon^{2N-2}.
]

We obtain the estimate for the last integral in (24) by multiplying (25) by (\Delta z) and taking into account that (\Delta z|_{\Gamma}=0),

[
\iint_Q\sum_{i=1}^{n}\left(\frac{\partial}{\partial z}(\Delta z)\right)^2\,d\Omega
\leq C\varepsilon^{2N-2}.
]

From (24) we obtain the estimates

[
\left{\iint_Q
\sum_{i,j,k=1}^{n}
\left(\frac{\partial^3 z}{\partial x_i\partial x_j\partial x_k}\right)^2
\,d\Omega\right}^{1/2}
\leq C\varepsilon^{N-1};
\tag{26}
]

[
\left{\iint_Q
\sum_{i,j,k,l=1}^{n}
\left(\frac{\partial^4 z}{\partial x_i\partial x_j\partial x_k\partial x_l}\right)^2
\,d\Omega\right}^{1/2}
\leq C\varepsilon^{N-1}.
\tag{27}
]

Remark. For the equation

[
L_\varepsilon u=\varepsilon^2\Delta\Delta u+L_2u=h,
]

where

[
L_2u=-\Delta u+\sum_{i=1}^{n}b_i(x_1,\ldots,x_n)\frac{\partial u}{\partial x_i}
+c(x_1,\ldots,x_n)u
]

under the boundary conditions

[
\left.\frac{\partial u}{\partial n}\right|{\Gamma}=0,
\qquad
\left.\frac{\partial}{\partial n}(\Delta u)\right|=0
]

there is an analogous representation of the solution (u=\tilde u+\varepsilon^3\tilde v), where

[
\tilde u=u_0+\varepsilon u_1+\ldots+\varepsilon^N u_N+\ldots,
]

[
\tilde v=v_0+\varepsilon v_1+\ldots+\varepsilon^N v_N+\ldots
]

For the residual (z) and its derivatives we obtain the same estimates (21), (22), (23), (26), (27).

Moscow State University
named after M. V. Lomonosov

Received
20 XI 1959

CITED LITERATURE

1. M. I. Vishik, L. A. Lyusternik, Uspekhi Mat. Nauk, 12, no. 5 (77) (1957).
2. O. A. Ladyzhenskaya, A Mixed Problem for a Hyperbolic Equation, Moscow, 1953, Ch. II, § 3.