On the Shapiro-Lopatinkii condition for elliptic problem
This paper is concerned with elliptic problems including a small parameter multiplying higher order derivatives. We found algebraic conditions on the operator and boundary conditions which guarantee the Fredholm property, and prove an a priori estimate for the solution with a constant independent of...
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Інститут прикладної математики і механіки НАН України
2014
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irk-123456789-1244472017-09-27T03:03:11Z On the Shapiro-Lopatinkii condition for elliptic problem Dyachenko, E. This paper is concerned with elliptic problems including a small parameter multiplying higher order derivatives. We found algebraic conditions on the operator and boundary conditions which guarantee the Fredholm property, and prove an a priori estimate for the solution with a constant independent of the small parameter. These results are known for elliptic boundary value problems with small parameter in the half space Rⁿ+. We extend them to the case of bounded domains with smooth boundary. The small parameter coercive conditions are formulated and two-sided estimate is proved. 2014 Article On the Shapiro-Lopatinkii condition for elliptic problem / E. Dyachenko // Український математичний вісник. — 2014. — Т. 11, № 1. — С. 49-68. — Бібліогр.: 12 назв. — англ. 1810-3200 2010 MSC. 58J37. http://dspace.nbuv.gov.ua/handle/123456789/124447 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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This paper is concerned with elliptic problems including a small parameter multiplying higher order derivatives. We found algebraic conditions on the operator and boundary conditions which guarantee the Fredholm property, and prove an a priori estimate for the solution with a constant independent of the small parameter. These results are known for elliptic boundary value problems with small parameter in the half space Rⁿ+. We extend them to the case of bounded domains with smooth boundary. The small parameter coercive conditions are formulated and two-sided estimate is proved. |
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Article |
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Dyachenko, E. |
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Dyachenko, E. On the Shapiro-Lopatinkii condition for elliptic problem Український математичний вісник |
| author_facet |
Dyachenko, E. |
| author_sort |
Dyachenko, E. |
| title |
On the Shapiro-Lopatinkii condition for elliptic problem |
| title_short |
On the Shapiro-Lopatinkii condition for elliptic problem |
| title_full |
On the Shapiro-Lopatinkii condition for elliptic problem |
| title_fullStr |
On the Shapiro-Lopatinkii condition for elliptic problem |
| title_full_unstemmed |
On the Shapiro-Lopatinkii condition for elliptic problem |
| title_sort |
on the shapiro-lopatinkii condition for elliptic problem |
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Інститут прикладної математики і механіки НАН України |
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2014 |
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http://dspace.nbuv.gov.ua/handle/123456789/124447 |
| citation_txt |
On the Shapiro-Lopatinkii condition for elliptic problem / E. Dyachenko // Український математичний вісник. — 2014. — Т. 11, № 1. — С. 49-68. — Бібліогр.: 12 назв. — англ. |
| series |
Український математичний вісник |
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AT dyachenkoe ontheshapirolopatinkiiconditionforellipticproblem |
| first_indexed |
2025-07-09T01:26:44Z |
| last_indexed |
2025-07-09T01:26:44Z |
| _version_ |
1837130745145982976 |
| fulltext |
Український математичний вiсник
Том 11 (2014), № 1, 49 – 68
On the Shapiro–Lopatinkii condition
for elliptic problems
Evgeniya Dyachenko
(Presented by M. M. Malamud)
Abstract. This paper is concerned with elliptic problems including
a small parameter multiplying higher order derivatives. We found alge-
braic conditions on the operator and boundary conditions which guaran-
tee the Fredholm property, and prove an a priori estimate for the solution
with a constant independent of the small parameter. These results are
known for elliptic boundary value problems with small parameter in the
half space Rn
+. We extend them to the case of bounded domains with
smooth boundary. The small parameter coercive conditions are formu-
lated and two-sided estimate is proved.
2010 MSC. 58J37.
Key words and phrases. Elliptic problems, small parameter, uniform
estimates, Shapiro–Lopatinskii condition.
Introduction
This paper studies linear elliptic differential equations with a small
parameter at the highest derivative when boundary conditions also con-
tain the same parameter. We consider operators acting on functions in
some domain D and depending on a small parameter ε ≥ 0 in a special
way. More precisely,
ε2m−2µA2m(x,D)u+ ε2m−2µ−1A2m−1(x,D)u+ · · · +A2µ(x,D)u = f,
supplemented by the boundary conditions
εbj−βjBj,bj (x
′, D)u+ εbj−βj−1Bj,bj−1(x
′, D)u+ · · · +Bj,βj (x
′, D)u = uj ,
for x′ ∈ ∂D, with j = 1, . . . ,m, where A2m−i(x,D) and Bj,bj−i(x
′, D) are
differential operators of order 2m− i and bj−i, respectively, with variable
coefficients. This family of operators is assumed to be elliptic for each
ε ≥ 0.
Received 7.05.2013
The work was supported by the Institut für Mathematik der Universität Potsdam.
ISSN 1810 – 3200. c© Iнститут математики НАН України
50 On the Shapiro–Lopatinkii condition...
The subject of our interest is the behaviour of solutions uε(x) when ε
tends to zero. It is known (see, for example, [4]) that solutions uε defined
on a manifold with boundary may have the so-called boundary layer. In
this case the solution uε(x) converges to u0(x), when ε → 0, uniformly
in each strictly inner bounded subdomain D̃ ⊂ D, but need not converge
at the boundary points x′ ∈ ∂D. Essentially this concept was introduced
by Prandtl in 1904 (for accurate historical background see e.g. [6]). He
studied fluid flow with small viscosity over a surface and explained how
insignificant friction forces influence on the main perfect fluid flow. His
idea is based on splitting a solution into two parts, namely a solution near
the boundary and a solution far away from the boundary, and stretching
the coordinates in the normal direction of the boundary.
Lyusternik and Vishik in the paper [2] extended this idea to differ-
ential equations which depend on a small parameter polynomially. Some
applications of the Lyusternik–Vishik method for partial differential equa-
tions (PDE) are discussed in [5]. This method suggests to look for a solu-
tion as the sum of a regular part, which depends uniformly on ε, and an
additional function, which grows rapidly when ε→ 0. The regular part is
found by using ordinary method of small parameter; the boundary layer
is supposed to be a solution of some ordinary differential equation (ODE)
in the direction normal to ∂D. This ODE is obtained using coordinate
stretching in the direction normal to the boundary, as it was proposed by
Prandtl.
In [2] the problem was considered in the case of Dirichlet boundary
conditions Bj and strong uniform ellipticity of the operator A. Then
the method was adapted to domains with conical points by Nazarov in
[8]. In [9] it was extended to pseudodifferential operators and general
elliptic boundary problems. In all these works uniform estimates in norms
depending on ε for solutions were found under some generalised coercivity
condition, and the estimates justify formal asymptotic series obtained by
the Lyusternik-Vishik method. But the comprehensive theory of elliptic
equations with small parameter was constructed by Volevich in [1]. For
the problem (A,B) in the half-space Rn
+ := {(x′, xn) ∈ Rn : xn > 0}
he introduced a Shapiro–Lopatinskii condition with small parameter and
proved its necessity and sufficiency for the existence of two-sided uniform
estimates for (A,B). Volevich used the norms proposed by Demidov
in [7].
The paper by Volevich falls short of providing complete arguments in
the case of arbitrary smooth bounded domains and the aim of the present
paper is to extend the results of Volevich to the case of bounded domains
D with smooth boundary ∂D using the local principle of elliptic theory
(see, for example, [3]).
E. Dyachenko 51
1. Asymptotic expansion
Now we apply the Vishik–Lyusternik method to the problem (A,B)
to find an asymptotic expansion of solution u. The domain D is required
to be bounded and have smooth boundary ∂D. Let us introduce new
coordinates (y1, y2, . . . , yn−1, z) in D, such that y ∈ ∂D is a variable
on the surface ∂D and z is the distance to ∂D. By A′(y, z,Dy, Dz, ε),
B′(y,Dy, Dz, ε) we denote operators A, B in the new variables.
We are looking for a solution of (A,B) in the form u(x, ε) = U(x, ε)+
V (y, z/ε, ε) where U is the regular part of u and V is the boundary
layer. Suppose that the function V (y, z/ε) satisfies the following three
conditions:
1. V (y, z/ε, ε) is a sufficiently smooth solution of the homogeneous
equation AV = 0;
2. V (y, z/ε, ε) depends on the “fast” variable t = z/ε;
3. V (y, z/ε, ε) differs from zero only in a small strip near the boundary
∂D.
The regular part and the boundary layer are looked for as formal
asymptotic series
U(x, ε) =
∞∑
k=0
εkuk(x), V (y, z/ε, ε) =
∞∑
k=0
εkvk(y, z/ε).
The first series is called the outer expansion and the second one is called
the inner expansion. The outer expansion is obtained using the standard
procedure of small parameter method. We substitute the series for U(x, ε)
into the equation A(x,D, ε)u = f and collect the terms with the same
power of ε. It gives us the system
A2µu0 = f, (1.1)
A2µuk = −
k∑
i=1
A2µ+iuk−i (1.2)
for unknowns uk.
To determine the coefficients vk(y, z/ε) of the inner expansion we
apply the operator A′(y, z,Dy, Dz, ε) to V (y, z/ε, ε). Condition 1 implies
∞∑
k=0
εkA′(y, z,Dy, Dz, ε)(vk(y, z/ε)) = 0.
52 On the Shapiro–Lopatinkii condition...
Let us rewrite the operator A′(y, z,Dy, Dz, ε) in the variables (y, z/ε = t).
For the homogeneous part A′
k of degree k we have
A′
k(y, z,Dy, Dz, ε) = ε−2µ+kA′
k(y, εt, εDy, Dt).
On expanding A′
k(y, εt, εDy, Dt) as Taylor series about the point (y, 0, 0,
Dt) we obtain
A′
k(y, z,Dy, Dz, ε) = ε−2µ+k
(
A′′
k(k, 0, 0, Dt) +
∞∑
l=1
εlAk,l(y, t,Dy, Dt)
)
,
where the operators Ak,l have smooth coefficients.
Therefore,
A′(y, z,Dy, Dz, ε) = ε−2µ
(
A′′(y, 0, 0, Dt) +
∞∑
l=1
εlAl(y, t,Dy, Dt)
)
,
Al depends on Ak,l linearly.
So it defines equations for vk
A′′(ξ, 0, 0, Dt)vk(y, t) = −
k∑
l=1
Al(y, t,Dy, Dt)vk−l.
Now we substitute the partial sums
Un =
n∑
k=0
εkuk(x),
Vn =
n∑
k=0
εkvk(y, t)
into the original equation and boundary conditions and find the discrep-
ancy. For A(x,D, ε), it looks like
A(x,D, ε)(u(x, ε) − Un − Vn)
= f −A2µ(x,D)u0 −
(
A(x,D, ε)Un −A2µ(x,D)u0
+A′(y, z,Dy, Dz, ε)Vn
)
= O(εn+1).
Hence, if we are able to find appropriate Banach spaces ‖u‖′, ‖u‖′∂D,
such that the operator (A(x,D, ε), B(x′, D, ε)) is bounded uniformly in
ε then the difference u(x, ε) − Un − Vn is small and so the formal series
approximates the solution u(x, ε) indeed.
E. Dyachenko 53
This problem was solved by Volevich [1] for the case where D is the
half-space Rn
+ = {(x′, xn) ∈ Rn : xn > 0}. He used the norms for
functions in D and their traces which had been introduced in [7]. They
are of the form
‖u;Hr,s(Rn)‖ = ‖(1 + |ξ|2)s/2(1 + ε2|ξ|2)(r−s)/2û‖L2 ,
‖u;Hρ,σ(Rn−1)‖ = ‖u‖L2(Rn−1) + ‖|η|σ(1 + ε2|η|2)(ρ−σ)/2u‖L2(Rn−1),
where σ ≥ 0. In these norms there is an estimate
‖u;Hr,s(Rn
+)‖ ≤ C
(
‖A(x,D)u;Hr,s(Rn
+)‖
+
m∑
j=1
‖Bj(D
′, ε)u;Hr−bj−1/2,s−βj−1/2(Rn−1)‖ + ‖u;L2(Rn
+)‖
)
, (1.3)
where C does not depend on ε. A trace theorem for the norms
‖u;Hr,s(Rn)‖ and ‖u;Hρ,σ(Rn−1)‖ is also proved in [1].
Theorem 1.1. For r > l + 1/2 and s ≥ 0, s 6= l + 1/2, we have
‖Dl
nu(·, 0);Hr−l−1/2,s−l−1/2(Rn−1)‖ ≤ c ‖u;Hr,s(Rn
+)‖
with c a constant independent of ε.
Volevich [1] also proves that estimate (1.3) holds true if the operator
(A,B) satisfies the small parameter ellipticity condition, the Shapiro–
Lopatinskii condition with small parameter and the system of equations
(1.1), (1.2) is correctly solvable. The problem (A,B) is called an elliptic
problem with parameter if it satisfies all these conditions. The first two
conditions are, of course, of greater interest than the last one. They read
as follows:
Small parameter ellipticity condition: The operator A(x,D, ε) is said
to be small parameter elliptic at some point x0 if its principal polynomial
A0(x0, ξ, ε) admits an estimate
|A0(x0, ξ, ε)| ≥ cx0 |ξ|2µ(1 + ε|ξ|)2m−2µ
from below.
Shapiro–Lopatinskii condition with small parameter: The problem
(A,B) satisfies the Shapiro–Lopatinskii condition for every ε ≥ 0.
As mentioned, this paper is aimed at extending the result of [1] to
the case of bounded domains D with smooth boundary ∂D. To this
end we develop the local principle that underlies elliptic theory (see for
example [3]) in the case of problems with parameter.
54 On the Shapiro–Lopatinkii condition...
2. The main spaces
Our first task is to introduce the main spaces. Hereinafter D stands
for a bounded domain with smooth boundary in Rn. The spaces Hr,s(Rn)
and Hr,s(Rn
+) are exactly the same as those used in the works of Demidov
(see for instance [7]). Namely, Hr,s(Rn) consists of all functions u ∈
Hr(Rn) which have finite norm ‖u‖r,s, and Hr,s(Rn
+) is the factor space
Hr,s(Rn)/Hr,s
− (Rn) whereHr,s
− (Rn) is the subspace of Hr,s(Rn) consisting
of all functions with support in {x ∈ Rn : xn ≤ 0}. As usual, the factor
space is endowed with the canonical norm
‖[u];Hr,s(Rn
+)‖ = inf
u∈[u]
‖u‖r,s.
When it does not cause misunderstanding we denote this norm simply
by ‖u‖r,s. Analogously, we introduce the spaces of functions defined in
some domain D. To wit,
Hr,s(D) := Hr,s(Rn)/Hr,s
Rn\D(Rn)
where functions of Hr,s
Rn\D(Rn) are supported outside the domain D. This
space is also given the canonical norm ‖u;Hr,s(D)‖, which we denote
sometimes by ‖u‖r,s for short.
Lemma 2.1. Let f be a smooth function in Rn, such that f(x) = 1 for
x ∈ D. Then ‖u;Hr,s(D)‖ = ‖fu;Hr,s(D)‖.
Lemma 2.2. If u ∈ Hr,s(Rn) and suppu ⊂ D, then ‖u;Hr,s(D)‖ =
‖u;Hr,s(Rn)‖.
For positive integer numbers s and r ≥ s the space Hr,s(D) proves to
be the completion of C∞(D̄) with respect to the norm ‖u;Hr,s(D)‖r,s.
The elliptic technique used in this paper includes the “rectification” of the
boundary. Therefore, the invariance of ‖ · ‖r,s with respect to a change of
variables is one of the key points. For every fixed ǫ ≥ 0, the norms ‖ · ‖r,s
are the ordinary Sobolev norms and the main question is what kind of
coordinate transformations save the form of the dependence of ‖ · ‖r,s on
ε. The following statement displays how ε enters into the norms ‖ · ‖r,s.
Lemma 2.3. For natural r and s satisfying r≥s, the norm ‖u;Hr,s(D)‖2
has a representation
r∑
i=0
i is even
ar,s,i(ε)‖∆i/2u‖2
L2(D) +
r∑
i=1
i is odd
ar,s,i(ε)‖∇iu‖2
L2(D),
where ar,s,i(ε) are polynomials of degree 2i and ar,s,0(ε) 6= 0.
E. Dyachenko 55
Proof. Applying the binomial formula we get
(1 + |ξ|2)s =
s∑
i=0
Ci
s|ξ|2i and (1 + ε2|ξ|2)r−s =
r−s∑
i=0
Ci
r−sε
2i|ξ|2i.
Hence, on multiplying the left-hand sides of these equalities we obtain
(1 + |ξ|2)s(1 + ε2|ξ|2)r−s =
r∑
i=0
ar,s,i(ε)|ξ|2i
where
ar,s,i(ε) =
i∑
j=0
Ci−j
s Cj
r−sε
2j . (2.1)
Here, we assume Ck
r = 0 when k > r. If ε = 0 or r = s, then ar,s,i = Ci
s.
Therefore, ar,s,i(ε) 6= 0 for all ε and 0 ≤ i ≤ r. As a consequence we get
‖u‖2
r,s =
r∑
i=0
ar,s,i(ε)‖|ξ|iû‖2
L2 .
Furthermore,
‖|ξ|iû‖2
L2 =
{∥∥∆i/2u
∥∥2
L2 , if i is even,∥∥∇iu
∥∥2
L2 , if i is odd,
which establishes the lemma.
Now everything is prepared for proving the invariance of the norm
‖ · ‖r,s with respect to local changes of variables x = T (y).
Lemma 2.4. Let r, s ∈ Z≥0 satisfy r ≥ s. The norm ‖u;Hr,s(D)‖ is
invariant with respect to any local changes of variables in D of the form
x = T (y), such that
1. T : U → U ′ is a Cr-diffeomorphism of domains U and U ′ in Rn,
both U and U ′ intersecting D;
2. T (U ∩D) = U ′ ∩D;
3. T (U ∩ ∂D) = U ′ ∩ ∂D.
Our task is to prove that there is a constant C > 0 independent of ε,
with the property that
‖T ∗u;Hr,s(D)‖ ≤ C ‖u;Hr,s(D)‖ (2.2)
56 On the Shapiro–Lopatinkii condition...
for all smooth functions u in the closure of D supported in some compact
set K ⊂ U ′ ∩ D̄). Here, by T ∗u(y) := u(T (y)) is meant the pullback of u
by the diffeomorphism T . If u is supported in K, then T ∗u is supported
in T−1(K), which is a compact subset of U ′ ∩ D̄ by the properties of T .
Since this applies to the inverse T−1 : U ′ → U , it follows from (2.2) that
the space Hr,s(D) survives under the local Cr -diffeomerphisms of D̄.
Proof. For the proof we make use of another norm in Hr,s(D) which is
obviously equivalent to ‖u;Hr,s(D)‖ and more convenient here. To wit,
‖u;Hr,s(D)‖ ∼=
∑
|α|≤r
ar,s,|α|(ε)‖∂αu;L2(D)‖ (2.3)
(or
‖u;Hr,s(D)‖ ∼=
r∑
i=0
ar,s,i(ε) ‖u;H i(D)‖,
as is easy to verify), where ar,s,i(ε) are the polynomials of Lemma 2.3.
Fix a compact set K in U ′∩D̄. As mentioned, if u is a smooth function in
D with support in K, then T ∗u is a smooth function in D with support
in T−1(K) ⊂ U ∩D. Obviously,
‖T ∗u;Hr,s(D)‖ = ‖u ◦ T ;Hr,s(U ∩D)‖
=
∑
|α|≤r
ar,s,|α|(ε)‖∂α(u ◦ T );L2(U ∩D)‖.
By the chain rule,
∂α
y (u(T (y))) =
∑
06=β≤α
cα,β(y)
(
∂β
xu
)
(T (y))
for any multiindex α with |α| ≤ r. Here, the coefficients cα,β(y) are
polynomials of degree |β| of partial derivatives of T (y) up to order |α| −
|β| + 1 ≤ r. Since T : U → U ′ is a diffeomorphism of class Cr, all
the cα,β(y) are bounded on the compact set T−1(K) and the Jacobian
detT ′(y) does not vanish on T−1(K). This implies
‖T ∗u;Hr,s(D)‖ ≤ c
∑
|α|≤r
ar,s,|α|(ε)
∑
β≤α
‖(∂β
xu) ◦ T ;L2(T−1(K))‖
≤ c
∑
|α|≤r
ar,s,|α|(ε)
∑
β≤α
‖∂β
xu;L
2(K)‖,
E. Dyachenko 57
where c = c(T, r,K) is a constant independent of u and different in
diverse applications. Interchanging the sums in α and β yields
‖T ∗u;Hr,s(D)‖ ≤ c
∑
|β|≤r
( ∑
|α|≤r
α≥β
ar,s,|α|(ε)
) ∑
06=β≤α
‖∂βu;L2(D)‖
for all smooth functions u in D with support in K.
Therefore, if there is a constant C > 0 such that
∑
|α|≤r
α≥β
ar,s,|α|(ε) ≤ C ar,s,|β|(ε)
for each multiindex β of norm |β| ≤ r, then the lemma follows. Since
∑
|α|≤r
α≥β
ar,s,|α|(ε) ≤ c
r∑
i=|β|
ar,s,i(ε)
with c a constant dependent only on r and n, we are left with the task
to show that there is a constant C > 0 independent of ε, such that
r∑
i=i0
ar,s,i(ε) ≤ C ar,s,i0(ε)
for all i0 = 0, 1, . . . , r. This latter estimate is in turn fulfilled if we show
that
ar,s,i(ε) ≤ C ar,s,i−1(ε) (2.4)
for all i = 1, . . . , r, where C is a constant independent of ε ∈ [0, 1]. By
formula (2.1),
ar,s,i(ε) =
i−s−1∑
j=0
Ci−j
s Cj
r−sε
2j ,
hence, estimate (2.4) is fulfilled for sufficiently small ε > 0 with any
constant C greater than Ci
s/C
i−1
s . Since (2.4) is valid for all ε in any
interval [ε0, 1] with ε0 > 0, the proof is complete.
Remark 2.1. The case of inner point is not singled out in the
Lemma 2.4. Clearly, the problem is easier away from the boundary,
for neither the condition 2 nor the condition 3 are no longer required.
Lemma 2.5. The spaces Hr,s(D) are invariant with respect to the local
changes variables described in Lemma 2.4, when r, s are positive real
numbers and r ≥ s.
58 On the Shapiro–Lopatinkii condition...
Proof. This follows from Lemma 2.4 by using standard interpolation
techniques (see e.g. [12]).
To use local techniques it is convenient to define the spaces Hρ,σ(∂D)
by locally rectifying the boundary surface. Since the boundary is com-
pact, there is a finite covering {Ui}N
i=1 of ∂D consisting of sufficiently
small open subsets Ui of Rn. Let {φi} be a partition of unity in a neigh-
bourhood of ∂D subordinate to this covering. If Ui is small enough,
there is a smooth diffeomorphism hi of Ui onto an open set Oi in Rn,
such that hi(Ui ∩ D) = Oi ∩ Rn
+ and hi(Ui ∩ ∂D) = Oi ∩ Rn−1, where
Rn−1 = {x ∈ Rn : xn = 0}. The transition mappings Ti,j = h−1
i ◦hj prove
to be local diffeomorphism of D, as explained in Lemma 2.4. For any
smooth function u on the boundary the norm ‖(h−1
i )∗(φiu);Hρ,σ(Rn−1)‖
is obviously well defined and we set
‖u;Hρ,σ(∂D)‖ :=
N∑
i=1
‖(h−1
i )∗(φiu);Hρ,σ(Rn−1)‖, (2.5)
where (h−1
i )∗(φiu) = (φiu) ◦ h−1
i . As usual, the space Hρ,σ(∂D) is intro-
duced to be the completion of C∞(∂D) with respect to the norm (2.5).
When combined with the trace theorem for the spaces Hr,s(Rn
+) and
Hρ,σ(Rn−1) proved in [1], and Lemma 2.4, a familiar trick readily shows
that the Banach spaces Hρ,σ(∂D) are actually independent of the partic-
ular choice of the covering of ∂D by coordinate patches {Ui} in Rn, the
special coordinate system hi : Ui → Rn in Ui and the partition of unity
{φi} in a neighbourhood of ∂D subordinate to the covering {Ui}. Any
other choice of these data leads to an equivalent norm (2.5) in C∞(∂D).
Lemma 2.6. As defined above, the spaces Hρ,σ(∂D) are invariant with
respect to local diffeomerphisms of the boundary surface ∂D.
The reader gives readily the concept of local diffeomorphisms of ∂D
a sense similar to that of Lemma 2.4.
3. Auxiliary results
When compared to the usual local techniques of elliptic theory, the
theory of elliptic boundary value problems with small parameter include
only three additional estimates uniform in the parameter. To wit,
1. the invariance of the norm with respect to local changes of variables
on the compact manifold D;
E. Dyachenko 59
2. estimates of the form εk‖∂αu‖r,s ≤ c ‖u‖r′,s′ with c independent
of ε;
3. inequalities like ‖u‖r,s ≤ δ ‖u‖r′,s′ + C(δ) ‖u‖L2 with r′ ≥ r, s′ ≥ s
and δ > 0 a fixed arbitrary small parameter.
As usual, we write α, β and γ for multiindices. By β ≤ α is meant
that βi ≤ αi for all i = 1, . . . , n. We first recall several basic inequalities
concerning Sobolev spaces. Directly from the multinomial theorem we
obtain
|ξα| ≤ 1
Cα
n
|ξ||α|/2, (3.1)
where Cα
n is the multinomial coefficient. This inequality, if combined with
the Plancherel theorem, yields
‖∂αu‖L2 ≤ 1
Cα
n
‖∆|α|/2u‖L2
for all u ∈ H |α| := H |α|(Rn), where ∆|α|/2 is a fractional power of the
Laplace operator in Rn.
Besides, we use the following consequence of the embedding theorem
for Sobolev spaces (see e.g. [10]).
Theorem 3.1. Suppose u is a square integrable function with compact
support in Rn and α ∈ Zn
≥0 is fixed. If, in addition, the weak derivatives
∂βu are square integrable for all β ≤ α, then
‖∂βu‖L2 ≤ C ‖∂αu‖L2 ,
where C = sup{|x|2 : x ∈ suppu}.
We also need some basic inequalities for the norms ‖ · ‖r,s.
Lemma 3.1. Let u ∈ Hr,s(Rn) be a function with compact support, k ≥ 1
an integer and α a multiindex. Then:
1. We have ε‖u‖r,s ≤ c ‖u‖r+1,s, where c depends on the support of u
but not on u and ε.
2. If k > |α|, then εk‖∂αu‖r,s ≤ c ‖u‖r+k,s, the constant c being inde-
pendent of u and ε.
3. If k ≤ |α|, then εk ‖∂αu‖r,s ≤ c ‖u‖r+|α|,s+|α|−k, where c is inde-
pendent of u and ε.
60 On the Shapiro–Lopatinkii condition...
Proof. Using the expression for the norm in Hr,s(Rn) we get
ε ‖∆1/2u‖r,s = ε ‖|ξ|(1 + |ξ|2)s/2(1 + ε2|ξ|2)(r−s)/2û‖L2 ≤ ‖u‖r+1,s.
As ‖u‖r,s ≤ c ‖∆1/2u‖r,s, the part 1 is true.
The part 2 is proved in much the same way if one applies k − |α|
times what has already been proved in the part 1.
To prove the part 3 we split the majorising factor as εk|ξ||α| =
(ε|ξ|)k |ξ||α|−k. The first factor contributes with order k to the terms
with ε while the second one does |α| − k to the others.
The part 2 actually holds for all function in Hr+k,s even if u fails to
be of compact support.
Lemma 3.2. Let δ be an arbitrary small positive number. Then there is
a constant C(δ), such that
‖u‖r−1,s−1 ≤ δ ‖u‖r,s + C(δ) ‖u‖L2
for all u ∈ Hr,s(Rn).
Proof. Set 〈ξ〉 =
√
1 + |ξ|2 for ξ ∈ Rn. Given any R > 0, we obtain
‖u‖2
r−1,s−1 =
∫
|ξ|>R
〈ξ〉2s
〈ξ〉2 〈εξ〉2(r−s)|û|2dξ +
∫
|ξ|≤R
〈ξ〉2(s−1)〈εξ〉2(r−s)|û|2dξ
≤ 1
1 +R2
‖u‖2
r,s + (1 +R2)s−1(1 + ε2R2)r−s ‖u‖2
L2 ,
Choosing R > 0 in such a way that δ2 ≤ (1 + R2)−1, we establish the
estimate, as is easy to check.
4. The main result
Now we are in a position to present the main result of this work.
We impose two restrictions on the boundary value problem under study,
namely, the condition of ellipticity and the Shapiro–Lopatinskii condition
with small parameter. To formulate these denote by A0 the principal part
of the operator A which is understood here as
A0(x,D, ε) := ε2m−2µA2m,0(x,D) + · · · + εA2µ+1,0(x,D) +A2µ,0(x,D),
where Aj,0(x, ξ) stands for the principal homogeneous symbol of the dif-
ferential operator Aj(x,D) of order j, with 2µ ≤ j ≤ 2m. Recall that
the differential operator A(x,D, ε) is said to satisfy the small parameter
E. Dyachenko 61
ellipticity condition in the domain D if n > 2 and for every x ∈ D the
polynomial A0(x, ξ, ε) admits an estimate
|A0(x, ξ, ε)| ≥ cx |ξ|2µ(1 + ε|ξ|)2m−2µ
for all ξ ∈ Rn and ε ∈ [0, 1], where cx > 0 is a constant which depends
only on the point x.
In the case n = 2 the polynomial A0(x, ξ
′, ξn, ε) considered with re-
spect to the variable ξn is assumed to possess exactly m roots in the
upper complex half-plane and m roots in the lower half-plane, for every
x ∈ D, ε > 0, ξ′ ∈ Rn−1.
As is well known in elliptic theory, the ellipticity condition guarantees
in the case n > 2 that the polynomial A0(x, ξ
′, ξn, ε) has m roots in the
upper half-plane and m roots in the lower one. So, this property can be
taken as basis for the small parameter ellipticity definition.
By the Shapiro–Lopatinskii condition with a small parameter is just
meant that the boundary value problem (A(x,D, ε), B(x′, D, ε)) satisfies
the usual Shapiro–Lopatinskii condition for each fixed x′ ∈ ∂D and ε ∈
[0, 1]. This latter condition means that the polynomials Bj(x
′, ξ, ε) are
linearly independent modulo A(x′, ξ, ε) for each point x′ ∈ ∂D and ε ≥ 0.
Theorem 4.1. Under the above conditions, if moreover r ≥ 2m and
s ≥ 2µ, then there is an estimate
‖u‖r,s ≤ C
(
‖A(x,D, ε)u‖r−2m,s−2µ
+
m∑
j=1
‖Bj(x
′, D, ε)u‖r−bj−1/2,s−βj−1/2 + ‖u‖L2(D)
)
(4.1)
with C a constant independent of u and ε.
The proof exploits localisation techniques. First, using a finite cov-
ering {Ui} of D by sufficiently small open sets (e.g. balls) in Rn, we
represent any function u ∈ Hr,s(D) as the sum of functions ui ∈ Hr,s(D)
compactly supported in Ui ∩D, just setting ui = φiu for a suitable par-
tition of unity {φi} in D subordinate to the covering {Ui}. Secondly, for
each summand ui we formulate its own elliptic problem and find a priori
estimates for its solutions. If Ui does not meet the boundary of D, then
the support of ui is a compact subset of D and the proof of (4.1) reduces
to global analysis in Rn considered in [1]. For those Ui which intersect
the boundary of D we choose a change of variables x = h−1
i (z) to rectify
the boundary surface within Ui. To wit, hi(Ui ∩D) = Oi ∩Rn
+, where Oi
is an open set in Rn, and so in the coordinates y estimate (4.1) reduces
62 On the Shapiro–Lopatinkii condition...
to that in the case D = Rn
+ treated in [1]. Thirdly, we glue together all
a priori estimates for ui thus obtaining a priori estimate (4.1) for u.
Perhaps the focus of local techniques is on the second and third steps.
Taking for granted the estimates of the second step, we complete the proof
of Theorem 4.1.
Proof. For each point x0 ∈ D we choose a neighbourhood Ux0 in D in
which the estimate of Theorem 5.1 holds. And for each point x0 ∈ ∂D we
choose a neighbourhood Ux0 in Rn, such that the estimate of Theorem 5
is valid. Shrinking Ux0 , if necessary, one can assume that the surface
Ux0 ∩ ∂D can be rectified by some diffeomorphism hi : Ux0 → Rn, as
explained above. The family {Ux0}x0∈D is an open covering of D, hence
it contains a finite family {Ui} which covers D. Fix a C∞ partition of
unity {φi} in a neighbourhood of D subordinate to the covering {Ui}.
Given any u ∈ Hr,s(D), we get
u =
∑
i
ui
in D, where ui := φiu belongs to Hs,r(D) and suppui ⊂ Ui ∩ D. By
assumption, for any function ui estimate (4.1) holds with a constant C
depending on i. As the family {Ui} is finite, there is no restriction of
generality in assuming that C does not depend on i. Hence,
‖u‖r,s ≤ C
∑
i
(
‖A(x,D, ε)ui‖r−2m,s−2µ
+
m∑
j=1
‖Bj(x
′, D, ε)ui‖r−bj−1/2,s−βj−1/2 + ‖ui‖L2(D)
)
.
By the Leibniz formula,
A(x,D, ε)ui = φiA(x,D, ε)u+ [A, φi]u,
Bj(x
′, D, ε)ui = φiBj(x,D, ε)u+ [Bj , φi]u,
where [A, φi]u = A(φiu)−φiAu is the commutator of A and the operator
of multiplication with φi, and similarly for [Bj , φi]. The commutators are
known to be differential operators of order less than that of A and Bj ,
respectively. From the structure of the operator A(x,D, ε) we see that
the summands of [A, φi]u are of the form
ε2m−2µ−kak,β(x)∂βu, (4.2)
where k = 0, 1, . . . , 2m − 2µ, |β| ≤ 2m − k − 1 and ak,β are smooth
functions in the closure of D independent of u.
E. Dyachenko 63
To estimate the norm of (4.2) in Hr−2m,s−2µ, we apply Lemma 3.1
and consider separately the cases
2m− 2µ− k > |β|,
2m− 2µ− k ≤ |β|.
If e.g. |β| ≥ 2m− 2µ− k, then
ε2m−2µ−k‖ak,β∂
βu‖r−2m,s−2µ≤c ε2m−2µ−k‖ak,β∂
βu‖r−2m+|β|,s−2m+|β|+k,
where |β| − 2m+ k ≤ −1. It follows that
ε2m−2µ−k‖ak,β∂
βu‖r−2m,s−2µ ≤ c ‖u‖r−1,s−1
with c a constant independent of u and ε. Such terms are handled by
Lemma 3.2. Analogously we estimate the summands (4.2) with 2m −
2µ− k > |β| and the commutators [Bj , φi], which establishes (4.1).
5. Local estimates in the interior
Theorem 5.1. For every x0 ∈ D there exists a neighbourhood Ux0 in D
and a constant C independent of ε, such that
‖u‖r,s ≤ C
(
‖A(x,D, ε)u‖r−2m,s−2µ + ‖u‖L2
)
(5.1)
for all functions u ∈ Hr,s(D) with compact support in Ux0, where r ≥
2m, s ≥ 2µ are integer.
This theorem is not contained in [1], for [1] focuses on differential
operators with constant coefficients in Rn.
Proof. If u ∈ Hr,s(D) is compactly supported in D, it can be thought
of as an element of Hr,s(Rn) as well. The norm of u in Hs,r(D) just
amounts to the norm of u in Hs,r(Rn). Hence, the paper [1] applies if
A(x,D, ε) has constant coefficients, as is the case e.g. for A0(x0, D, ε),
the principal part of A(x,D, ε) with coefficients frozen at x0. According
to [1], there is a constant C > 0 independent of ε, such that
‖u‖r,s ≤ C ‖A0(x0, D, ε)u‖r−2m,s−2µ (5.2)
for all functions u ∈ Hr,s(D) of compact support in D.
We are thus left with the task to majorise the right-hand side of (5.2)
by that of (5.1) uniformly in ε ∈ [0, 1] on functions with compact support
in Ux0 . To this end, we write
64 On the Shapiro–Lopatinkii condition...
A0(x0, D, ε) = A(x,D, ε) − (A(x,D, ε) −A0(x,D, ε))
− (A0(x,D, ε) −A0(x0, D, ε))
whence
‖A0(x0, D, ε)u‖r−2m,s−2µ
≤ ‖A(x,D, ε)u‖r−2m,s−2µ + ‖(A(x,D, ε) −A0(x,D, ε))u‖r−2m,s−2µ
+ ‖(A0(x,D, ε) −A0(x0, D, ε))u‖r−2m,s−2µ. (5.3)
Our next concern will be to estimate the last two summands on the
right-hand side of (5.3). We begin with the first of these two. By the very
structure of the operator A(x,D, ε), the difference A(x,D, ε)−A0(x,D, ε)
is the sum of terms of the form
ε2m−2µ−kak,β(x)∂βu,
where k = 0, 1, . . . , 2m − 2µ, |β| ≤ 2m − k − 1 and ak,β are smooth
functions in the closure of D (cf. (4.2)). Hence, the reasoning used
in the proof of Theorem 4.1 shows that the second summand on the
right-hand side of (5.3) is dominated uniformly in ε ∈ [0, 1] by the norm
‖u‖r−1,s−1. On applying Lemma 3.2 we conclude that
‖(A(x,D, ε)−A0(x,D, ε))u‖r−2m,s−2µ ≤ δ ‖u‖r,s + C(δ) ‖u‖L2 , (5.4)
where δ > 0 is an arbitrarily small parameter and C(δ) depends only
on δ but not on u and ε.
It remains to estimate the last summand on the right-hand side of
(5.3). Let us write
A0(x,D, ε) =
∑
2µ≤|β|≤2m
ε|β|−2µA0,β(x)∂β ,
where A0,β are smooth functions on the closure of D. Then
‖(A0(x,D, ε) −A0(x0, D, ε))u‖r−2m,s−2µ
≤
∑
2µ≤|β|≤2m
ε|β|−2µ‖(A0,β(x) −A0,β(x0))∂
βu‖r−2m,s−2µ.
To evaluate the summands we invoke the equivalent expression for the
norm in Hr−2m,s−2µ(D) given by (2.3). The typical term is
ar−2m,s−2µ,|α|(ε) ε
|β|−2µ
∥∥∂α
(
(A0,β(x) −A0,β(x0))∂
βu
)∥∥
L2(D)
E. Dyachenko 65
with |α| ≤ r−2m and ar−2m,s−2µ,|α|(ε) are the polynomials defined in of
(2.1). By the Leibniz formula,
∂α
(
(A0,β(x)−A0,β(x0))∂
βu
)
=(A0,β(x)−A0,β(x0))∂
α+βu+[∂α, A0,β ] ∂βu,
where the commutator [∂α, A0,β ] is a differential operator of order |α|−1
with smooth coefficients in D. Observe that |α| + |β| ≤ r. Arguing as
above we derive easily an estimate like (5.4) for the sum
∑
|α|≤r−2m
ar−2m,s−2µ,|α|(ε) ε
|β|−2µ ‖[∂α, A0,β ]u‖L2(D)
whenever u ∈ Hr,s(D) is of compact support in D.
It is the term
ar−2m,s−2µ,|α|(ε) ε
|β|−2µ ‖(A0,β(x) −A0,β(x0))∂
α+βu‖L2(D)
that admits a desired estimate only in the case if the support of u is
small enough. (Recall that u is required to have compact support in
Ux0 .) Since the coefficients A0,β(x) are Lipschitz continuous in D, for
any arbitrarily small δ′ > 0 there is a positive ̺ = ̺(δ′), such that
‖(A0,β(x) −A0,β(x0))∂
α+βu‖L2(D) ≤ δ′ ‖∂α+βu‖L2(D)
for all functions u ∈ Hr,s(D) with compact support in B(x0, ̺), the ball
of radius ̺ with center x0.
Summarising we conclude that for each δ > 0 there is a constant
C = C(δ) independent of ε, such that
‖(A0(x,D, ε) −A0(x0, D, ε))u‖r−2m,s−2µ ≤ δ ‖u‖r,s + C(δ) ‖u‖L2 (5.5)
for all functions u ∈ Hr,s(D) with compact support in B(x0, ̺), provided
that ̺ = ̺(δ) is sufficiently small. Needless to say that C(δ) need not
coincide with the similar constant of inequality (5.4), however, we may
assume this without loss of generality.
On gathering estimates (5.3) and (5.4), (5.5) and substituting them
into (5.1) we arrive at
(1 − 2Cδ) ‖u‖r,s ≤ C
(
‖A(x,D, ε)u‖r−2m,s−2µ + 2C(δ) ‖u‖L2
)
for all u ∈ Hr,s(D) with compact support in B(x0, ̺). Of course, this
latter inequality does not yield any estimate for ‖u‖r,s unless 1−2Cδ > 0.
Thus, choosing δ < 1/2C we get
‖u‖r,s ≤
2CC(δ)
1−2Cδ
(
‖A(x,D, ε)u‖r−2m,s−2µ + ‖u‖L2
)
,
if C(δ) ≥ 1/2.
66 On the Shapiro–Lopatinkii condition...
6. The case of boundary points
Localisation at a boundary point x0 ∈ ∂D requires not only small
parameter ellipticity of the operator A(x,D, ε) but also the Shapiro–
Lopatinskii condition with small parameter.
Theorem 6.1. For every point x0 ∈ ∂D there is a neighbourhood Ux0
in Rn, such that
‖u‖r,s ≤ C
(
‖A(x,D, ε)u‖r−2m,s−2µ
+
m∑
j=1
‖Bj(x
′, D, ε)u‖r−bj−1/2,s−βj−1/2 + ‖u‖L2(D)
)
(6.1)
for all functions u ∈ Hr,s(D) with compact support in Ux0 ∩D, where C
is a constant independent of both u and ε ∈ [0, 1].
Proof. Choose a neighbourhood U of x0 in Rn and a diffeomorphism z =
h(x) of U onto an a neighbourhood O of the origin 0 = h(x0) in Rn with
the property that h(U ∩D) = O∩Rn
+ and h(U ∩∂D) = {z ∈ O : zn = 0}.
If u ∈ Hr,s(D) is a function with compact support in U ∩ D, then the
pullback ũ = (h−1)∗u belongs to Hr,s(Rn
+) and has compact support in
O ∩ Rn
+, which is due to Lemma 2.4. On setting
A♯ := (h−1)∗Ah∗,
B♯
j := (h−1)∗Bjh
∗,
for j = 1, . . . ,m, we obtain the pullbacks of the operators A and Bj
under the diffeomorphism h : U ∩ D → O ∩ Rn
+. It is easily seen that
A♯ and B♯
j are differential operators with small parameter ε ∈ [0, 1] on
O ∩ Rn
+ in the sense explained above. We write à := A♯ and B̃j := B♯
j
for short. Since the spaces Hr,s(D) and Hρ,σ(∂D) are invariant under
local diffeomorphisms of D, it follows that estimate (6.1) is equivalent to
‖ũ‖r,s ≤ C
(
‖Ã(z,D, ε)ũ‖r−2m,s−2µ
+
m∑
j=1
‖B̃j(z
′, D, ε)ũ‖r−bj−1/2,s−βj−1/2 + ‖ũ‖L2(D)
)
(6.2)
for all functions ũ ∈ Hr,s(Rn
+) with compact support in O ∩ Rn
+, where
C is a constant independent of ũ and ε.
E. Dyachenko 67
From the transformation formula for principal symbols of differential
operators it follows that the problem
{
Ã0(0, D, ε)ũ = f̃ for zn > 0,
B̃j,0(0, D, ε)ũ = ũj for zn = 0,
where j = 1, . . . ,m, satisfies both the ellipticity condition and the Shapi-
ro–Lopatinskii condition with small parameter in the half-space. We now
apply the main result of [1] which says that there is a constant C > 0
independent of ε, such that the inequality
‖ũ‖r,s ≤ C
(
‖Ã0(0, D, ε)ũ‖r−2m,s−2µ
+
m∑
j=1
‖B̃j,0(0, D, ε)ũ‖r−bj−1/2,s−βj−1/2 + ‖ũ‖L2(Rn
+)
)
holds true for all functions ũ ∈ Hr,s(Rn
+) with compact support in the
closed half-space.
Estimate (6.2) follows from the latter estimate in much the same
way as estimate (5.1) does from (5.2), see the proof of Theorem 5.1.
The only difference consists in evaluating the boundary terms. However,
estimates on the boundary are reduced readily to those in the half-space
if one exploits the embedding theorem, see Theorem 1.1. Namely,
‖(B̃j(z
′, D, ε) − B̃j,0(0, D, ε))ũ; Hr−bj−1/2,s−βj−1/2(Rn−1)‖
≤ c ‖(B̃j(z
′, D, ε) − B̃j,0(0, D, ε))ũ; H
r−bj ,s−βj (Rn
+)‖
with c a constant independent of ũ and ε.
Conclusion
Theorem 4.1 answers the question about the Fredholm property of the
elliptic problem (A,B) in the case where D is a bounded domain with
smooth boundary. As is shown in Section 1, this allows one to apply
the boundary layer method. However, the smoothness of boundary is a
strong restriction, all arguments of this paper go through if the boundary
is of mere class Cr. Still the solvability and regularity in smooth bounded
domains lay the foundation for further researches and are a necessary step
in constructing the theory of small parameter ellipticity in domains with
singular points.
Acknowledgements. I thank Professor Nikolai Tarkhanov for the pro-
posed problem, his useful remarks and suggestions.
68 On the Shapiro–Lopatinkii condition...
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Contact information
Evgeniya
Dyachenko
Universität Potsdam,
Institut für Mathematik,
Am Neuen Palais 10,
14469 Potsdam,
Germany
E-Mail: dyachenk@uni-potsdam.de
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