Elliptic Boundary-Value Problems in the Sense of Lawruk on Sobolev and Hörmander Spaces
We study elliptic boundary-value problems with additional unknown functions in boundary conditions. These problems were introduced by Lawruk. We prove that the operator corresponding to a problem of this kind is bounded and Fredholm in appropriate couples of the inner product isotropic Hörmander spa...
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irk-123456789-1656092020-02-15T01:26:01Z Elliptic Boundary-Value Problems in the Sense of Lawruk on Sobolev and Hörmander Spaces Murach, A.A. Chepurukhina, I.S. Статті We study elliptic boundary-value problems with additional unknown functions in boundary conditions. These problems were introduced by Lawruk. We prove that the operator corresponding to a problem of this kind is bounded and Fredholm in appropriate couples of the inner product isotropic Hörmander spaces, which form the refined Sobolev scale. Досліджєно еліптичну крайову задачу з додатковими невідомими Функціями у крайових умовах. Ці задачi введеш Лавруком. Доведено, що оператор, відповідний такій задачі, є обмеженим i нетеровим у відповідних парах гільбертових ізотропних просторів Хермандера, які утворюють уточнену соболєвську шкалу. 2015 Article Elliptic Boundary-Value Problems in the Sense of Lawruk on Sobolev and Hörmander Spaces / A.A. Murach, I.S. Chepurukhina // Український математичний журнал. — 2015. — Т. 67, № 5. — С. 672–691. — Бібліогр.: 44 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165609 517.956.223 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Murach, A.A. Chepurukhina, I.S. Elliptic Boundary-Value Problems in the Sense of Lawruk on Sobolev and Hörmander Spaces Український математичний журнал |
| description |
We study elliptic boundary-value problems with additional unknown functions in boundary conditions. These problems were introduced by Lawruk. We prove that the operator corresponding to a problem of this kind is bounded and Fredholm in appropriate couples of the inner product isotropic Hörmander spaces, which form the refined Sobolev scale. |
| format |
Article |
| author |
Murach, A.A. Chepurukhina, I.S. |
| author_facet |
Murach, A.A. Chepurukhina, I.S. |
| author_sort |
Murach, A.A. |
| title |
Elliptic Boundary-Value Problems in the Sense of Lawruk on Sobolev and Hörmander Spaces |
| title_short |
Elliptic Boundary-Value Problems in the Sense of Lawruk on Sobolev and Hörmander Spaces |
| title_full |
Elliptic Boundary-Value Problems in the Sense of Lawruk on Sobolev and Hörmander Spaces |
| title_fullStr |
Elliptic Boundary-Value Problems in the Sense of Lawruk on Sobolev and Hörmander Spaces |
| title_full_unstemmed |
Elliptic Boundary-Value Problems in the Sense of Lawruk on Sobolev and Hörmander Spaces |
| title_sort |
elliptic boundary-value problems in the sense of lawruk on sobolev and hörmander spaces |
| publisher |
Інститут математики НАН України |
| publishDate |
2015 |
| topic_facet |
Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/165609 |
| citation_txt |
Elliptic Boundary-Value Problems in the Sense of Lawruk on Sobolev and Hörmander Spaces / A.A. Murach, I.S. Chepurukhina // Український математичний журнал. — 2015. — Т. 67, № 5. — С. 672–691. — Бібліогр.: 44 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT murachaa ellipticboundaryvalueproblemsinthesenseoflawrukonsobolevandhormanderspaces AT chepurukhinais ellipticboundaryvalueproblemsinthesenseoflawrukonsobolevandhormanderspaces |
| first_indexed |
2025-07-14T19:12:13Z |
| last_indexed |
2025-07-14T19:12:13Z |
| _version_ |
1837650761957244928 |
| fulltext |
UDC 517.956.223
A. A. Murach (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv; Chernihiv Nat. Pedagog. Univ.),
I. S. Chepurukhina (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
ELLIPTIC BOUNDARY-VALUE PROBLEMS IN THE SENSE OF LAWRUK
ON SOBOLEV AND HÖRMANDER SPACES
ЕЛIПТИЧНI КРАЙОВI ЗАДАЧI ЗА ЛАВРУКОМ
У ПРОСТОРАХ СОБОЛЄВА I ХЕРМАНДЕРА
We study elliptic boundary-value problems with additional unknown functions in boundary conditions. These problems
were introduced by Lawruk. We prove that the operator corresponding to a problem of this kind is bounded and Fredholm
in appropriate couples of the inner product isotropic Hörmander spaces Hs,ϕ, which form the refined Sobolev scale. The
order of differentiation for these spaces is given by a real number s and a positive function ϕ slowly varying at infinity
in Karamata’s sense. We consider this problem for an arbitrary elliptic equation Au = f in a bounded Euclidean domain
Ω under the condition that u ∈ Hs,ϕ(Ω), s < ordA, and f ∈ L2(Ω). We prove theorems on the a priori estimate and
regularity of the generalized solutions to this problem.
Дослiджено елiптичну крайову задачу з додатковими невiдомими функцiями у крайових умовах. Цi задачi вве-
денi Лавруком. Доведено, що оператор, вiдповiдний такiй задачi, є обмеженим i нетеровим у вiдповiдних парах
гiльбертових iзотропних просторiв Хермандера Hs,ϕ, якi утворюють уточнену соболєвську шкалу. Показник ди-
ференцiйовностi для цих просторiв задано дiйсним числом s i додатною функцiєю ϕ, яка повiльно змiнюється на
нескiнченностi за Караматою. Ця задача розглядається для довiльного елiптичного рiвняння Au = f в евклiдовiй
областi Ω за умов, що u ∈ Hs,ϕ(Ω), s < ordA i f ∈ L2(Ω). Доведено теореми про апрiорну оцiнку i регулярнiсть
узагальнених розв’язкiв цiєї задачi.
1. Introduction. In this paper, we investigate elliptic boundary-value problems with additional
unknown functions in boundary conditions. These functions are given on the smooth boundary of
a Euclidean domain Ω, where the problem is posed. Such problems were introduced by B. Lawruk
[1 – 3]. They appear naturally if we pass from a general (nonregular) elliptic boundary-value problem
to its formally adjoint problem. Moreover, the class of these problems are closed with respect to this
passage. Their various examples occur in hydrodynamics and the theory of elasticity [4 – 6].
This class has been investigated in some function spaces by V. A. Kozlov, V. G. Maz’ya, and
J. Rossmann [7] (Chapt. 3), mainly, for scalar elliptic equations and by I. Ya. Roitberg [8, 9] for mixed-
order elliptic systems. The results of I. Ya. Roitberg is expounded in Ya. A. Roitberg’s monograph [10]
(Chapt. 2). The main result obtained are the theorem on the Fredholm property of these problems
on two-sided scales of normed spaces, solutions of the elliptic equations being considered in the
spaces introduced by Ya. A. Roitberg [11, 12] (see also his monograph [13] (Sect. 2)). These spaces
coincide with the Sobolev spaces provided that their differentiation order is large enough. But, for
other orders, the Roitberg spaces contain elements that are not distributions on Ω.
The purpose of this paper is to prove a version of this theorem for Sobolev and more general
Hörmander spaces [14] (Sect. 2.2) of distributions on Ω. We use the Hörmander spaces
Hs,ϕ(Rn) :=
{
w ∈ S ′(Rn) : 〈ξ〉sϕ(〈ξ〉)ŵ(ξ) ∈ L2(Rn, dξ)
}
and their analogs for the domain Ω and its boundary. These spaces are parametrized with the number
s ∈ R and the function ϕ : [1,∞)→ (0,∞) that varies slowly at infinity in the sense of J. Karamata.
c© A. A. MURACH, I. S. CHEPURUKHINA, 2015
672 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 5
ELLIPTIC BOUNDARY-VALUE PROBLEMS IN THE SENSE OF LAWRUK . . . 673
Here, ŵ is the Fourier transform of the tempered distribution w, whereas 〈ξ〉 := (1 + |ξ|2)1/2. If
ϕ ≡ 1, then Hs,ϕ(Rn) becomes the Sobolev space Hs(Rn).
These Hörmander spaces form the refined Sobolev scale, which was selected and investigated by
V. A. Mikhailets and A. A. Murach [15 – 17]. This scale has an important interpolation property;
namely, every space Hs,ϕ(Rn) is a result of the interpolation with an appropriate function parameter
of the inner product Sobolev spaces Hs−ε(Rn) and Hs+δ(Rn) with ε, δ > 0.
V. A. Mikhailets and A. A. Murach [15, 16, 18 – 23] have elaborated the theory of solvability
of general elliptic boundary-value problems on the refined Sobolev scale and its modification by
Ya. A. Roitberg. The above-mentioned interpolation is a key method in their theory. In this
connection, we also note papers [24 – 26]. However, this theory does not involve the important class
of elliptic boundary-value problems in the sense of B. Lawruk.
In this paper, we consider these problems for an arbitrary scalar elliptic equation Au = f with
u ∈ Hs,ϕ(Ω) for s < ordA and f ∈ L2(Ω). This approach originates from the papers by J.-L. Lions
and E. Magenes [27, 28] (see also [29]). The case of s ≥ ordA was investigated in [30].
This paper consists of seven sections. In Section 2, we formulate an elliptic boundary-value
problem in the sense of B. Lawruk and discuss the corresponding Green formula and formally adjoint
problem. In Section 3, we give the definitions of Hörmander function spaces that form the refined
Sobolev scale. Section 4 contains the main results of the paper. They are the theorem on the Fredholm
property of the problem under investigation, the a priory estimate for its generalized solutions, and
the theorem about their regularity on the refined Sobolev scale. In Sections 5 and 6, we discuss
auxiliary results that we need to prove these theorems. The proofs of the main results are given in
Section 7.
2. Statement of the problem. Let Ω be a bounded domain in Rn, where n ≥ 2. We suppose
that its boundary Γ := ∂Ω is a closed C∞-manifold of dimension n−1, the C∞-structure on Γ being
induced by Rn. Let ν(x) denote the unit vector of the inward normal to Γ at a point x ∈ Γ.
Choose arbitrarily integers q ≥ 1, κ ≥ 1, m1, . . . ,mq+κ ∈ [0, 2q − 1] and r1, . . . , rκ. In the
paper, we consider the linear boundary-value problem on the domain Ω with κ additional unknown
functions on the boundary Γ:
Au = f on Ω, (1)
Bj u+
κ∑
k=1
Cj,k vk = gj on Γ, j = 1, . . . , q + κ. (2)
Here,
A := A(x,D) :=
∑
|µ|≤2q
aµ(x)Dµ
is a linear differential operator on Ω := Ω ∪ Γ of the even order 2q. Besides, each
Bj(x,D) :=
∑
|µ|≤mj
bj,µ(x)Dµ
is a linear boundary differential operator on Γ of order mj , and every Cj,k := Cj,k(x,Dτ ) is a
linear tangent differential operator on Γ with ordCj,k ≤ mj + rk. All the coefficients of these
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 5
674 A. A. MURACH, I. S. CHEPURUKHINA
differential operators are supposed to be infinitely smooth complex-valued functions given on Ω and
Γ respectively.
Here and below, we use the following standard notation: µ := (µ1, . . . , µn) is a multiindex with
|µ| := µ1 + . . . + µn, and Dµ := Dµ1
1 . . . Dµn
n with D` := i∂/∂x`, where i is imaginary unit and
x = (x1, . . . , xn) ∈ Rn. Besides, Dν := i∂/∂ν(x) and ξµ := ξµ1
1 . . . ξµnn for ξ = (ξ1, . . . , ξn) ∈ Cn.
The function u on Ω and all the functions v1, . . . , vκ on Γ are unknown in the problem (1), (2).
In the paper, all functions and distributions are supposed to be complex-valued.
We assume that the boundary-value problem (1), (2) is elliptic on Ω in the sense of B. Lawruk [1].
Recall the corresponding definition (see also [7] (Subsect. 3.1.3)).
Let A(0)(x, ξ) and B(0)
j (x, ξ) denote the principal symbols of the differential operators A(x,D)
and Bj(x,D) respectively. We recall that
A(0)(x, ξ) :=
∑
|µ|=2q
aµ(x)ξµ, with x ∈ Ω and ξ ∈ Cn,
is a homogeneous polynomial of order 2q in ξ and that
B
(0)
j (x, ξ) :=
∑
|µ|=mj
bj,µ(x)ξµ, with x ∈ Γ and ξ ∈ Cn,
is a homogeneous polynomial of order mj in ξ. Besides, let C(0)
j,k (x, τ) denote the principal symbol of
Cj,k(x,Dτ ) if ordCj,k = mj+rk. For every point x ∈ Γ, the expression C(0)
j,k (x, τ) is a homogeneous
polynomial of order mj + rk in τ, with τ being a tangent vector to the boundary Γ at the point x. (If
ordCj,k < mj + rk, then we put C(0)
j,k (x, τ) := 0.)
The boundary-value problem (1), (2) is said to be elliptic on Ω if the following three conditions
are fulfilled:
(i) The differential operator A(x,D) is elliptic at each point x ∈ Ω; i.e., A(0)(x, ξ) 6= 0 for an
arbitrary vector ξ ∈ Rn \ {0}.
(ii) The differential operator A(x,D) is properly elliptic at each point x ∈ Γ; i.e., for an arbitrary
vector τ 6= 0 tangent to Γ at x, the polynomial A(0)(x, τ + ζν(x)) in ζ ∈ C has q roots with positive
imaginary part and q roots with negative imaginary part (these roots are calculated with regard for
their multiplicity).
(iii) The system of boundary-value conditions (2) covers equation (1) at each point x ∈ Γ. This
means that, for an arbitrary vector τ 6= 0 from condition (ii), the boundary-value problem
A(0)(x, τ +Dt ν(x))θ(t) = 0 for t > 0,
B
(0)
j (x, τ +Dt ν(x))θ(t)
∣∣
t=0
+
κ∑
k=1
C
(0)
j,k (x, τ)λk = 0, j = 1, . . . , q + κ,
has only zero solution. This problem is considered relative to the unknown function θ ∈ C∞([0,∞))
with θ(t) → 0 as t → ∞ and the unknown complex-valued numbers λ1, . . . , λκ. Here, A(0)(x, τ +
+ Dt ν(x)) and B
(0)
j (x, τ + Dt ν(x)) are differential operators with respect to Dt := i∂/∂t. We
obtain them if we put ζ := Dt in the polynomials A(0)(x, τ + ζν(x)) and B(0)
j (x, τ + ζν(x)) in ζ.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 5
ELLIPTIC BOUNDARY-VALUE PROBLEMS IN THE SENSE OF LAWRUK . . . 675
It is worthwhile to note that condition (ii) follows from condition (i) in the n ≥ 3 case.
Examples of elliptic boundary-value problems in the sense of B. Lawruk are discussed in [7]
(Subsect. 3.1.5). Among them, we mention the following simple example:
∆u = f on Ω, u+ v = g1 and Dνu+Dτv = g2 on Γ.
Here, n = 2, κ = 1, ∆ is the Laplace operator, and Dτ := i∂/∂τ, with ∂/∂τ being the derivative
along the curve Γ. It is easy to check that this boundary-value problem is elliptic on Ω.
To the problem (1), (2) corresponds the linear mapping
Λ : (u, v1, . . . , vκ) 7→
(
Au, B1u+
κ∑
k=1
C1,k vk, . . . , Bq+κ u+
κ∑
k=1
Cq+κ,k vk
)
,
with u ∈ C∞(Ω) and v1, . . . , vκ ∈ C∞(Γ).
(3)
We will investigate properties of the extension (by continuity) of this mapping in suitable couples of
Hörmander function spaces and, in particular, Sobolev spaces.
In order to describe the range of this extension, we need the following Green formula (see [7]
(Theorem 3.1.1)):
(Au,w)Ω +
q+κ∑
j=1
(
Bj u+
κ∑
k=1
Cj,k vk, hj
)
Γ
=
= (u,A+w)Ω +
2q∑
j=1
(
Dj−1
ν u,Kj w +
q+κ∑
k=1
Q+
k,j hk
)
Γ
+
κ∑
j=1
(
vj ,
q+κ∑
k=1
C+
k,j hk
)
Γ
,
where u,w ∈ C∞(Ω), v = (v1, . . . , vκ) ∈ (C∞(Γ))κ, and h = (h1, . . . , hq+κ) ∈ (C∞(Γ))q+κ are
arbitrary, whereas (·, ·)Ω and (·, ·)Γ are the inner products in the Hilbert spaces L2(Ω) and L2(Γ) of
functions square integrable over Ω and Γ respectively.
Here, A+ stands for the differential operator that is formally adjoint to A with respect to (·, ·)Ω.
Besides, C+
k,j and Q+
k,j denote the tangent differential operators that are formally adjoint to Ck,j
and Qk,j respectively relative to (·, ·)Γ, the tangent differential operators Qk,j appearing in the
representation of the boundary differential operators Bj in the form
Bj(x,D) =
2q∑
k=1
Qj,k(x,Dτ )Dk−1
ν .
(If k > mj , then Qj,k = 0.) Finally, Kj := Kj(x,D) is a certain linear boundary differential operator
on Γ, with ordKj ≤ 2q − j.
Taking into account the Green formula, we consider the following boundary-value problem on Ω
with q + κ additional unknown function on Γ:
A+w = ω in Ω, (4)
Kj w +
q+κ∑
k=1
Q+
k,j hk = χj on Γ, j = 1, . . . , 2q, (5)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 5
676 A. A. MURACH, I. S. CHEPURUKHINA
q+κ∑
k=1
C+
k,j hk = χ2q+j on Γ, j = 1, . . . ,κ. (6)
This problem is formally adjoint to the problem (1), (2) with respect to the Green formula. It is
worthwhile to note that the problem (1), (2) is elliptic if and only if the problem (4) – (6) is elliptic as
well (see [7] (Theorem 3.1.2)).
3. Hörmander spaces and refined Sobolev scales. Here, we give the definitions of the
Hörmander inner product spaces that form the refined Sobolev scales over Rn, Ω, and Γ. We also
discuss some of their properties [23].
We should begin with the definition of the Hörmander spaces Hs,ϕ(Rn), with s ∈ R and ϕ ∈M.
Here and below, M stands for the set of all Borel measurable functions ϕ : [1,∞) → (0,∞) such
that both the functions ϕ and 1/ϕ are bounded on each compact interval [1, b], with 1 < b <∞, and
that the function ϕ is slowly varying at infinity in the sense of J. Karamata [31]. The latter property
means that ϕ(λt)/ϕ(t)→ 1 as t→∞ for every λ > 0.
Note that the slowly varying functions are well investigated and has various applications (see
monographs [32, 33]). We obtain an important example of these functions by setting
ϕ(t) := (log t)r1(log log t)r2 . . . (log . . . log︸ ︷︷ ︸
k times
t)rk of t� 1.
Here, k ∈ Z, with k ≥ 1, and r1, . . . , rk ∈ R are arbitrary parameters.
Let s ∈ R and ϕ ∈ M. By definition, the complex linear space Hs,ϕ(Rn), with n ≥ 1, consists
of all distributions w ∈ S ′(Rn) that the Fourier transform ŵ of w is locally Lebesgue integrable over
Rn and satisfies the condition ∫
Rn
〈ξ〉2sϕ2(〈ξ〉) |ŵ(ξ)|2 dξ <∞.
Here, as usual, S ′(Rn) is the linear topological space of all tempered distributions on Rn, and
〈ξ〉 := (1 + |ξ|2)1/2 is the smoothed modulus of ξ ∈ Rn. The space Hs,ϕ(Rn) is endowed with the
inner product
(w1, w2)Hs,ϕ(Rn) :=
∫
Rn
〈ξ〉2s ϕ2(〈ξ〉) ŵ1(ξ) ŵ2(ξ) dξ.
It naturally induces the norm
‖w‖Hs,ϕ(Rn) := (w,w)
1/2
Hs,ϕ(Rn).
The space Hs,ϕ(Rn) just defined is a special isotropic Hilbert case of the spaces Bp,µ introduced
and investigated by L. Hörmander in [14] (Sect. 2.2) (see also his monograph [34] (Sect. 10.1)).
Namely, Hs,ϕ(Rn) = Bp,µ if p = 2 and µ(ξ) ≡ 〈ξ〉sϕ(〈ξ〉). Note that the inner product spaces B2,µ
were also investigated by L. R. Volevich and B. P. Paneah in [35] (§ 2).
In the ϕ ≡ 1 case, the space Hs,ϕ(Rn) becomes the inner product Sobolev space Hs(Rn) of
order s ∈ R. Generally, we have the continuous and dense embeddings
Hs+ε(Rn) ↪→ Hs,ϕ(Rn) ↪→ Hs−ε(Rn) for every ε > 0. (7)
They show that, the numerical parameter s characterizes the main regularity of the distributions
w ∈ Hs,ϕ(Rn), whereas the function parameter ϕ defines a supplementary regularity, which is
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 5
ELLIPTIC BOUNDARY-VALUE PROBLEMS IN THE SENSE OF LAWRUK . . . 677
subordinate to s. Specifically, if ϕ(t) → ∞ [or ϕ(t) → 0] as t → ∞, then the parameter ϕ defines
the supplementary positive [or negative] regularity. Thus, we can say that ϕ refines the main regularity
s in the class of Hilbert separable spaces{
Hs,ϕ(Rn) : s ∈ R, ϕ ∈M
}
.
This class is selected by V. A. Mikhailets and A. A. Murach in [15] and is called the refined Sobolev
scale over Rn (see also monograph [23] (Subsect. 1.3.3)).
In the paper, we need the refined Sobolev scales over the domain Ω and its boundary Γ. These
scales are constructed in the standard way with the help of the Hörmander spaces Hs,ϕ(Rn). Let us
pass to the corresponding definitions [23] (Sect. 2.1 and 3.1).
The complex linear space Hs,ϕ(Ω) and the norm in it are defined as follows:
Hs,ϕ(Ω) :=
{
w �Ω : w ∈ Hs,ϕ(Rn)
}
,
‖u‖Hs,ϕ(Ω) := inf
{
‖w‖Hs,ϕ(Rn) : w ∈ Hs,ϕ(Rn), u = w �Ω
}
,
with u ∈ Hs,ϕ(Ω). Here, as usual, w �Ω stands for the restriction of the distribution w to the domain
Ω. The space Hs,ϕ(Ω) is Hilbert and separable with respect to this norm. The space Hs,ϕ(Ω) is
continuously embedded in the linear topological space S ′(Ω) := {w � Ω : w ∈ S ′(Rn)}. The set
C∞(Ω) is dense in Hs,ϕ(Ω).
In short, the space Hs,ϕ(Γ) consists of all distributions on Γ that belong, in local coordinates,
to Hs,ϕ(Rn−1). Let us dwell on the definition of Hs,ϕ(Γ). From C∞-structure on Γ, we arbitrarily
choose a finite collection of local charts αj : Rn−1 ↔ Γj , with j = 1, . . . , λ, that the open sets
Γ1, . . . ,Γλ form a covering of the manifold Γ. Besides, we arbitrarily choose functions χj ∈ C∞(Γ),
with j = 1, . . . , λ, that form a partition of unity on Γ which satisfies the condition suppχj ⊂ Γj .
Then the complex linear space Hs,ϕ(Γ) and the norm in it are defined as follows:
Hs,ϕ(Γ) :=
{
h ∈ D′(Γ) : (χjh) ◦ αj ∈ Hs,ϕ(Rn−1) for every j ∈ {1, . . . , λ}
}
,
‖h‖Hs,ϕ(Γ) :=
λ∑
j=1
‖(χjh) ◦ αj‖2Hs,ϕ(Rn−1)
1/2
.
Here, as usual, D′(Γ) is the linear topological space of all distributions on Γ, whereas (χjh) ◦ αj
is the representation of the distribution χjh in the local chart αj . The space Hs,ϕ(Γ) is Hilbert and
separable with respect to this norm. This space does not depend (up to equivalence of norms) on the
indicated choice of the local charts and partition of unity [23] (Theorem 2.3). The space Hs,ϕ(Γ) is
continuously embedded in D′(Γ). The set C∞(Γ) is dense in Hs,ϕ(Γ).
Thus, we have the refined Sobolev scales{
Hs,ϕ(Ω) : s ∈ R, ϕ ∈M
}
and
{
Hs,ϕ(Γ) : s ∈ R, ϕ ∈M
}
(8)
over Ω and Γ respectively. They contain the inner product Sobolev spaces Hs(Ω) := Hs,1(Ω) and
Hs(Γ) := Hs,1(Γ). (Here, of course, 1 denotes the function equaled identically to 1). It follows from
(7) that
Hs+ε(Ω) ↪→ Hs,ϕ(Ω) ↪→ Hs−ε(Ω) for every ε > 0, (9)
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678 A. A. MURACH, I. S. CHEPURUKHINA
Hs+ε(Γ) ↪→ Hs,ϕ(Γ) ↪→ Hs−ε(Γ) for every ε > 0. (10)
These embeddings are compact and dense.
It is worthwhile to note the following connection between the scales (8) (see [23] (Subsect. 3.2.1)).
Let s > 1/2 and ϕ ∈ M, then the trace mapping u 7→ u � Γ, with u ∈ C∞(Γ), extends uniquely
(by continuity) to a bounded surjective operator RΓ : Hs,ϕ(Ω) → Hs−1/2,ϕ(Γ). Thus, for each
distribution u ∈ Hs,ϕ(Ω), its trace RΓu on Γ is well defined. Moreover, we have the equivalence of
norms
‖h‖Hs−1/2,ϕ(Γ) � inf
{
‖u‖Hs,ϕ(Ω) : u ∈ Hs,ϕ(Ω), h = RΓu
}
on the class of all h ∈ Hs−1/2,ϕ(Γ). But it is impossible to define this trace reasonably in the s < 1/2
case.
4. Main results. To formulate them, we need to introduce the linear spaces N and N+ of
infinitely smooth solutions to the problem (1), (2) and its formally adjoint problem (4) – (6). Namely,
let N consist of all solutions
(u, v1, . . . , vκ) ∈ C∞(Ω)×
(
C∞(Γ)
)κ
to the problem (1), (2) in the case where f = 0 on Ω and all gj = 0 on Γ. Besides, let N+ consist of
all solutions
(w, h1, . . . , hq+κ) ∈ C∞(Ω)×
(
C∞(Γ)
)q+κ
to the problem (4) – (6) in the case where ω = 0 on Ω and all χj = 0 and all χ2q+j = 0 on Γ. Since
these problems are elliptic on Ω, the spaces N and N+ are finite-dimensional [7] (Lemma 3.4.2).
We are motivated by the following result.
Proposition 1. For arbitrary s > 2q − 1/2 and ϕ ∈ M, the mapping (3) extends uniquely (by
continuity) to a bounded operator
Λ : Hs,ϕ(Ω)⊕
κ⊕
k=1
Hs+rk−1/2,ϕ(Γ)→ Hs−2q,ϕ(Ω)⊕
q+κ⊕
j=1
Hs−mj−1/2,ϕ(Γ). (11)
This operator is Fredholm. Its kernel coincides with N, and its range consists of all vectors
(f, g1, . . . , gq+κ) ∈ Hs−2q,ϕ(Ω)⊕
q+κ⊕
j=1
Hs−mj−1/2,ϕ(Γ)
such that
(f, w)Ω +
q+κ∑
j=1
(gj , hj)Γ = 0 for all (w, h1, . . . , hq+κ) ∈ N+. (12)
The index of the operator (11) is equal to dimN − dimN+ and does not depend on s and ϕ.
Here and below, (·, ·)Ω and (·, ·)Γ denote the extension by continuity of the inner products in the
Hilbert spaces L2(Ω) and L2(Γ) respectively.
In the Sobolev case of ϕ(t) ≡ 1, Proposition 1 is proved in [7] (Theorem 3.4.1) for integer s
and in [10] (Theorems 2.4.1 and 2.7.3) for real s and general elliptic systems. For arbitrary ϕ ∈ M
and real s > 2q, this proposition is proved in [30] (Theorem 1). The case where ϕ ∈ M and
2q − 1/2 < s ≤ 2q is covered in [37] (Theorem 1).
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In connection with Proposition 1, we recall the definition of a Fredholm operator and its index.
A linear bounded operator T : E1 → E2 acting between certain Banach spaces E1 and E2 is said
to be Fredholm if its kernel kerT and co-kernel E2/T (E1) are finite-dimensional. If this operator
is Fredholm, then its range T (E1) is closed in E2; see, e.g., [36] (Lemma 19.1.1). The number
indT := dim kerT − dim(E2/T (E1)) is called the index of the Fredholm operator T.
Proposition 1 is not true if s ≤ 2q− 1/2. This is caused by the fact that the boundary differential
operators Bj cannot be reasonably applied to all distributions u ∈ Hs,ϕ(Ω) in this case. Namely, the
mapping u 7→ Bju, with u ∈ C∞(Ω), cannot be extended to a continuous operator from the whole
Hs,ϕ(Ω) to D′(Γ) if s ≤ mj + 1/2.
To obtain a version of Theorem 1 for arbitrary s ≤ 2q − 1/2, we restrict ourselves to the
solutions u ∈ Hs,ϕ(Ω) of the elliptic equation Au = f with f ∈ L2(Ω) = H0(Ω). This approach
originates from J.-L. Lions and E. Magenes [27, 28], who took it to the investigation of regular elliptic
boundary-value problems in Sobolev spaces.
Let s < 2q and ϕ ∈M. Consider the linear space
Hs,ϕ
A (Ω) :=
{
u ∈ Hs,ϕ(Ω) : Au ∈ L2(Ω)
}
endowed with the graph norm
‖u‖Hs,ϕ
A (Ω) :=
(
‖u‖2Hs,ϕ(Ω) + ‖Au‖2L2(Ω)
)1/2
. (13)
Here, Au is understood in the sense of the theory of distributions.
This space is Hilbert with respect to the norm (13). Indeed, this norm is generated by an inner
product because so are the norms on the right-hand side of (13). Besides, the space Hs,ϕ
A (Ω) is
complete with respect to this norm. In fact, if (uk) is a Cauchy sequence in this space, then there
exist the limits u := limuk in Hs,ϕ(Ω) and f := limAuk in L2(Ω) because the spaces Hs,ϕ(Ω) and
L2(Ω) are complete. The differential operator A is continuous in S ′(Ω), so Au = limAuk = f in
S ′(Ω). Here, recall, u ∈ Hs,ϕ(Ω) and f ∈ L2(Ω). Therefore, u ∈ Hs,ϕ
A (Ω) and limuk = u in the
space Hs,ϕ
A (Ω). Thus, this space is complete.
It is useful to note that even when all coefficients of A are constant, the space Hs,ϕ
A (Ω) depends
essentially on each of them. This follows from the result by L. Hörmander [38] (Theorem 3.1).
Consider the Hilbert spaces
Ds,ϕA (Ω,Γ) := Hs,ϕ
A (Ω)⊕
κ⊕
k=1
Hs+rk−1/2,ϕ(Γ),
E0,s,ϕ(Ω,Γ) := L2(Ω)⊕
q+κ⊕
j=1
Hs−mj−1/2,ϕ(Γ).
We investigate the properties of the elliptic boundary-value problem (1), (2) in the couple of these
spaces.
Theorem 1. Let s < 2q and ϕ ∈ M. Then the set C∞(Ω) is dense in the space Hs,ϕ
A (Ω), and
the mapping (3) extends uniquely (by continuity) to a bounded operator
Λ : Ds,ϕA (Ω,Γ)→ E0,s,ϕ(Ω,Γ). (14)
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680 A. A. MURACH, I. S. CHEPURUKHINA
This operator is Fredholm. Its kernel coincides with N, whereas its range consists of all vectors
(f, g) := (f, g1, . . . , gq+κ) ∈ E0,s,ϕ(Ω,Γ) (15)
that satisfy condition (12). The index of the operator (14) is equal to dimN − dimN+ and does not
depend on s and ϕ.
If N = {0} and N+ = {0}, then the operator (14) is an isomorphism between the spaces
Ds,ϕA (Ω,Γ) and E0,s,ϕ(Ω,Γ). Generally, this operator naturally induces an isomorphism
Λ : Ds,ϕA (Ω,Γ)/N ↔ R0,s,ϕ(Ω,Γ). (16)
Here,
R0,s,ϕ(Ω,Γ) :=
{
(f, g1, . . . , gq+κ) ∈ E0,s,ϕ(Ω,Γ) : (12) is true
}
is a subspace of E0,s,ϕ(Ω,Γ).
Let us discuss the properties of generalized solutions to the elliptic boundary-value problem (1),
(2). We first give the definition of such solutions.
Let us denote
S ′A(Ω) := {u ∈ S ′(Ω) : Au ∈ L2(Ω)}.
We interpret S ′A(Ω) as a subspace of the linear topological space S ′(Ω). Since the domain Ω is
bounded, we conclude that S ′A(Ω) is the union of all spaces Hs,ϕ
A (Ω) with s < 2q and ϕ ∈M.
Consider an arbitrary vector
(u, v) := (u, v1, . . . , vκ) ∈ S ′A(Ω)× (D′(Γ))κ. (17)
Since (u, v) ∈ Ds,ϕA (Ω,Γ) for certain s < 2q and ϕ ∈ M, we define its image (15) by the formula
Λ(u, v) = (f, g) if we apply the bounded operator (14). The vector (u, v) does not depend on s and
ϕ and is called a generalized solution of the boundary-value problem (1), (2) with the right-hand side
(15). This solution satisfies the following a priori estimate.
Theorem 2. Let s < 2q, ϕ ∈M, and σ > 0. Then there exists a number c > 0 such that
‖(u, v)‖Ds,ϕA (Ω,Γ) ≤ c
(
‖Λ(u, v)‖E0,s,ϕ(Ω,Γ) + ‖(u, v)‖Ds−σ,ϕA (Ω,Γ)
)
(18)
for an arbitrary vector (u, v) ∈ Ds,ϕA (Ω,Γ). Here, c = c(s, ϕ, σ) is independent of (u, v).
If N = {0}, then the second term on the right-hand side of the inequality (18) is absent. This
immediately follows from the isomorphism (16).
In connection with this theorem, we mention Peetre’s lemma [39] (Lemma 3) and remark that the
embedding Ds,ϕA (Ω,Γ) ↪→ Ds−σ,ϕA (Ω,Γ), with σ > 0, is not compact.
The next theorem says about the regularity properties of generalized solutions to the elliptic
boundary-value problem (1), (2).
Theorem 3. Let s ∈ R and ϕ ∈ M. Suppose that the vector (17) is a generalized solution to
the elliptic boundary-value problem (1), (2) whose right-hand side meets the conditions
f ∈ Hs−2q,ϕ(Ω) if s ≥ 2q, (19)
gj ∈ Hs−mj−1/2,ϕ(Γ) for each j ∈ {1, . . . , q + κ}. (20)
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ELLIPTIC BOUNDARY-VALUE PROBLEMS IN THE SENSE OF LAWRUK . . . 681
Then the solution satisfies the conditions
u ∈ Hs,ϕ(Ω), (21)
vk ∈ Hs+rk−1/2,ϕ(Γ) for each k ∈ {1, . . . ,κ}. (22)
We conclude that the solution inherits the (supplementary) regularity ϕ of the right-hand sides of
the elliptic problem.
We note that the restriction s ≥ 2q in condition (19) is caused by the following: if s < 2q, then
f = Au ∈ L2(Ω) ⊂ Hs−2q,ϕ(Ω)
in view of (17) and (9). Thus, the assumption f ∈ Hs−2q,ϕ(Ω) is superfluous in Theorem 3 in the
case of s < 2q.
5. Auxiliary results relating to the problem under consideration.. Our proof of Theorem 1
will be based on the result by V. A. Kozlov, V. G. Maz’ya and J. Rossmann [7] (Theorem 3.4.1).
They demonstrated that the elliptic boundary-value problem (1), (2) is Fredholm in the two-sided
scale of function spaces introduced by Ya. A. Roitberg [11, 12]. Being a certain modification of the
classical Sobolev scale, this two-sided scale proves to be useful in the theory of elliptic boundary-
value problems in the situation where their right-hand sides can be nonregular distributions (see
monographs by Yu. M. Berezansky [40] (Chapt. III, Sect. 6) and Ya. A. Roitberg [10, 13]).
Let us recall the definition of the inner product spaces Hs,(2q)(Ω) introduced by Ya. A. Roitberg.
For our purposes, we restrict ourselves to the case of s ∈ Z.
Beforehand, we need to define the space Hs,(0)(Ω). If s ≥ 0, then Hs,(0)(Ω) := Hs(Ω) is the
inner product Sobolev space of order s over Ω. But, if s < 0, then Hs,(0)(Ω) is the completion of
C∞(Ω) with respect to the Hilbert norm
‖u‖Hs,(0)(Ω) := sup
{
|(u,w)Ω|
‖w‖H−s(Ω)
: w ∈ H−s(Ω), w 6= 0
}
.
Thus, we have the rigging
Hs,(0)(Ω) ↪→ L2(Ω) ↪→ H−s,(0)(Ω), with s > 0,
of the space L2(Ω) with positive and negative Sobolev spaces (see [40] (Chapt. I, Sect. 3, Subsect. 1)).
Note that the negative space Hs,(0)(Ω), with s < 0, admits the following description: the mapping
u 7→ Ou, with u ∈ C∞(Ω), extends uniquely (by continuity) to an isometric isomorphism of the
space Hs,(0)(Ω) onto the subspace {w ∈ Hs(Rn) : suppw ⊆ Ω } of Hs(Rn). Here, Ou := u on Ω,
and Ou := 0 on Rn \ Ω.
We can now define the Hilbert space Hs,(2q)(Ω), with s ∈ Z. By definition, the space Hs,(2q)(Ω)
is the completion of the set C∞(Ω) with respect to the Hilbert norm
‖u‖Hs,(2q)(Ω) :=
(
‖u‖2
Hs,(0)(Ω)
+
2q∑
k=1
‖(Dk−1
ν u)�Γ‖2
Hs−k+1/2(Γ)
)1/2
.
This space admits the following description [13] (Sect. 2.2): the linear mapping
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682 A. A. MURACH, I. S. CHEPURUKHINA
T2q : u 7→
(
u, u�Γ, . . . , (D2q−1
ν u)�Γ
)
, with u ∈ C∞(Ω),
extends uniquely (by continuity) to an isometric operator
T2q : Hs,(2q)(Ω)→ Hs,(0)(Ω)⊕
2q⊕
k=1
Hs−k+1/2(Γ) =: Πs,(2q)(Ω,Γ);
the range of this operator consists of all vectors
(u0, u1, . . . , u2q) ∈ Πs,(2q)(Ω,Γ)
such that uk = RΓD
k−1
ν u0 for all integers k ∈ {1, . . . , 2q} satisfying s > k − 1/2. Thus, we
can interpret an arbitrary element u ∈ Hs,(2q)(Ω) as the vector (u0, u1, . . . , u2q) := T2qu. This
interpretation is taken in the mentioned monograph [7] (Sect. 3.2.1) as a definition of the space
Hs,(2q)(Ω).
Note that
Hs,(2q)(Ω) = Hs(Ω) for all s ≥ 2q (23)
as completions of C∞(Ω) with respect to equivalent norms.
We can now formulate the result by V. A. Kozlov, V. G. Maz’ya and J. Rossmann [7] (Theo-
rem 3.4.1). Consider the Hilbert spaces
Ds,(2q)(Ω,Γ) := Hs,(2q)(Ω)⊕
κ⊕
k=1
Hs+rk−1/2(Γ),
Es−2q,(0)(Ω,Γ) := Hs−2q,(0)(Ω)⊕
q+κ⊕
j=1
Hs−mj−1/2(Γ),
with s ∈ Z.
Proposition 2. For arbitrary s ∈ Z, the mapping (3) extends uniquely (by continuity) to a
bounded operator
Λ : Ds,(2q)(Ω,Γ)→ Es−2q,(0)(Ω,Γ). (24)
This operator is Fredholm. Its kernel coincides with N, and its range consists of all vectors
(f, g1, . . . , gq+κ) ∈ Es−2q,(0)(Ω,Γ) (25)
that satisfy (12). The index of the operator (24) is equal to dimN − dimN+ and does not depend
on s.
Note that if s ∈ Z, s ≥ 2q, and ϕ ≡ 1, then Proposition 1 coincides with Proposition 2 in view
of (23).
In the proof of Theorem 1, we will use Proposition 2 together with the following result. For an
arbitrary integer s < 2q, we consider the linear space
H
s,(2q)
A (Ω) :=
{
u ∈ Hs,(2q)(Ω) : Au ∈ L2(Ω)
}
endowed with the graph norm
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ELLIPTIC BOUNDARY-VALUE PROBLEMS IN THE SENSE OF LAWRUK . . . 683
‖u‖
H
s,(2q)
A (Ω)
:=
(
‖u‖2
Hs,(2q)(Ω)
+ ‖Au‖2L2(Ω)
)1/2
. (26)
Note that the imageAu ∈ Hs−2q,(0)(Ω) is well defined for each u ∈ Hs,(2q)(Ω) by closure because the
mapping u 7→ Au, with u ∈ C∞(Ω), extends by continuity to a bounded operator A : Hs,(2q)(Ω)→
→ Hs−2q,(0)(Ω) for every s ∈ Z [13] (Lemma 2.3.1). The space Hs,(2q)
A (Ω) is Hilbert with respect
to the norm (26), and the set C∞(Ω) is dense in it (see, e.g., [23] (Sect. 4.4.2, the proof of
Theorem 4.25)).
Proposition 3. For an arbitrary integer s < 2q, the identity mapping on C∞(Ω) extends
uniquely (by continuity) to an isomorphism between the spaces Hs,(2q)
A (Ω) and Hs
A(Ω). Thus, these
spaces are equal as completions of the set C∞(Ω) with respect to equivalent norms.
Here and below, Hs
A(Ω) is the space Hs,ϕ
A (Ω) in the Sobolev case of ϕ ≡ 1.
The proof of Proposition 3 is given in monograph [23] (Sect. 4.4.2, see the reasoning from
formula (4.196) to the end of the proof of Theorem 4.25).
6. Auxiliary results relating to interpolation between Hilbert spaces.. The refined Sobolev
scales possess an important interpolation property with respect to inner product Sobolev spaces.
Namely, each space Hs,ϕ(G), with s ∈ R, ϕ ∈ M, and G ∈ {Rn,Ω,Γ}, can be obtained by
the interpolation with an appropriate function parameter between the Sobolev spaces Hs−ε(G) and
Hs+δ(G) with ε, δ > 0. This property will play a key role in our proof of Theorem 1 for arbitrary
ϕ ∈M. Therefore, we recall the definition of this interpolation in the case of arbitrary Hilbert spaces
and discuss some of its properties that are necessary for us.
The method of interpolation with a function parameter between Hilbert spaces appeared first
in C. Foiaş and J.-L. Lions’ paper [41, p. 278]. We follow monograph [23] (Sect. 1.1), which
systematically expounds this method. For our purposes, it is sufficient to restrict ourselves to
separable Hilbert spaces.
Let X := [X0, X1] be a given ordered couple of separable complex Hilbert spaces X0 and X1
such that the continuous and dense embedding X1 ↪→ X0 holds. This couple is said to be admissible.
For X, there exists a self-adjoint positive-definite operator J on X0 that has the domain X1 and that
satisfies the condition ‖Jw‖X0 = ‖w‖X1 for each w ∈ X1. This operator is uniquely determined
by the couple X and is called a generating operator for X. It defines an isometric isomorphism J :
X1 ↔ X0.
Let B denote the set of all Borel measurable functions ψ : (0,∞) → (0,∞) such that ψ is
bounded on each compact interval [a, b], with 0 < a < b <∞, and that 1/ψ is bounded on every set
[r,∞), with r > 0.
For arbitrary ψ ∈ B, we consider the (generally, unbounded) operator ψ(J), which is defined on
X0 as the Borel function ψ of J and is built with the help of Spectral Theorem applied to the self-
adjoint operator J. Let [X0, X1]ψ or, simply, Xψ denote the domain of the operator ψ(J) endowed
with the norm ‖u‖Xψ = ‖ψ(J)u‖X0 . The space Xψ is Hilbert and separable with respect to this
norm.
A function ψ ∈ B is called an interpolation parameter if the following condition is fulfilled for
all admissible couples X = [X0, X1] and Y = [Y0, Y1] of Hilbert spaces and for an arbitrary linear
mapping T given on X0: if the restriction of T to Xj is a bounded operator T : Xj → Yj for each
j ∈ {0, 1}, then the restriction of T to Xψ is also a bounded operator T : Xψ → Yψ. In this case
we say that Xψ is obtained by the interpolation with the function parameter ψ of the couple X (or
between the spaces X0 and X1).
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684 A. A. MURACH, I. S. CHEPURUKHINA
The function ψ ∈ B is an interpolation parameter if and only if ψ is pseudoconcave on a
neighbourhood of infinity, i.e., ψ(t) � ψ1(t) with t� 1 for a certain positive concave function ψ1(t).
(As usual, ψ � ψ1 means that both the functions ψ/ψ1 and ψ1/ψ are bounded on the indicated set).
This fundamental property of the interpolation follows from J. Peetre’s results [42, 43]. Specifically,
if ψ(t) ≡ tσψ0(t) for some number σ ∈ (0, 1) and function ψ0 varying slowly at infinity (in the sense
of J. Karamata), then ψ is an interpolation parameter.
Let us formulate the above-mentioned interpolation property of the refined Sobolev scales [23]
(Theorems 1.14, 2.2, and 3.2).
Proposition 4. Let a function ϕ ∈M and positive real numbers ε, δ be given. Define a function
ψ ∈ B by the formula
ψ(t) =
t
ε/(ε+δ)ϕ(t1/(ε+δ)) if t ≥ 1,
ϕ(1) if 0 < t < 1.
(27)
Then ψ is an interpolation parameter, and[
Hs−ε(G), Hs+δ(G)
]
ψ
= Hs,ϕ(G) for each s ∈ R
with equivalence of norms. Here, G ∈ {Rn,Ω,Γ}. If G = Rn, then the equality of norms holds.
We will also use the following three general properties of the interpolation.
Proposition 5. Let X = [X0, X1] and Y = [Y0, Y1] be admissible couples of Hilbert spaces, and
let a linear mapping T be given onX0. Suppose that we have the bounded and Fredholm operators T :
Xj → Yj , with j = 0, 1, that possess the common kernel and the common index. Then, for an
arbitrary interpolation parameter ψ ∈ B, the bounded operator T : Xψ → Yψ is Fredholm, has the
same kernel and the same index, and, moreover, its range T (Xψ) = Yψ ∩ T (X0).
The proof of this proposition is given in [23] (Sect. 1.1.7).
Proposition 6. Let
[
X
(j)
0 , X
(j)
1
]
, with j = 1, . . . , r, be a finite collection of admissible couples
of Hilbert spaces. Then, for every function ψ ∈ B, we have r⊕
j=1
X
(j)
0 ,
r⊕
j=1
X
(j)
1
ψ
=
r⊕
j=1
[
X
(j)
0 , X
(j)
1
]
ψ
with equality of norms.
The proof of this proposition is given in [23] (Sect. 1.1.5).
The third property deals with the interpolation of subspaces connected with a linear bounded
operator. Beforehand, we admit the following notation.
LetH, Φ and Ψ be Hilbert spaces, with Φ ↪→ Ψ continuously, and let a bounded linear operator T :
H → Ψ be given. We put
(H)T,Φ := {u ∈ H : Tu ∈ Φ}.
The linear space (H)T,Φ is considered as a Hilbert space with respect to the graph norm
‖u‖(H)T,Φ :=
(
‖u‖2H + ‖Tu‖2Φ
)1/2
.
Note that this space does not depend on Ψ.
Proposition 7. Suppose that six separable Hilbert spaces X0, Y0, Z0, X1, Y1, and Z1 and three
linear mappings T, R, and S satisfy the following seven conditions:
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ELLIPTIC BOUNDARY-VALUE PROBLEMS IN THE SENSE OF LAWRUK . . . 685
(i) the couples X = [X0, X1] and Y = [Y0, Y1] are admissible;
(ii) the spaces Z0 and Z1 are subspaces of a certain linear space E;
(iii) the continuous embeddings Y0 ↪→ Z0 and Y1 ↪→ Z1 hold;
(iv) the mapping T is given on X0 and defines the bounded operators T : X0 → Z0 and T :
X1 → Z1;
(v) the mapping R is given on E and defines the bounded operators R : Z0 → X0 and R :
Z1 → X1;
(vi) the mapping S is given on E and defines the bounded operators S : Z0 → Y0 and S :
Z1 → Y1;
(vii) the equality TRω = ω + Sω holds for each ω ∈ E.
Then the couple of spaces
[
(X0)T,Y0 , (X1)T,Y1
]
is admissible, and[
(X0)T,Y0 , (X1)T,Y1
]
ψ
= (Xψ)T,Yψ
up to equivalence of norms for an arbitrary interpolation parameter ψ ∈ B.
An analog of Proposition 7 appeared first in J.-L. Lions and E. Magenes’ monograph [44]
(Theorem 14.3), where the case of holomorphic interpolation was considered. This proposition is
proved by V. A. Mikhailets and A. A. Murach in paper [18] (Sect. 4); see also monograph [23]
(Theorem 3.12).
7. Proofs of the main results. Let us prove Theorems 1 – 3.
Proof of Theorem 1. We will first prove this theorem in the Sobolev case where s ∈ Z, s < 2q,
and ϕ ≡ 1. Consider the bounded and Fredholm operator (24) from Proposition 2. Certainly, the
restriction of this operator to the space Hs,(2q)
A (Ω) is a bounded operator
Λ : Hs,(2q)
A (Ω)⊕
κ⊕
k=1
Hs+rk−1/2(Γ)→ L2(Ω)⊕
q+κ⊕
j=1
Hs−mj−1/2(Γ). (28)
Since C∞(Ω) is dense in Hs,(2q)
A (Ω), this operator is an extension by continuity of the mapping (3).
Let Λs,(2q) denote the operator (24), and let Λs denote the operator (28). Evidently, ker Λs =
= ker Λs,(2q) and
Ran Λs =
{
(f, g1, . . . , gq+κ) ∈ Ran Λs,(2q) : f ∈ L2(Ω)
}
,
with Ran denoting the range of the corresponding operator. It follows from this by Proposition 2 that
ker Λs = N and
Ran Λs =
{
(f, g1, . . . , gq+κ) ∈ L2(Ω)⊕
q+κ⊕
j=1
Hs−mj−1/2(Γ) : (12) is true
}
.
Therefore,
dim ker Λs = dimN <∞ and dim coker Λs = dimN+ <∞.
Thus, the bounded operator Λs is Fredholm. In view of Proposition 3, we conclude that Λs is the
required operator (14) in Theorem 1, the set C∞(Ω) being dense in Hs
A(Ω).
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686 A. A. MURACH, I. S. CHEPURUKHINA
We will now prove Theorem 1 in the general situation with the help of interpolation with a
function parameter between Sobolev spaces. Let s ∈ R, s < 2q, and ϕ ∈ M. Choose an integer
p ≥ 2 such that s > −2q(p−1). The mapping (3) extends by continuity to the bounded and Fredholm
operators
Λ : H−2q(p−1)
A (Ω)⊕
κ⊕
k=1
H−2q(p−1)+rk−1/2(Γ)→ L2(Ω)⊕
q+κ⊕
j=1
H−2q(p−1)−mj−1/2(Γ) (29)
and
Λ : H2q(Ω)⊕
κ⊕
k=1
H2q+rk−1/2(Γ)→ L2(Ω)⊕
q+κ⊕
j=1
H2q−mj−1/2(Γ). (30)
This has just been proved for the first operator and is said in Proposition 2 for the second. These
operators have the common kernel N and the same index equaled to dimN − dimN+. The range
of the first operator satisfies the equation
Λ
(
H
−2q(p−1)
A (Ω)
)
=
=
(f, g1, . . . , gq+κ) ∈ L2(Ω)⊕
q+κ⊕
j=1
H−2q(p−1)−mj−1/2(Γ) : (12) is true
, (31)
and an analogous formula holds for the second operator. Note that the second operator is a restriction
of the first.
Let us define the interpolation parameter ψ by formula (27) in which ε := s+ 2q(p− 1) > 0 and
δ := 2q − s > 0. Applying the interpolation with the function parameter ψ to the couples of spaces
in (29) and (30), we obtain the bounded operator
Λ :
[
H
−2q(p−1)
A (Ω)⊕
κ⊕
k=1
H−2q(p−1)+rk−1/2(Γ), H2q(Ω)⊕
κ⊕
k=1
H2q+rk−1/2(Γ)
]
ψ
→
→
[
L2(Ω)⊕
q+κ⊕
j=1
H−2q(p−1)−mj−1/2(Γ), L2(Ω)⊕
q+κ⊕
j=1
H2q−mj−1/2(Γ)
]
ψ
. (32)
According to Proposition 5, this operator is Fredholm with kernel N and index dimN − dimN+.
Let us describe the interpolation spaces in (32). Using Propositions 6 and 4 successively, we get[
H
−2q(p−1)
A (Ω)⊕
κ⊕
k=1
H−2q(p−1)+rk−1/2(Γ), H2q(Ω)⊕
κ⊕
k=1
H2q+rk−1/2(Γ)
]
ψ
=
=
[
H
−2q(p−1)
A (Ω), H2q(Ω)
]
ψ
⊕
κ⊕
k=1
[
H−2q(p−1)+rk−1/2(Γ), H2q+rk−1/2(Γ)
]
ψ
=
=
[
H
−2q(p−1)
A (Ω), H2q(Ω)
]
ψ
⊕
κ⊕
k=1
Hs+rk−1/2,ϕ(Γ). (33)
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ELLIPTIC BOUNDARY-VALUE PROBLEMS IN THE SENSE OF LAWRUK . . . 687
Here, we take into account the equalities
s− ε = −2q(p− 1) and s+ δ = 2q. (34)
Analogously,L2(Ω)⊕
q+κ⊕
j=1
H−2q(p−1)−mj−1/2(Γ), L2(Ω)⊕
q+κ⊕
j=1
H2q−mj−1/2(Γ)
ψ
=
= L2(Ω)⊕
q+κ⊕
j=1
Hs−mj−1/2,ϕ(Γ). (35)
These equalities of spaces hold true up to equivalence of norms.
Let us prove that [
H
−2q(p−1)
A (Ω), H2q(Ω)
]
ψ
= Hs,ϕ
A (Ω) (36)
up to equivalence of norms. To this end, we use Proposition 7 in which
X0 := H−2q(p−1)(Ω), X1 := H2q(Ω), Y0 := Y1 := Z1 := L2(Ω), Z0 := E := H−2qp(Ω),
and T := A. Then
H
−2q(p−1)
A (Ω) = (X0)T,Y0 and H2q(Ω) = (X1)T,Y1 . (37)
Note that the latter equality holds true up to equivalence of norms because A is a bounded operator
from Hσ(Ω) to Hσ−2q(Ω) for each σ ∈ R, specifically, from H2q(Ω) to L2(Ω).
Evidently, conditions (i) – (iv) of Proposition 7 are fulfilled. We also need to give certain operators
R and S satisfying conditions (v) – (vii). With this in mind, we use the known fact that the mapping
u 7→ ApAp+u+ u defines the isomorphism
ApAp+ + I : Hσ
D(Ω)↔ Hσ−4qp(Ω) for each σ ≥ 2qp (38)
(see, e.g., [23] (Lemma 3.1)). Here, of course, Ap is the p-th iteration of A, then Ap+ is the formally
adjoint operator to the differential operator Ap, and I is the identity operator. Besides,
Hσ
D(Ω) :=
{
u ∈ Hσ(Ω) : RΓD
j−1
ν u = 0 for each j ∈ {1, . . . , 2qp}
}
is a subspace of Hσ(Ω). The inverse of (38) defines the bounded linear operator
(ApAp+ + I)−1 : Hσ(Ω)→ Hσ+4qp(Ω) for each σ ≥ −2qp. (39)
We now let
R := Ap−1Ap+(ApAp+ + I)−1 and S = −(ApAp+ + I)−1.
Using (39), we get the bounded operators
R : Z0 = H−2qp(Ω)→ H2qp−2qp−2q(p−1)(Ω) = X0,
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688 A. A. MURACH, I. S. CHEPURUKHINA
R : Z1 = L2(Ω)→ H4qp−2qp−2q(p−1)(Ω) = X1,
S : Z0 = H−2qp(Ω)→ H2qp(Ω) ↪→ L2(Ω) = Y0,
S : Z1 = L2(Ω)→ H4qp(Ω) ↪→ L2(Ω) = Y1;
here, the embeddings are continuous. Besides,
AR = AAp−1Ap+(ApAp+ + I)−1 = (ApAp+ + I − I)(ApAp+ + I)−1 = I + S
on E = H−2qp(Ω). Thus, the rest conditions (v) – (vii) of Proposition 7 are satisfied.
Now, according to this proposition and in view of (37), we get[
H
−2q(p−1)
A (Ω), H2q(Ω)
]
ψ
=
[
(X0)T,Y0 , (X1)T,Y1
]
ψ
= (Xψ)T,Yψ .
Here, by Proposition 4 and (34), we have
Xψ =
[
H−2q(p−1)(Ω), H2q(Ω)
]
ψ
= Hs,ϕ(Ω).
Besides,
Yψ =
[
L2(Ω), L2(Ω)
]
ψ
= L2(Ω).
These three formulas immediately gives the required equality (36).
Note that
the set C∞(Ω) is dense in the space Hs,ϕ
A (Ω). (40)
This follows from the density of C∞(Ω) in H2q(Ω) and from the dense continuous embedding of
H2q(Ω) in Hs,ϕ
A (Ω). The latter embedding is a general property of the interpolation used in (36).
It follows from the interpolation formulas (33), (35), and (36) that the bounded and Fredholm
operator (32) acts between the spaces
Λ : Hs,ϕ
A (Ω)⊕
κ⊕
k=1
Hs+rk−1/2,ϕ(Γ)→ L2(Ω)⊕
q+κ⊕
j=1
Hs−mj−1/2,ϕ(Γ). (41)
(Recall that these spaces are denoted by Ds,ϕA (Ω,Γ) and E0,s,ϕ(Ω,Γ) respectively.) We have proved
that the operator (41) has the kernel N and the index dimN − dimN+. Furthermore, according to
Proposition 5 and formula (31), the range of this operator is equal to
E0,s,ϕ(Ω,Γ) ∩ Λ
(
H
−2q(p−1)
A (Ω)
)
=
{
(f, g1, . . . , gq+κ) ∈ E0,s,ϕ(Ω,Γ) : (12) is true
}
.
Finally, in view of (40), the operator (41) is an extension by continuity of the mapping (3). Thus,
this operator is the required operator (14).
Theorem 1 is proved.
Proof of Theorem 2. It follows from the isomorphism (16) that
inf
{
‖(u, v) + (u(0), v(0))‖Ds,ϕA (Ω,Γ) : (u(0), v(0)) ∈ N
}
≤ c0 ‖Λ(u, v)‖E0,s,ϕ(Ω,Γ) (42)
for each (u, v) ∈ Ds,ϕA (Ω,Γ), with c0 being the norm of the inverse operator to (16). Since N is a
finite-dimensional subspace of both the spaces Ds,ϕA (Ω,Γ) and Ds−σ,ϕA (Ω,Γ), the norms in them are
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ELLIPTIC BOUNDARY-VALUE PROBLEMS IN THE SENSE OF LAWRUK . . . 689
equivalent on N. Hence, for each (u(0), v(0)) ∈ N, we can write
‖(u(0), v(0))‖Ds,ϕA (Ω,Γ) ≤ c1‖(u(0), v(0))‖Ds−σ,ϕA (Ω,Γ)
with some number c1 > 0, which is independent of (u, v) and (u(0), v(0)). Besides,
‖(u(0), v(0))‖Ds−σ,ϕA (Ω,Γ) ≤ ‖(u, v) + (u(0), v(0))‖Ds−σ,ϕA (Ω,Γ) + ‖(u, v)‖Ds−σ,ϕA (Ω,Γ) ≤
≤ c2 ‖(u, v) + (u(0), v(0))‖Ds,ϕA (Ω,Γ) + ‖(u, v)‖Ds−σ,ϕA (Ω,Γ).
Here, c2 is the norm of the continuous embedding operator Ds,ϕA (Ω,Γ) ↪→ Ds−σ,ϕA (Ω,Γ). Hence,
‖(u, v)‖Ds,ϕA (Ω,Γ) ≤ ‖(u, v) + (u(0), v(0))‖Ds,ϕA (Ω,Γ) + ‖(u(0), v(0))‖Ds,ϕA (Ω,Γ) ≤
≤ ‖(u, v) + (u(0), v(0))‖Ds,ϕA (Ω,Γ) + c1‖(u(0), v(0))‖Ds−σ,ϕA (Ω,Γ) ≤
≤ (1 + c1c2) ‖(u, v) + (u(0), v(0))‖Ds,ϕA (Ω,Γ) + c1 ‖(u, v)‖Ds−σ,ϕA (Ω,Γ).
Passing in this inequality to the infimum over all (u(0), v(0)) ∈ N and using the bound (42), we arrive
at the estimate (18); namely, we get
‖(u, v)‖Ds,ϕA (Ω,Γ) ≤ (1 + c1c2) c0 ‖Λ(u, v)‖E0,s,ϕ(Ω,Γ) + c1 ‖(u, v)‖Ds−σ,ϕA (Ω,Γ)
)
.
Theorem 2 is proved.
Proof of Theorem 3. We first consider the case of s < 2q. By virtue of Theorem 1, the vector
(f, g) = Λ(u, v) satisfies (12). Note that (f, g) ∈ E0,s,ϕ(Ω,Γ) by the condition (20). Therefore,
according to Theorem 1, we get the inclusion (f, g) ∈ Λ(Ds,ϕA (Ω,Γ)). Thus, together with Λ(u, v) =
= (f, g), we have the equality Λ(u′, v′) = (f, g) for a certain vector (u′, v′) ∈ Ds,ϕA (Ω,Γ). Hence,
Λ(u− u′, v − v′) = 0, which implies that
(u− u′, v − v′) ∈ N ⊂ C∞(Ω)×
(
C∞(Γ)
)κ
.
Thus,
(u, v) = (u′, v′) + (u− u′, v − v′) ∈ Ds,ϕA (Ω,Γ).
Theorem 3 has been proved in the case of s < 2q.
Let us examine the case of s ≥ 2q. According to what has just been proved, the condition
gj ∈ Hs−mj−1/2,ϕ(Γ) ⊂ H2q−1/4−mj−1/2,ϕ(Γ) for each j ∈ {1, . . . , q + κ}
implies the inclusion
(u, v) ∈ H2q−1/4,ϕ(Ω)⊕
κ⊕
k=1
H2q−1/4+rk−1/2,ϕ(Γ).
Repeating the above reasoning and using Proposition 1 instead of Theorem 1, we deduce from this
inclusion and the condition of Theorem 3, that the vector (u, v) satisfies (21) and (22).
Theorem 3 is proved.
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690 A. A. MURACH, I. S. CHEPURUKHINA
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