Generalized solutions of elliptic boundary value problems with strong power singularities
The existence of a generalized solution of elliptic boundary value problems and its character of singularities depending on power of data singularities are established.
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| Опубліковано в: : | Нелинейные граничные задачи |
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| Дата: | 1999 |
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| Формат: | Стаття |
| Мова: | English |
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Інститут прикладної математики і механіки НАН України
1999
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/169272 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Generalized solutions of elliptic boundary value problems with strong power singularities / H.P. Lopushanska // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 55-59. — Бібліогр.: 6 назв. — англ. |
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Lopushanska, H.P. 2020-06-09T16:25:37Z 2020-06-09T16:25:37Z 1999 Generalized solutions of elliptic boundary value problems with strong power singularities / H.P. Lopushanska // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 55-59. — Бібліогр.: 6 назв. — англ. 0236-0497 https://nasplib.isofts.kiev.ua/handle/123456789/169272 The existence of a generalized solution of elliptic boundary value problems and its character of singularities depending on power of data singularities are established. en Інститут прикладної математики і механіки НАН України Нелинейные граничные задачи Generalized solutions of elliptic boundary value problems with strong power singularities Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Generalized solutions of elliptic boundary value problems with strong power singularities |
| spellingShingle |
Generalized solutions of elliptic boundary value problems with strong power singularities Lopushanska, H.P. |
| title_short |
Generalized solutions of elliptic boundary value problems with strong power singularities |
| title_full |
Generalized solutions of elliptic boundary value problems with strong power singularities |
| title_fullStr |
Generalized solutions of elliptic boundary value problems with strong power singularities |
| title_full_unstemmed |
Generalized solutions of elliptic boundary value problems with strong power singularities |
| title_sort |
generalized solutions of elliptic boundary value problems with strong power singularities |
| author |
Lopushanska, H.P. |
| author_facet |
Lopushanska, H.P. |
| publishDate |
1999 |
| language |
English |
| container_title |
Нелинейные граничные задачи |
| publisher |
Інститут прикладної математики і механіки НАН України |
| format |
Article |
| description |
The existence of a generalized solution of elliptic boundary value problems and its character of singularities depending on power of data singularities are established.
|
| issn |
0236-0497 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/169272 |
| citation_txt |
Generalized solutions of elliptic boundary value problems with strong power singularities / H.P. Lopushanska // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 55-59. — Бібліогр.: 6 назв. — англ. |
| work_keys_str_mv |
AT lopushanskahp generalizedsolutionsofellipticboundaryvalueproblemswithstrongpowersingularities |
| first_indexed |
2025-11-25T21:08:34Z |
| last_indexed |
2025-11-25T21:08:34Z |
| _version_ |
1850551438114029568 |
| fulltext |
GENERALIZED SOLUTIONS OF ELLIPTIC BOUNDARY VALUE
PROBLEMS WITH STRONG POWER SINGULARITIES
c© H.P.Lopushanska
The existence of a generalized solution of elliptic boundary value problems and its
character of singularities depending on power of data singularities are established.
In [1-3] and other articles the behavior of the generalized solutions to the elliptic
boundary value problems with power singularities on the right-side was studied. It was
established that the generalized solutions in the sense [4] exist if the power growth of
the problem data has the order λ > −n inside the domain and λ > max{1 − n,−n +
2m−1−m′} on its boundary (m′ denotes the maximum order of the normal derivatives
in the boundary conditions) and that these data require a regularization if their power
growth is more strong [1,2].
We trace the behavior of generalized solutions (in specific sense) for any power sin-
gularities on the right-side without any using of the data regularization. We start from
a representation of the solution.
Consider the following problem
A(x,D) = F0, x ∈ Ω, Bj(x, D)u |S = Fj , j = 1,m, (1)
where Ω denotes a bounded domain in Rn, with a closed boundary S of class C∞,
A(x,D) is an elliptic operator of order 2m, {Bj(x,D)}m
j=1 are some normal system of
boundary differential expressions satisfying Lopatinsky’s condition. We assume that the
coefficients of operators are infinitely differentiable.
Let B̂j , Tj , T̂j be such boundary differential operators with the infinitely differentiable
coefficients that Green’s formula
∫
Ω
(Auv − uA∗v)dx =
m∑
j=1
∫
S
(BjuT̂jv − TjuB̂jv)dS
holds.
We define now the following function spaces: D(Ω) = C∞(Ω), D(S) = C∞(S),
X(Ω) = {ϕ ∈ D(S) : B̂jϕ |S = 0, j = 1, m} and D′(Ω), D′(S), X ′(Ω) as spaces of linear
continious functionals defined respectively on D(Ω), D(S), X(Ω). Let (ϕ,F ) denote
the action of F ∈ D′(Ω)(X ′(Ω)) onto ϕ ∈ D(Ω)(X(Ω)) and < ϕ, F >–the action of
F ∈ D′(S) onto ϕ ∈ D(S).
For the case F0 ∈ X ′(Ω), Fj ∈ D′(S), j = 1,m, we define the solution of the problem
(1) as such generalized function u ∈ D′(Ω) that the equality
(A∗ψ, u) = (ψ,F0) +
m∑
j=1
< T̂jψ, Fj > (2)
is fulfilled for any ψ ∈ X(Ω).
In [4,6] there is established the existence and is studied the properties of Green’s
vector-function (G0(x, y), G1(x, y), . . . , Gm(x, y)) of the problem (1) on the class of the
functions u(x) which are orthogonal to the kernel N of the problem (1) (i.e. Pu = 0).
If
(ψ,F0) +
m∑
j=1
< T̂jψ, Fj >= 0 (3)
for any ψ ∈ N∗ (N∗ is the kernel of the adjoint problem), the solution of the problem
(1) in the sense (2) exists in D′(Ω)/N . It is defined by the formula
(ϕ, u) = (
∫
Ω
ϕ(x)G0(x, y)dx, F0) +
m∑
j=1
<
∫
Ω
ϕ(x)Gj(x, y), Fj >,
ϕ ∈ D(Ω).
(4)
The function with strong power singularities doesn’t belong to D′(Ω) or X ′(Ω) but
(4) can be extend to this case also. We consider special function spaces for it.
Let x0 denote a given point in Ω, %(x, x0) = %0(x − x0) be nonnegative compactly
supported infinitely differentiable function in Ω, wich has order d(x, x0) = |x − x0| in
neighborhood of the point x0 ∈ Ω, %(x0, x0) = 0.
For k ∈ R1 we define spaces Zk(Ω, x0) = {ϕ ∈ C∞(Ω \ x0) : %|α|(x, x0)Dαϕ(x) =
%k(x, x0)ϕα(x), ϕα(x) ∈ C(Ω) for arbitrary multi-index α}.
We shall say that the sequence ϕν → 0 in the space Zk(Ω, x0), if for all multi-index
α the sequence ϕαν(x) = %−k+|α|(x, x0)Dαϕν(x) uniformly tends to zero in Ω under
ν →∞.
We now notice the following main properties of the functions of these spaces:
1) Z0(Ω, x0) ⊂ C(Ω), C∞(Ω) ⊂ Zk(Ω, x0) for all k ≤ 0;
2) if ϕ ∈ Zk(Ω, x0), than, for all λ ∈ R1, |x− x0|λϕ ∈ Zk+λ(Ω, x0);
3) if ϕ ∈ Zk(Ω, x0), than Dγϕ ∈ Zk−|γ|(Ω, x0) for any multi-index γ;
4) if ϕ ∈ Zk(Ω, x0), ψ ∈ Zp(Ω, x0), than ϕψ ∈ Zk+p(Ω, x0);
5) Zk2(Ω, x0) ⊂ Zk1(Ω, x0) for k1 < k2;
6) Zk(Ω, x0) ⊂ C [k](Ω) for k ≥ 0 ( here [k] denotes the integral part of the number
k).
We denote the spaces of linear continuous functionals defined on Zk(Ω, x0) by Z ′k(Ω, x0).
Then
1) Z ′k1
(Ω, x0) ⊂ Z ′k2
(Ω, x0) for k1 < k2; if k ≥ 0, than (C [k](Ω))′ ⊂ Z ′k(Ω, x0); since
C∞(Ω) = D(Ω) ⊂ Z−k(Ω, x0) for k ≥ 0, Z ′−k(Ω, x0) ⊂ D′(Ω) for k ≥ 0.
2) If F ∈ Z ′k(Ω, x0), than D|α|F ∈ Z ′k+|α|(Ω, x0) for any multi-index α.
3) Z−k(Ω, x0) ⊂ Z ′k(Ω, x0).
Indeed, fα(x) = %k+|α|D|α|f ∈ C(Ω) ⊂ L1(Ω), if f ∈ Z−k(Ω, x0), and then the fol-
lowing linear continuous functionals fα on Zk(Ω, x0) are defined: (ϕ, fα) =
∫
Ω
ϕαfαdx =
∫
Ω
%−k+|α|Dαϕfαdx =
∫
Ω
%−k+|α|Dαϕ%k+|α|Dαfdx =
∫
Ω
Dαϕ%2|α|Dαfdx, ϕ ∈ Zk(Ω, x0),
for any multi-index α, in particular (ϕ, f) =
∫
Ω
ϕ0f0dx =
∫
Ω
%−kϕ%kfdx =
∫
Ω
ϕfdx, ϕ ∈
Zk(Ω, x0). Hence if f ∈ Z−k(Ω, x0), f will be a regular generalized function of Z ′k(Ω, x0).
4) Let gα ∈ L1(Ω) for any multi-index α, then for any ϕ ∈ Zk(Ω, x0) the expressions∫
Ω
%−k+|α|Dαϕgαdx exist and for any natural number N we have that the function
g(x) =
∑
|α|≤N
Dα((−1)|α|gα%−k+|α|) (derivatives are regarded in generalized sense) is
a linear continuous functional on Zk(Ω, x0): (ϕ, g) =
∑
|α|≤N
∫
Ω
%−k+|α|gαDαϕdx. In
particular g(x) = g0(x)(x− x0)−κ ∈ Z ′|κ|(Ω, x0).
5) For any multi-index α, ϕ ∈ Zk(Ω, x0), bounded functions gα(x) in Ω and any
numbers pα > −n, the expression
∫
Ω
gα%pα+|α|−kDαϕdx exists. Then g(x) =
∑
|α|≤N
Dα(gα%pα+|α|−k) ∈ Z ′k(Ω, x0).
In particular,
g(x) = g0(x)(x− x0)−κ ∈ Z ′|κ|−n+ε(Ω, x0)
for bounded function g0(x) in Ω and any ε > 0.
Notice, that g(x) ∈ D′(Ω) for |κ| < n and g(x) /∈ D′(Ω) for |κ| ≥ n.
Let f0(x) be a bounded function in Ω, F0(x) = f0(x)(x − x0)κ, |κ| ≥ 0,F1 = · · · =
Fm = 0, N∗ = {0}. We shall obtaine that the solution u(x) of the problem (1), such
as Pu = 0, belongs to Z ′|κ|−2m−n+ε(Ω, x0) for any ε > 0. It follows from the following
theorem.
Theorem 1 Let F0 ∈ Z ′p(Ω, x0), p > 2m− n, F1 = · · · = Fm = 0, N∗ = {0}, u(x) be
such solution of the problem (1) that Pu(x) = 0. Then u(x) ∈ Z ′p−2m(Ω, x0).
This conclusion is exact in the sense that there exists F0(x) = Au(x) ∈ Z ′p(Ω, x0),
for the solution u(x) ∈ Z ′p−2m−ε(Ω, x0) ⊂ Z ′p−2m(Ω, x0), ε > 0, of the problem, and it
is possibly, that doesn’t exist such F0 = Au(x) ∈ Z ′p(Ω, x0), for the solution u(x) ∈
Z ′p−2m+ε(Ω, x0). Really, for u(x) ∈ Z ′p−2m+ε(Ω, x0), we have Au(x) ∈ Z ′p+ε(Ω, x0), and
Z ′p(Ω, x0) ⊂ Z ′p+ε(Ω, x0), for ε > 0.
Proof. It is shown in [5] that ψ(y) = (G∗0ϕ)(y) =
∫
Ω
ϕ(x)G(x, y)dx ∈ X(Ω) ⊂ D(Ω)
for any ϕ ∈ D(Ω). Let us study its properties for ϕ ∈ Zk(Ω, x0), x0 ∈ Ω.
Let h(x) ∈ D(Ω), 0 ≤ h(x) ≤ 1, h(x) =
{
1, |x− x0| < η
0, |x− x0| > 2η
,ψ(y) = (G∗0ϕ)(y) =
∫
Ω
h(x)ϕ(x)G(x, y)dx +
∫
Ω
(1− h(x))ϕ(x)G(x, y)dx = ψ1(y) + ψ2(y).
The function (1 − h(y))ψ(y) = 0, if |y − x0| < η, therefore (1 − h(y))ψ(y) ∈
Zk+2m(Ω, x0) for any k ∈ R1, ϕ ∈ Zk(Ω, x0).
The function (1 − h(x))ϕ(x) = 0, if |x − x0| < η, therefore also h(y)ψ2(y) ∈
Zk+2m(Ω, x0) for any ϕ ∈ Zk(Ω, x0), k ∈ R1.
The function h(y)ψ1(y) = O
(
%2m+k(y, x0)
)
, if ϕ ∈ Zk(Ω, x0), k > −n. We shall ob-
taine that, for any multi-index α, the function vα(y) = %|α|(y, x0)Dα
∫
Ω
h(x)ϕ(x)G0(x, y)
h(y)dx = wα(y)%2m+k(y, x0), where wα ∈ C(Ω).
Indeed, we assume the function vα(y) in the form of the sum v1α(y)+v2α(y)+v3α(y)
of three items respectively to the partition of the domaine Ω into Ω1 = {x ∈ Ω :
|x − x0| < |y−x0|
2 }, Ω2 = {x ∈ Ω : |x − y| < |y−x0|
2 }, Ω3 = Ω \ (Ω1 ∪ Ω2). Later the
estimates of the derivatives of G0(x, y) are using.
As a result, (G∗0ϕ)(y) ∈ Zk+2m(Ω, x0), if ϕ ∈ Zk(Ω, x0) and k > −n, or (G∗0ϕ)(y) ∈
Zp(Ω, x0), for ϕ ∈ Zp−2m(Ω, x0) and p > 2m− n.
Then the map F0 → u, defined by (ϕ, u) = (G∗0ϕ,F0), determines u ∈ Z ′p−2m(Ω, x0)
for any F0 ∈ Z ′p(Ω, x0), if p > 2m − n. Since G∗0(A
∗ψ) = ψ, defined by that map
function u satisfies the condition (2) for any ψ ∈ Zp−2m(Ω, x0), and, therefor, any
ψ ∈ Zp−2m(Ω, x0) ∩X(Ω).
The spaces Zk(S, x0), x0 ∈ S, are defined similarly and we obtaine that ψj(y) =
(G∗jϕ)(y) ∈ Zk+mj+1(S, x0), j = 1,m, if ϕ ∈ Zk(Ω, x0), k > −n.
Therefore, the map Fj → uj , defined by (ϕ, uj) =<
∫
Ω
ϕ(x)Gj(x, y)dx, Fj >, deter-
mines uj ∈ Z ′p−mj−1(Ω, x0) for any Fj ∈ Z ′p(S, x0), if p > mj + 1− n.
Let F0 ∈ Z ′p(Ω, x0), Fj ∈ Z ′pj
(S, xj), x0 ∈ Ω, xj ∈ S, j = 1,m, p = (p, p1, . . . , pm),Xp =
Xp(Ω, x0, x1, . . . , xm) = {ϕ(x) ∈ Zp(Ω, x0)∩ (∩m
j=1Zpj (S, xj)) : B̂jϕ |S = 0} (if xi = xj ,
the space Zpi ∩ Zpj remaines by Zk(S, xi), k = max{pi, pj}).
It can be shown that, in the case N∗ = {0}, the function (4) satisfies the equality (2)
for any ψ ∈ Xp(Ω, x0, x1, . . . , xm), if p > 2m − n(x0 ∈ Ω), p > max{2m − n,m′′}(x0 ∈
S), pj > max{1 − n + mj ,mj + 1 − 2m + m′′}, j = 1,m, m′′ denotes the maximum of
the degrees of B̂l, l = 1, m.
So, we obtaine that, in the case N∗ = {0},
F0 = f0(x)(x− x0)−κ, Fj = fj(x)(x− xj)−κj ,
f0(x) ∈ L∞(Ω), fj(x) ∈ L∞(S)
(5)
(then F0 ∈ Z ′|κ|−n+ε|(Ω, x0), Fj ∈ Z ′|κj |−n+1+εj
(S, xj), ε, εj > 0), j = 1,m, |κ| >
2m − ε(x0 ∈ Ω), |κ| > max{2m, n + m′′} − ε(x0 ∈ S), |κj | > max{mj , n − 2m + mj +
m′′} − εj , j = 1,m, the formula (4) determines the solution of the problem (1)
u(x) = u0(x) +
m∑
j=1
uj(x),
u0(x) ∈ Z ′|κ|−n+ε−2m(Ω, x0), uj(x) ∈ Z ′|κj |−n−mj+εj
(Ω, xj),
in the sense of fulfilment (2) for any ψ ∈ Xp, where p = |κ|−n+ε, pj = |κj |−n+1+εj , j =
1,m. The existence of such function ψ may be proved.
This solution can have the power singularities of the order |κ| − 2m + ε inside Ω,
|κj | −mj + 1 + ε on its boundary S,j = 1,m.
We obtaine the similar results in the case of the power singularities of the right-hand
side data on any smooth closed manifold S1 inside Ω.
There are similar properties of the solutions of some boundary value problems for
the elliptic operators in fractional derivatives
Au(x) =
1
(2π)−n
∫
Rn
a(x, ξ)Fu(ξ)ei(x,ξ)dξ,
where Fu denotes Fourier transform of the function u(x),
a(x, ξ) =
N∑
j=1
∑
|α|=sj≤s
aα(x)(−iξ)α,
α = (α1, . . . , αn), αi, sj , s are nonnegative numbers, fractional in general, aα(x) ∈
C∞(Rn).
We assume the existence of the normal fundamental solution ω(x, y) ∈ C∞(x 6= y)
satisfying the following estimate
|ω(x, y)| ≤
{
C|x− y|s−n, n is odd
C|x− y|s−n(ln|x|+ 1), n is even
.
In the case of constant coefficients such fundamental solution exists.
References
1. .., , . . . 20 (1968), no. 3, 412-417.
2. Roitberg Y., Elliptic Boundary Value Problems in the Spaces of Distributions. Kluver Academic
Publishers. (1996).
3. .., , . ... 36 (1971), no. 1, 202-209.
4. .., , . ... 33 (1969), no. 1, 109-137.
5. .., .., , . 22 (1987), no. 3, 518-521.
6. .., .., ., . . . 19 (1967), no. 5, 3-32.
Ivan Franko Lviv State University
Universitetskaya str., 1
290001 Lviv, Ukraine
E-mail: diffeq@ franko.lviv.ua
Tel: 794-593
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