Green's formula and theorems on complete collection of isomorphisms for general elliptic boundary value problems for Douglis-Nirenberg systems
This work has been partially supported by Grant INTAS-94-2187.
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Інститут прикладної математики і механіки НАН України
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irk-123456789-1692732020-06-10T01:26:27Z Green's formula and theorems on complete collection of isomorphisms for general elliptic boundary value problems for Douglis-Nirenberg systems Roitberg, Ya. Roitberg, I. This work has been partially supported by Grant INTAS-94-2187. 1999 Article Green's formula and theorems on complete collection of isomorphisms for general elliptic boundary value problems for Douglis-Nirenberg systems / Ya. Roitberg, I. Roitberg // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 60-66. — Бібліогр.: 6 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169273 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
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This work has been partially supported by Grant INTAS-94-2187. |
| format |
Article |
| author |
Roitberg, Ya. Roitberg, I. |
| spellingShingle |
Roitberg, Ya. Roitberg, I. Green's formula and theorems on complete collection of isomorphisms for general elliptic boundary value problems for Douglis-Nirenberg systems Нелинейные граничные задачи |
| author_facet |
Roitberg, Ya. Roitberg, I. |
| author_sort |
Roitberg, Ya. |
| title |
Green's formula and theorems on complete collection of isomorphisms for general elliptic boundary value problems for Douglis-Nirenberg systems |
| title_short |
Green's formula and theorems on complete collection of isomorphisms for general elliptic boundary value problems for Douglis-Nirenberg systems |
| title_full |
Green's formula and theorems on complete collection of isomorphisms for general elliptic boundary value problems for Douglis-Nirenberg systems |
| title_fullStr |
Green's formula and theorems on complete collection of isomorphisms for general elliptic boundary value problems for Douglis-Nirenberg systems |
| title_full_unstemmed |
Green's formula and theorems on complete collection of isomorphisms for general elliptic boundary value problems for Douglis-Nirenberg systems |
| title_sort |
green's formula and theorems on complete collection of isomorphisms for general elliptic boundary value problems for douglis-nirenberg systems |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/169273 |
| citation_txt |
Green's formula and theorems on complete collection of isomorphisms for general elliptic boundary value problems for Douglis-Nirenberg systems / Ya. Roitberg, I. Roitberg // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 60-66. — Бібліогр.: 6 назв. — англ. |
| series |
Нелинейные граничные задачи |
| work_keys_str_mv |
AT roitbergya greensformulaandtheoremsoncompletecollectionofisomorphismsforgeneralellipticboundaryvalueproblemsfordouglisnirenbergsystems AT roitbergi greensformulaandtheoremsoncompletecollectionofisomorphismsforgeneralellipticboundaryvalueproblemsfordouglisnirenbergsystems |
| first_indexed |
2025-07-15T04:01:52Z |
| last_indexed |
2025-07-15T04:01:52Z |
| _version_ |
1837684084117078016 |
| fulltext |
GREEN’S FORMULA AND THEOREMS ON COMPLETE
COLLECTION OF ISOMORPHISMS FOR GENERAL
ELLIPTIC BOUNDARY VALUE PROBLEMS FOR
DOUGLIS-NIRENBERG SYSTEMS
c© Yakov Roitberg, Inna Roitberg
1 Green’s formula.
In the bounded domain G ∈ Rn with the boundary ∂G ∈ C∞ we consider the elliptic
boundary value problem for Douglis-Nirenberg elliptic (T, S)-system of order (T, S) =
(t1, . . . , tN , s1, . . . sN):
l(x,D)u(x) = (lrj(x,D))r,j=1,...Nu(x) = f(x) (x ∈ G), (1)
b(x,D)u(x) = (bhj(x,D))h=1,...,m
j=1,...,N
u(x) = ϕ(x) (x ∈ ∂G) (2)
Here ord lrj ≤ sr + tj (t1, . . . , tN , s1, . . . , sN are given integer), s1 + · · · + sN + t1 +
· · · + tN = 2m, t1 ≥ · · · ≥ tN ≥ 0 = s1 ≥ · · · ≥ sN , lrj(x,D) ≡ 0 for sr + tj < 0;
ord bhj ≤ σh + tj, bhj(x,D) ≡ 0 for σh + tj < 0, σ1, . . . , σm are given integer.
In the case of normal boundary conditions Green’s formula was deduced in [1] for Petro-
vskii elliptic systems. Without assumption of normality Green’s formula was obtained in
[2] for one equation and in [3] for Petrovskii elliptic systems. For general systems of equa-
tions Green’s formula was obtained in [4]; but there the scalar product was considered in
Sobolev space W k
2 with suffisiently large k in place of the scalar product in L2.
In the present work Green’s formula is obtained for the general elliptic boundary valye
problem (1)-(2) for the system of Douglis-Nirenberg structure; furthermore, the formally
adjoint problem with respect to the Green’s formula is studied too, and the solvability
conditions for the problem (1)-(2) are written more presisely. Green’s formula holds also
for parameter elliptic problems and, therefore, for general parabolic problems for general
systems of equations. For general parabolic boundary value problems for one equation
Green’s formula was deduced in [5].
Let κ = max{0, σ1 + 1, . . . , σm + 1}.
The function uj is differentiated τj := tj + κ(j = 1, . . . , N) times in the system and in
the boundary conditions. Thus, if uj ∈ H tj+κ,p(G) then lru =
∑N
j=1 lrjuj ∈ H−sr+κ,p(G).
Therefore there exist the traces
Dk−1
ν lru|∂G (k = 1, . . . ,−sr + κ)
of the function lru on ∂G.
Hence, if lru = fr, bu = ϕ, then the equalities
lru = fr(x) (x ∈ G, r = 1, . . . , N);
Dk−1
ν lru(x)|∂G = frk(x) (k = 1, . . . ,−sr + κ; frk = Dnuk−1fr|∂G);
bhu(x)|∂G = ϕh (h = 1, . . . ,m)
must be valid automatically.
It turns out that the boundary conditions (2) may be complemented by the matrix
c(x,D) = (chj(x,D)) h=1,...m
j=1,...,N
of boundary differential expressions of orders ord chj ≤ tj +
σ′h, (σ′h < 0) so that the system
Dk−1
ν lru(x)|∂G = frk(x) (r = 1, . . . N, k = 1, . . . ,−sr + κ);
bu|∂G = ϕ
cu|∂G = ψ
that is equivalent to the system
N∑
j=1
τj∑
s=1
Λhjs(x,D′)ηjs = Φh
(h = 1, . . . , |τ |; τj = tj + κ; |τ | = τ1 + . . . , τN ; ηjs = Ds−1
ν uj|∂G)
or to the system
Λη = Φ
is a Douglis-Nirenberg elliptic system on ∂G.
For simplicity, now let the defect of Λ be equal to {0}. Then
η = Λ−1Φ or Ds−1
ν uj|∂G =
|τ |∑
h=1
Γsj,h(x,D′)Φh.
It permit us to obtain Green’s formula.
If some additional condition holds (this condition will be given later) then the follow-
ing theorem is true.
Theorem 1. Green’s formula
(lu, v) +
m∑
h=1
〈bhu, c′hv〉+
N∑
r=1
−sr+κ∑
k=1
〈Dk−1
ν lru, e′rk(x,D)v〉 = (u, l+v) +
m∑
h=1
〈chu, b′hv〉 (3)
(u, v ∈ (C∞(G))N)
holds.
Here Dν = i∂/∂ν, ν is an interior normal to ∂G; (·, ·) and 〈·, ·〉 denote the scalar products
in (L2(G))N and L2(∂G), respectively.
Definition. The problem
l+v = g (in G), b′hv|∂G = ψh (h = 1, . . . ,m) (4)
is called a formally adjoint to the problem (1)-(2) with respect to Green’s formula (3).
They are proved the following theorems.
Theorem 2. The problem (4) is elliptic if the problem (1), (2) is elliptic.
Theorem 3. The problem (1), (2) is solvable if and only if the equality
(f0, v) +
m∑
h=1
〈ϕh, c
′
hv〉+
N∑
r=1
−sr+κ∑
k=1
〈frk, e
′
kr(x,D)v〉 = 0 (∀v ∈ N+) (5)
holds.
Here N+ = {v ∈ (C∞(G))N : l+v = ′, b′〈v|∂G = ′ (〈 = ∞, . . . ,m)} denotes the kernel of
the problem (4); f0 = f |G, frkD
k−1
ν fr|∂G (r = 1, . . . , N, k = 1, . . . ,−sr + κ).
In order to precise Theorem 3 we introduce corresponding functional spaces.
For every s ≥ 0 and p ∈ (1,∞) we denote by Hs,p(G) the space of Bessel potentials
(Liouville classes); H−s,p(G) = (Hs,p′(G))∗ (1/p + 1/p′ = 1); || · ||s,p denotes the norm in
Hs,p(G) (s ∈ R).
By Bs,p(∂G) (s ∈ R) denote the Besov space with the norm 〈〈·〉〉s,p; B−s,p(∂G) =
(Bs,p′(∂G))∗.
For every positive integer r and s ∈ R (s 6= k+1/p, k = 0, . . . , r−1) denote by H̃s,p,(r)
the completion of C∞(G) in the norm
|||u|||s,p,(r) := (||u||ps,p +
r∑
j=1
〈〈D/nuj−1u〉〉ps−j+1−1/p,p)
1/p. (6)
For s = k + 1/p (k = 0, . . . , r − 1) the space H̃s,p,(r) and the norm |||u|||s,p,(r) (6) are
defined by method of complex interpolation. Finally, if r = 0 then H̃s,p,(r) = Hs,p(G);
||| · |||s,p,(0) = || · ||s,p.
Let τ = (τ1, . . . τN), |τ | = τ1 + . . . τN , τj = tj + κ (j = 1, . . . , N).
For any s ∈ R, p ∈ (1,∞) the closure A = As,p of the mapping
u 7→ ((lju|G : j = 1, . . . , N), (Dk−1
ν lju|∂G : j = 1, . . . , N, k = 1, . . . ,−sJ + κ),
(bju|∂G : j = 1, . . .m)) =: Au
acts continuously in the pair of spaces
H̃T+s,p,(τ) :=
N∏
j=1
H̃ tj+s,p,(τj) → Ks,p :=
N∏
j=1
H̃s−sj ,p,(κ−sj) ×
m∏
h=1
Bs−σh−1/p,p(∂G).
Definition. The element u ∈ H̃T+s,p,(τ) such that
Au = F = (f, ϕ1, . . . , ϕm), f = (f1, . . . , fN), fj = (fj0, . . . , fj,κ−sj
) (7)
is called a generalized solution of the problem (1), (2).
It turns out (see, for example, [6]) that for any s ∈ R, p ∈ (1, +∞) the operator
A = As,p is Noetherian, the kernel N and the cokernel N ∗ are finitedimentional and do
not depend on s and p and consist of infinitely smooth functions.
This theorem on complete collection of isomorphisms gives us the more precise
formulation of Theorem 3.
Theorem 3’. The problem (6) As,pu = F = (f, ϕ) ∈ Ks,p is solvable in H̃T+s,p,(τ) if
and only if relation (5) holds.
It was above mentioned that the Theorem 1 is true if the some additional condition
is valid. Now we write this condition.
Consider the equalities
bhu|∂G = ϕh ∈ C∞(∂G) (h = 1, . . . , m)
Dk−1
ν lju|∂G = fjk ∈ C∞(∂G) (j = 1, . . . , N, k = 1, . . . ,−s + κ)
chu|∂G = χh ∈ C∞(∂G) (h = 1, . . . , m)
(8)
We will find the elements Dk−1
ν uj|∂G = ηjk ∈ C∞(∂G) (j = 1, . . . , N, k = 1, . . . , τj)
from these relations (8). Thus, we construct the system of |τ | = τ1 + · · · + τN equations
with |τ | variables ηjk (we will call this system as (8’)).
This system (8’) is elliptic in the sense of Douglis-Nirenberg on the manifold ∂G
without border.
The Theorem 1 is true if the next condition is valid: the problem (8’) must be uniquely
solvable for any smooth right-hand sides; in orther words, the defect of this problem must
be equal to {0}.
If the problem (1), (2) is parameter elliptic or if it is a Dirichlet system then this
condition is always taking place.
If this condition does not hold then Green’s formula has additional terms with pro-
jective operators. Newetheless, the statements of the Theorem 1 (And Theorem 2, and
Theorem 3) remain true.
2 Theorems on complete collection of isomorphisms.
For simplisity, again let N = N ∗ = {′}. Recall that in these case the operator A = As,p
realizes an isomorphism
A : H̃T+s,p,(τ) → Ks,p. (9)
Green’s formula permit us to obtain a number of different theorems on complete collection
of isomorphisms from the theorem on isomorphisms (9).
To obtain them we need the following too simple statements.
”Pasting” (or factorization) method. Let B1 and B2 be Banach spaces, and let
T be a linear operator isomorphically mapping the space B1 onto the space B2. Let E1
be a subspace of the space B1, and let E2 = TE1. Then it is clear that the operator T by
the natural way defines the linear operator T1 isomorphically mapping the factor space
B1/E1 onto the factor space B2/E2.
Graph method. Let Q2 be a Banach space, and let Q2 ⊂ B2 (this imbedding is
algebraic and topological). Then Q1 = T−1Q2 is a linear (generally speaking, nonclosed)
subset of B1. However, the space Q1 becams the Banach space denoted by Q1T with
respect to the graph norm
||x||Q1T
= ||x||B1 + ||Tx||Q2 (x ∈ Q1).
The restriction of the operator T onto Q1 establishes an isomorphism Q1T → Q2.
Different elements u ∈ H̃T+s,p,(τ) may have the same components
(u0, c1u|∂G, . . . , cmu|∂G).
”Pasting” tham and doing the corresponding factorization in the space of images, we ob-
tain new statement on isomorphisms from isomorphism (9).
Theorem 4. The closure A1 = A1,s,p of the mapping
u 7→ (lu, {bhu|∂G : s− σ − 1/p > 0}) (u ∈ (C∞(G))N)
which is considered acting in the pair of spaces
∏N
j=1 H tj+s,p(G)×∏
h:s−σc
h
−1/p<0 Bs−σc
h−1/p,p(∂G) →
(
∏N
j=1 Hs−sj ,p(G)×∏
Bs−σh−1/p,p(∂G))/M1
s,p
(10)
(here the subspace M1
s,p dascribed by Greens formula), realizes an isomorphism (10).
Corollary. There holds the next estimate
N∑
j=1
||uj||Htj+s,p(G) +
m∑
j=1
〈〈cju〉〉
B
s−σc
j
−1/p,p
(∂G)
≤ c1(||lu||∏N
j=1
Hs−sj ,p(G)
+
+
∑
h:s−σh−1/p>0
〈〈bhu〉〉Bs−σh−1/p,p(∂G) +
N∑
j=1
||uj||Htj+sk,p(G)) (u ∈ (C∞(G))N),
where k > 0 may be choosen arbitrary large.
Different elements u ∈ H̃T+s,p,(τ) may have the same components
(u0, c1u|∂G, . . . , cmu|∂G, b1u|∂G, . . . , bmu|∂G).
We ”paste” them and make a corresponding factorization in the space of images. The
obtained space of preimages, denoted by H̃
T+s,p,(τ)
c,b is the completion of (C∞(G))N in the
morm
|||u|||T+s,p,(c,b) := (||u||pT+s,p +
m∑
j=1
〈〈cju〉〉ps−σc
h
−1/p,p +
m∑
j=1
〈〈bju〉〉ps−σj−1/p,p)
1/p.
The obtained space of images is:
(
N∏
j=1
Hs−sj ,p(G)×
m∏
h=1
Bs−σh−1/p,p(∂G))/M2
s,p := K2
s,p,
where M2
s,p described by means of the Green’s formula (3).
Theorem 5. The closure A2 = A2,s,p of the mapping
u 7→ (lu, bu) (u ∈ (C∞(G))N)
which is considered acting in the pair of spaces
H̃
T+s,p,(τ)
c,b → K2
s,p (11)
is an isomorphism between the spaces (13).
Corollary. The following estimate is true:
|||u|||T+s,p,(c,b) ≤ c1(||lu||s−S,p +
∑〈〈bju〉〉s−σj−1/p,p + ||u||T+s−k,p)
(u ∈ (C∞(G))N),
where k > 0 may be choosen arbitrary large.
Different elements u ∈ H̃T+s,p,(τ) may have the same components
(u0, b1u|∂G, . . . , bmu|∂G).
Using the ”pasting” method we obtain new Theorem 6 on complete collection of iso-
morphisms:
H̃
T+s,p,(τ)
b → K3
s,p.
Here H̃
T+s,p,(τ)
b denotes a completion of of (C∞(G))N in the morm
|||u|||T+s,p,(b) := (||u||pT+s,p +
m∑
j=1
〈〈bju〉〉ps−σj−1/p,p)
1/p,
and the space of images is
(
N∏
j=1
Hs−sj ,p(G)×
m∏
h=1
Bs−σh−1/p,p(∂G))/M3
s,p := K3
s,p,
where M3
s,p described by means of Green’s formula (3). The corresponding estimate is
also true.
Let
C∞
B := {u ∈ (C∞(G))N : bu|∂G = 0}
and let
HT+s
B = {u ∈ H̃T+s,p,(τ) : bu|∂G = 0} =
= {u ∈ HT+s,p(G) : bhu|∂G = 0/, (∀h : s− σh − 1/p > 0)} ⊂ H̃
T+s,p,(τ)
b .
The Theorem 6 implies the following theorem on isomorphisms.
Theorem 7. The closure A3(B) of the mapping
u 7→ lu (u ∈ C∞
(B))
which we consider acting in the pair of spaces
HT+s
B → Hs−S,p/M4
s,p, (12)
where
M4
s,p = {f ∈ Hs−S,p : (f, v) +
∑
k,r:s−sr−k+1−1/p>0
〈Dk−1
ν fr, e
′
krv〉 = 0 (∀v ∈
(C∞(G))N : e′krv|∂G = 0(k, r : s− sr − k + 1− 1/p < 0), b′v|∂G = 0)} ⊂ Hs−S,p
realizes an isomorphism between the spaces (12).
In the special case of one equation with normal boundary conditions the isomorphism
(12) was obtained by Yu. Berezanskii - S. Krein - Ya. Roitberg (1963).
One can obtain a number of the theorems on complete collection of isomorphisms by
means of ”graph method”. In the special case of one equation with normal boundary
conditions by using this method one can obtain the isomorphism of Lions - Magenes.
3 Generalizations.
All these results remain true for elliptic with a parameter systems. But now instead of
Noetherity the unique solvability is taking place for sufficiently large parameter. It implies
that the results remain true for parabolic problems. In addition, one can consider the
multy-times parabolic problems.
This work has been partially supported by Grant INTAS-94-2187.
References
1. Roitberg Ya.A., Sheftel Z.G., Mat. Sbornik, 78 (1969), No. 3, p. 446-472.
2. Roitberg Ya.A., Mat. Sbornik, 83 (1970), No. 6, p. 181-213.
3. Roitberg Ya.A., Sheftel Z.G., Math. Investigations, Kishinjov, 1972, 7 (1972), No. 2, p.
143-157.
4. Lvin S.Ja., Green’s formula and solvability of elliptic problems with boundary condi-
tions of arbitrary order, VINITI, No. 3318-78, p. 1-30.
5. Roitberg I., Roitberg Ya., Differentsialnye Uravneniya, 31 (1995), No. 8, p. 1437-1444.
6. Roitberg Ya.A., Elliptic Boundary Value Problems in the Spaces of Distributions,
Kluwer Akad. Publishers, Dordrecht/Boston/London, 1996, 427 pp.
Chernigov State Pedagogical Institute,
Department of Mathematics,
Sverdlova Street 53,
250038 Chernigov, Ukraine.
Fax: 04622-32069
Phone: 0462-959055
E-mail: alex@elit.chernigov.ua
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