ISSN 1064–5624, Doklady Mathematics, 2006, Vol. 73, No. 3, pp. 349–353. © Pleiades Publishing, Inc., 2006. Original Russian Text © Yu.A. Dubinskii, 2006, published in Doklady Akademii Nauk, 2006, Vol. 408, No. 2, pp. 160–164.
MATHEMATICS m, k
m
Decomposition of the Spaces W p and D p into Sums of Solenoidal and Potential Subspaces and Factorization Inequalities Yu. A. Dubinskii Presented by Academician V.A. Il’in November 28, 2005 Received January 30, 2006
DOI: 10.1134/S1064562406030094 m
We decompose the Sobolev scale of W p and the m, k
gradient-divergence scale of D p , where p > 1 and m = 0, ±1, …, k ≥ 0, into direct sums of complex-solenoidal and complex-potential subspaces. The smoothing property for the solenoidal factorization is proved. The potential is represented in divergence form. 1. NOTATION AND SPACES Suppose that z ∈ n, i.e., z = (z1, z2, …, zn), where n ≥ 1 and zj = xj + iyj for xj, yj ∈ and j = 1, 2, …, n. Let G ⊂ n be a bounded domain with boundary Γ. Identifying the space n with the real space 2n of variables x = (x1, x2, …, xn) and y = (y1, y2, …, yn), we can regard the domain G as a domain in 2n with sufficiently smooth boundary. Let u(z) be a complex-valued scalar or vector function u(z) = (u1(z), u2(z), …, un(z)) defined on the domain G ⊂ n. We denote it also by u(x, y), when regarding G as a domain in 2n. We consider the following spaces. 1. = G) (m ≥ 0) is the Sobolev space of vector functions u(z) considered as functions of 2n variables x and y, i.e., m Wp
m Wp
W p = { u: u ∈ L p , ∇u ∈ L p, …, ∇ u ∈ L p }, m
m
m
u
=
〈 u, φ〉 = sup ----------------- , – m φ p', – m φ∈W p'
p where p' = ------------ . The duality between the spaces in the p–1 –m
m
pair ( W p , W p' ) extends the duality between the Lebesgue spaces Lp and Lp', which correspond to m = 0; i.e., 0
W p = Lp. Consider the complex divergence and gradient operators div z u ( z ) = ∂ z1 u 1 ( z ) + ∂ z2 u 2 ( z ) + … + ∂ zn u n ( z ), ∇ z r ( z ) = ( ∂ z1 r ( z ), ∂ z2 r ( z ), …, ∂ zn r ( z ) ) [here, r(z) is a scalar function], where ∂ ∂ ∂ z j = ------- – i -------, ∂x j ∂y j
∂ ∂ ∂ z j = ------- + i ------- , ∂x j ∂y i
j = 1, 2, …, n.
For m ≥ 1, these operators act as usual operators in the m sense of Sobolev spaces on the functions from W p . If m ≤ 0, then they act as 〈 div z u ( z ), φ ( z )〉 = – 〈 u ( z ), ∇ z φ ( z )〉 , 〈 ∇ z u ( z ), φ ( z )〉 = – 〈 u ( z ), div z φ ( z )〉 ,
m
u
p, m
φ≠0
endowed with the norm p m, p
–m
For m < 0, the spaces W p = ( W p' )* are their duals with norms
∑
∇u k
p 0, p .
k=0
Moscow Power Institute, ul. Krasnokazarmennaya 17, Moscow, 111250 Russia e-mail:
[email protected]
01 p –m where φ(z) ∈ W p' ∩ W p' , p' = ------------ , and, as usual, the p–1 zero over the symbol of a space means that the functions vanish on the boundary Γ.
2. In the spaces W p , where m = 0, ±1, …, we consider the subspaces of solenoidal and cosolenoidal m
349
350
DUBINSKII m
functions. The solenoidal subspace in the space W p is defined as S p = { u ∈ W p : div z u = 0 }. m
m
m
phisms for elliptic problems is exposed in the monograph [1] of Roitberg, which contains also relevant bibliography. In what follows, we apply the main theorem of this theory to the Dirichlet problem for the Poisson equation
m
∆u ( z ) = h ( z ),
It is easy to see that S p is a closed subspace in W p and the inclusion ⊂ is strict. The cosolenoidal subspace is the “orthogonal” comm plement to S p in the sense of the duality between the m Sp
spaces
m Wp
and
m Wp
–m W p' ;
m ⊥
i.e., by definition,
( S p ) = { u ∈ W p : 〈 u ( z ), φ s ( z )〉 = 0, ∀φ s ∈ S p' }. m
m
–m
∞
3. Let V 0 = C∞( G ) ∩ C0( G ) be the space of functions infinitely differentiable on the closure of the domain G and vanishing on its boundary Γ. We set ∞
def
W p, 0 = complV 0 ,
m = 0, ± 1, …;
∞
m
this is the completion of V 0 in the norm of W p . Obviously, for m ≥ 1, the spaces W p, 0 can be idenm
m
01
∇z : ( W p ∩ W p ) → W p m
m–1
is continuous. If m ≤ 0, then the ideal elements of the completion [we denote them by the same symbol u(z)] vanish on Γ in the weak sense, which corresponds to the definition of the gradient operator in the theory of weak solutions to elliptic problems, i.e., in the sense of the “integral identity” 〈 ∇ z r ( z ), φ ( z )〉 = – 〈 r ( z ), div z φ ( z )〉 ,
(1)
where r(z) ∈ W p, 0 is a scalar function and the test vector functions φ(z) are arbitrary elements of the space –m+1 . This means that the operator ∇ z acts from W p' m
m
m–1
(2)
= 0.
According to this theorem, for any right-hand side 01
h(z) ∈ ( W p'
∩ W p' )*, problem (2) has a unique
–m+1
solution u(z) ∈ W p, 0 , and the mapping ∆ 0 : h(z) → u(z) is continuous. m+1
–1
, where m = 0, –1, …, and the mapping ∇ z : W p, 0 → W p m
2. DECOMPOSITION OF THE SCALE m OF W p IN A SUM OF SOLENOIDAL AND COSOLENOIDAL (POTENTIAL) SUBSPACES The main result of this section is the following theorem. Theorem 1. The following assertions are valid: (i) for any m = 0, ±1, … the operator ∇ z determines an elliptic isomorphism
m–1
m ⊥
∇ z : W p, 0 → ( S p ) ; m+1
0 1 Wp
tified with the Sobolev spaces W p ∩ whose elements vanish on Γ in the sense of usual Sobolev spaces. Clearly, the mapping
W p, 0 to W p
Γ
–m
Thus, ( S p )⊥ is the annihilator of the subspace S p' .
m
u(z)
(3)
m
(ii) the space W p can be represented as the direct sum m m . m ⊥ (4) W p = S p + ( S p ) , m ∈ , 1 < p < ∞, or, equivalently [see (i)], m m . m+1 W p = S p + ∇ z W p, 0 .
(5)
Remark 1. Obviously, (5) implies that any vector m function u ∈ W p uniquely decomposes as u ( z ) = u s ( z ) + ∇ z r ( z ), where us(z) is the solenoidal component of the function u(z) and ∇ z r(z) is its cosolenoidal component, which is the complex gradient of a function r(z) ∈ W p, 0 ; this function is uniquely determined by u(z). More precisely, considering the elliptic isomorphism (3), we take into account the fact that the function q(z) belongs m to the “orthogonal” complement ( S p )⊥ if and only if m+1
there exists a function r(z) ∈ W p, 0 such that m+1
is continuous. 1. Spaces of type W p, 0 with m ≤ 0 naturally arise in the theory of weak solutions to elliptic problems. It is for these spaces that the theorem about a complete set of Lp-isomorphisms corresponding to coercive elliptic problems in the weak sense, i.e., in the sense of integral identities, is valid. A complete theory of Lp-isomorm
∇z r ( z ) = q ( z ) and M
–1
q
m, p
≤ r
m + 1, p
≤M q
m, p ,
where M > 0 is a constant. DOKLADY MATHEMATICS
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No. 3
2006
m, k
m
351
DECOMPOSITION OF THE SPACES W p AND D p
Assertions (i) and (ii) in the theorem are proved by constructing the projectors Ps :
m Wp
→
⊥ Ps :
m Sp ,
m Wp
→
m ⊥ (Sp ) .
⊥
⊥
P s = ∇ z ∆ 0 div z , –1
(6)
where ∆ 0 is the solving operator for the Dirichlet problem –1
div z ∇ z r ( z ) = h ( z ), r(z)
Γ
u
L p /S p
≤ M div z u
(10)
– 1, p ,
01
Namely, we set Ps = I – Ps ,
2. If m = 0, then
z ∈ G,
where ||·||–1 is the norm of the dual space ( W p )*. This inequality says that the norms ||u||0, p and ||divzu||–1, p on the quotient space Lp/Sp are equivalent. It is a divergence analogue of the well-known LBB inequality in the theory of Stokes and Navier–Stokes equations. The following inequality, which is intermediate between (9) and (10), is also useful:
(7) (8)
= 0.
The proof proceeds as follows:
u
L p /S p
≤ M div z u
Lp;
this ia an analogue of the Poincaré inequality for functions with zero mean.
⊥
(i) It is shown that Ps and P s are bounded projectors onto the subspaces S p and ( S p )⊥, respectively; m
m
(ii) It is verified that the sum in formula (4) is direct; (iii) The formula ( S p )⊥ = W p, 0 and the corresponding estimate are proved. Remark 2. For the real space L2, decomposition (5) is known (see [2]–[4]). m
m+1
Theorem 1 says that any function u(z) ∈ W p can be uniquely represented as m
u ( z ) = u s ( z ) + ∇ z r ( z ), where us(z) is its solenoidal part and the potential r(z) ∈ m+1
W p, 0 is a solution to the Dirichlet problem ∆r ( z ) = div z u ( z ),
r(z)
Γ
u – us
m, p
m, p
≤ M div z u
≤ M div z u
–m+1
( W p'
01
∩ W p' )*
( m ≤ 0 ),
or, equivalently, the factorization inequalities u u
m m W p /S p
≤ M div z u
m m ≤ M div z u W /S p
p
m – 1, p
–m+1
( W p'
0
1
p
p
w (9)
Lp.
on the quotient space
DOKLADY MATHEMATICS
are equivalent. Vol. 73
No. 3
Γ
= 0
n
w ∈ L 2 &div z w ∈ L 2 }
with norm
( m ≤ 0 ).
This inequality says that the norms ||u||1, p and ||divzu||0, p 1 1 W p /Sp
= { w: G →
2006
0, 1
D2
= w
0, 2
+ div z w
0, 2 .
0, 1
Obviously, W 2 ⊂ D 2 ; however, decomposition (11), 1
1 1 ≤ M div z u W /S
r(z)
(in fact, this was mentioned at the end of Section 2). This suggests that, as far as decomposition (11) is concerned, the scale of spaces determined by both the gradient and divergence operators might be as natural as the Sobolev scale. We start with the simplest example, namely, with the 1 analogue of W 2 , which is the space 0, 1
Examples. 1. If m = 1, then u
(11)
the determining role in finding the potential r(z) is played not so much by the function u(z) itself as by its divergence divzu(z), because r(z) is a solution to the Dirichlet problem
D2
( m ≥ 1 );
∩ W p' )*
u ( z ) = u s ( z ) + ∇ z r ( z ),
∆r ( z ) = div z u ( z ),
( m ≥ 1 ),
m – 1, p
The main result of Section 2 refers to a decomposim tion of the Sobolev scale of W p generated by the operator ∇x, y (or, equivalently, by the operators ∇ z and ∇z). However, in the representation of the function u(z) ∈ m W p in the form
= 0.
This and the known Lp-estimates for the solutions to the Dirichlet problem for the Poisson equation imply the inequalities u – us
3. DECOMPOSITION OF THE SCALE m, k OF D p . THE SOLENOIDAL FACTORIZATION AS A SMOOTHING OPERATION
which was obtained above for u(z) ∈ W 2 , remains valid 1
0, 1
for any function w(z) ∈ D 2 with the same divergence divzw(z). Moreover, this decomposition has the following
352
DUBINSKII
useful property. Since divzw(z) ∈ L2, it follows that the potential r(z), being a solution to the Dirichlet problem ∆r ( z ) = div z w ( z ),
r(z)
Γ
= 0,
m ⊥
m, k
( S p ) = { w ∈ W p : 〈 w ( z ), φ ( z )〉 = 0 },
01
belongs to the space W 2 ∩ W 2 . Thus, ∇ z r(z) ∈ W 2 2
Thus, their “orthogonal” complements in the spaces m, k D p are
1
0, 1
and, therefore, for any function w(z) ∈ D 2 , the decomposition
where φ(z) ∈ S p' (of course, these subspaces depend on k, but we do not indicate this dependence in the notation). It is easy to verify that the operators –m
w ( z ) = ws ( z ) + ∇z r ( z )
⊥
⊥
P s = I – P s and P s = ∇ z ∆ 0 div z ⊥
–1
implies that the “factored” function w(z) – ws(z) ∈ S 2
are projectors of D p
is smoother; namely, w(z) – ws(z) ∈
( S p )⊥; thereby, they determine the decompositions
1 W2 .
m, k
m
0, 1
Thus, the factorization of the space D 2 by the subspace S2 of solenoidal functions yields a subspace of 1
W 2 ; i.e., 0, 1
m, k
Dp
m, k
Thus, each function w(z) ∈ D p
1
≤ M div z w
0, 2 ,
∆r ( z ) = div z w ( z ), m, k
Now, consider the complete scale of spaces D p , where k ≥ 0 – is an integer. We set m, k
= { w: G →
w ∈ W p &div z w ∈ V p
n
m
m–1+k
Since divzw(z) ∈ V p
m–1+k
m+1+k W p, 0
r
}, r
where m–1+k
m–1+k
=
–( m – 1 + k )
∩ W p'
m, k
m
)*
m – 1 + k, p ( m –( m – 1 + k )
( W p'
0
1
∩ W p' )*
+ k – 1 ≥ 0 ); ( m – 1 + k < 0 ).
gives the factorization inequalities
m+k
m + k, p
0, 0
m
= W p ; in particular, D p n
= Lp and k–1
m, k
m
}.
Note that all spaces D p have subspaces S p of solenoidal vector functions not depending on k = 0, 1, …, namely, = { ws ∈
m Wp:
≤ M div z w
m + k, p
m – 1 + k, p ( m
≤ M div z w
+ k – 1 ≥ 0 );
–( m – 1 + k )
( W p'
0
1
∩ W p' )*
( m – 1 + k < 0 ).
.
= { w: G → w ∈ L p &div z w ∈ W p
m Sp
≤ M div z w
≤ M div z w
w – ws W p ⊃ Dp ⊃ W p
0, k
m + 1 + k, p
m + 1 + k, p
Obviously,
Dp
, it follows that r(z) ∈
; moreover, we have
w – ws
m, 0
= 0.
w ( z ) = ws ( z ) + ∇z r ( z ) 0 1 ( W p'
if m – 1 + k < 0.
This implies D p
Γ
m, k
= Wp
if m – 1 + k ≥ 0 and Vp
r(z)
Therefore, for any function w(z) ∈ D p , the decomposition
m–1+k
Vp
can be represented as
m
where M > 0 is a constant.
Dp
(12)
where ws(z) ∈ S p is the solenoidal component of w(z) and the cosolenoidal potential r(z) is a solution to the Dirichlet problem
Moreover, we have 1, 2
m . m+1+k = S p + ∇ z W p, 0 .
w ( z ) = w s ( z ) + ∇ z r ( z ),
D 2 /S 2 ⊂ W 2 .
w – ws
m
onto the subspaces S p and
div z w s = 0 }.
(In Section 2, these inequalities were obtained for k = 0, m, k i.e., in the case where the scale of D p coincides with m
that of W p .) We regard decomposition (12) and the factorization inequalities given above as main results of this paper. m m–1+k Note that if w(z) ∈ W p and divzw(z) ∈ V p , then w(z) – ws(z) ∈ W p
m+k
m, k Dp
m /Sp
. This means that the coset in
that contains this function contains also a
DOKLADY MATHEMATICS
Vol. 73
No. 3
2006
m
m, k
DECOMPOSITION OF THE SPACES W p AND D p
function of higher degree of smoothness than the function w(z) itself. We call this fact the smoothing property of the solenoidal factorization. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (project no. 05-01-00526).
DOKLADY MATHEMATICS
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REFERENCES 1. Y. Roitberg, Elliptic Boundary Value Problems in the Spaces of Distributions (Klüwer, Dordrecht, 1996). 2. R. Temam, Navier–Stokes Equations (North-Holland, Amsterdam, 1979; Mir, Moscow, 1981). 3. O. A. Ladyzhenskaya, Mathematical Questions of Viscous Incompressible Fluids (Fizmatlit, Moscow, 1961). 4. V. I. Yudovich, The Linearization Method in Stability Theory (Rostov. Gos. Univ., Rostov-on-Don, 1984).
