The Helmholtz decomposition in weighted Lp spaces in cones
Serge Nicaise∗ June 19, 2017
Abstract
In this paper we prove the Helmholtz decomposition in conical domains ofR3 in weighted Lpβ spaces under a spectral condition onα, p. The basic bricks are the transformation of the original problem into a problem set in cylindrical domains and the combination of a priori bounds from [5] with the vector-valued multiplier theorem [29].
AMS (MOS) subject classification35J05, 35J20.
Key WordsHelmholtz decomposition, cones.
1 Introduction
The Helmholtz decomposition, namely the decomposition of vector fields into a solenoidal field and a gradient part, is an indispensable tool in the analysis of incompressible fluid flows. While in the L2-setting such a decomposition is easily obtained for any domains ofRd, d≥2 using elementary Hilbert space properties [20, Lemma 2.5.1], the situation becomes more delicate in theLp-setting.
Nevertheless, there are numerous results for bounded domains [3, 10, 12, 20] or exterior domains (complement of a bounded domains) [17, 18, 28]; see also [6, 7, 23, 22] for such results in weighted spaces. A common feature of such domains is that they have a compact boundary. On the other hand there exist domains with non compact boundary (sector-like) for which the Lp-Helmholtz decomposition fails [1, 2, 16]. Nevertheless it is still valid for basic domains with a smooth boundary like half-spaces or perturbations of them [11, 13, 27], aperture domains [4, 17, 6], cylinders or layers [17, 21, 5, 19, 26, 25], or domains whose boundary is the graph of smooth functions [2, 11]. But to the best of our knowledge, the question of the validity of theLp-Helmholtz decomposition for domains with a non compact and non-smooth boundary remains open. Nevertheless let us mention the paper [15] that obtains a Helmholtz decomposition in anisotropicLpspaces for domains whose boundary is the graph of a globally Lipschitz functions (hence conical domains are admitted).
The goal of this paper is therefore to prove such a Helmholtz decomposition in conical domains of R3in weightedLpβspaces (Lp-spaces being a particular one whenβ= 0) under a condition between α, pand the eigenvalues of the Laplace-Beltrami operator with Neumann boundary conditions on the intersection of the cone with the unit sphere. The basic tools are first to transform the problem into a problem set in cylindrical domains using the Euler change of variables, second by applying a formal Fourier transformation we found a problem that is a lower order perturbation of a problem
∗Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, F-59313 - Valenciennes Cedex 9 France, Serge.Nicaise@univ-valenciennes.fr
considered in [5], hence combining a priori bounds from this paper with the vector-valued multiplier theorem [29], we conclude the result. Note that this result can be derived via a duality argument, we refer to [24] in this respect.
The paper is organized as follows. The second section deals with the weak Neumann problem in the Fourier space, namely we study the resolvent of the operator obtained after Euler change of variables and Fourier transform and show itsR-boundedness. In section 3, we show the well- posedness of the weak Neumann problem in a cone and conclude with the Helmholtz decomposition.
Acknowledgement: We are grateful to Professor P. Deuring (Univ. Littoral) to have pointed out the problem considered in this paper and for various discussions about it.
2 The weak Neumann problem in the Fourier space
Let us fix an open subsetS of the unit sphere of R3 with a C1,1 boundary. On such a domain, we want to consider a Neumann type problem with complex parametersξ and onLp(S)spaces.
Hence as in [5], in order to have uniform estimates in ξ and to use the vector-valued multiplier theorem [29], we enlarge the setting to weightedLp spaces.
Before let us fix some notations. Let us denote by (µk)k≥0, the sequence of the eigenvalues (repeated according to their multiplicites) of the (non negative) Laplace-Beltrami operator LNeuLB onS with Neumann boundary conditions (obviously µ0= 0andµ1>0).
Let us further fix an atlas (U`,Φ`), ` = 1,· · · , L of S in the sense that for each `, U` is a bounded open set ofR2 and
Φ`:U`→S is an injective andC1mapping such that
∪`=1Φ`(U`) =S.
Now a non-negative function ω ∈ L1(S) is called an Ap-weight if and only if ω◦Φ` is the restriction toU`of anAp-weight of R2. Finally we introduce the spaceLpω(S)defined by
Lpω(S) :={u∈L1(S) :kukp,ω= Z
S
|u|pω dϑ 1p
<∞}.
In the same manner, we define
LpT ,ω(S) :={u∈Lpω(S)3:u·ϑ= 0}, as well as
Hp,ω1 (S) :={u∈Lpω(S) :∇Tu∈LpT ,ω(S)}, equipped with their natural norm.
The main goal of this section is to study the next problem: given two real parametersη, δ, and functionsg∈LpT ,ω(S)andg∈Lpω(S), we look for a solutionψ∈Hp,ω1 (S)to
Z
S
∇Tψ· ∇Tϕ − (iξ−η)(iξ+δ)ψϕ¯ (2.1) dϑ
= Z
S
(g(ϑ)· ∇Tϕ¯−g(ϑ)(iξ+δ) ¯ϕ)dϑ,∀ϕ∈Hp10,ω0(S),
wherep0 is the conjugate ofpand ω0 =ω−p−11 .
The case whenδ=−η with|η|< µ1is treated in [5, Theorem 3.6] (since Theorem 3.2 of [5] is valid by replacing∆0 by the operator
X
i,j
aij∂i∂j,
with constant coefficientsaij=ajithat is elliptic in the (usual) sense that the matrix(aij)1≤i,j≤2 is positive definite). We here extend this result under one condition onη, δ. Namely we have the next result.
Theorem 2.1 Assume that
(2.2) ηδ6=−µk,∀k∈N.
Then for allg∈LpT ,ω(S)andg∈Lpω(S), there exists a unique solutionψ∈Hp,ω1 (S)of (2.1)with the estimate
(2.3) k∇ψkp,ω+ (1 +|ξ|)kψkp,ω ≤C(kgkp,ω+kgkp,ω),
with a positive constantC that may depend on theAp-constantAp(ω)but is independent of ξ.
Proof.We first prove the existence and uniqueness of the solution. For that purpose, we introduce the family of operators (compare with [5, p. 375])
Tp,ω(η, δ, ξ) :Hp,ω1,0(S)→(Hp1,00,ω0(S))∗:ψ→Tp,ω(η, δ, ξ)ψ, where
Hp,ω1,0(S) :={ψ∈Hp,ω1,0(S) : Z
S
ψ dϑ= 0}, and
hTp,ω(η, δ, ξ)ψ, ϕi= Z
S
∇Tψ· ∇Tϕ−(iξ−η)(iξ+δ)ψϕ¯
dϑ,∀ϕ∈Hp1,00,ω0(S).
The proof of Theorem 3.6 of [5] shows thatTp,ω(0,0, ξ)is an isomorphism. Now as (Tp,ω(0,0, ξ)−Tp,ω(η, δ, ξ))ψ= (ξ2+ (iξ−η)(iξ+δ))ψ,
and since by Proposition 2.5 of [5], Hp,ω1 (S) is compactly embedded intoLpω(S), we deduce that Tp,ω(0,0, ξ)−Tp,ω(η, δ, ξ)is a compact operator. ThereforeTp,ω(η, δ, ξ)is a Fredholm operator of index 0 and it will be an isomorphism if and only if it is injective. So letψ∈kerTp,ω(η, δ, ξ)or equivalentlyψ∈Hp,ω1,0(S)satisfies
(2.4)
Z
S
∇Tψ· ∇Tϕ¯−(iξ−η)(iξ+δ)ψϕ¯
dϑ= 0,∀ϕ∈Hp1,00,ω0(S).
But asψis of mean zero, we have Z
S
∇Tψ· ∇T1−(iξ−η)(iξ+δ)ψ1
dϑ=−(iξ−η)(iξ+δ) Z
S
ψ dϑ= 0, and therefore (2.4) extends to the wholeHp10,ω0(S), namely
(2.5)
Z
S
∇Tψ· ∇Tϕ¯−(iξ−η)(iξ+δ)ψϕ¯
dϑ= 0,∀ϕ∈Hp10,ω0(S).
But Lemma 2.2 of [8] guarantees thatHp,ω1 (S) is continuously embedded into W1,s(S)for some s >1and by the Sobolev embedding theorem,W1,s(S)being continuously embedded intoL2(S), we deduce thatψ∈L2(S).
Now for anyh∈L2(S), as by our assumption(−iξ−η)(−iξ+δ)is never an eigenvalue ofLNeuLB, there exists an unique solutionϕ∈H2(S)solution of
LNeuLB −(−iξ−η)(−iξ+δ) ϕ=h.
SinceH2(S)is continuously embedded intoW1,q(S)for allq > 1, by Lemma 2.4 of [5],H2(S)is continuously embedded intoHp10,ω0(S). Consequently the above identity implies that
Z
S
∇Tψ· ∇Tϕ¯−(iξ−η)(iξ+δ)ψϕ¯ dϑ=
Z
S
ψ¯h dϑ.
Comparing with (2.5), we see that
Z
S
ψ¯h dϑ= 0, for anyh∈L2(S)and we conclude thatψ= 0.
At this stage it remains to show the estimate (2.3). Let us then fixg∈LpT ,ω(S)andg∈Lpω(S) and the unique solutionψ∈Hp,ω1 (S)of (2.1). Thenψcan be viewed as the solution of
Z
S
∇Tψ· ∇Tϕ+ (ξ+iα)2ψϕ dϑ¯ = Z
S
(g(ϑ)· ∇Tϕ¯−g(ϑ)(iξ+δ) ¯ϕ) + (iξ(δ−η+α)−ηκ−α2)ψϕ¯
dϑ,∀ϕ∈Hp10,ω0(S), with a fixedα∈(0,√
µ1), or equivalently Z
S
∇Tψ· ∇Tϕ+ (ξ+iα)2ψϕ dϑ¯ = Z
S
(g(ϑ)· ∇Tϕ¯−i(ξ+iα)h(ϑ) ¯ϕ)dϑ,∀ϕ∈Hp10,ω0(S), wherehis given by
h(ϑ) = −iξ(δ−η+α)−ηκ−α2
iξ−α ψ(ϑ) + iξ+δ iξ−αg(ϑ).
Hence using Theorem 3.6 of [5] to this last problem, we find that k∇ψkp,ω+|ξ|kψkp,ω ≤C(kgkp,ω+khkp,ω),
with a positive constant C that may depend on the Ap-constant Ap(ω) but is independent of ξ.
From the form ofh, we find that
k∇ψkp,ω+|ξ|kψkp,ω≤C(kgkp,ω+kgkp,ω+kψkp,ω), since the ratios iξ(δ−η+α)−ηκ−α2
iξ−α and iξ−αiξ+δ are bounded for any ξ ∈R. This proves (2.3) for |ξ|
large. For |ξ| small, the previous estimate (compare with the estimate (3.11) of [5]) allows to use the arguments of the proof of Theorem 3.6 of [5] to deduce that (2.3) is also valid (using our assumption (2.2)).
At this stage for a fixed pair of real numbers (δ, η) fulfilling (2.2), for any ξ ∈ R and any (g, g)∈Xp,ω :=LpT ,ω(S)×Lpω(S), we set
(2.6) Mδ,η(ξ)(g, g) = (∇Tψ, iξψ),
whereψ∈Hp,ω1 (S)is the unique solution of (2.1). According to Theorem 2.1, the operatorMδ,η(ξ) is bounded fromXp,ω into itself. For T ∈ L(Xp,ω)denote by |||T||| its operator norm. Then the previous Theorem mainly allows to obtain the next result.
Corollary 2.2 Givenp∈(1,∞), fix a pair of real numbers(δ, η)fulfilling (2.2). ThenMδ,η(ξ)is Fréchet differentiable with respect toξ∈Rand there exists anAp-consistent constantc(depending on(δ, η)) such that
(2.7) |||Mδ,η(ξ)|||+|ξ||||M0δ,η(ξ)||| ≤c,∀ξ∈R.
Proof. The estimate of |||Mδ,η(ξ)||| is nothing else than (2.3). But it is easy to check that the Fréchet derivative ∂ξ∂Mδ,η(ξ)(g, g) = (∇Tψ0, iξψ0) + (0, iψ), whereψ∈Hp,ω1 (S)is solution of (2.1) andψ0∈Hp,ω1 (S)is solution of (compare with (2.1))
Z
S
∇Tψ0· ∇Tϕ − (iξ−η)(iξ+δ)ψ0ϕ¯ dϑ
= −i Z
S
(g(ϑ) + (2iξ+δ−η)ψϕ)¯ dϑ,∀ϕ∈Hp10,ω0(S).
Hence applying Theorem 2.1 to this problem, we find that k∇ψ0kp,ω+|ξ|kψ0kp,ω≤C(kψkp,ω+ 1
1 +|ξ|kgkp,ω),
with a positive constant C that may depend on the Ap-constant Ap(ω) but is independent of ξ.
By the estimate (2.3) satisfied byψ, we find that k∇ψ0kp,ω+|ξ|kψ0kp,ω ≤ C1
1 +|ξ|(kgkp,ω+kgkp,ω),
with a positive constantC1 that may depend on theAp-constant Ap(ω)but is independent of ξ, which yields the requested on|||M0δ,η(ξ)|||.
Remark 2.3 Notice that the statement of Corollary 2.2 remains valid for the operator Mfδ,η(ξ) defined by
Mfδ,η(ξ)(g, g) = (∇Tψ,(1 +iξ)ψ), for anyξ∈Rand any(g, g)∈Xp,ω.
To end up our preliminary results, we need an extrapolation property on Xp,ω and its con- sequence concerning R-boundedness, see Theorem 4.3 of [9] in Lpω(Ω) or Theorem 4.3 of [5] in Lpω(Ω)n for an open set ofRn,n∈N∗.
Theorem 2.4 Let1< p, s <∞,ω∈Ap. Furthermore, letT be a family ofL(Xp,ω)that satisfies (2.8) kT Fks,ν ≤CkFks,ν,∀T ∈ T, F ∈Xp,ω∩Xs,ν,
for everyν ∈As with a constant C that depends only on the As-constantAs(ν). Then T isR- bounded onL(Xp,ω), in the sense that there exists C >0 such that for any N ∈N,Tj ∈ T and Fj ∈Xp,ω, it holds
k
N
X
j=1
|TjFj|2
1 2
kp,ω≤Ck
N
X
j=1
|Fj|2
1 2
kp,ω.
Proof. First by using an atlas of S, we notice that the result is valid inLpω(S)4, consequently it suffices to use an extension argument from Xp,ω to Lpω(S)4. For that purpose, for T ∈ T, we introduce the operatorT˜ as follows: for any(g, g)∈Lpω(S)4, we denote by
T(g, g) :=˜ T(g−(g·θ)θ, g).
Note thatT˜is well-defined sinceg−(g·θ)θ belongs toLpT ,ω(S). Hence we find a familyTe ={T˜: T ∈ T }of elements ofL(Lpω(S)4). Furthermore, by our assumption (2.8), one has
kT˜(g, g)ks,ν ≤Ck(g−(g·θ)θ, g)ks,ν,
with C >0 from (2.8). As there exists a positive constant C1 depending only on s and S (and that we may suppose to be≥1) such that
kg−(g·θ)θks,ν ≤C1kgks,ν, we deduce that
kT(g, g)k˜ s,ν ≤CC1k(g, g)ks,ν.
HenceT˜ satisfies the statement of the Theorem inLpω(S)4, hence it isR-bounded.
Let us now show that this property implies a similar property forT. Indeed, taking anyN∈N, Tj ∈ T andFj∈Xp,ω, we notice that
TjFj = ˜TjFj, and therefore
k
N
X
j=1
|TjFj|2
1 2
kp,ω = k
N
X
j=1
|T˜jFj|2
1 2
kp,ω
≤ Ck
N
X
j=1
|Fj|2
1 2
kp,ω,
for someC >0due to theR-boundedness of the familyTe.
This Theorem and Corollary 2.2 directly implies the next result.
Theorem 2.5 Given p∈ (1,∞), fix a pair of real numbers (δ, η) fulfilling (2.2). Then the sets {Mδ,η(ξ) :ξ∈R}and{ξM0δ,η(ξ) :ξ∈R} areR-bounded onL(Xp,ω).
3 The weak Neumann problem in a cone
In the whole section letΓ be a three-dimensional cone with a sectionS with aC1,1 boundary, in the sense that
Γ ={x∈R3:ϑ= x
|x| ∈S}.
On such domain, we recall that the weighted Sobolev space of Kondrat’ev’s typeVαk,p(Ω)with k∈N,p∈(1,∞)andβ∈Ris defined by
Vβk,p(Γ) ={u∈L2loc(Γ) :kukVk,p
β (Γ)<∞}, where
kukp
Vβk,p(Γ)=X
|α|≤kkrβ+|α−kDαukpLp(Γ),
r(x) =|x|being the distance fromxto the origin, that is the corner point ofΓ. In particular, we writeLpβ(Γ) =Vβ0,p(Γ). Let us also recall that the singular functions of the Laplace operator with Neumann boundary conditions are in the form
rλk±ϕk, whereϕk is the eigenvector ofLNeuLB of eigenvalueµk and
(3.1) λk± =−1±p
1 + 4µ2k
2 .
In this section, we prove the next result.
Theorem 3.1 Let p∈(1,∞) andβ∈Rbe such that
(3.2) 1−3
p−β6=λk±,∀k∈N. Then for allf ∈Lpβ(Γ)3 there exists a unique solution u∈Vβ1,p(Γ) of (3.3)
Z
Γ
∇u· ∇¯v dx= Z
Γ
f · ∇¯v dx,∀v∈V−β1,q(Γ),
whereq >1is the conjugate ofp, i.e., 1p+1q = 1. Furthermore there existsC >0(independent of uandf) such that
(3.4) kukV1,p
β (Γ)≤CkfkLp
β(Γ)3,
Proof. Assume that such a solution uexists. We perform the Euler change of variablesr =et that transformΓinto the stripB=R×S and for anyv∈V−β1,q(Γ), set
U(t, ϑ) =eηtu(et, ϑ), V(t, ϑ) =eδtv(et, ϑ),
withη =−(1−3p−β)and δ= 1−η. By the arguments of [14, p. 193] (easily extended to the Lp-setting), we deduce equivalently thatU belongs toW1,p(B)and satisfies
Z
B
∇TU· ∇TV + (∂tU −ηU)(∂tV¯ −δV¯) dϑ (3.5)
= Z
B
hT(t, ϑ)· ∇TV¯ +hr(t, ϑ)(∂tV¯ −δV¯)
dϑ,∀V ∈W1,q(B),
where we have set
hr(t, ϑ) =e(2−δ)tf(et, ϑ)·ϑ, hT(t, ϑ) =e(2−δ)tf(et, ϑ)−hr(t, ϑ)ϑ, that respectively belong toLp(B)andLpT(B) :={h∈Lp(B)3:h·ϑ= 0}with
(3.6) khTkLp
T(B)+khrkLp(B)≤CkfkLp
β(Γ)3,
for some positive constantC. Hence it suffices to show that (3.5) has a unique solutionU ∈W1,p(B) for(hT, hr)∈LpT(B)×Lp(B).
Performing a formal Fourier transformF int, we see thatUˆ(ξ, ϑ) = (Ft→ξU(·, ϑ))(ξ)satisfies (3.7)
Z
S
∇TUˆ· ∇Tϕ+ (iξ−η)(−iξ−δ) ˆUϕ¯ dϑ=
Z
S
hˆT(ξ, ϑ)· ∇Tϕ¯−hˆr(ξ, ϑ)(iξ+δ) ¯ϕ dϑ,
for all suitable test-functions ϕ. Comparing with (2.1), we see that formally (∇TU , iξˆ Uˆ) = {Mδ,η(ξ)(ˆhT,ˆhr). To be more correct, reminding the definition (2.6), as our assumption (3.2) guar- antees that the pair(δ, η), defined above, satifies (2.2), by Theorem 2.5, the sets{Mδ,η(ξ) :ξ∈R} and{ξM0δ,η(ξ) :ξ ∈R} areR-bounded on Lp(S). Consequently by the vector-valued multiplier Theorem (see [29, Theorem 3.4]), we deduce thatF−1Mδ,η(·)F defines a bounded linear operator onLp(R;Xp,1)into itself. Consequently the vector-vaued function (u, v) = (F−1Mδ,ηF)(hT, hr) belongs toLp(R;LpT(S)×Lp(S)). By taking a sequence of functionsϕn ∈ D(R, Xp,1) that con- verges to(hT, hr), using Corollary 2.2, Remark 2.3 and passing to the limit the next results can be obtained. There existsU ∈W1,p(B)such that
u=∇TU, v=i∂tU, with
kUkW1,p(B)≤C(khTkLp
T(B)+khrkLp(B)),
for someC >0. Furthermore it will satisfy (3.5) sinceD(R;W1,q(S))is dense inW1,q(B). Finally as
kukV1,p
β (Γ)≤CkUkW1,p(B), for someC >0, with (3.6) we conclude that (3.4) holds.
Corollary 3.2 Letp∈(1,∞)andβ ∈Rsatisfy (3.2). Then anyf ∈Lpβ(Γ)3admits the Helmholtz decomposition
f =f0+∇u, withf0∈Lpβ(Γ)3 divergence free andu∈Vβ1,p(Γ) such that
(3.8) kukV1,p
β (Γ)+kf0kLp
β(Γ)3 ≤CkfkLp
β(Γ)3, for some positive constantC independent of f.
Proof. Givenf ∈ Lpβ(Γ)3, the previous Theorem furnishes u∈Vβ1,p(Γ) solution of (3.3) and we conclude by setting
f0=f − ∇u,
that is obviously divergence free by taking test-functionsv∈ D(Γ)in (3.3) and satisfies (3.8) owing to (3.4).
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