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L
rsolutions of elliptic equation in a complete
riemannian manifold.
Eric Amar
To cite this version:
Eric Amar. Lr solutions of elliptic equation in a complete riemannian manifold.. Journal of Geometric
The LIR Method.
L
r
solutions of elliptic equation in a
complete riemannian manifold.
Eric Amar
Contents
1 Introduction. 2
2 The Local Increasing Regularity Method (LIRM). 6
3 Application to elliptic PDE. 7
4 Case of compact manifold with a smooth boundary. 12
5 Relations with the local existence of solutions. 14
6 The non compact case. 16
7 Appendix. 22
7.1 Vitali covering. . . 22
7.2 Sobolev spaces. . . 23
Abstract
Let X be a complete metric space and Ω a domain in X. The Local Increasing Regularity Method (LIRM) allows to get from local a priori estimates on solutions u of a linear equation Du= ω global ones in Ω.
As an application we shall prove that if D is an elliptic linear differential operator of order m with C∞ coefficients operating on p-forms in a compact Riemannian manifold M without
boundary and ω∈ Lr
p(M )∩ (kerD∗)⊥,then there is a u∈ Wpm,r(M ) such that Du = ω on M.
Next we investigate the case of a compact manifold with boundary.
1
Introduction.
Let X be a complete metric space with a positive σ-finite measure µ. Let Ω be a relatively compact domain in X. We shall denote Ep(Ω) the set of Cp valued fonctions on Ω. This means that
ω ∈ Ep(X) ⇐⇒ ω(x) = (ω
1(x), ..., ωp(x)). We put a punctual norm on ω in Ep(Ω) the following
way: for any x ∈ Ω, |ω(x)|2 :=Pp
j=1|ωj(x)| 2
and we consider the Lebesgue space Lr
p(Ω), i.e. ω ∈ Lr p(Ω) ⇐⇒ kωk r Lr p(Ω) := R Ω|ω(x)| r dµ(x) <∞. The space L2
p(Ω) is a Hilbert space with the scalar product hω, ω′i :=
R Ω Pp j=1ωj(x)¯ω′j(x) dµ(x). We are interested in solutions of a linear equation Du = ω, where D = Dp is a linear operator
acting on Ep. This means that D is a matrix whose entries are linear operators on functions.
We shall suppose that there is a threshold s > 1 such that we have a global solution u on Ω of Du = ω with estimates Ls
p(Ω)→ Lsp(Ω). It may happen that we have a constrain:
∃K ⊂ Ls′
p(Ω) ::∀u, Du ∈ Lsp(Ω) ⇒ ∀h ∈ K, hDu, hi = 0 (Du ⊥ K),
where s′ is the conjugate exponent to s, 1
s′ := 1−1s.
If there is no constrain, we set K ={0}. So we make the threshold hypothesis: ∀ω ∈ Ls
p(Ω), ∃u ∈ Lsp(Ω) :: Du = ω, provided that ω⊥ K.
The aim is to find local conditions in order to have the existence of global solutions of Du = ω with estimates Lrp(Ω)→ Lrp(Ω) for r6= s.
In the study of solutions of classical partial differential equations, in particular elliptic ones, there are, at least, two important points:
(i) local existence of solutions with estimates; (ii) local a priori interior estimates.
• Associated to the first point (i) we introduced the Raising Steps Method (RSM) [4].
We have local existence of solutions with strict increase of regularity i.e. there is a δ > 0 such that for 1 < r < s with 1 t = 1 r − δ, we have ∀x ∈ ¯Ω, ∀ω ∈ Lr p(Ω), ∃B(x, R) :: ∃u ∈ Ltp(B(x, R/2)), Du = ω.
Then the RSM gives the existence of a global solution Dv = ω with essentially v ∈ Ltp(Ω) if ω ∈ Lr
p(Ω) provided that ω ⊥ K.
• Associated to the second point (ii) we introduce now another method, the Local Increasing Regularity Method (LIRM) to perform the same goal.
We ask for a strict increase in interior regularity i.e. there is a τ > 0 such that for r > s with 1
t = 1
r − τ and for any x ∈ X, there is a R > 0 such that
∀u ∈ Lr(B(x, R)), kukLt(B(x,R/2)) ≤ C(kDukLr(B(x,R))+kukLr(B(x,R))).
Then the LIRM will give us the existence of a global solution Dv = ω with essentially v ∈ Lt(Ω) if
ω ∈ Lr(Ω) provided that ω ⊥ K.
These two methods are based on complementary assumptions and give complementary results: the first one deals with r < s and the second one with r > s.
m with C∞ coefficients. There is a formal adjoint D∗ : Λp(M) → Λp(M) defined by the identity
hD∗f, gi = hf, Dgi. This implies the existence of a constrain: let K := kerD∗, we get:
∀h ∈ K, hDu, hi = hu, D∗hi = 0
hence, in order to find a u such that Du = ω, we need ω ⊥ K.
On (M, g) we can define Sobolev spaces Wk,r(Ω) (see [18]) and if M is compact these spaces are
in fact independent of the metric. Moreover the Sobolev embeddings are true in this case and a chart diffeomorphism makes a correspondence between Sobolev spaces in Rn and Sobolev spaces in
M.
We shall see that the conditions to apply the LIRM will be fulfilled and we shall prove:
Theorem 1.1. Let (M, g) be a C∞ smooth compact riemannian manifold without boundary. Let
D : Λp(M)→ Λp(M) be an elliptic linear differential operator of order m with C∞ coefficients. Let
ω ∈ Lr
p(M)∩(kerD∗)⊥ with r ≥ 2. Then there is a bounded linear operator S : Lrp(M)∩(kerD∗)⊥→
Wm,r
p (M) such that DS(ω) = ω on M. So, with u := Sω we get Du = ω and u∈ Wpm,r(M).
By duality we get the range r < 2 as we did in [3], using an avatar of the Serre duality [26]. The "Riemannian double" Γ := Γ(N) of N, obtained by gluing two copies of (a slight extension of) N along ∂N, is a compact Riemannian manifold without boundary. Moreover, by its very construction, it is always possible to assume that Γ contains an isometric copy N of the original domain N. See Guneysu and Pigola [17, Appendix B].
We shall need:
Definition 1.2. We shall say that D has the weak maximum property, WMPp, if, for any smooth D-harmonic p-form h in N, smooth up to the boundary ∂N, which is flat on ∂N, i.e. zero on ∂N with all its derivatives, then h is zero in N.
This definition has to be linked to the Definition [21, Introduction, p. 948]:
Definition 1.3. We shall say that an operator D has the Unique Continuation Property, UCP, if Du = 0 on Γ and u = 0 in an open set O 6= ∅ of Γ implies that u = 0 in Γ.
WMP is weaker than the UCP, because if D has the UCP and if h is flat on ∂N, then we can extend h by zero in Nc in Γ, which makes h still a D-harmonic form, and apply the UCP to get
that h is zero in N.
Theorem 1.4. Let N be a smooth compact riemannian manifold with smooth boundary ∂N. Let ω ∈ Lrp(N). There is a form u∈ Wm,r
p (N), such that Du = ω andkukWpm,r(N ) ≤ ckωkLrp(N ), provided that the operator D has the WMP for the D-harmonic p-forms.
In the special, but fundamental, case of the Hodge laplacian, these results are quite well known: see CB Morrey [22] for the basic L2 case, C. Scott [25] for the Lp case in a compact riemannian
manifold without boundary and G. Schwarz [24], for the Lp case in a compact riemannian manifold
with boundary. They all used a Gaffney type inequality which is not the case here. See also [4] and the references therein.
The general case of a linear elliptic operator of order m was done in the L2 case for instance in
The case Lr for a general linear elliptic operator of order m on p-forms in a riemannian compact
manifold with or without boundary seems new as for the results in the case of a complete non compact riemannian manifold.
We shall use the same ideas as we did in [5] to go from the compact case to the non compact one. First we have to define a ǫ, m-admissible ball centered at x ∈ M. Its radius R(x) must be small enough to make that ball like its euclidean image. Precisely:
Definition 1.5. Let (M, g) be a riemannian manifold and x ∈ M. We shall say that the geodesic ball B(x, R) is m, ǫ admissible if there is a chart ϕ : (x1, ..., xn) defined on it with
1) (1− ǫ)δij ≤ gij ≤ (1 + ǫ)δij in B(x, R) as bilinear forms,
2) X
|β|≤m−1
sup i,j=1,...,n, y∈Bx(R)
∂βgij(y) ≤ ǫ.
Definition 1.6. Let x∈ M, we set R′(x) = sup {R > 0 :: B(x, R) is ǫ admissible}. We shall say that Rǫ(x) := min (1, R′(x)) is the m, ǫ admissible radius at x.
Our admissible radius is bigger than the harmonic radius rH(1 + ǫ, m− 1, 0) defined in the book
[18, p. 4], because we do not require the coordinates be harmonic. I was strongly inspired by this book.
When comparing non compact M to the compact case treated above, we have four important issues:
(0) we have no longer, in general, a global solution u∈ L2
p(M) of Du = ω for a form ω ∈ L2p(M)
verifying ω ⊥ kerD∗. So we have to make this "threshold" hypothesis, which depends on the degree p of the form. In case the elliptic operator D is essentially self adjoint, this amounts to ask that its spectrum has a gap near 0: i.e. ∃δ > 0 such that D has no spectrum in ]0, δ[. We shall note this hypothesis (THL2p). Moreover, because L2p(M) is a Hilbert space, we have that the u∈ L2
p(M), Du = ω with the smallest norm is given linearly with respect to ω. This means that the
hypothesis (THL2p) gives a bounded linear operator S : L2p(M) → L2p(M) such that D(Sω) = ω provided that ω ⊥ kerD∗.
(1) The "ellipticity" constant may go to zero at infinity and we prevent this by asking that this constant is bounded below uniformly. To be sure that the constants in the local elliptic inequalities are uniform, we make also the hypothesis that the coefficients of D are in C1(M). This is the
hypothesis (UEAB) in Definition 6.1.
(i) The "admissible" radius may go to 0 at infinity, which is the case, for instance, if the canonical volume measure dvg of (M, g) is finite and M is not compact.
(ii) If dvg is not finite, which is the case, for instance, if the "admissible" radius is bounded below,
then p forms in Ltp(M) are generally not in Lrp(M) for r < t.
We address these two last problems by use of adapted weights on (M, g). These weights are relative to a Vitali type covering Cǫ of "admissible balls": the weights are positive functions which
vary slowly on the balls of the covering Cǫ.
Definition 1.7. We shall define the Sobolev exponents Sk(r) by 1 Sk(r) := 1 r − k n where n is the dimension of the manifold M.
Now we suppose we have an elliptic operator D with C1(M) smooth coefficients, of order m,
oper-ating on the p-forms in M. We set tl := Sml(2). We suppose that tl−1 ≤ r < tl, and, in order to
avoid triviality, we suppose also that tl−1 <∞. Now we can state the main result of this section:
Theorem 1.8. Under hypotheses (THL2p) and (UEAB), setting wl(x) = Rlmtl−1, and vr(x) :=
R(x)(tlr−1)+(l+2)mr, we have, provided that ω∈ L2(M)∩ Ltl−1(M, w
l), ω ⊥ kerD∗, tl−1 ≤ r < tl,
kukLr(M,vr)≤ max(kωkLtl−1(M,wl),kωkL2(M )) with u := Sω ⇒ Du = ω.
We also have with the same u:
kukWm,r(M,vr) ≤ c1kωkLtl(M,vr)+ c2max(kωkLtl−1(M,wl),kωkL2(M )).
Remark 1.9. If the admissible radius R(x) is uniformly bounded below, we can forget the weights and we get
kukLr(M ) ≤ max(kωkLtl−1(M ),kωkL2(M )).
kukWm,r(M ) ≤ c1kωkLtl(M )+ c2max(kωkLtl−1(M ),kωkL2(M )). In the spirit of [5], we introduce a weight wt such that, with t > 2,
Z
M
wt(x)
t
2−tdv(x) <∞, then we get the easy to read, but less precise, corollaries:
Corollary 1.10. Under hypotheses (THL2p) and (UEAB), with vr(x) := R(x)(
r
tl−1)+(l+2)mr and for any weight wt(x)≥ 1 such that
Z M wt(x) t 2−tdv(x) <∞, with t := t l ≥ r > tl−1, we have: kukWm,r(M,vr) .kωkLt(M,wt/2 t )
provided that ω ⊥ kerD∗, with u := Sω⇒ Du = ω.
Corollary 1.11. Let ∀j ∈ N, 1 tj = 1 2 − jm n . With r ≥ 2 and vr(x) := R(x) (r tl−1)+(l+2)mr, we have, provided that ω ∈ Lt(M, wtt/2), with t := tl−1≤ r,
kukLr(M,vr).kωkLt(M,wt/2 t )
provided that ω ⊥ kerD∗, with u := Sω⇒ Du = ω.
An advantage of this method is that it separates cleanly the geometry and the analysis.
The geometry controls the behavior of the admissible radius R(x) as a function of x in M. For instance by theorem 1.3 in Hebey [18], we have that the harmonic radius rH(1 + ǫ, m, 0) is bounded
below if the Ricci curvature Rc verifies ∀j ≤ m, ∇jRc
∞ < ∞ and the injectivity radius is
bounded below. This implies that the m, ǫ admissible radius R(x) is also bounded below.
I am indebted to A. Bachelot, B. Helffer, G. Métivier and J. Sjöstrand for clearing strongly my knowledge on the local existence of solutions to system of elliptic equations needed in the study of elliptic equations acting on p-forms.
2
The Local Increasing Regularity Method (LIRM).
We shall deal with the following situation: we have a complete metric space X admitting a positive σ-finite measure µ. We shall denote Ep(Ω) the set of Cp valued fonctions on Ω as in the
introduction. We defined the Lebesgue spaces Lr
p(Ω) for any measurable set Ω in X.
We shall make the following hypotheses.
Let Ω be a relatively compact connected domain in X. Let Bx := B(x, Rx) be a ball in X and
B1
x := B(x, Rx/2). There is a τ > 0 with
1 t =
1
r − τ, and a positive constant cl such that: (i) Local Increasing Regularity (LIR), we have:
∀x ∈ ¯Ω, ∃Rx > 0 ::∀r ≥ s, ∀u ∈ Lrp(Bx), kukLt
p(Bx1) ≤ cl(kDukLrp(Bx)+kukLrp(Bx)). It may append, in the case X is a manifold, that we have a better regularity locally:
(i’) Local Increasing Regularity (LIR) with Sobolev estimates: there is α > 0 such that ∀x ∈ ¯Ω, ∃Rx > 0 ::∀r ≥ s, ∀u ∈ Lrp(Bx), kukWpα,r(Bx1) ≤ cl(kDukLrp(Bx)+kukLrp(Bx)). (ii) Global resolvability: let K be a subspace of Ls′
p(Ω), s′ the conjugate exponent of s. In case
with no constrain, we set K = {0}. We can solve Dw = ω globally in Ω with Ls− Ls estimates, i.e.
∃cg > 0, ∃w s.t. Dw = ω in Ω and kwkLs
p(Ω) ≤ cgkωkLsp(Ω), provided that ω⊥ K.
It may append, in the case X is a manifold, that we have a better regularity for the global existence:
(ii’) Sobolev regularity: We can solve Dw = ω globally in Ω with Ls− Wα,s estimates, i.e.
∃cg > 0, ∃w s.t. Dw = ω in Ω and kwkWα,s
p (Ω) ≤ cgkωkLsp(Ω), provided that ω ⊥ K. Then we have:
Theorem 2.1. Under the assumptions (i), (ii) above, there is a positive constant cf such that
for r ≥ s, if ω ∈ Lr
p(Ω), ω ⊥ K there is a u ∈ Ltp(Ω) with
1 t =
1
r − τ, such that Du = ω and kukLt
p(Ω) ≤ cfkωkLrp(Ω).
If moreover we have (i’) and (ii’) and the manifold X admits the Sobolev embedding theorems, then u∈ Wα,r
p (Ω) with control of the norm.
Proof. Let ω ∈ Lr
p(Ω), r > s. Because Ω is relatively compact and µ is σ-finite, we have that ω ∈ Lsp(Ω).
The global resolvability, condition (ii), gives that there is a u∈ Ls
p(Ω) such that Du = ω, provided
that ω ⊥ K.
The LIR, condition (i), gives that, for any x ∈ ¯Ω there is a ball B := B(x, R) and a smaller ball B1 := B(x, R/2) such that, with 1
t1
= 1
s − τ (we often forget the subscript p for simplicity): kukLt1(B1) ≤ C(kDukLs(B)+kukLs(B)) = C(kωkLs(B)+kukLs(B))≤ C(kωkLr(B)+kukLs(B)), because kωkLs(B) .kωkLr(B), since r≥ s and ¯Ω is compact.
Then applying again the LIR we get, with the smaller ball B2 := B(x, R/4) and with t
kukLt2(B2) ≤ C(kωkLt1(B1)+kukLt1(B1)) . (kωkLr(B)+kukLs(B)). • If t1 ≥ r ⇒ t2 = r, and kukLr(B1) .(kDukLr(B)+kukLs(B)) and with
1 t =
1 r − τ, kukLt(B2) .(kωkLr(B1)+kukLr(B1)) . (kωkLr(B)+kukLs(B)).
It remains to cover ¯Ω by a finite set of balls B2 := B(x, R
x/4) to be done, because
P
B2kukLs(B).kukLs(Ω) and kukLs(Ω) .kωkLs(Ω) by the threshold hypothesis. • If t1 < r, we still have:
kukLt2(B2) .(kωkLr(B1)+kukLt1(B1)).
Then applying again the LIR we get, with the smaller ball B3 := B(x, R/8) and with t
3 := min(r, t2),
kukLt3(B3) .(kωkLr(B2)+kukLt2(B2)) . (kωkLr(B1)+kukLt1(B1)) . (kωkLr(B)+kukLs(B)). Hence if t2 ≥ r we are done as above, if not we repeat the process. Because t1k = 1s−kτ after a finite
number k ≤ 1 + 1
τ(r−s2s ) of steps we have tk ≥ r and we get, with B
k := B(x, R/2k) and another
constant C, kukLtk(Bk) ≤ C(kωkLr(B)+kukLs(B)).
It remains to cover ¯Ω with a finite number of balls Bk(x) to prove the first part of the theorem.
For the second part, the global resolvability, condition (ii), gives that there is a global solution u ∈ Ls(Ω) such that Du = ω in Ω with kuk
Ls(Ω) .kωkLs(Ω). Now if we have the LIR with Sobolev estimates, condition (i’), then
∀x ∈ ¯Ω, ∃Rx > 0 ::∀r ≥ s, ∀v ∈ Lr(B(x, Rx)), kvkWα,r(B1
x) ≤ C(kDvkLr(Bx)+kvkLr(Bx)). So, because r ≥ s, and ¯Ω is compact, ω ∈ Ls(Ω) and we get
kukWα,s(B1
x) .(kDukLs(Bx)+kukLs(Bx)) . (kωkLr(Bx)+kukLs(Bx)). The Sobolev embedding theorems, true by assumption here, give kukLτ(B1
x) ≤ ckukWα,s(B1x) with 1 τ = 1 s− α n.
So applying again the LIR condition in a ball B2
x relatively compact in B1x, we get, with t1 :=
min(τ, r),
kukWα,t1(B2
x) .(kωkLt1(Bx)+kukLt1(Bx1)) . (kωkLr(Bx)+kukLs(Bx)).
Now we proceed as above. If τ ≥ r ⇒ t1 = r, then we apply again the LIR condition to a smaller
ball B3
x we get
kukWα,r(B3
x) .(kωkLr(Bx)+kukLr(Bx2)) . (kωkLr(Bx)+kukLs(Bx)). and we are done by covering ¯Ω by a finite set of balls B3 as above.
If τ < r, then we iterate the process as in the previous part, adding the use of the Sobolev embedding theorem to increase the exponent, up to the moment we reach r. Remark 2.2. We notice that in fact the solution u in Theorem 2.1 is the same as the one given by condition (ii). It is a case of "self improvement" of estimates.
3
Application to elliptic PDE.
Let (M, g) be aC∞smooth connected compact riemannian manifold without boundary. We shall
denote Λp(M) the set of C∞ p forms on M. We can define punctually, for ω, ϕ ∈ Λp(M), a scalar
product (ω, ϕ)(x) hence a modulus: for x ∈ M, |ω| (x) := p(ω, ω)(x). By use of the canonical volume dvg on M we get a scalar product:
hω, ϕi := Z
M
(ω, ϕ)(x)dvg(x),
for p forms in L2
kωk2L2(M ) := Z
M|ω| 2
(x)dvg(x) < ∞.
The same way we define the spaces Lr p(M).
Let D : Λp(M) → Λp(M) be a linear differential operator of order m withC∞ coefficients. There
is a formal adjoint D∗ : Λp(M)→ Λp(M) defined by the identity hD∗f, gi = hf, Dgi.
We suppose that D is elliptic in the sense of Warner [28, Definition 6.28, p. 240]. Then for s = 2, Warner [28, Exercice 21, p. 257] (see also Donaldson [12, Theorem 4, p. 16]) we have:
Theorem 3.1. Let D be an operator of order m on Λp(M) in the connected compact riemannian
manifold M without boundary. Suppose that D is elliptic and with C∞ smooth coefficients.
1. In L2
p(M), kerD, kerD∗ are finite dimensional vector spaces.
2. We can solve the equation Du = ω in L2p(M) if and only if ω is orthogonal to kerD∗.
Moreover, because L2p(M) is a Hilbert space, we have that there is a bounded linear operator S : L2p(M) → L2
p(M) such that D(Sω) = ω provided that ω ⊥ kerD∗.
On the other hand we have local interior regularity by Hörmander[ [20], Theorem 17.1.3, p. 6] in the case of 0 forms, i.e. functions. We quote it in the weakened form we need:
Theorem 3.2. (LIR) Let D be an operator of order m on Λp(M) in the complete riemannian
manifold M. Suppose that D is elliptic and with C∞ smooth coefficients. Then, for any x∈ M there
is a ball Bx := B(x, R) and a smaller ball Bx′ relatively compact in Bx, such that:
kukWm,r(B′
x)≤ C(kDukLr(Bx)+kukLr(Bx)).
For the case of p forms, we need to use Agmon, Douglis and Nirenberg[ [2], Theorem 10.3]:
Theorem 3.3. Positive constants r1 and K1 exist such that, if r ≤ r1 and the kujktj, j = 1, ..., N,
are finite, then kujkl+tj also is finite for j = 1, ..., N, and
kujkl+tj ≤ K1 X j kFjkl−sj + X j kujk0 ! .
The constants r1, K1 depend on n, N, t′, A, b, p, k, and l and also on the modulus of continuity of the
leading coefficients in the lij.
From this theorem we get quite easily what we want (in the case r = 2 and in its global version, F.W. Warner [28, Theorem 6.29, p. 240] quotes it as Fundamental Inequality):
Theorem 3.4. (LIR) Let D be an operator of order m on Λp(M) in the complete riemannian
manifold M. Suppose that D is elliptic and with C1(M) smooth coefficients. Then, for any x∈ M
there is a ball Bx := B(x, R) and a smaller ball Bx′ relatively compact in Bx, such that:
kukWm,r(B′
x)≤ c1kDukLr(Bx)+ c2R
−mkuk
Lr(Bx)).
Moreover the constants are independent of the radius R of the ball Bx.
Proof.
Let x∈ M we choose a chart (V, ϕ := (y1, ..., yn)) so that gij(x) = δij and ϕ(V ) = B where B = Be
We denote by Dϕ the operator D read in the map (V, ϕ). This is still an elliptic system operating
on p forms in Bein Rn. Let χ∈ D(Be) such that χ = 1 in Be1 ⋐Be. Let u be a p form in Lr(ϕ−1(Be))
such that Du is also in Lr(ϕ−1(Be)). Denote by uϕ the p form u read in (V, ϕ). We can apply the
Agmon, Douglis and Nirenberg Theorem 3.3 to χuϕ and we get, with the constant K independent
of the radius Re of Be,
kχuϕkWm,r(Be) ≤ K(kD(χuϕ)kLr(Be)+ R−me kχuϕkLr(Be)). (3.1) We have that D(χuϕ) = χD(uϕ) + uϕDχ + ∆ϕ, with ∆ϕ := D(χuϕ)− χD(uϕ)− uϕDχ. The point
is that ∆ϕ contains only derivatives of the jth component of uϕ of order strictly less than in the jth
component of uϕ in Du. So we have
k∆ϕkLr(Be) ≤ k∂χk∞kχuϕkWm−1,r(Be) ≤ R−1e kχuϕkWm−1,r(Be).
We can use the "Peter-Paul" inequality [16, Theorem 7.28, p. 173] (see also [28, Theorem 6.18, (g) p. 232] for the case r = 2.)
∀ǫ > 0, ∃Cǫ > 0 ::kχuϕkWm−1,r(Be)≤ ǫkχuϕkWm,r(Be)+ Cǫ−m+1kχuϕkLr(Be). We choose ǫ = Reη and we get
Re−1kχuϕkWm−1,r(Be) ≤ ηkχuϕkWm,r(Be)+ Cη−m+1R−me kχuϕkLr(Be). Putting this in (3.1) we get
kχuϕkWm,r(Be)≤ K(kχDuϕkLr(Be)+ ηkχuϕkWm,r(Be)
+Cη−m+1R−me kχuϕkLr(Be)+kuϕDχkLr(Be)). But againkDχk∞≤ R−m
e so, choosing η small enough to get ηK ≤ 1/2, we have with new constants
still independent of Re: 1
2kχuϕkWm,r(Be)≤ c1kχDuϕkLr(Be)+ c2R−me kχuϕkLr(Be). Now χ = 1 in B1
e and χ≤ 1 gives, changing the constants suitably:
kuϕkWm,r(B1
e)≤ c1kDuϕkLr(Be)+ c2R
−m
e kuϕkLr(Be).
It remains to go back to the manifold M to end the proof.
We deduce the local elliptic inequalities:
Corollary 3.5. Let D be an operator of order m on Λp(M) in the complete riemannian manifold
M. Suppose that D is elliptic and with C1(M) smooth coefficients. Then, for any x ∈ M there is
a ball Bx := B(x, R) and a smaller ball Bx1 relatively compact in Bx, such that, ∀k ∈ N, with D in
Ck+1(M) here, we get:
kukWm+k,r(B1
x) ≤ Ck(c1kDukWk,r(Bx)+ c2R
−m−kkuk
Lr(Bx)). Moreover the constants are independent of the radius R of the ball Bx.
Proof.
We apply Theorem 3.4 to the p form ∂αu for |α| ≤ k and in B2 := B(x, R
x/4), B1 := B(x, Rx/2), k∂αuk Wm,r(B2 x)≤ c1kD(∂ αu)k Lr(B1 x)+ c2R −mk∂αuk Lr(B1 x)).
We use the fact that the coefficients of D are Ck+1 smooth to be able to compute kDuk
W|α|,r(B1 x): k∂αukWm,r(B2 x)≤ Ck(c1kDukW|α|,r(Bx1)+ c2R −mkuk W|α|,r(B1 x)). (3.2)
For k≤ m we get, using that kukW|α|,r(B1
x) ≤ kukWm,r(Bx1) ≤ K(c1kDukLr(Bx)+ c2R
−mkuk
Putting it in (3.2), we deduce: k∂αuk
Wm,r(B2
x)≤ Ck(kDu)kW|α|,r(Bx1)+ K(kDukLr(Bx)+kukLr(Bx)). But in B, kDukLr(Bx) ≤ kDu)kW|α|,r(Bx) so we get, changing the constant Ck,
k∂αuk
Wm,r(B2
x)≤ Ck(kDu)kW|α|,r(Bx)+kukLr(Bx)). This implies clearly:
kukWm+k,r(B2
x) ≤ Ck(kDukWk,r(Bx)+kukLr(Bx)).
For k > m, we iterate the process.
Remark 3.6. We stress here the dependence in R because we shall need it to study the case of non compact riemannian manifolds.
Now we can prove
Theorem 3.7. Let (M, g) be a C∞ smooth compact riemannian manifold without boundary. Let
D : Λp(M)→ Λp(M) be an elliptic linear differential operator of order m with C∞(M) coefficients.
Let ω ∈ Lr
p(M)∩ (kerD∗)⊥ with r ≥ 2. Then there is a u ∈ Wpm,r(M) such that Du = ω on M.
Moreover u is given linearly w.r.t. to ω. Proof.
Because M is compact, we have ω ∈ L2
p(M). Theorem3.1gives us the Global Resolvability, condition
(ii), with the threshold s = 2, and with K := kerD∗, i.e. provided that ω⊥ K:
u := Sω ∈ L2
p(M) :: Du = ω, kuk2 ≤ Ckωk2.
The Theorem 3.4 of Agmon, Douglis and Nirenberg gives us the Local Interior Regularity with the Sobolev estimates for α = m.
So we can apply Theorem 2.1 and we use Remark2.2 to have that u = Sω so u is given linearly
w.r.t. to ω.
By duality we get the range r < 2. For this case we shall proceed as we did in [3], using an avatar of the Serre duality [26].
Let g ∈ Lr′
p(M) ∩ kerD⊥, because D∗ has the same elliptic properties than D, we can solve
D∗v = g, with r′ < 2 and r′ conjugate to r the following way.
We know by the previous part that:
∀ω ∈ Lrp(M)∩ (kerD∗)⊥, ∃u ∈ Lrp(M), Du = ω. (3.3)
Consider the linear form ∀ω ∈ Lr
p(M), L(ω) := hu, gi,
where u is a solution of (3.3); in order for L(ω) to be well defined, we need that if u′ is another
solution of Du′ = ω, thenhu − u′, gi = 0; hence we need that g must be "orthogonal" to p forms ϕ
such that Dϕ = 0 which is precisely our assumption.
Hence we have thatL(f) is well defined and linear; moreover |L(f)| ≤ kukLr(M )kgkLr′(M ) ≤ ckωkLr(M )kgkLr′(M ). So this linear form is continuous on ω ∈ Lr
p(M)∩ (kerD∗)⊥. By the Hahn Banach Theorem there is
a form v ∈ Lr′
p(M) such that:
∀ω ∈ Lr
But ω = Du, so we have hω, vi = hDu, vi = hu, D∗vi = hu, gi, for any u ∈ C∞
p (M). Hence we solved
D∗v = g in the sense of distributions with v ∈ Lr′
p(M). So we proved:
Theorem 3.8. For any r, 1 < r ≤ 2, if g ∈ Lr
p(M)∩ (kerD)⊥ there is a v ∈ Lrp(M) such that
D∗v = g, kvk Lr
p(M ) ≤ ckgkLrp(M ). Moreover the solution is in Wm,r(M).
It remains to prove the "moreover" and for this we use the LIR Theorem 3.4: for any x ∈ M there is a ball B := B(x, R) and a smaller ball B1 := B(x, R/2) such that:
kukWm,r(B1)≤ C(kDukLr(B)+kukLr(B)). We cover M with a finite number of balls B1
x to prove the theorem.
Set H2
p := kerD∗∩ L2p(M).
Because D and D∗ have the same elliptic properties, we finally end with:
Theorem 3.9. Let (M, g) be a C∞ smooth compact riemannian manifold without boundary. Let
D : Λp(M) → Λp(M) be an elliptic linear differential operator of order m with C1 coefficients. Let
ω ∈ (H2
p)⊥ with r > 1. Then there is a u∈ Lrp(M) such that Du = ω on M. Moreover the solution
is in Wm,r(M).
Now we make the hypothesis that D has C∞ smooth coefficients. The Theorem 3.1 of Warner
gives, on a compact manifold M without boundary, that dimRH2p <∞.
Lemma 3.10. We have H2p ⊂ C∞(M). Proof.
This is classical. Take x∈ M, h ∈ H2
p. The fundamental inequalities, Corollary 3.5, gives, applied
to D∗, that there is a ball B := B(x, R) and a smaller ball B1 := B(x, R/2) such that:
∀k ∈ N, khkWm+k,2(B1) ≤ Ck(kD∗hkWk,2(B)+khkL2(B))≤ CkkhkL2(B).
The Sobolev embedding theorems, valid in a compact smooth manifold, give that, for any l ∈ N, h∈ Cl(B1). Covering M by a finite number of balls B1
x we get that:
∀l ∈ N, h ∈ Cl(M) ⇒ h ∈ C∞(M).
Lemma 3.11. There is a linear projection from Lr
p(M) to H2p. Proof. We set ∀v ∈ Lrp(M), H(v) := N X j=1 hv, ejiej
where {ej}j=1,...,N is an orthonormal basis for H2p. This is meaningful because v ∈ Lr(M) can be
integrated against ej ∈ H2p ⊂ C∞(M). Moreover we have v− H(v) ∈ Lrp(M)∩ H⊥p in the sense that
∀h ∈ H2
p, hv − H(v), hi = 0; it suffices to test on h := ek. We get
hv − H(v), eki = hv, eki − * N X j=1 hv, ejiej, ek + =hv, eki − hv, eki = 0.
Proposition 3.12. We have a direct decomposition: Lr
p(M) =H2p⊕ ImD(Wp2,r(M)).
Proof. Let v ∈ Lr
p(M). Set h := H(v) ∈ H2p, and ω := v − h. We have that ∀k ∈ H2p, hω, ki =
hv − H(v), ki = 0. Hence we can solve Du = ω with u ∈ W2,r
p (M)∩ L2p(M). So we get v = h + Du
which means: Lrp(M) =H2
p ⊕ ImD(Wp2,r(M)).
We have a direct decomposition because if ω ∈ H2
p∩ ImD(Wp2,r(M)), then ω∈ C∞(M) and
ω = Du⇒ ∀k ∈ H2
p, ω⊥ k,
so choosing k = ω ∈ H2
p we get hω, ωi = 0 hence ω = 0. The proof is complete.
In the special case where D is the Hodge Laplacian, we already seen [4] that we recover this way the strong Lr Hodge decomposition without using Gaffney’s inequalities.
4
Case of compact manifold with a smooth boundary.
Let N be a C∞ smooth connected riemannian manifold compact with a C∞ smooth boundary
∂N. We want to show how the results in case of a compact boundary-less manifold apply to this case.
First we know that a neighborhood V of ∂N in N can be seen as ∂N×[0, δ] by [23, Theorem 5.9 p. 56] or by [11, Théorème (28) p. 1-21]. This allows us to "extend" slightly N :
we have N = (N\V ) ∪ V ≃ (N\V ) ∪ (∂N×[0, δ]). So we set M := (N\V ) ∪ (∂N×[0, δ + ǫ]). Then M can be seen as a riemannian manifold with boundary ∂M ≃ ∂N and such that ¯N ⊂ M.
Now a classical way to get rid of a "annoying boundary" of a manifold is to use its "double". For instance: Duff [14], Hörmander [20, p. 257]. Here we copy the following construction from [17, Appendix B].
The "Riemannian double" Γ := Γ(M) of M, obtained by gluing two copies, M and M2, of M along
∂M, is a compact Riemannian manifold without boundary. Moreover, by its very construction, it is always possible to assume that Γ contains an isometric copy of the original manifold N. We shall also write N for its isometric copy to ease notation.
We extend the operator D to M smoothly by extending smoothly its coefficients, and because D is strictly elliptic, choosing ǫ small enough, we get that the extension is still an elliptic operator on M. Then we take a C∞ function χ with compact support on M ⊂ Γ such that: 0 ≤ χ ≤ 1; χ ≡ 1
on N; and we consider ˜D := χD + (1− χ)D2 where D2 is the operator D on the copy M2 of M.
Then ˜D≡ D on N and is elliptic on Γ. And we make the following definition:
Definition 4.1. We shall say that D has the weak maximum property, WMP, if, for any smooth D-harmonic form h in N, C∞ smooth up to the boundary ∂N, which is flat on ∂N, i.e. zero with
all its derivatives, then h is zero in N.
Definition 4.2. We shall say that an operator D has the Unique Continuation Property, UCP, if Du = 0 on Γ and u = 0 in an open set O 6= ∅ of Γ implies that u = 0 in Γ.
WMP is weaker than the UCP, because if D has the UCP and if h is flat on ∂N, then we can extend h by zero in Nc in Γ, which makes h still a D-harmonic form, and apply the UCP to get
that h is zero in N.
Of course if there is a maximum principle for D then WMP is true. This is the case for smoothly bounded open sets in Rn by a Theorem of S. Agmon [1] for 0 forms and by [2, Theorem 4.2, p. 59]
for p-forms.
Because this maximum principle is not local, I don’t know what happen on a compact riemannian manifold with smooth boundary in general.
The Hodge laplacian in a riemannian manifold has the UCP by a difficult result by N. Aronszajn, A. Krzywicki and J. Szarski [7].
The UCP is also true for all second order elliptic operator [6], with C∞ coefficients, acting on
functions, i.e. 0 forms and, by Hörmander [19], for instance in the case where D = P (x, ∂) is a linear, elliptic partial differential operator of order m with Lipschitz continuous coefficients and for homogeneous P with simple (complex) characteristics. Also in the same paper there are extended results of those of Calderon [10]. See also [20, Theorem 17.2.1, p. 10].
The main lemma of this section is: Lemma 4.3. Let ω ∈ Lr
p(N), then we can extend it to ω′ ∈ Lrp(Γ) such that: ∀h ∈ Hp(Γ), hω′, hiΓ =
0 provided that the operator D has the WMP for the D-harmonic p-forms. Proof.
Recall thatHp(Γ) := kerD∗∩L2p(Γ), it is of finite dimension KpandHp(Γ)⊂ C∞(Γ) by Lemma3.10.
Make an orthonormal basis {e1, ..., eKp} of Hp(Γ) with respect to L
2
p(Γ), by the Gram-Schmidt
procedure. We get hej, ekiΓ :=
R
Γejekdv = δjk.
Set λj :=hω1N, eji = hω, ej1Ni, j = 1, ..., Kp, which makes sense since ej ∈ C∞(Γ)⇒ ej ∈ L∞(Γ),
because Γ is compact.
We shall see that the system {ek1Γ\N}k=1,...,Kp is a free one. Suppose this is not the case, then it will exist γ1, ..., γKp, not all zero, such that
PKp
k=1γkek1Γ\N = 0 in Γ\N. But the function h := Kp X
k=1
γkek
is in Hp(Γ) so if h is zero in Γ\N which is non void, hence h is flat on ∂N, then h ≡ 0 in Γ by
UCP or by the WMP. This is not possible because the ek make a basis for Hp(Γ). So the system
{ek1Γ\N}k=1,...,Kp is a free one.
We set γjk:=ek1Γ\N, ej1Γ\N hence we have that det{γjk} 6= 0. So we can solve the linear system
to get {µk} such that
∀j = 1, ..., Kp, Kp X k=1 µkek1Γ\N, ej = λj. (4.4) We put ω′′:=PKp
j=1µjej1Γ\N and ω′ := ω1N − ω′′1Γ\N = ω − ω′′. From (4.4) we get
∀j = 1, ..., Kp, hω′, ejiΓ =hω, eji − hω′′, eji = λj − Kp X
k=1
So the p-form ω′ is orthogonal to H
p. Moreover ω|N′ = ω and clearly ω′′ ∈ Lrp(Γ) being a finite
combination of ej1Γ\N, so ω′ ∈ Lrp(Γ) because ω itself is in Lrp(Γ). The proof is complete.
Now let ω ∈ Lr(N) and see N as a subset of Γ; then extend ω as ω′ to Γ by Lemma 4.3.
By the results on the compact manifold Γ, because ω′ ⊥ H
p(Γ), we get that there exists u′ ∈
Wm,r
p (Γ), u′ ⊥ Hp(Γ), such that Du′ = ω′; hence if u is the restriction of u′ to N we get u ∈
Wm,r
p (N), Du = ω in N.
Hence we proved
Theorem 4.4. Let N be a smooth compact riemannian manifold with smooth boundary ∂N. Let ω ∈ Lrp(N). There is a form u∈ Wm,r
p (N), such that Du = ω andkukWpm,r(N ) ≤ ckωkLrp(N ), provided that the operator D has the WMP for the D-harmonic p-forms.
Remark 4.5. The hope was great that the WMP condition be also necessary, but this is not the case as the Theorem 5.2 shows.
5
Relations with the local existence of solutions.
Let (M, g) be a C∞ smooth compact riemannian manifold without boundary. We shall denote
Λp(M) the set of C∞ p forms on M.
Let D : Λp(M)→ Λp(M) be a linear differential operator of order m with C∞ coefficients.
As above we suppose that D is elliptic in the sense of Warner [28, Definition 6.28, p. 240]. Let x∈ M and a ball B := B(x, R). We have ω ∈ L2
p(B) and we want to solve Du = ω. For this
we shall extend ω as ω′ ∈ L2
p(M) in the whole of M with ω′ ⊥ Hp(M) := kerD∗ in order to apply
Theorem 3.1.
Consider ω := ω1B the trivial extension of ω to M. We have, with Ph the orthogonal projection
on Hp(M), h := Phω. Take an orthonormal basis {e1, ..., eN} of Hp(M), then we have
h := PN
j=1hjej.
If h = 0, we set ω′ = ω and we are done. If not let R be small enough to have
ke11Bk ≤ 4√1N, ..., keN1Bk ≤ 4√1N.
This is possible because the ej arC∞(M) so if B is small enough we have kej1Bk ≤
1
4√N, and we have a finite number of such conditions.
We set ω1 := 1BcPN j=1hjej. Then kω1k2 := R Bc PN j=1hjej 2 dv≤R M PN j=1hjej 2 dv≤khk2. And kh − ω1k ≤PNj=1|hj| k1Bejk ≤ 14PNj=1|hj| ≤ √ Nkhk4√1 N = 1 4khk.
Hence, because Ph has norm one,
kh − Phω1k = kPhh− Phω1k ≤ kh − ω1k ≤ 14khk.
Now we set h1 := h− Phω1. Then kh1k ≤ 14khk and we have h1 := N
X
j=1
h1jej. So we set ω2 :=
1BcPNj=1h1jej. We have the same way: kω2k ≤ kh1k ≤ 1
4khk and kh1− Phω2k ≤
1
So at the step k we get khk− Phωk+1k ≤ 1 4khkk ≤ 1 4kkhk and kωk+1k ≤ 1 4kkhk. Now we set ω′′ :=P∞
j=1ωj. We get that the series converges in norm L2(M) and Phω′′= h.
So setting ω′ := ω− ω′′, we get that ω′ = ω on B and P
h(ω′) = 0, which means that ω′ ⊥ Hp(M).
So we can apply Theorem 3.1 to get Du′ = ω′ in L2
p(M) because ω′ ⊥ Hp. We set u := u′|B in B
to have Du = ω in B. So we proved:
Theorem 5.1. Let x in M. There is a R0(x) > 0 such that for any R ≤ R0 if ω ∈ L2p(B) with
B := B(x, R) there is a u∈ L2p(B) such that Du = ω and kukL2
p(B) .kωkL2p(B). To get the Lrp(B) case for r > 2, we proceed as in the proof of Theorem 2.1.
Theorem 5.2. Under the assumptions above, for any x∈ M, there is a positive constant cf such
that for r ≥ 2, if ω ∈ Lr(B), there is a u ∈ Lt(B1) with 1
t = 1
r − τ, such that Du = ω and kukLt(B1) ≤ cfkωkLr(B).
Moreover we have u ∈ Wm,r
p (B1) with control of the norm.
Proof. Let ω ∈ Lr
p(B), r > 2. Because B is relatively compact and dv is σ-finite, we have that ω ∈ L2p(B).
Theorem 5.1 gives that there is a u∈ L2
p(B) such that Du = ω.
Diminishing R if necessary, the LIR, condition (i), gives that there is a ball B := B(x, R) and a smaller ball B1 := B(x, R/2) such that, with 1
t1
= 1 2− τ,:
kukLt1(B1).(kDukL2(B)+kukL2(B)) = (kωkL2(B)+kukL2(B)) .kωkLr(B), (5.5) because kωkL2(B) .kωkLr(B), since r≥ 2 and ¯B is compact and kukL2
p(B).kωkL2p(B). Then applying again the LIR we get, with the smaller ball B2 := B(x, R/4) and with t
2 := min(r, t1),
kukLt2(B2) .(kωkLt1(B1)+kukLt1(B1)) .kωkLr(B), by (5.5).
• If t1 ≥ r ⇒ t2 = r, and kukLr(B1) .(kDukLr(B)+kukL2(B)) .kωkLr(B) and with 1 t = 1 r − τ, kukLt(B2) .kωkLr(B1)+kukLr(B1)) .kωkLr(B). We are done. • If t1 < r, we still have: kukLt2(B2) ≤ C(kωkLr(B1)+kukLt1(B1)).
Then applying again the LIR we get, with the smaller ball B3 := B(x, R/8) and with t
3 := min(r, t2),
kukLt3(B3) ≤ C(kωkLr(B2)+kukLt2(B2)).
Hence if t2 ≥ r we are done as above, if not we repeat the process. Because t1k = 12 − kτ after
a finite number k ≤ 1 + 1
τ(r−24 ) of steps we have tk ≥ r and we get, with B
k := B(x, R/2k) and
kukLtk(Bk).kωkLr(B).
So we prove the first part of the theorem.
For the second part, we have the LIR with Sobolev estimates, then ∀r ≥ 2, ∀v ∈ Lr(B(x, R)), kvk
kukWm,s(B1) .(kDukL2(B)+kukL2(B)) . (kωkLr(B)+kukL2(B)) .kωkLr(B).
The Sobolev embedding theorems, true here, give kukLτ(B1)≤ ckukWm,2(B1) with τ1 = 12 − mn. So applying again the LIR condition in a ball B2 relatively compact in B1, we get, with t
1 :=
min(τ, r),
kukWα,t1(B2
x) .(kωkLt1(Bx)+kukLt1(Bx1)) .kωkLr(Ω).
Now we proceed as above. If τ ≥ r ⇒ t1 = r, then we apply again the LIR condition to a smaller
ball B3
x we get
kukWm,r(B3).(kωkLr(B)+kukLr(B2)) .kωkLr(B), and we are done.
If τ < r, then we iterate the process as in the previous part, adding the use of the Sobolev embedding theorem to increase the exponent, up to the moment we reach r. So we proved the local existence of solutions with estimates; this is an already known theorem (see for instance [13]). This means also that the Raising Steps Method can be applied to this case and that the LIR condition is stronger than the local existence of solutions with estimates.
6
The non compact case.
We shall use the same ideas as in [5] to go from the compact case to the non compact one. In subsection 7.1 we define a Vitali type covering Cǫ by balls suited to our "admissible balls" (see
Definition 1.5). We use these notions now.
Definition 6.1. We shall say that the hypothesis (UEAB) is fulfilled for the operator D if D has smooth C1(M) coefficients.
Moreover we ask that the "ellipticity" constant of D be bounded below also uniformly in M.
When comparing non compact M to the compact case treated above, we have seen four important issues in the introduction.
We start with ω ∈ L2
p(M) so, by the (THL2p) hypothesis, provided that ω ⊥ kerD∗, we have
that there is a p-form u ∈ L2
p(M) such that Du = ω. Moreover, because L2p(M) is a Hilbert space,
we have that the u ∈ L2
p(M), Du = ω with the smallest norm is given linearly with respect to ω.
This means that we have a bounded linear operator S : L2p(M) → L2
p(M) such that D(Sω) = ω
provided that ω ⊥ kerD∗.
We have the local elliptic inequalities by Corollary 3.5 which become uniform by the hypothesis (UEAB):
Corollary 6.2. Let D be an operator of order m on Λp(M) in the complete riemannian manifold
M. Suppose that D verifies (UEAB). Then, for any Bx := B(x, R) ∈ Cǫ and a smaller ball Bx1 :=
B(x, R/2), we have, with D with C1(M) coefficients:
kukWm,r(B1
x)≤ c1kDukLr(Bx)+ c2R
−mkuk Lr(Bx).
The hypotheses (UEAB) are precisely done to warranty that the constants c1, c2 depend only on
So with t = Sm(r), we get, by Lemma 7.7 from the Appendix,
kukLt(B(x,R)) ≤ CR−m kukWm,r(B(x,R)).
Lemma 6.3. We have, with Bl:= B(x, 2−lR) and t
0 = 2, B0 = B(x, R), the a priori estimates:
R(l+1)mkukLtl(Bl) ≤ l X j=1 cjR(l−j+1)mkDukLl−j(Bl−j)+ cl+1kukL2(B). And R(l+2)mkukWm,tl(Bl+1) ≤ c0R(l+2)mkDukLtl(Bl)+ l X j=1 cjR(l−j+1)mkDukLtl−j(Bl−j)+ cl+1kukL2(B). Proof.
From the LIR we have
∀B ∈ Cǫ, kukWm,2(B1)≤ c1kD(u)kL2(B)+ c2R−mkukL2(B). Now we shall use the local Sobolev embedding theorem to get
∀B ∈ Cǫ, kukLt1(B1) ≤ CR−mkukWm,2(B) so we get
∀B ∈ Cǫ, kukLt1(B1)≤ c1R−mkDukL2(B)+ c2R−2mkukL2(B) with 1 t1 := 1 2 − m n ⇐⇒ t1 := Sm(2).
• If t1 ≥ r, then we get still by the LIR:
∀B ∈ Cǫ, kukWm,t1(B2) ≤ c1kDukLt1(B1)+ c2R−mkukLt1(B1). (6.6) Putting the estimate of kukLt1(B1) in (6.6) we get
kukWm,t1(B2)≤ c1kDukLt1(B1)+ c2R−m
c1kDukL2(B)+ c2R−mkukL2(B)
so with suitable constants
kukWm,t1(B2)≤ c1kDukLt1(B1)+ c2R−mkDukL2(B)+ c3R−2mkukL2(B). Putting the powers of R on the other side to isolate kukL2(B), we get
R2mkukWm,t1(B2) ≤ c1R2mkDukLt1(B1)+ c2RmkDukL2(B)+ c3kukL2(B). We iterate, using again the local Sobolev embedding theorem,
u∈ Lt2(B2), kuk
Lt2(B2)≤ cR−mkukWm,t1(B2) so
R3mkukLt2(B2) ≤ c1R2mkDukLt1(B1)+ c2RmkDukL2(B)+ c3kukL2(B). with 1 t2 := 1 t1 − m n = 1 2 − 2m
n ⇐⇒ t2 := S2m(2). The LIR gives again: kukWm,t2(B3)≤ c1kDukLt2(B2)+ c2R−mkukLt2(B2)
so
R4mkukWm,t2(B3) ≤ c1R4mkDukLt2(B2)+ c2R3mkukLt2(B2) hence
R4mkukWm,t2(B3) ≤ c1R4mkDukLt2(B2)+ c2R2mkDukLt1(B1)+ c3RmkDukL2(B)+ c4kukL2(B). So iterating the same way we get
R(l+1)mkukLtl(Bl) ≤ c1RlmkDukLtl−1(Bl−1)+ c2R(l−1)mkDukLt(l−2)(B(l−2))+· · ·+ +clRmkDukL2(B)+ cl+1kukL2(B).
kukWm,tl(Bl+1)≤ c1kDukLtl(Bl)+ c2R−mkukLtl(Bl) so
R(l+2)mkukWm,tl(Bl+1) ≤ c1R(l+2)mkDukLtl(Bl)+ c2R(l+1)mkukLtl(Bl) and
R(l+2)mkukWm,tl(Bl+1) ≤ c1R(l+2)mkDukLtl(Bl)+ c2RlmkDukLt(l−1)(B(l−1))+ +c3R(l−1)mkDukLt(l−2)(B(l−2))+···+clR
mkDuk
L2(B)+cl+1kukL2(B).
Which proves the lemma.
Lemma 6.4. We have for r < t, B := B(x, R), ∀f ∈ Lr(B), kfk Lr(B) ≤ R 1 r− 1 tkfk Lt(B). Proof.
Because the measure dµ(x) := 1B(x)
|B| dm(x) is a probability measure, using that r < t, we have
kfkLr(µ) ≤ kfkLt(µ) which implies readily the lemma.
Corollary 6.5. Let ∀j ∈ N, 1 tj =
1 2 −
jm
n . Fix r ≥ 2, we have, for tl−1 < r < tl,
R(tl1−1r)+(l+1)mkuk Lr(Bl)≤ l X j=1 cjR(l−j+1)mkDukLl−j(Bl−j)+ cl+1kukL2(B). Proof.
By Lemma 6.4 we get kukLr(Bl) ≤ R 1 r− 1 tlkuk Lt(Bl) so by Lemma 6.3 we have R(l+1)mkukLr(Bl)≤ R 1 r− 1 tlkuk Lt(Bl) ≤ R 1 r− 1 tl l X j=1 cjR(l−j+1)mkDukLl−j(Bl−j)+cl+1R 1 r− 1 tlkuk L2(B). Isolating kukL2(B) we get
R(tl1− 1 r)+(l+1)mkuk Lr(Bl)≤ l X j=1 cjR(l−j+1)mkDukLl−j(Bl−j)+ cl+1kukL2(B).
Now we have a finite number of terms, so changing the values of the constants, we get R(tlr−1)+(l+1)mrkukr Lr(Bl) ≤ l X j=1 cjR(l−j+1)mrkDukrLl−j(Bl−j)+ cl+1kukrL2(B).
which ends the proof of the corollary.
Theorem 6.6. Under hypotheses (THL2p) and (UEAB), with tj := Sjm(2) i.e. t1j = 12 − jmn and
tl−1 < r < tl, vr(x) := R(x) (1 tl−1r)+(l+1)m, w j(x) = R(l+1−j)m, and kωktl−j Ltl−j(M,wtl−jj ) := R M|ω(x)| tl−jw
j(x)tl−jdv(x), we have, provided that ω⊥ kerD∗ that there is
a u := Sω linearly given from ω such that Du = ω and: kukLr(M,vr r) ≤ l X j=1 cjkωkLtl−j(M,wtl−j j ) + cl+1kωkL2(M ). Proof. By hypothesis (THL2p) for ω ∈ L2
We have, with hypothesis (UEAB) and using the covering of M by the Bl, hence a fortiori by the Bj, j < l, with t l−1 < r < tl, vr(x) := R(x)( 1 tl− 1 r)+(l+1)m, kukrLr(M,vr r)≤ X B∈Cǫ R(tl1−1r)+(l+1)mrkukr Lr(Bl). (6.7)
Using that the overlap of the covering is bounded by T, X B∈Cǫ R(l+1−j)mtl−jkωktl−j Lt(l−j)(Bl−j) ≤ T kωk tl−j Ltl−j(M,wtl−jj ). (6.8)
with wj(x) = wj,l(x) = R(l+1−j)m, and for any γ, kγksLs(M,ws k):= R M |γ(x)wk(x)| s dv(x). Now if r ≥ tl−1 ≥ tl−j, j ≤ l − 1, we have P j∈Narj ≤ P j∈Na tl−j j r/tl−j so X B∈Cǫ R(l+1−j)mrkωkrLt(l−j)(Bl−j) ≤ X B∈Cǫ R(l+1−j)mtl−jkωktl−j Lt(l−j)(Bl−j) !r/tl−j . So, using (6.8) we get
X
B∈Cǫ
R(l+1−j)mrkωkrLt(l−j)(Bl−j) ≤ T
r/tl−jkωkr
Ltl−j(M,wtl−jj ).
So, grouping with (6.7) we get kukrLr(M,vr r) ≤ l X j=1 cjTr/tl−jkωkrLtl−j(M,wtl−j j ) + cl+1kukrL2(M ).
Changing the constants, we take the r root to get, using the hypothesis (THL2p), which says also that kukL2(M ) ≤ ckωkL2(M ), kukLr(M,vr r) ≤ l X j=1 cjkωkLtl−j(M,wtl−j j ) + cl+1kωkL2(M )
which finishes the proof.
Lemma 6.7. Provided that ω ∈ L2(M)∩ Ltk(M, R(x)αk), with αj := k + 1 k m×jtj, βj := (j + 1)m×tj, we have: ∀j ≤ k, ω ∈ Ltj(M, Rβj), kωk Ltj(M,Rβj) ≤ C max(kωkLtk(M,Rαk), kωkL2(M )). Proof.
So, because the function x+1x is decreasing, we get k+1k ≤ j+1j for j ≤ k so, R(x) ≤ 1 ⇒ R(x)αj ≥
R(x)βj with α
j := k+1k m×jtj, βj := (j + 1)m×tj and αj ≤ βj.
Using this we get
kωkLtj(M,Rβj)≤ kωkLtj(M,Rαj). (6.9)
So by interpolation we have that ω ∈ L2(M)∩ Ltk(M, Rαk)⇒ ω ∈ Ltj(M, Rαj), with kωkLtj(M,Rαj) ≤ C max(kωkLtk(M,Rαk), kωkL2(M )).
Now using (6.9) we get
∀j ≤ k, ω ∈ Ltj(M, Rβj), kωk
Ltj(M,Rβj) ≤ C max(kωkLtk(M,Rαk), kωkL2(M )).
This proves the lemma.
Corollary 6.8. Let ∀j ∈ N, 1 tj = 1 2− jm n . With w1(x) = w1,l(x) = R
lm, fix r≥ 2, we have, provided
that ω ∈ L2(M)∩ Ltl−1(M, wtl−1
1 ), tl−1 ≤ r < tl, and that ω ⊥ kerD∗, with u := Sω⇒ Du = ω,
kukLr(M,vr
r) ≤ C max(kωkLtl−1(M,wtl−11 ),kωkL2(M )). Proof.
Clear.
To get an estimate for kukWm,r(B) we use again the LIR: kukWm,tl(Bl+1)≤ c1kDukLtl(Bl)+ c2R−mkukLtl(Bl). Replacing kukLtl(Bl) by use of Corollary 6.5, we get:
R(tl1− 1 r)+(l+2)mkuk Wm,r(Bl+1) ≤ c1R( 1 tl− 1 r)+(l+2)mkωk Ltl(Bl)+ c2R( 1 tl− 1 r)+(l+1)mkuk Ltl(Bl), so R(tl1− 1 r)+(l+2)mkuk Wm,r(Bl+1) ≤ c1R (1 tl− 1 r)+(l+2)mkωk Ltl(Bl)+ + l X j=1 cjR(l−j+1)mkωkLl−j(Bl−j)+ cl+1kukL2(B). Now we cover the manifold M the same way as for the proof of Lemma 6.7 and we proved, with vr′(x) := R(x)(tlr−1)+(l+2)mr and wj(x) = wj,l(x) = R(l+1−j)m, kukWm,r(M,v′ r)≤ c1kωkLtl(M,v′r)+ Pl j=1cjkωkLtl−j(M,wtl−j j ) + cl+1kωkL2(M ). Using again Lemma 6.7, we end with
kukWm,r(M,v′
r)≤ c1kωkLtl(M,v′r)+ c2max(kωkLtl−1(M,wtl−11 ),kωkL2(M )).
So we proved:
Theorem 6.9. Under hypotheses (THL2p) and (UEAB), let ∀j ∈ N, t1
j =
1 2 −
jm
n and fix r ≥ 2
and l such that tl−1 ≤ r < tl. With v′r(x) := R(x) (r
tl−1)+(l+2)mr and w′
1(x) = Rlmtl−1, and provided
that ω ⊥ kerD∗: then u := Sω ⇒ Du = ω verifies:
kukWm,r(M,v′
r) ≤ c1kωkLtl(M,v′r)+ c2max(kωkLtl−1(M,w′1),kωkL2(M )).
Remark 6.10. We always ask that tl−1 < ∞ to have, because tl−1 ≤ r < tl, r < ∞ and this
implies that 2(l− 1)m < n. This condition in turn implies that (r
tl − 1) + (l + 2)mr ≥ 0 so, if the
kukWm,r(M ) ≤ c1kωkLtl(M )+ c2max(kωkLtl−1(M ),kωkL2(M )). In the spirit of [5], we introduce a weight wt such that, with t > 2,
Z
M
wt(x)
t
2−tdv(x) <∞, then Hölder inequality gives:
R M|f| 2 dv =R M|f| 2 ww−1dv≤ R M |f| 2α wαdv1/α R Mw−βdv 1/β so with 2α = t and 2 t + 1 β = 1⇒ β = t t−2 we get, if Z M w(x)2−tt dv(x) < ∞, that f ∈ Lt(M, wt/2 t ) implies that f ∈ L2(M).
Lemma 6.11. Suppose that ω ∈ Lt(M, wt/2
t ) then, for 2≤ s ≤ t:
max(kωkLt(M ), kωkLs(M ), kωkL2(M )) .kωkLt(M,wt/2 t ). Proof.
We already have that ω ∈ Lt(M, wt/2
t )⇒ ω ∈ L2(M). We get, because we choose wt(x)≥ 1,
kωktLt(M ) = Z M|ω(x)| t dv(x)≤ Z M |ω(x)| t wt/2t dv(x).
Recall that kωkL2(M ) <∞ which implies, with Ω := {x ∈ M :: |ω(x)| ≥ 1}, ∞ >R M|ω(x)| 2 dv≥R Ω|ω(x)| 2 dv ≥ |Ω| , hence |Ω| < ∞. So Z M|ω(x)| s dv(x) = Z Ω|ω(x)| s dv(x) + Z M \Ω|ω(x)| s dv(x). But because |Ω| < ∞ we get R
Ω|ω(x)| s dv1/s . R Ω|ω(x)| t dv(x)1/t
provided that s≤ t. And, for |ω(x)| ≤ 1, we get Z M \Ω|ω(x)| s dv 1/s . Z M \Ω|ω(x)| 2 dv(x) 1/2
provided that s≥ 2. So finally kωkLs(M ) .kωkLt(M )+kωkL2(M ), which implies the lemma. Now we get the easy to read, but less precise, corollary:
Corollary 6.12. Under hypotheses (THL2p) and (UEAB), with vr(x) := R(x) (r
tl−1)+(l+2)mr and for any weight wt(x) ≥ 1 such that
Z
M
wt(x)
t
2−tdv(x) < ∞, with t := t
l ≥ r > tl−1, and provided that
ω ⊥ kerD∗: then u := Sω ⇒ Du = ω verifies:
kukWm,r(M,vr) .kωkLt(M,wt/2 t ). Proof.
We use that R(x) ≤ 1 implies w′
1(x) ≤ 1 to get kωkLtl−1(M,w′
1) ≤ kωkLtl−1(M ). The same way we havekωkLtl(M,v′
r) ≤ kωkLtl(M ). The lemma 6.11 gives max(kωkLt(M ), kωkLs(M ), kωkL2(M )) .kωkLt(M,wt/2
t ), hence
max(kωkLt(M,v′
r), kωkLtl−1(M,w′1), kωkL2(M )) .kωkLt(M,wtt/2).
So, applying Theorem 6.9 we are done.
Corollary 6.13. Let ∀j ∈ N, 1 tj = 1 2 − jm n . With r ≥ 2 and vr(x) := R(x) (r tl−1)+(l+2)mr, we have, provided that ω ∈ Lt(M, wt/2
t ), with t := tl−1 ≤ r, and that ω ⊥ kerD∗: then u := Sω ⇒ Du = ω
verifies:
kukLr(M,vr).kωkLt(M,wt/2 t ).
7
Appendix.
Of course, without any extra hypotheses on the riemannian manifold M, we have ∀m ∈ N, m ≥ 2, ∀ǫ > 0, ∀x ∈ M, taking gij(x) = δij in a chart on B(x, R) and the radius R small enough, the
ball B(x, R) is m, ǫ admissible, using here Definition 1.5. We shall use the following lemma.
Lemma 7.1. Let (M, g) be a riemannian manifold then with R(x) = Rǫ(x) = the ǫ admissible
radius at x∈ M and d(x, y) the riemannian distance on (M, g) we get: d(x, y)≤ 1
4(R(x) + R(y))⇒ R(x) ≤ 4R(y). Proof.
Let x, y ∈ M :: d(x, y) ≤ 1
4(R(x) + R(y)) and suppose for instance that R(x) ≥ R(y). Then y ∈ B(x, R(x)/2) hence we have B(y, R(x)/4) ⊂ B(x,3
4R(x)). But by the definition of R(x), the ball B(x,3
4R(x)) is admissible and this implies that the ball B(y, R(x)/4) is also admissible for exactly the same constants and the same chart; this implies that R(y) ≥ R(x)/4.
7.1
Vitali covering.
Lemma 7.2. Let F be a collection of balls {B(x, r(x))} in a metric space, with ∀B(x, r(x)) ∈ F, 0 < r(x) ≤ R. There exists a disjoint subcollection G of F with the following property:
every ball B in F intersects a ball C in G and B ⊂ 5C. This is a well known lemma, see for instance [15], section 1.5.1.
So fix ǫ > 0 and let ∀x ∈ M, r(x) := Rǫ(x)/120, where Rǫ(x) is the admissible radius at x, we
built a Vitali covering with the collection F := {B(x, r(x))}x∈M. So the previous lemma gives a disjoint subcollection G such that every ball B in F intersects a ball C in G and we have B ⊂ 5C. We set G′ := {x
j ∈ M :: B(xj, r(xj)) ∈ G} and Cǫ := {B(x, 5r(x)), x ∈ G′}: we shall call Cǫ the
m, ǫ admissible covering of (M, g).
We shall fix m ≥ 2 and we omit it in order to ease the notation. Then we have:
Proposition 7.3. Let (M, g) be a riemannian manifold, then the overlap of the ǫ admissible covering Cǫ is less than T =
(1 + ǫ)n/2
(1− ǫ)n/2(120) n, i.e.
∀x ∈ M, x ∈ B(y, 5r(y)) where B(y, r(y)) ∈ G for at most T such balls. So we have
∀f ∈ L1(M), P
j∈N
R
Proof.
Let Bj := B(xj, r(xj))∈ G and suppose that x ∈ k \ j=1 B(xj, 5r(xj)). Then we have ∀j = 1, ..., k, d(x, xj)≤ 5r(xj) hence d(xj, xl)≤ d(xj, x) + d(x, xl)≤ 5(r(xj) + r(xl))≤ 1 4(R(xj) + R(xl))⇒ R(xj)≤ 4R(xl) and by exchanging xj and xl, R(xl)≤ 4R(xj).
So we get
∀j, l = 1, ..., k, r(xj)≤ 4r(xl), r(xl)≤ 4r(xj).
Now the ball B(xj, 5r(xj) + 5r(xl)) contains xl hence the ball B(xj, 5r(xj) + 6r(xl)) contains the
ball B(xl, r(xl)). But, because r(xl)≤ 4r(xj), we get
B(xj, 5r(xj) + 6×4r(xj)) = B(xj, r(xj)(5 + 24)) ⊃ B(xl, r(xl)).
The balls in G being disjoint, we get, setting Bl := B(xl, r(xl)), k
X
j=1
Vol(Bl)≤ Vol(B(xj, 29r(xj))).
The Lebesgue measure read in the chart ϕ and the canonical measure dvg on B(x, Rǫ(x)) are
equivalent; precisely because of condition 1) in the admissible ball definition, we get that: (1− ǫ)n ≤ |detg| ≤ (1 + ǫ)n,
and the measure dvg read in the chart ϕ is dvg =p|detgij|dξ, where dξ is the Lebesgue measure in
Rn. In particular:
∀x ∈ M, Vol(B(x, Rǫ(x)))≤ (1 + ǫ)n/2νnRn,
where νn is the euclidean volume of the unit ball in Rn.
Now because R(xj) is the admissible radius and 4×29r(xj) < R(xj), we have
Vol(B(xj, 29r(xj))) ≤ 29n(1 + ǫ)n/2vnr(xj)n.
On the other hand we have also
Vol(Bl)≥ vn(1− ǫ)n/2r(xl)n ≥ vn(1− ǫ)n/24−nr(xj)n, hence k X j=1 (1− ǫ)n/24−nr(xj)n ≤ 29n(1 + ǫ)n/2r(xj)n, so finally k ≤ (29×4)n(1 + ǫ)n/2 (1− ǫ)n/2,
which means that T ≤ (1 + ǫ)
n/2
(1− ǫ)n/2(120) n.
Saying that any x∈ M belongs to at most T balls of the covering {Bj} means that Pj∈N1Bj(x)≤ T, and this implies easily that:
∀f ∈ L1(M), X j∈N Z Bj |f(x)| dvg(x)≤ T kfkL1(M ).
7.2
Sobolev spaces.
First define the covariant derivatives by (∇u)j := ∂ju in local coordinates, while the components
of ∇2u are given by
(∇2u)
ij = ∂iju− Γkij∂ku, (7.10)
with the convention that we sum over repeated index. The Christoffel Γkij verify [8]: Γkij = 1 2g il(∂gkl ∂xj + ∂glj ∂xk − ∂gjk ∂xl ). (7.11)
If k ∈ N and r ≥ 1 are given, we denote by Cr
k(M) the space of smooth functions u∈ C∞(M) such
that ∇ju ∈ Lr(M) for j = 0, ..., k. Hence Cr k(M) :={u ∈ C∞(M), ∀j = 0, ..., k, Z M ∇ju r dvg <∞} Now we have [18]
Definition 7.4. The Sobolev space Wk,r(M) is the completion of Cr
k(M) with respect to the norm:
kukWk,r(M ) = k X j=0 Z M ∇ju r dvg 1/r .
We extend in a natural way this definition to the case of p forms.
Let the Sobolev exponents Sk(r) as in the Definition 1.7, then the k th Sobolev embedding is true
if we have
∀u ∈ Wk,r(M), u∈ LSk(r)(M).
This is the case in Rn, or if M is compact, or if M has a Ricci curvature bounded from below and
inf x∈Mvg(Bx(1))≥ δ > 0, due to Varopoulos [27], see Theorem 3.14, p. 31 in [18].
Lemma 7.5. We have the Sobolev comparison estimates where B(x, R) is a ǫ admissible ball in M and ϕ : B(x, R) → Rn is the admissible chart relative to B(x, R),
∀u ∈ Wm,r(B(x, R)), kuk Wm,r(B(x,R)) ≤ (1 + ǫC) u◦ ϕ−1 Wm,r(ϕ(B(x,R))), and, with Be(0, t) the euclidean ball in Rn centered at 0 and of radius t,
kvkWm,r(Be(0,(1−ǫ)R)) ≤ (1 + 2Cǫ)kukWm,r(B(x,R)). Proof.
We have to compare the norms of u, ∇u, ..., ∇mu with the corresponding ones for v := u◦ ϕ−1 in
Rn.
First we have because (1− ǫ)δij ≤ gij ≤ (1 + ǫ)δij in B(x, R):
Be(0, (1− ǫ)R) ⊂ ϕ(B(x, R)) ⊂ Be(0, (1 + ǫ)R).
Because X
|β|≤m−1
sup i,j=1,...,n, y∈Bx(R)
∂βgij(y)
≤ ǫ in B(x, R), we have the estimates, with ∀y ∈ B(x, R), z := ϕ(y),
∀y ∈ B(x, R), |u(y)| = |v(z)| , |∇u(y)| ≤ (1 + Cǫ) |∂v(z)| . Because of (7.11) and (7.10) we get
