Weak compactness of solutions for fourth order elliptic systems with critical growth

0.26 MB·English

by  Paweł Goldstein

WEAK COMPACTNESS OF SOLUTIONS FOR FOURTH ORDER
ELLIPTIC SYSTEMS WITH CRITICAL GROWTH
PAWEŁ GOLDSTEIN, PAWEŁ STRZELECKI, AND ANNA ZATORSKA-GOLDSTEIN
3
1 Abstract. We consider a class of fourth order elliptic systems which include the Euler–
0 Lagrangeequationsofbiharmonicmappings indimension 4 andwe provethatweak limit
2 of weak solutions to such systems is again a weak solution to a limit system.

n
a
J

5
1. Introduction

]
P In this paper we consider fourth order elliptic systems of equations of the form
A
. (1.1) ∆2u = ∆(D ·∇u)+div(E ·∇u)+G·∇u+∆Ω·∇u
h
at for an unknown map u: B4 ⊂ R4 → Rm, i.e., in components,
m
[ (1.2) ∆2ui = ∆ Dij∂ uj +∂ Eij ∂ uj +Gij∂ uj +∆(Ωij)∂ uj
α α α αβ β α α α α
(cid:16) (cid:17) (cid:16) (cid:17)
1
v where α,β = 1,2,3,4 and i,j = 1,...,m. The coefficient functions D,E,G,Ω are assumed
7 to satisfy
2
9 (1.3) D ∈ W1,2, E ∈ L2, G ∈ L4/3, Ω ∈ W1,2(B4;so(m)⊗Λ1R4).
0
.
1 We study compactness of the space of solutions in the weak sequential topology of the
0 Sobolev space W2,2.
3
1 Let us note immediately that under the above assumptions G · ∇u is just in L1, as
: ∇u ∈ W1,2 ⊂ L4 by Sobolev imbedding in dimension 4. Thus, (1.1) is critical. However,
v
i the whole point is that we allow D,E,G,Ω to depend nonlinearly on u. The class of
X
systems we consider contains, in particular, the Euler–Lagrange equations of biharmonic
r
a maps from domains in R4 into compact Riemannian manifolds. Our approach relies in a
crucialwayontheantisymmetryofthe1-formΩandontheuseofnonlinearcounterpartsof
the Hodgedecomposition, originating in gaugetheory. This key idea is due to T. Rivière; it
has been first used in his pioneering paper [7] on conformally invariant second order elliptic
systems in the plane, with harmonic maps from planar domains into compact Riemannian
manifolds serving as the crucial example. Later on, Rivière–Struwe [8], Lamm– Rivière [3],
and Struwe [10] extended this approach to stationary harmonic maps in higher dimensions
and to biharmonic maps.
Date: December 11, 2013.
All authors partially supported by MNiSW Grant no N N201 397737, Nonlinear partial differential
equations: geometric and variational problems.
1
2 PAWEŁ GOLDSTEIN,PAWEŁ STRZELECKI, AND ANNAZATORSKA-GOLDSTEIN
Let us also note that (1.1) in full generality is wider than the class of biharmonic maps.
It can happen in dimension 4 that a solution of (1.1) is continuous, even C1,λ, but still not
C2. We shall comment on that later on; let us now state the main result.
Theorem 1.1. Suppose (u ) is a sequence in W2,2(B4,Rm) of weak solutions to the system
k
(1.4) ∆2u = ∆(D ·∇u )+div(E ·∇u )+G ·∇u +∆Ω ·∇u in B4.
k k k k k k k k k
Suppose u ⇀ u in W2,2(B4,Rm). If the coefficients D , E , G , Ω are weakly convergent
k k k k k
in their respective Sobolev spaces, i.e.
W1,2 W1,2 L2 L4/3
(1.5) D −−−⇀ D, Ω −−−⇀ Ω, E −⇀ E, G −−⇀ G,
k k k k
then u is a weak solution to the limit system
(1.6) ∆2u = ∆(D ·∇u)+div(E ·∇u)+G·∇u+∆Ω·∇u in B4.
Relatedcompactness resultsforbiharmonicmaps, alongwithanenergyidentityaccount-
ingforthepossible‘energyloss’underthelimitpassagehavebeenobtainedbyHornungand
Moser [2] and Laurain and Rivière [4]. (For second order elliptic systems ∆u = Ω ·∇u ,
k k k
Sharp and Topping [9] provide a compactness theorem under an assumption that rules out
the concentration of the energy |∇u |2 but allows for concentration of |∇2u |.)
k k
One of our main points is thRat the proof in this paper is different Rfrom [2] and [4]:
contrary to these two papers, we do not rely at all either on continuity of solutions (or
their first andsecond order derivatives) oronany other improved regularity properties (like
higher integrability of ∇2u), working all the time just in W2,2 and using the imbedding
W2,2 ⊂ W1,4. 1 Besides K. Uhlenbeck’s Theorem 2.1, the main tool is the concentration–
compactness method of P.-L. Lions. The combination of the two allows us, very roughly
speaking, to reduce the complexity of (1.4)–(1.6) to the case
∆2u = G ·∇u , u ⇀ u in W2,2,
k k k k
where G is a bounded sequence in L4/3, with G ⇀ G. Then, a passage to the limit (in
k k
dimension 4) can be justified by an application of Sobolev inequality, the concentration–
compactness lemma [5, Lemma 1.2], and a standard capacity type argument.
Remark 1.2. Conditions (1.3) imposed on the coefficients allow for nonlinear dependence
on u. Roughly speaking, the class (1.1) contains systems of the form ∆2u = R(u,∇u,∇2u)
where the right hand side R depends polynomially on (the entries of) ∇u and ∇2u, with
coefficients that are smooth and bounded in u, so that
|R(u,∇u,∇2u)| . |∇2u|2 +|∇u|4 +1.
The point is that the terms of R depending quadratically on ∇2u need to have some
structure, whereas all the other terms are allowed to be arbitrary. A model case looks as
1In Section 4 we provide anexample showing that (1.1) admits weak solutions that are C1 but not C2.
WEAK COMPACTNESS FOR 4-TH ORDER SYSTEMS 3
follows (summation over repeated indices is understood):
(1.7) D = D(u,∇u) ∈ W1,2, Dij = dij(u)∂ ul;
α l α
(1.8) E = E(u,∇u,∇2u) ∈ L2, Eij = eij,(1)(u)∂ (∂ ul)+eij,(2)(u)∂ ul∂ us;
αβ l α β ls α β
(1.9) G = G(u,∇u,∇2u) ∈ L4/3, Gij = gij (u) ∂ (∂ ul)+∂ ul∂ us ∂ up;
α lsp,βγ β γ β γ α
finally, the antisymmetric term Ω = Ω(u,∇u) ∈ W1,2(B4;(cid:0)so(m)⊗Λ1R4) is giv(cid:1)en by
(1.10) Ωij = w ∂ wj −wj∂ wi = −Ωji.
α i α α α
where w = ν ◦u of u for some bounded smooth map ν: Rm → Rm.
All the coefficients d,e,g (with various indices) in (1.7)–(1.9) are assumed to be of class
C1 ∩W1,∞ on Rm. Struwe [10, Section 2] explains that the biharmonic map equation can
be written in that form, with ν being the normal to the target manifold. In that case,
d,e,g depend explicitly on ν and its derivatives, and the growth estimates
|D|+|Ω| . |∇u|,
(1.11) |E|+|∇D|+|∇Ω| . |∇2u|+|∇u|2,
|G| . |∇2u||∇u|+|∇u|3
follow from (1.7)–(1.10). Under these assumptions, for every weakly convergent sequence
u ⇀ u in W2,2 we have convergence (1.5), with
k
L2 ∋ E = E(u ,∇u ,∇2u ) ⇀ E = E(u,∇u,∇2u)
k k k k
in Lp for p < 2, and L4/3 ∋ G = G(u ,∇u ,∇2u ) ⇀ G = G(u,∇u,∇2u) in Ls for
k k k k
s < 4/3. Since E is bounded in L2 and G is bounded in L4/3, and we deal with a bounded
k k
domain, it is an exercise to see that in fact E ⇀ E and G ⇀ G also for the limiting
k k
exponents p = 2 and s = 4/3.
We do not need the full strength of (1.7)–(1.11).
Remark 1.3. Our proof depends in a crucial way on Sobolev imbedding in dimension 4.
It would be interesting to know what happens in higher dimensions. For example: is the
convergence u → u in BMO, combined with the boundedness of (u ) in W2,2, sufficient
k k
to guarantee (a) that u solves the limiting system, (b) that u → u strongly in W2,2? It
k
is possible to check, using the sharp version of Gagliardo–Nirenberg inequality k∇wk2 .
L4
kwkBMOkwkW2,2 (cf. Meyer–Rivière [6], or [11]), that both answers would be positive for
uniformly bounded weak solutions of the simplified system ∆2u = G ∇u , with G =
k k k k
G(u ,∇u ,∇2u ), where G is of the form (1.9). It seems plausible that convergence of,
k k k
say, biharmonic maps in BMO prevents bubbling and loss of energy in the limit.
The rest of the paper is organized as follows. In Section 2, we recall Uhlenbeck’s result
in the form that is used later on. Section 3 forms the bulk of the paper. There, we first
explain the strategy ofthe proofin moredetail, and then carryout the necessary estimates,
pass to the limit and remove the singularities of the limiting system. Finally, in Section 4
we give an example showing that (1.1) can have solutions in C1 \C2.
4 PAWEŁ GOLDSTEIN,PAWEŁ STRZELECKI, AND ANNAZATORSKA-GOLDSTEIN
2. Uhlenbeck’s result
We remind that in our case we consider a mapping u going from a ball B4 ⊂ R4 into
Rm. Below we state the theorem of Uhlenbeck in a form adjusted to the situation. Note
that if Ω ∈ W1,2(B4) then Ω ∈ L4(B4) because of the Sobolev imbedding theorem.
Theorem 2.1. There exist a number ε = ε(m) > 0 and a constant C > 0 such that for
any Ω ∈ W1,2(B4;so(m)⊗Λ1R4) which satisfies
k∇ΩkL2 +kΩk2L4 ≤ ε
there exist P ∈ W2,2(B4,SO(m)) and ξ ∈ W2,2(B4,so(m)⊗Λ2R4) such that
(2.1) (dP)P−1+PΩP−1 = ∗dξ on B4
and d(∗ξ) = 0 on B4, ξ|∂B4 = 0.
Moreover
(2.2) k∇2PkL2 +k∇PkL4 +k∇2ξkL2 +k∇ξkL4 . k∇ΩkL2 +kΩk2L4.
Uhlenbeck’s Theorem is, in fact, a local theorem in the sense that we can use it not
only on the unit ball B4, but on any ball, and, as long as we consider balls with uniformly
bounded radii, we can choose the constant ε in an uniform way (i.e. independently on the
radius of the ball). This is in accordance with the original use of this theorem to prove the
existence of global Coulomb gauges on compact manifolds. Indeed, a look at the proof of
Lemma 2.5 in [13] shows that we can choose ε = (2(C r+1)C )−1, where C and C are
P S P S
the constant in the Poincaré and Sobolev inequalities for the unit ball, and r denotes the
radius of the ball B. Thus ε = (2(C R+1)C )−1 can be chosen as a uniform estimate for
P S
all balls with radius bounded by R.
Corollary 2.2. Theorem 2.1 holds for any ball B ⊂ B4 in place of B4, and the constant ε
can be chosen uniformly for all such balls.
Another corollary deals with the problem of weak continuity of P and ξ with respect to
Ω. Note that Theorem 2.1 does not claim that either of them is defined uniquely.
Corollary 2.3. Suppose {Ω } is a sequence in W1,2(B4;so(m)⊗Λ1R4). Assume
k
(2.3) Ω ⇀ Ω in W1,2.
k
Assume that P and ξ are chosen so that (2.1) and (2.2) of Theorem 2.1 hold with Ω in
k k k
place of Ω.
Then both P and ξ are uniformly bounded in W2,2, and for any subsequence of (Ω ) for
k k k
which P and ξ are weakly convergent in W2,2 to P and ξ, respectively, conditions (2.1)
k k
and (2.2) of Uhlenbeck’s Theorem do hold for Ω, P and ξ.
In other words, the decomposition for the limit matrix Ω can be effectuated with (any)
weak limit of the transformations P and forms ξ .
k k
Indeed, assume Ω ⇀ Ω w W1,2 and that Ω satisfy, uniformly with respect to k,
k k
assumptions of Theorem 2.1. We obtain then, by the theorem, transformations P and ξ
k k
WEAK COMPACTNESS FOR 4-TH ORDER SYSTEMS 5
such that
(2.4) dP +P Ω = ∗dξ P on B4
k k k k k
and
(2.5) k∇2PkkL2 +k∇PkkL4 +k∇2ξkkL2 +k∇ξkkL4 . k∇ΩkkL2 +kΩkk2L4 ≤ M,
where M > 0 is a constant which is independent of k.
Since, by (2.5), the sequences (ξ ) and P are bounded in W2,2, we can choose subse-
k k
quences (for simplicity still indexed by k) such that
ξ ⇀ ξ weakly in W2,2,
k
P ⇀ P weakly in W2,2 and a.e.
k
Thus (after again choosing subsequence) we may assume that
dξ → dξ strongly in L4−δ,
k
dP → dP strongly in L4−δ,
k
Ω → Ω strongly in L4−δ.
k
for any small δ > 0. Since P are also uniformly bounded in L∞, we can take the L4−δ-limit
k
on both sides of (2.4) obtaining
(2.6) dP +PΩ = ∗dξ on B4
in the sense of distributions, which proves that P, ξ and Ω indeed satisfy (2.1). The
remaining estimates and boundary conditions in Theorem 2.1 are obvious.
This sort of continuity of the decomposition of Ω, i.e. the fact that the transformation
of the limit Ω may be attained by taking the weak limits of the elements of decomposition
of Ω allows us later to estimate the W2,2-norm of differences between the elements of
k
decomposition of Ω and Ω.
k
3. Proof
Let us first give a rough sketch and plan of the proof. The key idea is to prove that u
solves the limiting system (1.6) outside a countable set of points and then remove these
possible singularities with theuse of a properly chosen test function. Astandardargument,
cf. Section 3.1, shows that B4 \ A , where A is finite, can be covered by balls B such
1 1 j
that kΩkkW1,2(Bj) is small, so that Uhlenbeck’s decomposition can be applied inside each
B separately. Next, in Section 3.2, we fix one of the B and, following the crucial ideas
j j
of Lamm-Rivière [3] and Struwe [10], we use the equation (dP )P−1+P Ω P−1 = ∗dξ on
k k k k k k
B to rewrite (1.4) as
j
(3.1) ∆(P ∆u )+a perturbation = H ·∇u +∗d∆ξ ·(P ∇u ) on B ,
k k k k k k k j
where H depends on D ,E ,G ,P ,Ω and their derivatives.
k k k k k k
It is a purely routine matter to pass to the weak limit on the left hand side of (3.1). On
the right hand side, after passing to subsequences, H is bounded in L4/3 and converges
k
6 PAWEŁ GOLDSTEIN,PAWEŁ STRZELECKI, AND ANNAZATORSKA-GOLDSTEIN
strongly in all Ls for s < 4, and ∇u is bounded in L4 by the Sobolev imbedding, and
3 k
converges strongly in Lq for all q < 4. Thus, H ·∇u is bounded in L1 but a priori does
k k
not have to converge in D′. The second term, ∗d∆ξ ·(P ∇u ), presents a similar difficulty.
k k k
To cope with that, we apply in Section 3.3 P.-L. Lions’ concentrated compactness method
[5], following earlier work of Freire, Müller and Struwe [1] on wave maps and harmonic
maps, and the second and third author [12] on H-systems. The key idea is to exploit the
existence of second order derivatives of u. This yields
H ·∇u +∗d∆ξ ·(P ∇u ) −→ H ·∇u+∗d∆ξ ·(P∇u)+S on B ,
k k k k k j j
where H·∇u+∗d∆ξ·(P∇u) is the desired term of the limit system (1.6) rewritten in the
(P,ξ) gauge, and S is a combination of Dirac delta measures, supported on a countable
j
subset A of B .
j
Tocomplete theproof,inSection3.4we show thateachS must bezero, using acapacity
j
argument, based on the fact that W2,2(R4) contains unbounded functions. Thus, the limit
u of u satisfies (1.6) in B4 \A ; another application of the same argument shows that A
k 1 1
must be empty.
3.1. Preparation to Uhlenbeck’s transformation. Since the sequence Ω is weakly
k
convergent, it is bounded in W1,2; we shall denote the bound on its norm by M.
In what follows, we want to cover the ball B4 by balls B in such a way, that in every of
j
the Bj we may, after passing to a subsequence, assume that kΩkkW1,2(Bj) ≤ ε, where ε is as
in Corollary 2.2 to Uhlenbeck’s Theorem. This might not be possible for the whole B4, but
it is outside a finite set of points. To visualize this better, replace B4 by a four-dimensional
cube and consider its dyadic decomposition into cubes C , where the second subscript j
i,j
counts subsequent cubes of a specified generation i.
A cube C is bad, if one cannot choose a subsequence of Ω such that its W1,2-norm is
k
bounded on C by ε, i.e. if, for all k sufficiently large, kΩkkW1,2(C) > ε. Notice that, in
every generation of the dyadic decomposition, the number of bad cubes C is bounded by
i,j
the same constant N = ⌈Mε−1⌉, and that if a cube is not bad (i.e. it is good), neither are
all its descendants. The intersection of all bad cubes,
∞
(3.2) A := bad cubes C
1 i,j
(cid:18) (cid:19)
i\=1 1≤[j≤N
is a finite set of (at most N) points, and any point not in A lies in a cube that is good.
1
These points are, in fact, accumulation points of the W1,2-normof the weakly convergent
sequence (Ω ), more precisely — these accumulation points, for which the energy loss
k
exceeds ε.
For the next two subsections, let us fix an arbitrary good ball B = B contained in
j
B4 \ A . We consider (not changing the notation, to keep things simple), instead of the
1
whole sequence u , only the subsequence of u (and of D , E , G and Ω ) for which
k k k k k k
kΩkkW1,2(Bj) ≤ ε. By Corollary 2.2 we can assume that the Uhlenbeck’s theorem holds for
B, so there exist P and ξ such that dP P−1 +P Ω P−1 = ∗dξ .
k k k k k k k k
WEAK COMPACTNESS FOR 4-TH ORDER SYSTEMS 7
3.2. Transformation of the equation: calculations. The calculations below follow
closely and with more detail the brief calculations by Struwe in [10]. We provide them
for the reader’s convenience, and also because we shall need some more knowledge on the
structure of certain terms in our reasoning.
We recall the indices of the multidimensional objects that appear in our system:
u = (ui), ∇u = (∂ ui), P = (Pij), D = (Dij),
α α
E = (Eij ), G = Gij, Ω = (Ωij),
αβ α α
with i,j = 1,...,n and α,β = 1,2,3,4. To simplify the notation without making the
calculations ambiguous we shall use the standard summation conventions.
Furthermore, it is often convenient to omit at least some of the indices. In that case,
• multiplication of tensor objects that is denoted by a dot (·) is a standard scalar
product in the R4 ×Rm space, e.g.
4 n i
E ·∇u = Eij ∂ uj i = Eij ∂ uj
αβ β α (cid:18) αβ β (cid:19)
(cid:0) (cid:1) Xβ=1Xj=1 α
4 n i
G·∇u = Gij∂ uj i = Gij∂ uj
α α (cid:18) α α (cid:19)
(cid:0) (cid:1) Xα=1Xj=1
• multiplication of tensor objects that is not denoted by any operatorsign is standard
matrix multiplication, e.g.
n
P∆2u = Pij∆2uj = Pij∆2uj
i (cid:18) (cid:19)
(cid:0) (cid:1) Xj=1 i
• tensor multiplication (⊗) denotes tensor product in R4 (and then, possibly, matrix
multiplication in the coordinates), e.g.
i
∇P ⊗∇u = ∇Pij ⊗∇uj = ∂ Pij ⊗ ∂ uj
α α β β
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
n i
= ∂ Pij∂ uj .
α β
(cid:18) (cid:19)
Xj=1 αβ
Below, we transform the system (1.4) for u . In the calculations that follow we omit the
k
index k (one should not confuse this temporary notational simplification with the claim
that u – a weak limit of (u ) – satisfies (1.6); proving this is the goal of our paper).
k
Applying P to the Laplacian on the left-hand side of the system, we obtain
∆(P∆u)i = ∆(Pij∆uj) = ∂ ∂ (Pij∂ ∂ uj)
α α β β
= Pij∂ ∂ ∂ ∂ uj +∂ ∂ ∂ Pij∂ uj −∂ (∂ ∂ Pij∂ uj)
α α β β β α α β β α α β
+2∂ ∂ (∂ Pij∂ uj)−2∂ (∂ ∂ Pij∂ uj),
α β α β α β α β
8 PAWEŁ GOLDSTEIN,PAWEŁ STRZELECKI, AND ANNAZATORSKA-GOLDSTEIN
which can be rewritten shortly as
∆(P∆u)i = (P∆2u)i +(∇(∆P)·∇u)i −div(∆P∇u)i
(3.3)
+2div2(∇P ⊗∇u)i −2div(∇2P ·∇u)i.
By the equation (1.4) satisfied by u we have, still omitting the index k for the sake of
k
simplicity,
(P∆2u)i = Pij∆(D ·∇u)j +Pijdiv(E ·∇u)j +Pij(G·∇u)j +Pij(∆Ω·∇u)j,
which can be rewritten as
P∆2u = ∆(PD·∇u)+div((PE −2∇PD)·∇u)
(3.4)
+(∆PD +PG+P∆Ω−∇P ·E)·∇u.
Substituting (3.4) into (3.3), after some rearranging, yields
∆(P∆u) = ∆(PD·∇u)+2div2(∇P ⊗∇u)
(3.5) +div (PE −2∇PD−2∇2P ·∇u)·∇u−∆P∇u
+(∆P(cid:0)D+PG+P∆Ω+∇(∆P)−∇P ·E)·∇u.(cid:1)
Define
D˜il = PijDjl
α α
(3.6) E˜il = PijEjl −2∂ PijDjl −2∂ ∂ Pil −δ ∆Pil
αβ αβ α β α β αβ
H = ∆PD +PG+P∆Ω+∇(∆P)−∇P ·E −∗d∆ξP,
where, as before, the Roman lowercase indices run from 1 to n, and the Greek indices from
1 to 4; δ denotes Kronecker’s delta.
αβ
With this notation, we rewrite (3.5) as
∆(P∆u) = ∆(D˜ ·∇u)+2div2(∇P ⊗∇u)
(3.7)
+div(E˜ ·∇u)+H ·∇u+∗d∆ξ ·P∇u.
We shall need the precise form and integrability properties of terms that appear in H. By
Theorem 2.1,
∗d∆ξP = ∆ (∇P +PΩ)P−1 P,
thus (cid:0) (cid:1)
H = ∆PD +PG+P∆Ω+∇(∆P)−∇P ·E −∆ (∇P +PΩ)P−1 P,
and after some simple reductions we get (cid:0) (cid:1)
Hij = ∆PilDlj +PilGlj −∂ PilElj −∆PilΩlj
β β β α αβ β
−2 ∂ Pil∂ Ωlj +∂ ∂ Pil∂ (P−1)lsPsj
α α β α β α
(cid:16)
+∂ PilΩlt∂ (P−1)tsPsj +Pil∂ Ωlt∂ (P−1)tsPsj
α β α α β β
(cid:17)
−∂ Pil∆(P−1)lsPsj −PilΩlt∆(P−1)tsPsj,
β β
WEAK COMPACTNESS FOR 4-TH ORDER SYSTEMS 9
or, in the simplified notation, without the jungle of indices,
H = ∆PD +PG−∇P ·E −∆PΩ
−2 ∇P ·∇Ω+(∇2P ·∇(P−1))P
(3.8) (cid:16)
+(∇PΩ)·∇(P−1)P +P(∇Ω·∇(P−1))P
(cid:17)
−∇P∆(P−1)P −PΩ∆(P−1)P.
Please bear in mind that in fact we shall use equations (3.6)–(3.8) for each k, adding the
subscript k to all letters H,D,E,G,P,Ω,ξ,D˜,E˜.
From now on we shall return to using the index k, where appropriate.
The bounds for D , E , G and Ω together with the estimates for P and ξ given by
k k k k k k
Theorem 2.1 imply that H ∈ L4/3 and
k
(3.9) kHkkL4/3 ≤ C = C supmax kDkkW1,2,kEkkL2,kGkkL4/3,kΩkkW1,2 .
(cid:16) k (cid:0) (cid:1)(cid:17)
To check this, one just uses Hölder’s inequality, and – when it is appropriate – the Sobolev
imbedding W1,2 ⊂ L4 in dimension 4; we leave the details to the reader.
Remark. If the coefficients D = D(u ,∇u ), E = E(u ,∇u ,∇2u ), G = G(u ,∇u )
k k k k k k k k k k
and Ω = Ω(u ,∇u ) are given by the composition of fixed smooth functions with the u
k k k k
and their derivatives, and satisfy the growth conditions (1.11), then a computation yields
kH k . ku k3/2 +k∇u k3 .
k L4/3 k W2,2 k L4
We do not rely on that particular estimate, though.
2
Let H be defined analogously to H , i.e. by formula (3.8). The convergence of all the
k
terms in the right hand side of the formula for H is such that, up to a subsequence, we
k
can assume
4
(3.10) H ⇀ H in Lq for all 1 ≤ q < .
k
3
To see this, we just use the elementary observation: if 1+1 = 1, p,q,r > 1, and f → f in
p q r k
Lp and g ⇀ g in Lq, then f g ⇀ fg in Lr, combining it with the imbedding W2,2 ⊂ W1,4
k k k
and Rellich-Kondrashov’s compactness theorem.
Moreover, by the estimates on H , we can once again choose a subsequence of H that
k k
is convergent weakly in L4/3.
2This time the subscript k in (3.8) is really omitted, not just for the sake of simplicity!
10 PAWEŁ GOLDSTEIN,PAWEŁ STRZELECKI, AND ANNAZATORSKA-GOLDSTEIN
We write out the weak formulation of (1.4) in B , using its transformed form (3.7), and
j
separating the terms into ‘easy’ (left hand side) and ‘hard’ (right hand side): the identity
˜
P ∆u ∆ψ − (D ·∇u )∆ψ
k k k k
Z Z
(3.11) + 2div(∇P ⊗∇u )+(E˜ ·∇u ) ·∇ψ
k k k k
Z h i
= (H ·∇u )ψ + ∗d∆ξ ·(P ∇u )ψ
k k k k k
Z Z
holds for each smooth map ψ compactly supported in B = B . (Since ξ is only of class
j k
W2,2, the last term has to be interpreted using one integration by parts.)
3.3. Convergence of (3.11). We consider the left hand side and the right hand side
separately. (The key difficulty is to prove that right hand sides converge to the appropriate
limit).
3.3.1. Convergence of the left hand side of (3.11). By assumption, u ⇀ u in W2,2, thus,
k
after passing to a subsequence, we may assume that
u → u in Ls for all 1 ≤ s < ∞,
k
∇u ⇀ ∇u in L4,
k
∇u → ∇u in Lp for all 1 ≤ p < 4,
k
∇2u ⇀ ∇2u in L2.
k
Taking into account the bounds for D , E , G and Ω we can also assume that (again,
k k k k
after passing to a subsequence) there exist D, E, G and Ω such that
D ⇀ D and Ω ⇀ Ω in W1,2,
k k
E ⇀ E in Lp for all 1 ≤ p < 2, G ⇀ G in Ls for all 1 ≤ s < 4/3.
k k
Similarly, Theorem 2.1 gives uniform estimates on P and ξ , which allow us to assume
k k
(after passing to subsequences) that
(P ,ξ ) ⇀ (P,ξ) in W2,2,
k k
and, by the Rellich–Kondrashov’s compactness theorem, ∇P → ∇P and ∇ξ → ∇ξ
k k
strongly in Ls for all 1 ≤ s < 4, and P → P in Ls for all 1 ≤ s < ∞. The limits P and ξ
k
of P and ξ satisfy the claim of Corollary 2.3 on B = B . In particular,
k k j
dPP−1+PΩP−1 = ∗dξ.
ByHölder’sinequality, itfollowsfromalltheseconvergence assumptionsthat∇P ⊗∇u →
k k
∇P ⊗∇u in D′,
D˜ ·∇u = P D ·∇u → PD·∇u = D˜ ·∇u in D′,
k k k k k
and finally,
E˜ ·∇u → E˜ ·∇u in D′
k k