On the optimal constants in Korn's and geometric rigidity estimates, in bounded and unbounded domains, under Neumann boundary conditions

0.24 MB·English

by  Marta Lewicka

ON THE OPTIMAL CONSTANTS IN KORN’S AND
GEOMETRIC RIGIDITY ESTIMATES,
IN BOUNDED AND UNBOUNDED DOMAINS,
5
UNDER NEUMANN BOUNDARY CONDITIONS
1
0
2
MARTALEWICKAANDSTEFANMU¨LLER

n
a
J

8
Abstract. We are concerned with the optimal constants: in the Korn in-
] equality under tangential boundary conditions on bounded sets Ω Rn, and
P in the geometric rigidityestimate on the whole R2. We prove that⊂the latter
A
constant equals √2, and wediscuss the relation of the formerconstants with
. theoptimalKorn’sconstantsunderDirichletboundaryconditions,andinthe
h wholeRn,whicharewellknowntoequal√2. Wealsodiscusstheattainability
t
a oftheseconstantsandthestructureofdeformations/displacementfieldsinthe
m optimalsets.
[

1
v 1. Introduction and the main results
7
InthispaperweareconcernedwiththeoptimalconstantsintheKorninequality
1
9 [10, 11] and in the Friesecke-James-Mu¨ller geometric rigidity estimate [7, 8].
1 Let Ω be anopen, bounded, andconnectedsubsetofRn withLipschitz continu-
0 ous boundary. The Korn inequality [10, 11, 13] states that there exists a constant
1. C(Ω) depending only on Ω, such that for all u W1,2(Ω,Rn) there holds:
∈
0
5 (1.1) min u A L2(Ω) C(Ω) D(u) L2(Ω),
1 A∈so(n)k∇ − k ≤ k k
:
v where by D(u)= 1 u+( u)T we mean the symmetric part of u.
i 2 ∇ ∇ ∇
X Let now ~n denote the outward unit normal on ∂Ω. Given (1.1), it is not hard
(cid:0) (cid:1)
r to deduce (see Lemma 2.1) the following version of Korn’s inequality subject to
a
tangential boundary conditions. Namely, there exists a constant κ(Ω), depending
only onΩ, suchthat for all u W1,2(Ω,Rn) satisfying u ~n=0 on∂Ω there holds:
∈ ·
(1.2) min u A L2(Ω) κ(Ω) D(u) L2(Ω),
A∈LΩk∇ − k ≤ k k
wherebyL abovewedenotethe linearspaceofskew-symmetricmatricesthatare
Ω
gradients of affine maps tangential on the boundary of Ω:
L = A so(n); a Rn x ∂Ω (Ax+a) ~n(x)=0 .
Ω
{ ∈ ∃ ∈ ∀ ∈ · }
1991 Mathematics Subject Classification. 74B05.
Keywordsandphrases. Korn’sinequality,geometricrigidityestimate,optimalconstant,linear
elasticity,nonlinearelasticity.
1
2 MARTALEWICKAANDSTEFANMU¨LLER
The optimal constant in (1.2) is given by:
κ(Ω)=sup min u A L2(Ω); u W1,2(Ω,Rn), u ~n=0 on ∂Ω
(A∈LΩk∇ − k ∈ ·
(1.3)
and D(u) L2(Ω) =1 ,
k k )
and we aim to study its relation to Korn’s constant in the whole Rn, which is √2
(see Lemma 2.2):
(1.4) κ(Rn)=sup u L2(Rn); u W1,2(Rn,Rn), D(u) L2(Rn) =1 =√2.
(k∇ k ∈ k k )
In this setting, our first set of main results is:
Theorem 1.1. For any open, bounded, Lipschitz, connected Ω Rn:
⊂
(1.5) κ(Ω) κ(Rn)=√2.
≥
In fact, κ(Ω) may be arbitrarily large. In Example 3.3 we will recall our con-
struction in [12] which implies that for a sequence of thin shells around a sphere,
the Korn constants go to as the thickness goes to 0. On the other hand, as we
∞
show in Example 3.2, there is: κ([0,1]2)=√2. 1 We however have:
Theorem 1.2. Assume that there exists a sequence u ∞ , u W1,2(Ω,Rn),
{ k}k=1 k ∈
u ~n=0 on ∂Ω, with the following properties:
k
·
(i) u converges to 0 weakly in W1,2(Ω,Rn),
k
(ii) D(uk) L2(Ω) =1,
k k
(iii) limk→∞ uk L2(Ω) =κ(Ω).
k∇ k
Then κ(Ω)=√2.
Theorem 1.3. Ifκ(Ω)>√2then thesupremuminthedefinition (1.3)is attained.
More precisely, for every A L there exists u W1,2(Ω,Rn), such that u ~n=0
0 Ω
∈ ∈ ·
on ∂Ω, D(u) 0, and:
6≡
(1.6) min u A L2(Ω) = u A0 L2(Ω) =κ(Ω) D(u) L2(Ω).
A∈so(n)k∇ − k k∇ − k k k
Theorem 1.4. The vector fields u for which Korn’s constant κ(Ω) is attained:
(1.7) u W1,2(Ω,Rn); u ~n=0 on ∂Ω, u satisfies (1.6) for some A L ;
0 Ω
∈ · ∈
form anclosed linear subspace of W1,2(Ω,Rn). Moreover, if κ(Ω) > √2 thenothis
space is of finite dimension.
InthesecondpartofthispaperweconcentrateonthenonlinearversionofKorn’s
inequality, namely the Friesecke-James-Mullergeometric rigidity estimate [7, 8]. It
states that for an open, bounded, smooth and connected domain Ω Rn, there
exists aconstantκ (Ω) depending only onΩ, suchthat for everyu ⊂W1,2(Ω,Rn)
nl
∈
there holds:
(1.8) min u R L2(Ω) κnl(Ω) dist( u,SO(n)) L2(Ω).
R∈SO(n)k∇ − k ≤ k ∇ k
1Weareabletoprovethatforsmoothdomainstherealwaysholds: κ(Ω)>√2. Theproofof
thisfactwillappearelsewhere.
OPTIMAL CONSTANTS IN RIGIDITY ESTIMATES 3
Define:
κ (Rn)=sup min k∇u−RkL2(Rn) ; u W1,2(Rn,Rn),
nl (R∈SO(n) dist( u,SO(n)) L2(Rn) ∈ loc
k ∇ k
(1.9)
dist( u,SO(n)) L2(Rn) 0 .
∇ ∈ \{ })
Our results in this context are restricted to dimension 2:
Theorem 1.5. We have: κ (R2)=√2. In particular:
nl
u W1,2(R2,R2) dist( u,SO(2)) L2(R2) =
∀ ∈ loc ∇ ∈ ⇒
min u R L2(R2) √2 dist( u,SO(2)) L2(R2).
R∈SO(n)k∇ − k ≤ k ∇ k
Theorem 1.6. For every rotation R SO(2) there exists u W1,2(R2,R2) with
dist( u,SO(2)) L2(R2) 0 such0th∈at: ∈ loc
∇ ∈ \{ }
min u R L2(R2) = u R0 L2(R2)
R∈SO(2)k∇ − k k∇ − k
(1.10)
=√2 dist( u(x),SO(2)) L2(R2).
k ∇ k
Theorem 1.7. The vector fields for which the nonlinear Korn constant in (1.9) is
attained, namely:
u W1,2(R2,R2); dist( u,SO(2)) L2(R2),
∈ loc ∇ ∈
n
u satisfies (1.10) for some R SO(2)
0
∈
o
have the defining property that their gradients are of the form:
(1.11)
a(x) b(x) cosα sinα
∇u(x)=R0R(α(x))+ b(x) a(x) with R(α)= sinα −cosα ,
(cid:20) − (cid:21) (cid:20) (cid:21)
for some α,a,b L2(R2). Conversely, for every α L2(R2) there exists a,b
∈ ∈ ∈
L2(R2) and u W1,2(R2,R2) such that (1.10) and (1.11) hold.
∈ loc
The proofsofthe three Theoremsaboveareindependent fromthe proofof(1.8)
in [7]. They rely on the conformal-anticonformal decomposition of 2 2 matrices,
×
andit is not clear how this constructionand methods could be extended to yield a
result in higher dimensions n>2.
ThereisanextensiveliteraturerelatingtoKorn’sinequalityanditsapplications,
notablyinlinearelasticity[2,3,10,13]. Onthe otherhand,the nonlinearestimate
(1.8) plays crucial role in models in nonlinear elasticity [8, 7]. Indeed, the relation
betweenthesetwoestimatesisclearifwerecallthatthetangentspacetoSO(n)at
Idisso(n). Theblow-uprateandpropertiesofκ(Ω)forthinspherical-likedomains
aroundagivensurfacewerestudiedin[12]. Therelationsofκ(Ω)withthemeasure
of axisymmetry of Ω have been discussed in [5]. An interesting extension of both
Korn’s and the geometric rigidity estimates under mixed growth conditions has
been recently established in [4].
4 MARTALEWICKAANDSTEFANMU¨LLER
2. Preliminaries
Recall that the linear space of skew-symmetric matrices is:
so(n)= A Rn×n; A= AT
{ ∈ − }
while SO(n) stands for the group of proper rotations:
SO(n)= R Rn×n;RT =R−1 and detR=1 .
{ ∈ }
The scalar product and the (Frobenius) norm in the space of n n (real) matrices
Rn×n are given by: ×
A:B =tr(ATB) A2 =A:A.
| |
We first notice the following characterizationof the minimiser in (1.2):
Lemma 2.1. Let u W1,2(Ω,Rn), u ~n=0 on ∂Ω. Then the minimum in the left
∈ ·
hand side of (1.2) is attained, uniquely, at:
A =P u,
0 LΩ ∇
Ω
where P denotes the orthogonal projection of Rn×n on L .
LΩ Ω
Proof. LetA L beaminimiserof u A 2 overL . Takingthederivative
0 ∈ Ω k∇ − kL2(Ω) Ω
in the direction of A L , one obtains:
Ω
∈
A L ( u A ):A=0.
∀ ∈ Ω ˆ ∇ − 0
Ω
Equivalently, there holds:
u A L⊥,
∇ − 0 ∈ Ω
(cid:18) Ω (cid:19)
which implies the lemma.
For convenience of the reader, we now sketch the proof of (1.4).
Lemma 2.2. For every open, Lipschitz, connected Ω Rn, the Korn constant
⊂
under Dirichlet boundary conditions equals √2:
(2.1) κ0(Ω)=sup k∇ukL2(Ω); u∈W01,2(Ω,Rn), kD(u)kL2(Ω) =1 =√2.
n o
Proof. For every u W1,2(Ω,Rn) we have:
∈
2 D(u)2 = u2+ u:( u)T = u2+ tr( u)2
ˆ | | ˆ |∇ | ˆ ∇ ∇ ˆ |∇ | ˆ ∇
(2.2) Ω
= u2+ div u2+ tr( u)2 (tr u)2 .
ˆ |∇ | ˆ | | ˆ ∇ − ∇
(cid:0) (cid:1)
When, additionally, Ω is bounded and u W1,2(Ω,Rn), this implies that:
∈ 0
2 D(u)2 = u2+ div u2,
ˆ | | ˆ |∇ | ˆ | |
Ω
because tr( u)2 (tr u)2 is a null-Lagrangean, i.e. its integral depends only
∇ − ∇
on the boundary value of u on ∂Ω. We therefore conclude that, in this case:
(cid:0) (cid:1)
u L2(Ω) √2 D(u) L2(Ω). The same inequality is also true on unbounded
k∇ k ≤ k k
domains, because of the density of ∞(Ω,Rn) in W1,2(Ω,Rn).
Cc 0
OPTIMAL CONSTANTS IN RIGIDITY ESTIMATES 5
To prove that √2 is optimal and that it is attained, it is enough to take u
∞(Ω,Rn) with div u = 0 (when n = 3, take u = curl v for any compactly sup∈-
Cc
ported v). This achieves the proof.
We now recall the Poincar´e inequality for tangential vector fields. The proof,
whichcanbefoundin[1],isdeducedthroughastandardargumentbycontradiction.
Lemma 2.3. Let Ω Rn be an open, bounded, Lipschitz set. For every u
W1,2(Ω,Rn), u ~n=0⊂on ∂Ω, there holds: ∈
·
(2.3) u L2(Ω) C(Ω) u L2(Ω),
k k ≤ k∇ k
where the constant C(Ω) depends only on Ω (it is independent of u).
3. The optimal Korn constant κ(Ω): a proof of Theorem 1.1 and two
examples
In the course of proof of Theorem 1.1, we will use the following observation:
Proposition 3.1. For any f L2(Rn) there holds:
∈
Rl→im∞R−n/2kfkL1(BR) =0,
on the ball B = x Rn, x R .
R
{ ∈ | |≤ }
Proof. Fixǫ>0. Formsufficientlylarge,onehaskfkL2(Rn\Bm) <ǫ. Denotebyωn
m n/2
thevolumeoftheunitballB1inRn. TakeanyR>msothat: R kfkL2(Rn) ≤
ǫ. Then: (cid:16) (cid:17)
R−n/2kfkL1(BR) =R−n/2 ˆBR\Bm|f|+ˆBm|f|!
R−n/2 BR 1/2ǫ+R−n/2 Bm 1/2 f L2(Rn)
≤ | | | | k k
m n/2
≤ωn1/2ǫ+ R ωn1/2kfkL2(Rn) ≤2ωn1/2ǫ,
(cid:16) (cid:17)
which achieves the proof.
Proof of Theorem 1.1
1. Without loss of generality we may assume that 0 Ω. Let u W1,2(Rn,Rn)
with D(u) L2(Rn) = 1. Define the sequence uk ∈W1,2(Rn,Rn∈) by: uk(x) =
k k ∈
kn/2−1u(kx). One has:
uk L2(Rn) = u L2(Rn), D(uk) L2(Rn) = D(u) L2(Rn) =1.
k∇ k k∇ k k k k k
Let now φ ∞(Ω) be a nonnegative function, equal identically to 1 in a neigh-
∈ Cc
borhood of 0, and define: v =φu . Clearly v W1,2(Ω,Rn) and:
k k k ∈ 0
v =φ u +u φ.
k k k
∇ ∇ ⊗∇
We claim that:
(3.1) lim vk L1(Ω) =0,
k→∞k∇ k
(3.2) lim vk L2(Ω) = u L2(Rn),
k→∞k∇ k k∇ k
(3.3) lim D(vk) L2(Ω) = D(u) L2(Rn) =1.
k→∞k k k k
6 MARTALEWICKAANDSTEFANMU¨LLER
To prove the claim, notice first that:
lim uk φ L2(Ω) lim φ L∞k−1 u L2(Rn) =0.
k→∞k ⊗∇ k ≤k→∞k∇ k k k
On the other hand, for all i,j :1...n:
2 2 2
∂ ∂ ∂
lim φ uj = lim φ(x/k) uj(x) dx= uj .
k→∞(cid:13) ∂xi k(cid:13)L2(Ω) k→∞ˆRn(cid:12) ∂xi (cid:12) (cid:13)∂xi (cid:13)L2(Ω)
(cid:13) (cid:13) (cid:12) (cid:12) (cid:13) (cid:13)
Thus we o(cid:13)btain (3.(cid:13)2) and (3.3). Simil(cid:12)arly: (cid:12) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:12) (cid:12) (cid:13) (cid:13)
kl→im∞kφ∇ukkL1(Ω) ≤kl→im∞kφkL∞k−n/2k∇ukL1(kΩ) =0,
wherethe lastequalityfollowsbyProposition3.1. Hence weconclude(3.1)aswell.
2. Notice that by Lemma 2.1:
Am∈iLnΩk∇vk−AkL2(Ω) =(cid:13)∇vk−PLΩ Ω∇vk(cid:13)L2(Ω)
(cid:13) (cid:13)
(cid:13) (cid:13)
≥k∇vkkL2(Ω)− PL(cid:13)Ω ∇vk ≥k(cid:13)∇vkkL2(Ω)−|Ω|−1/2k∇vkkL1(Ω).
(cid:13) Ω (cid:13)L2(Ω)
(cid:13) (cid:13)
Now, by (3.1) and (3.2),(cid:13)the right ha(cid:13)nd side of the above inequality converges to
(cid:13) (cid:13)
u L2(Rn) as k . On the other hand, by (1.2) and (1.3), the left hand side is
k∇ k →∞
bounded by κ(Ω) D(vk) L2(Ω). Therefore,passing to the limit and using (3.3), we
k k
obtain:
u L2(Rn) κ(Ω) D(u) L2(Rn) =κ(Ω).
k∇ k ≤ k k
Recalling the definition (1.4) the theorem follows.
Example 3.2. We now show that κ(Q)=√2 for Q=[0,1]2 R2.
⊂
Firstly, observe that (see Theorem 9.4 [12]) L = 0 if and only if Ω has a
Ω
6 { }
rotational symmetry. When this is not the case, then:
(3.4) κ(Ω)=sup k∇ukL2(Ω) ; u W1,2(Ω,Rn), u ~n=0 on ∂Ω .
( D(u) L2(Ω) ∈ · )
k k
In view of (3.4) and Theorem 1.1, it is hence enough to prove that for every u
W1,2(Q,R2) satisfying u1(0,x ) = u1(1,x ) = 0 and u2(x ,0) = u2(x ,0) = 0 fo∈r
2 2 2 1
all x ,x [0,1], there holds:
1 2
∈
(3.5) u2 2 D(u)2.
ˆ |∇ | ≤ ˆ | |
Q Q
Consider first a regular vector field u 2(Q¯,R2). As in (2.2), we obtain:
∈C
1 1
(3.6) D(u)2 = u2+ div u2+ ∂ u2∂ u1 ∂ u1∂ u2 .
ˆ | | 2ˆ |∇ | 2ˆ | | ˆ 1 2 − 1 2
Q Q Q Q
(cid:0) (cid:1)
Note that:
∂ u2∂ u1 ∂ u1∂ u2 = ∂ (u2∂ u1) ∂ (u2∂ u1),
ˆ 1 2 − 1 2 ˆ 1 2 −ˆ 2 1
Q Q Q
(cid:0) (cid:1)
and that both terms in the right hand side of the above equality integrate to 0 on
Q,because of the assumedboundary condition. Thus,(3.6)yields (3.5) for u 2.
∈C
It now suffices to check that every u W1,2(Q,R2) with u ~n = 0 on ∂Q,
can be approximated by a sequence of 2(∈Q¯,R2) vector fields sati·sfying the same
C
OPTIMAL CONSTANTS IN RIGIDITY ESTIMATES 7
boundary condition. To this end, define the extensionu¯1 W1,2([0,1] [ 1,2],R)
of the component u1 W1,2(Q,R), by: ∈ × −
∈
u1(x) if x [0,1]
2
∈
x [0,1] x [ 1,2] u¯1(x)= u1(x , x ) if x [ 1,0]
∀ 1 ∈ ∀ 2 ∈ −  1 − 2 2 ∈ −
u1(x ,2 x ) if x [1,2].
 1 2 2
− ∈
Let φ : ( 1,2) R be a nonnegative, smooth and compactly supported function,
− → 
equal to 1 on [0,1]. Then φu¯1 W1,2([0,1] [ 1,2],R), and thus φu¯1 can be ap-
proximatedinW1,2 byasequen∈ceu01 ∞([0×,1−] [ 1,2],R). Clearly,u1 converges
k ∈Cc × − k
to u1 on Q, and each u1(x)=0 whenever x 0,1 .
k 1 ∈{ }
In a similar manner, we construct smooth approximating sequence u2 .
Writing u =(u1,u2) ∞(Q¯,R2), we obtain the desired approximations{ofku}k.≥1
k k k ∈C
Example 3.3. Wenowrecalltheconstruction[12]ofafamilyofdomainsΩh Rn
⊂
parametrised by 0<h 1, with the property that:
≪
κ(Ωh) as h 0.
→∞ →
Let S denote the (n 1)-dimensional unit sphere in Rn and let g :S (0,1) be a
− → 3
smooth function on S. Define:
Ωh = (1+t)x; x S, t hg(x) h,hg(x) .
∈ ∈ −
Clearly, we may requesnt from function g to b(cid:0)e such that no Ωh(cid:1)ohas any rotational
symmetry, and hence L = 0 implies (3.4) for all h.
Let now v :S Rn bΩeh a t{an}gent vector field given by a rotation: v(x)=a x,
for some a Rn.→Define uh W1,2(Ωh,Rn): ×
∈ ∈
uh(x+tx)= (1+t)Id+hx g(x) v(x)=(1+t)(a x)+ a,x g(x) x.
⊗∇ × h ×∇ i
(cid:16) (cid:17)
One can check that uh is tangent at ∂Ωh and that:
k∇uhkL2(Ωh) ≥Ch1/2, kD(uh)kL2(Ωh) ≤Ch3/2.
Henceweconcludetheblow-upofKorn’sconstant: κ(Ωh) Ch−1 inthevanishing
≥
thickness h 0.
→
4. The optimal Korn constant κ(Ω): proofs of Theorems 1.2, 1.3 and
1.4
Proof of Theorem 1.2
1. From (ii) and (iii) we see that the sequences:
u 2χ dx ∞ and D(u )2χ dx ∞
{|∇ k| Ω }k=1 {| k | Ω }k=1
are bounded in the space of Radon measures (Rn). Therefore (possibly passing
to subsequences), they converge weakly in M(Rn) to some µ,ν, concentrated on
Ω¯. That is: M
φ ∞(Rn) lim φ2 u 2 dx= φ2 dµ,
(4.1) ∀ ∈Cc k→∞ˆΩ |∇ k| ˆRn
lim φ2 D(u )2 dx= φ2 dν.
k→∞ˆΩ | k | ˆRn
8 MARTALEWICKAANDSTEFANMU¨LLER
In particular, one has:
(4.2) µ(Ω¯)=κ(Ω)2, ν(Ω¯)=1.
We now assume that:
(4.3) κ(Ω)>κ(Rn),
and derive a contradiction. We will distinguish two cases: when µ(Ω) > 0 and
µ(Ω)=0.
2. First, notice that:
(4.4) φ ∞(Rn) φ2 dµ κ(Ω)2 φ2 dν.
∀ ∈Cc ˆRn ≤ ˆRn
Indeed, for a given φ as above consider the sequence v = φu W1,2(Ω,Rn).
k k
∈
Clearly v ~n=0 on ∂Ω and by Lemma 2.1 we have:
k
·
(4.5) ∇vk−PLΩ ∇vk ≤κ(Ω)kD(vk)kL2(Ω).
(cid:13) Ω (cid:13)L2(Ω)
(cid:13) (cid:13)
Since v = φ u(cid:13) +u φ, the (cid:13)sequence v converges to 0 in Rn×n by
∇ k ∇(cid:13)k k ⊗∇ (cid:13) Ω∇ k
(i). The same convergence must be true for thefflrespective sequence of projections.
Similarly, limk→∞ uk φ L2(Ω) = 0 by (i). Hence (4.5), after passing to the
k ⊗∇ k
limit with k yields:
→∞
lim φ uk L2(Ω) κ(Ω) lim φD(uk) L2(Ω),
k→∞k ∇ k ≤ k→∞k k
which in view of (4.1) proves (4.4).
Assume now that µ(Ω)>0. In this case we are ready to derive a contradiction.
Let B be an open ball, compactly contained in Ω, with µ(B)>0. By (4.4):
(4.6) µ(Ω¯ B) κ(Ω)2ν(Ω¯ B).
\ ≤ \
On the other hand, recalling the definition (1.4) and reasoning exactly as in the
proof of (4.4), we get:
φ ∞(B) φ2 dµ κ(Rn)2 φ2 dν,
∀ ∈Cc ˆ ≤ ˆ
B B
which implies:
(4.7) µ(B) κ(Rn)2ν(B).
≤
Now, both sides of (4.7) are positive, so by (4.3): µ(B) < κ(Ω)2ν(B). Together
with (4.6) this yields:
µ(Ω¯)<κ(Ω)2ν(Ω¯),
contradicting (4.2).
3. It remains to consider the case µ(Ω) = 0, when the measure µ concentrates
on ∂Ω, due to the lack of the equiintegrability of the sequence u 2 ∞ close
{|∇ k| }k=1
to ∂Ω. We will prove that:
(4.8) µ(∂Ω) κ(Rn)2ν(∂Ω).
≤
Both sides of (4.8) are positive, and so (4.3) in view of the assumption µ(Ω) = 0
implies:
µ(Ω¯)=µ(∂Ω)<κ(Ω)2ν(∂Ω) κ(Ω)2ν(Ω¯),
≤
contradicting (4.2). This will end the proof of the theorem.
OPTIMAL CONSTANTS IN RIGIDITY ESTIMATES 9
Towards establishing (4.8), let θ : [0, ) [0,1] be a smooth, non-increasing
∞ −→
function such that:
θ(t)=1 for t [0,1], θ(t)=0 for t 2.
∈ ≥
Define: φ (x) = θ(kdist(x,∂Ω)). For large k we have φ ∞((∂Ω) ) on a small
k k ∈ Cc ǫ
open neighborhood (∂Ω) of ∂Ω. By (4.1), for some increasing sequence n ∞ :
ǫ { k}k=1
(4.9) µ(∂Ω)= lim φ u 2 , ν(∂Ω)= lim φ D(u ) 2 .
k→∞k k∇ nkkL2(Ω) k→∞k k nk kL2(Ω)
To simplify the notation, we will pass to subsequences and write n =k.
k
Define the extension of u on (∂Ω) by reflecting the normal components oddly
k ǫ
and tangential components evenly, across ∂Ω. That is, denoting by π : (∂Ω)
ǫ
−→
∂Ω the projection onto ∂Ω along the normal vectors~n, so that:
x π(x) ~n(π(x)) x (∂Ω) ,
ǫ
− k ∀ ∈
let, for all x (∂Ω)ǫ Ω(cid:0): (cid:1)
∈ \
u (x) ~n(π(x))= u (2π(x) x) ~n(π(x)),
k k
(4.10) · − − ·
u (x) τ =u (2π(x) x) τ τ T ∂Ω.
k k π(x)
· − · ∀ ∈
Since u ~n = 0 on ∂Ω, the above defined extension u is still W1,2 regular. By
k k
·
(1.4) there holds:
(φkuk) L2(Rn) κ(Rn) D(φkuk) L2(Rn).
k∇ k ≤ k k
Again, by taking n in (4.9) converging to sufficiently fast, we may without
k
{ } ∞
loss of generality assume that uk L2(Ω) 1/k2. Therefore:
k k ≤
(4.11) lim φk uk L2(Rn) κ(Rn) lim φkD(uk) L2(Rn).
k→∞k ∇ k ≤ k→∞k k
Consider the quantity:
I = lim φ u 2 φ u 2 .
k→∞(ˆRn\Ω| k∇ k| −ˆΩ| k∇ k| )
After changing the variables in the first integral and noting that:
det (2π(x) x)=det (2 π(x) Id)= 1+ (1)x π(x),
∇ − ∇ − − O | − |
we obtain:
I = lim φ (x) u (2π(x) x)2 φ (x) u (x)2 dx
k→∞ˆΩ | k ∇ k − | −| k ∇ k |
(4.12) n o
= lim u (2π(x) x)2 u (x)2 dx.
k→∞ˆΩ∩{dist(x,∂Ω)<1/k} |∇ k − | −|∇ k |
n o
The definition of extension (4.10) yields now the following identities, for each x
∈
(∂Ω) and each τ,η T ∂Ω:
ǫ π(x)
∈
∂ (u η)(2π(x) x)= 1+ (1)x π(x) ∂ (u η)(x),
τ k τ k
· − O | − | ·
∂ (u η)(2π(x) x(cid:16))= ∂ (u η(cid:17))(x),
~n(π(x)) k ~n(π(x)) k
(4.13) · − − ·
∂ (u ~n(π(x))(2π(x) x)= 1+ (1)x π(x) ∂ (u ~n(π(x))(x),
τ k τ k
· − − O | − | ·
∂ (u ~n(π(x))(2π(x) x(cid:16))=∂ (u ~n(π(x(cid:17)))(x).
~n(π(x)) k ~n(π(x)) k
· − ·
10 MARTALEWICKAANDSTEFANMU¨LLER
Since η∂ v = ∂ (v η) v ∂ η, we see that equating the contribution of all com-
τ k τ k k τ
−
ponents in (4.12) and recalling (iii) we have:
I =0.
In the same manner, (4.13) implies that D(u )(2π(x) x)2 equals to D(u )(x)2
k k
| − | | |
plus lower orderterms whose integrals onΩ dist(x,∂Ω)<1/k vanish, as k
∩{ } −→
. Hence also:
∞
II = lim φ D(u )2 φ D(u )2 =0.
k→∞(ˆRn\Ω| k k | −ˆΩ| k k | )
Therefore:
lim φk uk L2(Rn) =2 lim φk uk L2(Ω),
k→∞k ∇ k k→∞k ∇ k
(4.14)
lim φkD(uk) L2(Rn) =2 lim φkD(uk) L2(Ω).
k→∞k k k→∞k k
Combining (4.14), (4.11) with (4.9) proves (4.8).
Proof of Theorem 1.3
It is enough to assume that A = 0. Let u ∞ be a maximizing sequence
of (1.3), that is: uk W1,2(Ω,0Rn), uk ~n ={ k0}ko=n1∂Ω, D(uk) L2(Ω) = 1 and
∈ · k k
lim u P u =κ(Ω).
k→∞ ∇ k− LΩ Ω∇ k L2(Ω)
By modifying u wffle may, without loss of generality, assume that:
(cid:13) k (cid:13)
(cid:13) (cid:13)
(4.15) PLΩ ∇uk =0, kl→im∞k∇ukkL2(Ω) =κ(Ω).
Using Lemma 2.3 (after possibly passing to a subsequence), we have:
(4.16) u ⇀u weakly in W1,2(Ω,Rn),
k
for some u satisfying u ~n=0 on ∂Ω.
·
Wenowshowthat(1.6)holdswithA =0. Firstofall,byapplyingTheorem1.2
0
to the sequence u , we see that u= 0. Further, (4.16) implies that P u =
lim P {u k=} 0, so: 6 LΩffl ∇
k→∞ LΩ ∇ k
ffl
(4.17) u L2(Ω) κ(Ω) D(u) L2(Ω).
k∇ k ≤ k k
Since P (u u)=0, there also holds:
LΩ ∇ k−
ffl
(uk u) L2(Ω) κ(Ω) D(uk u) L2(Ω).
k∇ − k ≤ k − k
Squaring both sides of the above inequality, passing to the limit with k and
→ ∞
recalling (4.15) and (4.16), we obtain:
κ(Ω)2 u 2 κ(Ω)2 1 D(u) 2 .
−k∇ kL2(Ω) ≤ −k kL2(Ω)
(cid:16) (cid:17)
Together with (4.17) this proves:
u L2(Ω) =κ(Ω) D(u) L2(Ω),
k∇ k k k
yielding the result.
Proof of Theorem 1.4
1. Let E be the set in (1.7). It is clear that u E implies λu E, for all λ R.
∈ ∈ ∈
If u ,u E, then by Lemma 2.1:
1 2
∈
(4.18) ∇ui−PLΩ ∇ui =κ(Ω)kD(ui)kL2(Ω) ∀i=1,2.
(cid:13) Ω (cid:13)L2(Ω)
(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)