Twist maps as energy minimisers in homotopy classes: symmetrisation and the coarea formula

0.63 MB·

by  Charles Morris

Twist maps as energy minimisers in homotopy
classes: symmetrisation and the coarea formula
7
1
C. Morris, A. Taheri†
0
2

n
a Abstract
J Let X=X[a,b]={x:a<|x|<b}⊂Rn with 0<a<b<∞ fixed be

7 an open annulus and consider the energy functional,
2
1(cid:90) |∇u|2
F[u;X]= dx,
] 2 |u|2
P X
A over the space of admissible incompressible Sobolev maps
. (cid:26) (cid:27)
th Aφ(X)= u∈W1,2(X,Rn):det∇u=1 a.e. in X and u|∂X =φ ,
a
m
whereφistheidentitymapofX. Motivatedbytheearlierworks[15,16]in
[ thispaperweexaminethetwistmapsasextremisersofFoverA (X)and
φ
investigatetheirminimalitypropertiesbyinvokingthecoareaformulaand
1
asymmetrisationargument. Inthecasen=2whereA (X)isaunionof
v φ
infinitely many disjoint homotopy classes we establish the minimality of
7
8 theseextremisingtwistsintheirrespectivehomotopyclassesaresultthat
9 then leads to the latter twists being L1-local minimisers of F in Aφ(X).
7 We discuss variants and extensions to higher dimensions as well as to
0 related energy functionals.
.
1
0
7
1 1 Introduction and preliminaries
:
v
i Let X = X[a,b] = {(x ,...,x ) : a < |x| < b} with 0 < a < b < ∞ fixed be an
X 1 n
open annulus in Rn and consider the energy functional
r
a 1(cid:90) |∇u|2
F[u;X]= dx, (1.1)
2 |u|2
X
over the space of incompressible Sobolev maps,
(cid:26) (cid:27)
Aφ(X)= u∈W1,2(X,Rn):det∇u=1 a.e. in X and u|∂X =φ . (1.2)
Here and in future φ denotes the identity map of X and so the last condition in
(1.2) means that u≡x on ∂X in the sense of traces.
1
2 Morris, Taheri
ByatwistmapuonX⊂Rn wemeanacontinuousself-mapofXontoitself
which agrees with the identity map φ on the boundary ∂X and has the specific
spherical polar representation (see [16]-[18] for background and further results)
u:(r,θ)(cid:55)→(r,Q(r)θ), x∈X. (1.3)
Here r = |x| lies in [a,b] and θ = x/|x| sits on Sn−1 with Q ∈ C([a,b],SO(n))
satisfyingQ(a)=Q(b)=I. ThereforeQformsaclosedloopinSO(n)basedat
I and for this in sequel we refer to Q as the twist loop associated with u. Also
note that (1.3) in cartesian form can be written as
u:x(cid:55)→Q(r)x=rQ(r)θ, x∈X. (1.4)
Next subject to a differentiability assumption on the twist loop Q it can be
verified that u∈A (X) with its F energy simplifying to
φ
1(cid:90) |∇u|2 1(cid:90) |∇Q(r)x|2
F[Q(r)x;X]= dx= dx
2 |u|2 2 |x|2
X X
n(cid:90) dx ω (cid:90) b
= + n |Q˙|2rn−1dr, (1.5)
2 |x|2 2
X a
wherethelastequalityuses|∇[Q(r)x]|2 =n+r2|Q˙θ|2. Nowastheprimarytask
here is to search for extremising twist maps we first look at the Euler-Lagrange
equationassociatedwiththeloopenergyE=E[Q]definedbythelastintegralin
(1.5) over the loop space {Q ∈ W1,2([a,b];SO(n)) : Q(a) = Q(b) = I}. Indeed
this can be shown to take the form (see below for justification)
d (cid:104)(cid:16) (cid:17) (cid:105)
rn−1Q˙ Qt =0, (1.6)
dr
with solutions Q(r) = exp[−β(r)A]P, where P ∈ SO(n), A ∈ Rn×n is skew-
symmetric and β =β(|x|) is described for a≤r ≤b by
(cid:40)
ln1/r n=2,
β(r)= (1.7)
r2−n/(n−2) n≥3.
Now to justify (1.6) fix Q ∈ W1,2([a,b],SO(n)) and for F ∈ W1,2([a,b],Rn×n)
0
set H =(F −Ft)Q and Q =Q+(cid:15)H. Then QtQ =I+(cid:15)2HtH and
(cid:15) (cid:15) (cid:15)
dd(cid:15)(cid:90) b2−1|Q˙(cid:15)|2rn−1dr(cid:12)(cid:12)(cid:12)(cid:12) =(cid:90) b(cid:104)Q˙,(F˙ −F˙t)Q+(F −Ft)Q˙(cid:105)rn−1dr
a (cid:15)=0 a
(cid:90) b
= (cid:104)Q˙,(F˙ −F˙t)Q(cid:105)rn−1dr
a
(cid:90) b d
= (cid:104) (rn−1Q˙Qt),(F −Ft)(cid:105)dr =0,
dr
a
and so the arbitrariness of F with an orthogonality argument gives (1.6).
Twist maps as energy minimisers in homotopy classes 3
Returning to (1.1) it is not difficult to see that the Euler-Lagrange equation
associated with F over A (X) is given by the system (cf. Section 4)
φ
|∇u|2 (cid:26)∇u (cid:27)
u+div −p(x)cof∇u =0, u=(u ,...,u ), (1.8)
|u|4 |u|2 1 n
wherep=p(x)isasuitableLagrangemultiplier. Hereafurtheranalysisreveals
thatoutofthesolutionsQ=Q(r)to(1.6)justdescribedonlythosetwistloops
in the form
Q(r)=Rdiag[R[g](r),...,R[g](r)]Rt, R∈SO(n), (1.9)
when n is even and Q(r)≡I (a≤r ≤b) when n is odd can grant extremising
twist maps u for the original energy (1.1). For clarification R[g] denotes the
SO(2) matrix of rotation by angle g:
(cid:20) (cid:21)
cosg sing
R[g]= .
−sing cosg
Indeed direct computations give the angle of rotation g =g(r) to be
log(r/a)
g(r)=2πk +2πm, k,m∈Z, (1.10)
log(b/a)
when n=2 and
(r/a)2−n−1
g(r)=2πk +2πm, k,m∈Z, (1.11)
(b/a)2−n−1
when n ≥ 4 even. (See also [13], [16], [18] for complementing and further
results.)
Our point of departure is (1.4)-(1.9) and the aim is to study the minimising
properties the twist maps calculated above. Of particular interest is the case
n = 2 where the space A (X) admits multiple homotopy classes (A : k ∈ Z).
φ k
Here direct minimisation of the energy over these classes gives rise to a scale of
associated minimisers (u ). Using a symmetrisation argument and the coarea
k
formulaweshowthatthetwistmapsu withtwistanglegaspresentedin(1.10)
k
areindeedenergyminimisersinA andasaresultalsoL1 localminimisersofF
k
overA (X). Wediscussvariantsandextensionsincludingalargerscaleofener-
φ
gies where similar techniques can be applied to establish minimality properties
in homotopy classes.
2 The homotopy structure of the space of self-
maps A = A(X)
The rich homotopy structure of the space of continuous self-maps of the an-
nulus X = X[a,b] ⊂ Rn will prove useful later on in constructing local energy
4 Morris, Taheri
minimisers. For this reason here we give a quick outline of the main tools and
resultsandreferthereaderto[15]forfurtherdetailsandproofs. Tothisendset
A=A(X)={f ∈C(X,X):f =φ on ∂X} equipped with the uniform topology.
A pair of maps f ,f ∈ A are homotopic iff there exists H ∈ C([0,1]×X;X)
0 1
such that, firstly, H(0,x) = f (x) for all x ∈ X, secondly, H(1,x) = f (x) for
0 1
allx∈XandfinallyH(t,x)=xforallt∈[0,1], x∈∂X. Theequivalenceclass
consistingofallg ∈Ahomotopictoagivenf ∈Aisreferredtoasthehomotopy
class of f and is denoted by [f]. Now the homotopy classes {[f] : f ∈ A} can
be characterised as follows depending on whether n=2 or n≥3.
• (n=2) Using polar coordinates, for f ∈A and for θ ∈[0,2π] (fixed), the
S1-valued curve γ defined by
θ
γ :[a,b]→S1 ⊂R2, γ :r (cid:55)→f|f|−1(r,θ),
θ θ
hasawell-definedindexorwindingnumberabouttheorigin. Furthermore,
due to continuity of f, this index is independent of the particular choice
of θ ∈ [0,2π]. This assignment of an integer (or index) to a map f ∈ A
will be denoted by
f (cid:55)→deg(f|f|−1). (2.1)
NotefirstlythatthisintegeralsoagreeswiththeBrouwerdegreeofthemap
resultingfromidentifyingS1 ∼=[a,b]/{a,b},justifiedasaresultofγ (a)=
θ
γ (b)andsecondlythatforadifferentiablecurve(takingadvantageofthe
θ
embedding S1 ⊂C) we have the explicit formulation
1 (cid:90) dz
deg(f|f|−1)= . (2.2)
2πi z
γ
Proposition 2.1. (n=2). The map deg:{[f]:f ∈A}→Z is bijective.
Moreover, for any pair of maps f ,f ∈A, we have
0 1
[f ]=[f ] ⇐⇒ deg(f |f |−1)=deg(f |f |−1). (2.3)
0 1 0 0 1 1
• (n≥3) Using the identification X∼=[a,b]×Sm where for ease of notation
we have set m=n−1 it is plain that for f ∈A the map 1
ω :[a,b]→C (Sm,Sm), ω :r (cid:55)→f|f|−1(r,·),
φ
uniquelydefinesanelementofthefundamentalgroupπ [C (Sm,Sm)]. By
1 φ
consideringtheactionofSO(n)onSm –viewedasitsgroupoforientation
preserving isometries, i.e., through the assignment,
E:ξ ∈SO(n)(cid:55)→ω ∈C(Sm,Sm), (2.4)
where
ω(x)=E[ξ](x)=ξx, x∈Sm, (2.5)
1Here as usual φ denotes the identity map of the m-sphere and C (Sm,Sm) is the path-
φ
connectedcomponentofC(Sm,Sm)containingφ.
Twist maps as energy minimisers in homotopy classes 5
it can be proved that the latter assignment induces a group isomorphism
on the level of the fundamental groups, namely,
E(cid:63) :π [SO(n),I ]∼=π [C (Sm,Sm),φ]∼=Z . (2.6)
1 n 1 φ 2
Thus, summarising, we are naturally lead to the assignment of an integer
mod 2 to any f ∈A which will be denoted by
f (cid:55)→deg (f|f|−1)∈Z . (2.7)
2 2
Proposition 2.2. (n≥3) The degree mod 2 map deg :{[f]:f ∈A}→
2
Z is bijective. Moreover, for a pair of maps f ,f ∈A, we have
2 0 1
[f ]=[f ] ⇐⇒ deg (f |f |−1)=deg (f |f |−1). (2.8)
0 1 2 0 0 2 1 1
3 A countable family of L1 local minimisers of F
when n = 2
When n = 2 by Lebesgue monotonicity and degree theory (see [15] as well as
[9],[11],[14],[19])everymapuinA=A (X)hasarepresentative(againdenoted
φ
u) in A. As a consequence we can introduce the components – hereafter called
the homotopy classes,
(cid:26) (cid:27)
A := u∈A:deg(u|u|−1)=k , k ∈Z. (3.1)
k
Evidently A are pairwise disjoint and their union (over all k ∈ Z) gives A.
k
Furthermore it can be seen without difficulty that each A is W1,2-sequentially
k
weakly closed and that for u∈A and s>0 there exists δ =δ(u,s)>0 with
k
{v :F[v]<s}∩BL1(u)⊂A . (3.2)
δ k
Here BL1(u)={v ∈A:||v−u|| <δ}, that is, the L1-ball in A centred at u.
δ L1
Indeed for the sequential weak closedness fix k and pick (u : j ≥ 1) ⊂ A so
j k
that u (cid:42)u in W1,2. Then by a classical result of Y. Reshetnyak
j
∗
det∇u (cid:42)det∇u (3.3)
j
(as measures) and so u∈A while u →u uniformly on X gives by Proposition
j
2.1 that u ∈ A . For the second assertion arguing indirectly and assuming the
k
contrary there exist u ∈ A , s > 0 and (v : j ≥ 1) in A such that F[v ;X] < s
k j j
and ||v −u|| →0 while v ∈/ A . However by passing to a subsequence (not
j L1 j k
re-labeled) v (cid:42)u in W1,2(X,R2) and as above v →u uniformly on X. Hence
j j
againbyProposition2.1,v ∈A forlargeenoughj whichisacontradiction. (cid:3)
j k
NowinviewofthesequentialweaklowersemicontinuityofFinA(seebelow)
an application of the direct methods of the calculus of variations leads to the
following existence and multiplicity result.
6 Morris, Taheri
Theorem 3.1. (Local minimisers) Let X=X[a,b]⊂R2 and for k ∈Z consider
the homotopy classes A as defined by (3.1). Then there exists u=u(x;k)∈A
k k
such that
F[u;X]= inf F[v;X]. (3.4)
v∈Ak
Furthermore for each such minimiser u there exists δ =δ(u)>0 such that
F[u;X]≤F[v;X], (3.5)
for all v ∈A (X) satisfying ||u−v|| <δ.
φ L1
Proof. Fix k and pick (v )⊂A an infimizing sequence: F[v ]↓L:=inf F[·].
j k j Ak
Then as L < ∞ and a ≤ |v(x)| ≤ b for v ∈ A it follows that by passing to a
subsequence (not re-labeled) v (cid:42) u in W1,2(X,R2) and uniformly in X where
j
by the above discussion u∈A . Now
k
(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)X |∇|vvkk|2|2 −(cid:90)X |∇|uv|k2|2(cid:12)(cid:12)(cid:12)(cid:12)≤(cid:90)X|∇vk|2(cid:18)||v|ku||22|−vk||u2|2|(cid:19)
||v |2−|u|2|(cid:90)
≤sup k |∇v |2 →0
|u|2|v |2 k
X k X
as k (cid:37)∞ together with
(cid:90) |∇u|2 (cid:90) |∇v |2
≤lim k (3.6)
|u|2 |u|2
X X
givesthedesiredlowersemicontinuityoftheFenergyonA (X)asclaimed,i.e.,
φ
(cid:90) |∇u|2 (cid:90) |∇v |2
F[u;X]= ≤lim k =limF[v ;X]. (3.7)
2|u|2 2|v |2 k
X X k
As a result L ≤ F[u] ≤ liminfF[v ] ≤ L and so u is a minimiser as required.
j
To justify the second assertion fix k ∈Z and u as above and with s=1+F[u]
pick δ >0 as in the discussion prior to the theorem. Then any v ∈A satisfying
||u−v|| <δalsosatisfies(3.5)[otherwiseF[v]<F[u]<simplyingthatv ∈A
L1 k
and hence in view of u being a minimiser, F[v]≥F[u] which is a contradiction.]
(cid:3)
4 Twist maps and the Euler-Lagrange equation
associated with F
The purpose of this section is to formally derive the Euler-Lagrange equation
associated with F over A (X). Note that F[u] can in principle be infinite if |u|
φ
is too small or zero, however, for twist maps or more generally Ln-integrable
maps in A , |u| is bounded away from zero as u is a self-map of X onto itself.
φ
Moreoverdet∇uisL1-integrableforthelattermapsbutnotingeneralformaps
u of Sobolev class W1,2 (with n≥3).
Twist maps as energy minimisers in homotopy classes 7
Now the derivation uses the Lagrange multiplier method and proceeds for-
mally by considering the unstrained functional
(cid:90) (cid:20)|∇u|2 (cid:21)
K[u;X]= −p(x)(det∇u−1) dx, (4.1)
2|u|2
X
for suitable p = p(x) where evidently K[u;X] = F[u;X] when u ∈ A (X). We
φ
can calculate the first variation of this energy by setting d/dεK[u ;X]| =0,
ε ε=0
whereu∈A (X)issufficientlyregularandsatisfies|u|≥c>0inX,u =u+εϕ
φ ε
for all ϕ∈C∞(X,Rn) and ε∈R sufficiently small, hence obtaining,
c
0=ddε(cid:90)X(cid:20)|2∇|uuεε||22 −p(x)(det∇uε−1)(cid:21) dx(cid:12)(cid:12)(cid:12)(cid:12)ε=0
 
=(cid:90) (cid:88)n (cid:20) 1 ∂ui −p(x)[cof∇u] (cid:21)∂ϕi −(cid:88)n |∇u|2u ϕ dx
|u|2∂x ij ∂x |u|4 i i
X j j 
i,j=1 i=1
 
=(cid:90) − (cid:88)n ∂ (cid:20) 1 ∂ui −p(x)[cof∇u] (cid:21)ϕ −(cid:88)n |∇u|2u ϕ dx
∂x |u|2∂x ij i |u|4 i i
X j j 
i,j=1 i=1
 
=(cid:90) −(cid:88)n |∇u|2u +(cid:88)n ∂ (cid:20) 1 ∂ui −p(x)[cof∇u] (cid:21)ϕ dx.
|u|4 i ∂x |u|2∂x ij i
X  j j 
i=1 j=1
As this is true for every compactly supported ϕ as above an application of the
fundamental lemma of the calculus of variation results in the Euler-Lagrange
system for u=(u ,...,u ) in X:
1 n
|∇u|2 (cid:26)∇u (cid:27)
u+div −p(x)cof∇u =0, (4.2)
|u|4 |u|2
where the divergence operator is taken row-wise. Proceeding further an appli-
cation of the Piola identity on the cofactor term gives (with 1≤i≤n)
|∇u|2u +(cid:88)n (cid:26) ∂ (cid:18) 1 ∂ui(cid:19)−[cof∇u] ∂p (cid:27)=0. (4.3)
|u|4 i ∂x |u|2∂x ij∂x
j j j
j=1
Next expanding the differentiation further allows us to write
0= |∇u|2u +(cid:88)n (cid:40) 1 ∂2ui − 2 (cid:88)n ∂ui ∂uku −[cof∇u] ∂p (cid:41)
|u|4 i |u|2 ∂x2 |u|4 ∂x ∂x k ij∂x
j=1 j k=1 j j j
= |∇u|2u +(cid:88)n (cid:40) 1 ∂2ui − 2 ∂ui[∇utu] −[cof∇u] ∂p (cid:41). (4.4)
|u|4 i |u|2 ∂x2 |u|4∂x j ij∂x
j=1 j j j
Finally transferring back into vector notation and invoking the incompress-
ibilty condition det∇u=1 it follows in turn that
∆u |∇u|2 2
+ u− ∇u(∇u)tu=(cof∇u)∇p, (4.5)
|u|2 |u|4 |u|4
8 Morris, Taheri
and subsequently
(∇u)t (cid:20) |∇u|2 2 (cid:21)
∆u+ u− ∇u(∇u)tu =∇p. (4.6)
|u|2 |u|2 |u|2
Thus the Euler-Lagrange system (4.2) is equivalent to (4.6) that in particular
asksforthenonlineartermontheleftof (4.6)tobeagradientfieldinX. Recall
from earlier discussion that restricting F to the class of twist maps results in
the Euler-Lagrange equation (1.6) where the solution Q = Q(r) as explicitly
computed is the twist loop associated with the map
u:(r,θ)(cid:55)→(r,Q(r)θ), x∈X, (4.7)
withQ(r)=exp[−β(r)A]P,P ∈SO(n),A∈Rn×nskew-symmetric(At =−A).
The boundary condition u=φ on ∂X gives, 2
exp[−β(a)A]P =I, exp[−β(b)A]P =I. (4.8)
Therefore it must be that P = exp[β(a)A] and exp([β(b)−β(a)]A) = I. Now
as A lies in so(n) it must be conjugate to a matrix S in the Lie algebra of the
standard maximal torus of orthogonal 2-plane rotations in SO(n). This means
that there exists R∈SO(n) such that A=RSRT for some S as described and
soS ∈(β(b)−β(a))−1LwhereL={T ∈t:exp(T)=I},thatis,Listhelattice
in the Lie subalgebra t ⊂ so(n) consisting of matrices sent by the exponential
maptotheidentityI ofSO(n). HenceQ(r)=Rexp(−[β(r)−β(a)]S)Rt. Next
upon noting that the derivatives of β =β(r) are given by
1 n−1
β˙(r)=− , β¨(r)= ,
rn−1 rn
we can write
AQ A2Q AQ
Q˙ = , Q¨ = −(n−1) . (4.9)
rn−1 r2n−2 rn
Now, moving forward, a set of straightforward calculations show that for a
twist map u with a twice continuously differentiable twist loop Q = Q(r) we
have the differential relations
(∇u)t =Qt+rθ⊗Q˙θ,
|∇u|2 =tr[(∇u)t(∇u)]=n+r2|Q˙θ|2, (4.10)
and likewise
(cid:104) (cid:105)
∆u= (n+1)Q˙ +rQ¨ θ. (4.11)
2Thefunctionβ=β(r)wasintroducedearlierinSection1.
Twist maps as energy minimisers in homotopy classes 9
Thus for the particular choice of a twist map with twist loop arising from a
solutionto(1.6)theabovequantitiescanbeexplicitlydescribedbytherelations
(∇u)t =Qt+r2−nθ⊗AQθ, (4.12)
|∇u|2 =tr[(Qt+r2−nθ⊗AQθ)(Q+r2−nAQθ⊗θ)]
|AQθ|2
=n+ , (4.13)
r2(n−2)
and likewise
(cid:20) AQ (cid:18) A2Q AQ(cid:19)(cid:21)
∆u= (n+1) +r −(n−1) θ
rn−1 r2n−2 rn
(cid:20)2AQ A2Q (cid:21)
= + θ. (4.14)
rn−1 r2n−3
For the ease of notation we shall hereafter write ω =Qθ. Proceeding now with
the calculations and using (4.13)-(4.14) we have
|∇u|2 (cid:20) A A2 1(cid:18) |Aω|2(cid:19) (cid:21)
∆u+ u= 2 + + n+ I ω (4.15)
|u|2 rn−1 r2n−3 r r2n−4
and in a similar way
∇u(∇u)tu=(cid:2)Q+r2−nAω⊗θ(cid:3)(cid:2)Qt+r2−nθ⊗Aω(cid:3)ω
|u|2 r
(cid:20) Aω⊗θQt+Qθ⊗Aω Aω⊗Aω(cid:21)ω
= I+ +
rn−2 r2n−4 r
(cid:20) (cid:21)
Aω⊗ω+ω⊗Aω Aω⊗Aω ω
= I+ + .
rn−2 r2n−4 r
1
= (I+r2−nA)ω, (4.16)
r
where the last identity here uses (x⊗y)z = (cid:104)y,z(cid:105)x and (cid:104)ω,ω(cid:105) = (cid:104)Qθ,Qθ(cid:105) = 1
along with (cid:104)Aω,ω(cid:105) = 0 for skew-symmetric A. Hence putting together (4.15)
and (4.16) gives
|∇u|2 ∇u(∇u)t (cid:20)A2+|Aω|2I (cid:21)ω
∆u+ u−2 u= +(n−2)I , (4.17)
|u|2 |u|2 r2n−2 r
which when combined with (4.12) results in
(∇u)t (cid:20) |∇u|2 ∇u(∇u)t (cid:21)
(4.6)= ∆u+ u−2 u
|u|2 |u|2 |u|2
1 (cid:20) θ⊗Aω(cid:21)(cid:20)A2+|Aω|2I (cid:21)ω
= Qt+ +(n−2)I
r2 rn−2 r2n−4 r
(cid:20)A2+|Aω|2I n−2 (cid:21) (θ⊗Aω)A2ω
=Qt + I ω+ . (4.18)
r2n−1 r3 r3n−3
10 Morris, Taheri
Noting (θ⊗Aω)A2ω =(cid:104)Aω,A2ω(cid:105)θ =(cid:104)µ,Aµ(cid:105)=0 with A skew-symmetric and
µ=Aω the last set of equations give
(∇u)t (cid:20) |∇u|2 ∇u(∇u)t (cid:21)
(4.6)= ∆u+ u−2 u
|u|2 |u|2 |u|2
(cid:20)A2+|Aω|2I n−2 (cid:21)ω
=Qt + I =I. (4.19)
r2n−2 r2 r
Therefore to see if (4.6) admits twist solutions it suffices to verify if the
quantity described by (4.19) is a gradient field in X. Towards this end recall
that here we have Q(r)=exp(−β(r)A)P where as seen P =exp(β(a)A). Thus
a basic calculation gives
|Aω|2 =|AQθ|2 =θtQtAtAQθ =−θtQtA2Qθ
=−θtPtexp(β(r)A)A2exp(−β(r)A)Pθ
=−θPtA2Pθ =θPtAtAPθ
=|APθ|2,
and likewise by substitution we have
QtA2ω =PtA2Pθ.
Hence using the above we can proceed by writing the Euler-Lagrange equa-
tion (4.19) upon substitution as,
(∇u)t (cid:20) |∇u|2 ∇u(∇u)t (cid:21)
I= ∆u+ u−2 u
|u|2 |u|2 |u|2
=Pt(cid:0)A2+|APθ|2I(cid:1)P θ +(n−2) θ . (4.20)
r2n−1 r3
Now as for a fixed skew-symmetric matrix B by basic differentiation we have
∇(cid:0)|By|2(cid:1)=−2B2y, ∇|y|2n =2n|y|2n−2y,
it is evident that we can write
(cid:18) |By|2 (cid:19) B2y |By|2y
−∇ = + . (4.21)
2n|y|2n n|y|2n |y|2n+2
In particular with B = PtAP being skew-symmetric, (4.20) can be written in
the form
(∇u)t (cid:20) |∇u|2 ∇u(∇u)t (cid:21)
I= ∆u+ u−2 u
|u|2 |u|2 |u|2
(cid:18)|PtAPx|2(cid:19) PtA2Px 1
=−∇ +(n−1) +−(n−2)∇ . (4.22)
2n|x|2n n|x|2n |x|