+ u2vi dx(i).

•V = u v

n’2 n

We now have a representative for each of the classical conservation laws

corresponding to a conformal symmetry of our equation

n+2

∆u = Cu n’2 . (3.64)

n+2

3.4. CONSERVATION LAWS FOR ∆u = Cu n’2 115

We say “representative” because a conservation law is actually an equivalence

class of (n ’ 1)-forms. In fact, our •V is not the (n ’ 1)-form classically taken

to represent the conservation law corresponding to V ; our •V involves second

derivatives of the unknown u(x), while the classical expressions are all ¬rst-

order. We can ¬nd the ¬rst-order expressions by adding to •V a suitable exact

(n ’ 1)-form, obtaining

def 2 j

•g = •V + n’2 d(uux v dx(ij))

i

2n

j

u||2 + vi

4 2 2C n’2

’ n’2 ||

= n’2 ux ux v u

i j

n

j

+u2vi + i

2

’ uxj vxj ) dx(i).

u(uxi vxj

n’2

This turns out to give the classical expressions for the conservation laws

associated to our equation (3.64), up to multiplicative constants. It could have

been obtained more directly using the methods of Section 1.3. For this, one

would work on the usual J 1 (Rn, R), with standard coordinates (xi, u, qi) in

which the contact structure is generated by

θ = du ’ qidxi,

and then consider the Monge-Ampere system generated by θ and

n+2

Ψ = ’dqi § dx(i) + Cu n’2 dx.

A little experimenting yields a Lagrangian density

2n

||q||2 n’2

L dx = + 2n Cu dx,

n’2

2

so the functional

Λ = L dy + θ § Lqi dy(i)

induces the Poincar´-Cartan form

e

Π = θ § Ψ = dΛ.

One can then determine the Lie algebra of the symmetry group of Π by solving

an elementary PDE system, with a result closely resembling that of Proposi-

tion 3.4. Applying the Noether prescription to these vector ¬elds and this Λ

yields (n ’ 1)-forms which restrict to transverse Legendre submanifolds to give

•g above.

Returning to our original situation, we now compute •g explicitly for various

choices of g as in Proposition 3.4. These choices of g correspond to subgroups

of the conformal group.

116 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

Translation: g = w ipi .

In this case, we have v i = ’wi , vi = 0, so we ¬nd on a transverse Legendre

submanifold of J 1 (Rn, R+ ) that

2n

u||2wi ’ u i u j wj 2C n’2 i

2 4

||

•g = + u w dx(i).

n’2 x x

n’2 n

The typical use of a conservation law involves its integration along the smooth

(n ’ 1)-dimensional boundary of a region „¦ ‚ Rn. To make more sense of

the preceding expression, we take such a region to have unit normal ν and

area element dσ (with respect to the Euclidean metric), and using the fact that

‚

qi dx(i)|‚„¦ = q, ν dσ for a vector q = q i ‚xi , we have

2n

2

u||2w ’ 4 2C n’2

•g |‚„¦ = n’2 || u, w u+ u w, ν dσ.

n’2 n

‚

Here, we have let w = w i ‚xi be the translation vector ¬eld induced on ¬‚at

conformal space R = Rn ∪ {∞} by the right-invariant vector ¬eld on P which

gives this conservation law.

Rotation: g = ai pixj , ai + aj = 0.

j j i

In this case, we have v i = ’ai xj , vi = aj pj . On a transverse Legendre sub-

j i

manifold of J 1(Rn , R+ ), we have from (3.63) that pi = n’2 u’1uxi , and we ¬nd

2

that

2n j

u||2 + a i xj ’ kj

2 2C n’2 4 2

n’2 ||

•g = u n’2 ux uxk aj x + n’2 uux ai dx(i).

i j

j

n

In this formula, the last term represents a trivial conservation law; that is,

d(uuxj aj dx(i)) = 0 on any transverse Legendre submanifold, so it will be ignored

i

below. Restricting as in the preceding case to the smooth boundary of „¦ ‚ Rn

with unit normal ν and area element dσ, this is

2n

u||2 +

2 2C n’2 4

•g |‚„¦ = || a’

u u, a u, ν dσ.

n’2 n n’2

‚

Here, we have let a = ai xj ‚xi be the rotation vector ¬eld induced on ¬‚at

j

conformal space R by the right-invariant vector ¬eld on P which gives this

conservation law.

Dilation: g = 1 + xipi .

This generating function gives the right-invariant vector ¬eld whose value at the

identity is the Lie algebra element (in blocks of size 1, n, 1)

«

20 0

0 0 0 ,

0 0 ’2

n+2

3.4. CONSERVATION LAWS FOR ∆u = Cu n’2 117

which generates a 1-parameter group of dilations about the origin in ¬‚at con-

formal space R. In this case, we have v i = ’xi , vi = pi , and on a transverse

Legendre submanifold with pi = n’2 u’1 uxi , we ¬nd that

2

2n

u||2 + xi ’ u i u j xj

2 2C n’2 4

|| ’ 2uuxi dx(i).

•g = u (3.65)

n’2 x x

n’2 n

For this conservation law, it is instructive to take for „¦ ‚ Rn the open ball of

radius r > 0 centered at the origin, and then

2n

u||2 + 2

2 2C n’2 4

•g |‚„¦ = r n’2 || ’ ’ 2u

nu u, ν u, ν dσ. (3.66)

n’2

A simple consequence of this conservation law is the following uniqueness theo-

rem.5

n+2

¯

Theorem 3.1 (Pohoˇaev) If u(x) ∈ C 2(„¦) is a solution to ∆u = Cu n’2 in

z

the ball „¦ of radius r, with u ≥ 0 in „¦ and u = 0 on ‚„¦, then u = 0.

Proof. We will ¬rst use the conservation law to show that u = 0 everywhere

on ‚„¦. If we decompose u = u„ + uν ν into tangential and normal compo-

nents along ‚„¦, so that in particular u„ = 0 by hypothesis, then the conserved

integrand (3.66) is

•g |‚„¦ = ’ n’2 u2 dσ,

2r

ν

so the conservation law ‚„¦ •g = 0 implies that uν = 0 on ‚„¦.

Now with u = 0 on ‚„¦, we can compute

—du

0 =

‚„¦

d — du

=

„¦

= ∆u dx.

„¦

But it is clear from the PDE that ∆u cannot change sign, so it must vanish

identically, and this implies that u = 0 throughout „¦.

In fact, looking at the expression (3.65) for •g for a more general region, it is

not hard to see that the same proof applies whenever „¦ ‚ Rn is bounded and

star-shaped.

Inversion: g = ’ 1 pj bj ||x||2 + bj xj (1 + pi xi ).

2

This is the generating function for the vector ¬eld in R = Rn ∪{∞} which is the

conjugate of a translation vector ¬eld by inversion in an origin-centered sphere.

5 See[Poh65], where a non-existence theorem is proved for a more general class of equations,

for which dilation gives an integral identity instead of a conservation law.

118 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

In this case, we have v i = 2 bi||x||2 ’bj xj xi, vi = bixj pj ’bj xipj +bj xj pi +bi ,

1

and on a transverse Legendre submanifold, we ¬nd after some tedious calculation

that

2n

( n’2 || u||2 + (bk xk xj ’ 1 bj ||x||2)

2 2C n’2 4

’

•g = u )δij n’2 uxi uxj

n 2

’2ubj xj uxi + u2bi dx(i).

Again taking „¦ ‚ Rn to be the open ball of radius r > 0 centered at the origin,

we have

2n

(r2 (’4 2

+ || u||2 + + u2)b

C n’2

(n ’ 2)•g |‚„¦ = u, ν u )

n

+2(r2 b, u ’ (n ’ 2)ru b, ν ) u, ν dσ,

‚

where b = bi ‚xi is the vector ¬eld whose conjugate by a sphere-inversion is the

vector ¬eld generating the conservation law.

3.5 Conservation Laws for Wave Equations

In this section, we will consider non-linear wave equations

z = f(z), (3.67)

which are hyperbolic analogs of the non-linear Poisson equations considered

previously. Here, we are working in Minkowski space Ln+1 with coordinates

(t, y1 , . . . , yn ), and the wave operator is

2 2

‚ ‚

=’ + .