Singular metrics and associated conformal groups underlying ...

4 K. R. Payne
conservation laws. The infinitesimal generators of the nontrivial symmetries
are given by the vector fields
v T k = ∂
, k = 1, . . . , N (2.10)
∂xk
v D = (m + 2)x · ∇x + 2y ∂
∂y
v R jk = xk
v I k = −d(x, y) ∂
N(m + 2) − 2
− u
2
∂
, (2.11)
∂u
∂ ∂
− xj , 1 ≤ j < k ≤ N (2.12)
∂xj ∂xk
+ 2xkx · ∇x +
∂xk
4 ∂
xky
m + 2 ∂y
N(m + 2) − 2
− xku
m + 2
∂
,
∂u
k = 1, . . . , N (2.13)
and together with the trivial symmetries generate the complete symmetry
group (cf. Theorem 2.5 of [15]). In particular, we recall that the proof exploits
well known infinitesimal techniques for classifying the infinitesimal generators
v =
N
ξ i (x, y, u) ∂
i=1
∂xi
+ η(x, y, u) ∂
∂
+ ϕ(x, y, u)
∂y ∂u
(2.14)
of symmetries, where v is thought of as a vector field which acts an open
subset M of the 0-jet space, R N+1 × U (0) R N+1 × R (the space of values
for independent and dependent variables) together with the action of their
prolongations onto higher order jet spaces (which includes the values of higher
order derivatives of u). In the smooth coefficient case for L where K(y) =
y m , which is degenerate elliptic for m even and of mixed type for m odd,
the complete theory can be applied globally on all of R N+1 . Tedious but
elementary calculations show that the coefficients of each generator v in (2.14)
satisfies: ξ i = ξ i (x, y) and η = η(x, y) are independent of u; ϕ = α(x, y)u +
β(x, y) is linear in u; and
Lα = Lβ = 0 (2.15)
Lξ i = 2y m αxi , i = 1, . . . , N (2.16)
Lη = 2αy
(2.17)
2y m ξ i xi − 2ym ηy − my m−1 η = 0, i = 1, . . . , N (2.18)
m
y ξ i xj + ξj
xi
= 0, 1 ≤ i < j ≤ N (2.19)
y m ηxi + ξ i y = 0, i = 1, . . . , N. (2.20)
For m ∈ R + \ N, the lack of regularity in the coefficients creates no essential
difficulty as each group action preserves the half-spaces R N+1
± . See the
Appendix of [15] for details.