Singular metrics and associated conformal groups underlying ...

2 K. R. Payne
L = K(y)∆x + ∂ 2 y
(1.1)
where (x, y) ∈ R N × R, ∆x is the Laplace operator on R N with N ≥ 1, and
the coefficient K ∈ C 0 (R) satisfies
K(0) = 0 and K(y) = 0 for y = 0. (1.2)
The equation degenerates along the hypersurface y = 0 and our main interest
will be cases in which K yields a change of type; that is, K also satisfies
yK(y) > 0 for y = 0 (1.3)
so that the operator (1.1) is of mixed type (elliptic for y > 0 and hyperbolic
for y < 0). However, much of what will be discussed depends only on the
form of the degeneracy in (1.1) − (1.2).
In the paper [15], we classified the symmetry groups and calculated the
associated conservation laws for the equation Lu = 0, which is the Euler-
Lagrange equation for the Lagrangian
L(y, u, ∇u) = 1
K(y)|∇xu|
2
2 + u 2 y . (1.4)
In fact, the class defined by (1.1) − (1.3) represents the simplest examples
of second order equations of mixed type associated to a Lagrangian with
degeneracy on a hypersurface. As is to be expected, the largest possible
symmetry groups occur when K takes a pure power form
K(y) = y|y| m−1 , m > 0 (1.5)
in the mixed type case, or ±|y| m in the purely elliptic/hyperbolic but degenerate
cases. The operator in (1.1) with (1.5) is known as the Gellerstedt
operator and gives the Tricomi operator when K(y) = y, while the choice
K(y) = y 2 yields the degenerate elliptic Grushin operator. Such operators
arise in many physical and geometrical problems with a particular structure,
such as: transonic fluid flow [3] [16], quantum cosmology [11], and the imbedding
of manifolds with curvature that changes sign [13]. See also section 6 of
[15] for a brief discussion.
If one takes the limiting case, m = 0 in (1.5), one arrives at the Laurentiev-
Bitsadze operator which glues the Laplacian for y > 0 to the D’Alembertian
for y < 0. It is well known that the symmetry groups for these classical
operators are given by the group of conformal transformations with respect
to the Euclidian and Minkowski metrics respectively (cf. [17] and [5]). We will
show that this also holds in a suitably interpreted sense for the mixed type
operators satisfying (1.1) and (1.5) with respect to a suitable singular metric
of mixed Riemannian-Lorentzian signature as announced in [15]. Moreover,
the analogous result holds in the purely elliptic/hyperbolic but degenerate
setting of (1.1) with K(y) = ±|y| m , m > 0.