Ultrahyperbolic equation

In the mathematical field of partial differential equations, the ultrahyperbolic equation is a partial differential equation for an unknown scalar function u of 2n variables x1, ..., xn, y1, ..., yn of the form

\frac{\partial^2 u}{\partial x_1^2}%2B\cdots%2B\frac{\partial^2 u}{\partial x_n^2}-\frac{\partial^2 u}{\partial y_1^2}-\cdots-\frac{\partial^2 u}{\partial y_n^2}=0.\qquad\qquad(1)

More generally, if a is any quadratic form in 2n variables with signature (n,n), then any PDE whose principal part is a_{ij}u_{x_ix_j} is said to be ultrahyperbolic. Any such equation can be put in the form 1. above by means of a change of variables.[1]

The ultrahyperbolic equation has been studied from a number of viewpoints. On the one hand, it resembles the classical wave equation. This has led to a number of developments concerning its characteristics, one of which is due to Fritz John: the John equation.

The equation has also been studied from the point of view of symmetric spaces, and elliptic differential operators.[2] In particular, the ultrahyperbolic equation satisfies an analog of the mean value theorem for harmonic functions

Notes

  1. ^ See Courant and Hilbert.
  2. ^ See, for instance, Helgasson.

References