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}+\cdots+\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.

Walter Craig and Steven Weinstein recently (2008) proved that under a nonlocal constraint, the initial value problem is well-posed for initial data given on a codimension-one hypersurface.[2]

The equation has also been studied from the point of view of symmetric spaces, and elliptic differential operators.[3] In particular, the ultrahyperbolic equation satisfies an analog of the mean value theorem for harmonic functions

Notes

  1. ↑ See Courant and Hilbert.
  2. ↑ Craig, Walter; Weinstein, Steven. "On determinism and well-posedness in multiple time dimensions". Proc. R. Soc. A vol. 465 no. 2110 3023-3046 (2008). Retrieved 5 December 2013.
  3. ↑ See, for instance, Helgasson.

References


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