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, ..., x_n, y₁, ..., y_n 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

• David Hilbert and Richard Courant (1962). Methods of Mathematical Physics, Vol. 2. Wiley-Interscience. pp. 744–752. ISBN 978-0-471-50439-9.
• Lars Hörmander (2001). "Asgeirsson's Mean Value Theorem and Related Identities". Journal of Functional Analysis (184): 377–401.
• Lars Hörmander (1990). The Analysis of Linear Partial Differential Operators I. Springer-Verlag. pp. Theorem 7.3.4. ISBN 3-540-52343-X.
• Sigurdur Helgason (2000). Groups and Geometric Analysis. American Mathematical Society. pp. 319–323.
• Fritz John (1938). "The Ultrahyperbolic Differential Equation with Four Independent Variables". Duke Math. J. 4 (2): 300–322. doi:10.1215/S0012-7094-38-00423-5.