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 $a_{ij}u_{x_{i}x_{j}}$ is said to be ultrahyperbolic. Any such equation can be put in the form 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.

In 2008, Walter Craig and Steven Weinstein 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[]

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

References[]

Richard Courant; David Hilbert (1962). Methods of Mathematical Physics, Vol. 2. Wiley-Interscience. pp. 744–752. ISBN 978-0-471-50439-9.
Lars Hörmander (20 August 2001). "Asgeirsson's Mean Value Theorem and Related Identities". Journal of Functional Analysis. 2 (184): 377–401. doi:10.1006/jfan.2001.3743.
Lars Hörmander (1990). The Analysis of Linear Partial Differential Operators I. Springer-Verlag. Theorem 7.3.4. ISBN 978-3-540-52343-7.
Sigurdur Helgason (2000). Groups and Geometric Analysis. American Mathematical Society. pp. 319–323. ISBN 978-0-8218-2673-7.
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.

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[See_Courant_and_Hilbert-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.

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