Fritz John pp 79-101 | Cite as

# The Ultrahyperbolic Differential Equation with Four Independent Variables

Chapter

## Abstract

The properties of a linear homogeneous partial differential equation of the second order for a function, .

$$\sum\limits_{i,k} {{a_{ik}}} \frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_k}}} + \sum\limits_i {{b_i}} \frac{{\partial u}}{{\partial {x_i}}} + cu = 0$$

(1)

*u*(*x*_{1}, • • •,*x*_{n}) are known to depend largely on the index of the quadratic form*Q*(*ξ*) =$$\sum\limits_{i,k} {{a_{ik}}{\xi _i}} {\xi _k}$$

^{1}If by a suitable real linear transformation*Q*can be brought into the form ±(ξ_{1}^{2}+• • • + ξ_{ n }^{2}, i.e., if*Q*is definite, (1) is called an*elliptic*equation. If*Q*can be transformed into ±(ξ_{1}^{2}+ • • • + ξ_{ n-1}^{2}- ξ_{ n }^{2}), the equation is called*normal hyperbolic*. Elliptic and normal hyperbolic equations constitute the two types which have been studied more extensively, besides the case of a parabolic equation for which det (*a*_{ ik }) = 0. Equations which are neither elliptic, nor parabolic, nor normal hyperbolic, i.e., equations for which the corresponding quadratic form*Q*can be written in the form ξ_{1}^{2}+ ξ_{2}^{2}± • • • ± ξ_{ n-2}^{2}- ξ_{ n-1}^{2}- ξ_{ n }^{2}, have scarcely been treated, at least not without restriction to solutions which are analytic in all or some of the variables. For such equations the notation*ultrahyperbolic*has been introduced by R. Courant.## Preview

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## Copyright information

© Springer Science+Business Media New York 1985