Front cover image for Plane waves and spherical means applied to partial differential equations

Plane waves and spherical means applied to partial differential equations

Fritz John
Elementary and self-contained, this heterogeneous collection of results on partial differential equations employs certain elementary identities for plane and spherical integrals of an arbitrary function, showing how a variety of results on fairly general differential equations follow from those identities. The author demonstrates that integrals over ordinary spheres and planes can be used to advantage, even for equations unrelated to the ordinary Euclidean metric. The first chapter deals with the decomposition of arbitrary functions into functions of the type of plane waves; i.e., into functions that have parallel planes as level surfaces. Chapter II introduces the first application of the Radon transformation, the solution of the initial value problem for homogeneous hyperbolic equations with constant coefficients. Examined in Chapter III is the construction of the fundamental solution for a linear elliptic equation with analytic coefficients -- and, more generally, for a linear elliptic system. Succeeding chapters derive expressions for an arbitrary function f in terms of spherical integrals of f and bring the identity of Asgeirsson together with a somewhat more general identity due to A. Howard. The final chapters concern the problem of determining a function from its integrals over spheres of radius 1 and extend the results to linear non-elliptic equations.
Print Book, English, 2004
Dover Publications, Mineola, N.Y., 2004