On partial spherical means formulas and radiation boundary conditions for the 3+1 wave equation

Lau, Stephen

doi:10.1090/S0033-569X-10-01160-0

Author: Stephen R. Lau
Journal: Quart. Appl. Math. 68 (2010), 179-212
MSC (2000): Primary 35L05, 78A40
DOI: https://doi.org/10.1090/S0033-569X-10-01160-0
Published electronically: February 17, 2010
MathSciNet review: 2662998
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Abstract: We derive various partial spherical means formulas for the 3+1 wave equation. Such formulas, considered earlier by both Weston and Teng, involve only partial integration over a solid angle in addition to history-dependent boundary terms, and are appropriate for faces, edges, and corners of “computational domains”. For example, a hemispherical means formula corresponds to a face (plane boundary). Exploiting the theory of wave front sets for linear operators developed by Hörmander, Warchall has proved theorems which suggest the existence of “one-sided update formulas” for wave equations. We attempt to realize such an update formula via an explicit construction based on our hemispherical means formula. We focus on face points and plane boundaries, but also introduce one-fourth and one-eighth spherical means formulas with most of our arguments going through for a point located on either a domain edge or a corner. Throughout our analysis we encounter a number of, we believe, heretofore unknown identities for classical solutions to the wave equation.

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