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The Asymptotic Theory of Solutions of [delta] U+k2u

Willard L. Miranker

出版	New York University, Institute of Mathematical Sciences, Division of Electromagnetic Research, 1957
URL	http://books.google.com.hk/books?id=PY70929MzY4C&hl=&source=gbs_api
EBook	FULL_PUBLIC_DOMAIN

註釋

The subject of this report is the asymptotic theory of solutions, u, of the reduced wave equation, [delta] u+k2u = 0, defined in infinite domains. In Section 1 we furnish new proofs of three well-known theorems concerning u. These are Rellich's growth estimate, the uniqueness theorem for the exterior boundary-value problem, and the representation theorem. A new result, the representation theorem for u when the boundary of the domain of definition of u is infinite, is also given. In Section 2 Rellich's growth estimate is extended to solutions of the equation [delta] v+k2(x)v = 0. From this result we are able to deduce various uniqueness and representation theorems for solutions of this equation. In Section 3 we show that the normal boundary values of a radiating solution, u, of [delta] u+k2u = 0 is bounded by a homogenous quadratic functional of its boundary values. This result combined with the representation theorem for u yields an L2-maximum principle for u. Finally, in section 4 the behavior of u when the parameter k becomes large is considered. We explain the method of G. Birkhoff for obtaining formal asymptotic expansions for u, and deduce several results concerning the existence and validity of these formal expansions.