Symmetric Green's Function for a Class of CIV Boundary Value Problems

Abstract

Generalized boundary value problems are considered for hyperbolic equations of the form utt — uss + λp(s, t)u = 0. By constructing symmetric Green's functions appropriate to such problems the existence of eigenvalues is established.

A boundary value problem for the wave equation

“…Furthermore, in the case where F(x,t) = λu, a complete set of eigenfunctions and corresponding eigenvalues are explicitly computed. In [7] on the other hand, where F(x,t) = λp(x, t)u, such explicit computations are not possible. The selfadjointness of L in L p 2 (R), the set of weighted L 2 functions in R with weight p, and thereby the existence of a complete set of eigenfunctions, is established by constructing a symmetric Hilbert Schmidt kernel [9] for L −1 .…”

“…In this regard, prescribing data along characteristics as formulated by Kalmenov [5] is of special interest. The most recent works in this area have resulted in a number of interesting discoveries [3,4,5,7,8]. Our aim here is to extend some of these results to a more general domain which includes the characteristics of the underlying wave equation as a part of its boundary.…”

“…Also, we had chosen the positive semi axis t = 0 instead of s = f (t), we would have had y = x as a part of the boundary of T . If, in addition, datum on t = 0 was chosen to be zero then the result of [7] would be applicable to v.…”

“…Introduction. The well-known two point boundary value problem for the massspring system has an analog in the continuum case which was first formulated in [5,7] as follows Lu = u tt − u xx = F(x, t), (x, t) ∈ R, (1.1)…”

A characteristic initial value problem for a strictly hyperbolic system

“…In [3], Kreith generalized the result of Kal'menov [1] to the case where separation of variables was not necessarily possible, that is, the problem…”