Home / Math and Physics Files / Math / Sneddon / Sned p203 Ku=mu u and the Lk=delta problem Jim et al / A 2D integral EV problem, first efforts
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Short explanatory note, signed PhL and dated 1.5.11, describing the documents in this folder. They try, without success, to solve the eigenvalue problem Ku = mu u, where K is a 2D integral operator with a 1/distance kernel over a flat surface such as a disk. Phil suspects the problem is ill-posed and notes that no differential operator Lxy with delta-function output has been found; Shankar doubts it exists and Jim found two candidates only for the origin case. He calls it a Stakgold test problem and lists possible follow-ups, though a closing remark says he did not stop working on it.
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Each doc in this folder has a one paragraph summary at the start.
This is the first series of docs in which I try but fail to solve a certain 2D integral eigenvalue problem
∫∫S dx'dy' u(x',y') 1/( ) = μ u(x,y) Ku = μu
where S is some flat surface like a disk. This is the problem suggested by Sneddon on his page 203.
I think there is "something fishy" going on with this problem, it is not well-posed in some sense. I don't know what the boundary conditions should be or not be, and I cannot find a corresponding differential EV equation which ought to involve the operator Lxy where
Lxy(1/) = 4πδ(x-x')δ(y-y')
and this is the subject of the "math problem" doc. Shankar thinks Lxy does not exist. Jim found two candidate Lxy operators for the case x'=y'=0, but not for the equation as stated above.
If I want to pursue it further, I would probably have to go look up Sned's p 203 references and perhaps ask other people like Carleton or Richard Price.
The problem is only of academic interest to me as a Stakgold test problem. It does not have the interest say of the charged bowl problem. Since I still have that bowl problem not fully resolved, and other pending problems and readings as well, I think I will cease and desist here probably forever. Maybe in my readings I will come across something related to this problem. [ But I did not cease and desist...]
-PhL 1.5.11