This has been a question I've been asking myself for quite some time now. Is there a physical Interpretation of the Hypersingular Boundary Operator?
First, let me give some motivation why I think there could be one. There is a rather nice physicial interpretation of the Single and Double Layer potential (found here: Physical Interpretation of Single and Double Layer Potentials).
To give a short summary of the Article:
- We can think of the Single Layer Potential as a potential induced by a distribution of charges on the Boundary.
- And the Double Layer Potential of two parallel distributions (as in the single layer case) of opposite sign.
As the Hypersingular Operator arises from the Double Layer, I would think that it would have an analog interpretation.
The Hypersingular Operator W is (most commonly) defined as:
Wφ(x):=−∂nxKφ(x)
for some x∈Γ, where ∂nx is the normal derivative at x and K denotes the double layer boundary integral operator:
Kφ(x):=−14π∫Γφ(y)∂ny1|x−y|dsy
I was wondering, is there some physical, intuitiv or geometric way of thinking about this (or perhaps a paper that could help me gain some intuition)? Or is it merely an Operator meant to "tidy" things up a bit?
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