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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:

$W \varphi (x) := -\partial_{n_x} K \varphi(x)$

for some $x\in\Gamma$, where $\partial_{n_x}$ is the normal derivative at $x$ and $K$ denotes the double layer boundary integral operator:

$K\varphi(x) := -\frac{1}{4\pi} \int_\Gamma \varphi(y)\partial_{n_y} \frac{1}{\vert x-y\vert} ds_y$

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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question on notation: do $n_z$ and $n_y$ refer to normals to the surfaces? –  rcollyer Sep 22 '11 at 4:50
Yes, they correspond to the normals. Sorry about that, I edited the post to be more specific. Also edited the variable notation as there was a type so that it makes more sense –  Michael Sep 22 '11 at 13:05
I think I've come up with something, and I should have some time later to post it. –  rcollyer Sep 22 '11 at 13:20
I don't understand: from your question, it is clear that the hypersingular operator is the electric field at x from dipoles of magnitude $\phi(y)$ at each point y on the surface $\Gamma$. What more is there to know about it? It is the also the rate of change of the electrostatic energy with respect to changing the dipole moment, the value of $\phi$, at y, and this is probably how it arises while studying K, but I am guessing. Can you give a more specific reference for the applications? Also, is this specialized thing really called the hypersingular boundary operator, or are there others? –  Ron Maimon Oct 7 '11 at 6:08
@RonMaimon, are you sure the Hypersingular Operator can be viewed as the electric field? That was my interpretation at first, but I've been having doubts. Here is one application: sciencedirect.com/science/article/pii/S0377042700002697 . Judging from the literature, it does seem to be the hypersingular boundary operator, differing only be the kernel. My example above uses the Kernel for the 2-D Laplacian –  Michael Oct 9 '11 at 11:05
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