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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 physical 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 analoguous 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, intuitive 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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  • $\begingroup$ question on notation: do $n_z$ and $n_y$ refer to normals to the surfaces? $\endgroup$
    – rcollyer
    Commented Sep 22, 2011 at 4:50
  • $\begingroup$ 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 $\endgroup$
    – Michael
    Commented Sep 22, 2011 at 13:05

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