# Hypersingular Boundary Operator in Physics

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?

• question on notation: do $n_z$ and $n_y$ refer to normals to the surfaces? Commented Sep 22, 2011 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 Commented Sep 22, 2011 at 13:05
• I think I've come up with something, and I should have some time later to post it. Commented Sep 22, 2011 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? Commented Oct 7, 2011 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: doi.org/10.1016/S0377-0427(00)00269-7 . 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 Commented Oct 9, 2011 at 11:05