• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,345 answers , 22,720 comments
1,470 users with positive rep
818 active unimported users
More ...

  Hypersingular Boundary Operator in Physics

+ 10 like - 0 dislike

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?

This post has been migrated from (A51.SE)

asked Sep 21, 2011 in Theoretical Physics by Michael Kissner (230 points) [ revision history ]
retagged Mar 24, 2014 by dimension10
question on notation: do $n_z$ and $n_y$ refer to normals to the surfaces?

This post has been migrated from (A51.SE)
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

This post has been migrated from (A51.SE)
I think I've come up with something, and I should have some time later to post it.

This post has been migrated from (A51.SE)
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?

This post has been migrated from (A51.SE)
@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: http://www.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

This post has been migrated from (A51.SE)

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights