.. _lpotdef:

Layer potential definitions
============================


- `Helmholtz, Laplace, Yukawa <layer_pot.html#hlylpotdef>`__
- `Stokes <layer_pot.html#stokeslpotdef>`__


.. _hlylpotdef:

Helmholtz, Laplace, Yukawa
~~~~~~~~~~~~~~~~~~~~~~~~~~~

For $k \in \mathbb{C}$, let $\mathcal{S}_{k}$, and $\mathcal{D}_{k}$
denote the Helmholtz single and double layer potentials given by

.. math::
   
   \mathcal{S}_{k}[\sigma](x) &= \frac{1}{4\pi}\int_{\Gamma}
   \frac{e^{ik\|x-y\|}}{\|x-y\|} \sigma(y) dS_{y} \\
   \mathcal{D}_{k}[\sigma](x) &= \frac{1}{4\pi}\int_{\Gamma}
   \nabla_{y} \frac{e^{ik\|x-y\|}}{\|x-y\|} \cdot n(y) \sigma(y) dS_{y} \, ,

and $n(y)$ is the normal to the surface $\Gamma$ at $y$. 

The Laplace and Yukawa layer potentials are special cases of the 
Helmholtz layer potentials corresponding to the cases $k=0$, and $k$ 
being purely imaginary respectively.

The operators $\mathcal{S}_{k}'[\sigma]$ and $\mathcal{D}_{k}'[\sigma]$ 
denote the principal value or the finite part of the 
Neumann data $\frac{\partial u}{\partial n}$
associated with the layer potentials $u = \mathcal{S}_{k}[\sigma]$ 
and $u = \mathcal{D}_{k}[\sigma]$ respectively.

.. _stokeslotdef:

Stokes
~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let $\mathcal{S}^{\textrm{stok}}$, and $\mathcal{D}^{\textrm{stok}}$
denote the Stokes single and double layer potentials given by

.. math::

   \mathcal{S}^{\textrm{stok}}[\sigma](x) &= \int_{\Gamma}
   \mathcal{G}^{\textrm{stok}}(x,y) \sigma(y) \, dS_{y} \\
   \mathcal{D}^{\textrm{stok}}[\sigma](x) &= \int_{\Gamma}
   \mathcal{T}^{\textrm{stok}} \cdot n(y) \sigma(y) dS_{y} \, ,

where $n(y)$ as before is the normal to the surface $\Gamma$ at $y$, 
$\mathcal{G}^{\textrm{stok}}(x,y)$ is the Stokeslet given by, 

.. math::
   \mathcal{G}^{\textrm{stok}}(x,y)=\frac{1}{8\pi \|x-y\|^3}
   \begin{bmatrix}
   (x_{1}-y_{1})^2 + \|x-y \|^2 & (x_{1}-y_{1})(x_{2}-y_{2}) &
   (x_{1}-y_{1})(x_{3}-y_{3}) \\ 
   (x_{2}-y_{2})(x_{1}-y_{1}) & (x_{2}-y_{2})^2 + \|x-y \|^2 & 
   (x_{2}-y_{2})(x_{3}-y_{3}) \\ 
   (x_{3}-y_{3})(x_{1}-y_{1})  & (x_{3}-y_{3})(x_{2}-y_{2}) & 
   (x_{3}-y_{3})^2 + \|x-y \|^2 
   \end{bmatrix} \, ,

and $\mathcal{T}^{\textrm{stok}}(x,y)$ is the Stresslet whose action on
a vector $v$ is given by

.. math::
   \mathcal{T}^{\textrm{stok}}(x,y) \cdot v = 
   \frac{3(x-y) \cdot v}{4\pi \|x-y \|^5}
   \begin{bmatrix}
   (x_{1}-y_{1})^2 & (x_{1}-y_{1})(x_{2}-y_{2}) &
   (x_{1}-y_{1})(x_{3}-y_{3}) \\ 
   (x_{2}-y_{2})(x_{1}-y_{1}) & (x_{2}-y_{2})^2 & 
   (x_{2}-y_{2})(x_{3}-y_{3}) \\ 
   (x_{3}-y_{3})(x_{1}-y_{1})  & (x_{3}-y_{3})(x_{2}-y_{2}) & 
   (x_{3}-y_{3})^2  
   \end{bmatrix} \, .

The operators $(\mathcal{S}^{\textrm{stok}})'[\sigma]$ 
and $(\mathcal{D}^{\textrm{stok}})'[\sigma]$ 
denote the principal value or the finite part of the 
surface traction
associated with the layer potentials $u = \mathcal{S}^{\textrm{stok}}[\sigma]$ 
and $u = \mathcal{D}^{\textrm{stok}}[\sigma]$ respectively.