Layer potential definitions

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

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

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.

Stokes

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

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

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,

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

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

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

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.