Calderón projector

In applied mathematics, the Calderón projector is a pseudo-differential operator used widely in boundary element methods. It is named after Alberto Calderón.

Definition

The interior Calderón projector is defined to be:

${{𝒞}}_{\mathrm{\Omega }}=\left(\begin{array}{cc}(1-\sigma )\mathsf{I}\mathsf{d}-\mathsf{K}& \mathsf{V}\\ \mathsf{W}& \sigma \mathsf{I}\mathsf{d}+{\mathsf{K}}^{\prime }\end{array}\right),$

where $\sigma$ is $\frac{1}{2}$ almost everywhere, $\mathsf{I}\mathsf{d}$ is the identity boundary operator, $\mathsf{K}$ is the double layer boundary operator, $\mathsf{V}$ is the single layer boundary operator, ${\mathsf{K}}^{\prime }$ is the adjoint double layer boundary operator and $\mathsf{W}$ is the hypersingular boundary operator ; on a given bounded domain $\mathrm{\Omega }$ with compact boundary $\mathrm{\Gamma }$.

The exterior Calderón projector on a complementary domain ${\mathrm{\Omega }}^c={\mathbb{ℝ}}^n-\mathrm{\Omega }$ (taking its interior) is defined to be:

${{𝒞}}_{{\mathrm{\Omega }}^c}=\left(\begin{array}{cc}\sigma \mathsf{I}\mathsf{d}+\mathsf{K}& -\mathsf{V}\\ -\mathsf{W}& (1-\sigma )\mathsf{I}\mathsf{d}-{\mathsf{K}}^{\prime }\end{array}\right).$

That is also linked to the interior Calderón projector by ${{𝒞}}_{{\mathrm{\Omega }}^c}={ℐ}-{{𝒞}}_{\mathrm{\Omega }}$.

Calderón identities

As the Calderón operator is a projector ; that means that ${{𝒞}}_{\mathrm{\Omega }}^2={{𝒞}}_{\mathrm{\Omega }}$, the following identities holds :

$VW=\frac{1}{4}Id-K^2$

$WV=\frac{1}{4}Id-K{\text{'}}^2$

$KV=V{K}^{\prime }$

$WK=K^{\prime }W$