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:Template:Rp

$𝒞=\left(\begin{array}{cc}(1-\sigma )𝖨𝖽-𝖪& 𝖵\\ 𝖶& \sigma 𝖨𝖽+{𝖪}^{\prime }\end{array}\right),$

where $\sigma$ is $\frac{1}{2}$ almost everywhere, $𝖨𝖽$ is the identity boundary operator, $𝖪$ is the double layer boundary operator, $𝖵$ is the single layer boundary operator, ${𝖪}^{\prime }$ is the adjoint double layer boundary operator and $𝖶$ is the hypersingular boundary operator.

The exterior Calderón projector is defined to be:Template:Rp

$𝒞=\left(\begin{array}{cc}\sigma 𝖨𝖽+𝖪& -𝖵\\ -𝖶& (1-\sigma )𝖨𝖽-{𝖪}^{\prime }\end{array}\right).$