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:

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

where $$\sigma$$ is $$\tfrac12$$ almost everywhere, $$\mathsf{Id}$$ is the identity boundary operator, $$\mathsf{K}$$ is the double layer boundary operator, $$\mathsf{V}$$ is the single layer boundary operator, $$\mathsf{K}'$$ is the adjoint double layer boundary operator and $$\mathsf{W}$$ is the hypersingular boundary operator.

The exterior Calderón projector is defined to be:


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