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.

The interior Calderón projector is defined to be:^[1]: 137

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

where ${\displaystyle \sigma }$ is ${\displaystyle {\tfrac {1}{2}}}$ almost everywhere, ${\displaystyle {\mathsf {Id}}}$ is the identity boundary operator, ${\displaystyle {\mathsf {K}}}$ is the double layer boundary operator, ${\displaystyle {\mathsf {V}}}$ is the single layer boundary operator, ${\displaystyle {\mathsf {K}}'}$ is the adjoint double layer boundary operator and ${\displaystyle {\mathsf {W}}}$ is the hypersingular boundary operator ; on a given bounded domain ${\displaystyle \Omega }$ with compact boundary ${\displaystyle \Gamma }$.

The exterior Calderón projector on a complementary domain ${\displaystyle \Omega ^{c}=\mathbb {R} ^{n}-\Omega }$ (taking its interior) is defined to be:^[1]: 182

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

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

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

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

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

${\displaystyle KV=VK'}$

${\displaystyle WK=K'W}$