Closed-Form Exact Inverses of the Weakly Singular and Hypersingular Operators On Disks

We introduce new boundary integral operators which are the exact inverses of the weakly singular and hypersingular operators for the Laplacian on flat disks. Moreover, we provide explicit closed forms for them and prove the continuity and ellipticity of their corresponding bilinear forms in the natural Sobolev trace spaces. This permit us to derive new Calder\'on-type identities that can provide the foundation for optimal operator preconditioning in Galerkin boundary element methods.