Sobolev regularity of bivariate isogeometric finite element spaces in case of a geometry map with degenerate corner

We investigate Sobolev regularity of bivariate functions obtained in Isogeometric Analysis when using geometry maps that are degenerate in the sense that the first partial derivatives vanish at isolated points. In particular, we show how the known \(C^1\) $C^1$ -conditions for D-patches have to be tightened to guarantee square integrability of second partial derivatives, as required when computing finite element approximations of elliptic fourth order PDEs like the biharmonic equation.