  We consider the elliptic estimates for Dirichlet-Neumann operator related to
the water-wave problem on a two-dimensional corner domain in this paper. Due to
the singularity of the boundary, there will be singular parts in the solution
of the elliptic problem for D-N operator. To begin with, we study elliptic
problems with mixed boundary condition to derive singularity decompositions and
estimates. Based on the analysis, we present the estimates for both D-N
operator and its shape derivative with the existence of singular parts.
