We consider the elliptic estimates for the Dirichlet–Neumann operator related to the water waves 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 the D–N operator. To begin with, we study elliptic problems with mixed boundary conditions to derive singularity decompositions and estimates. Based on this analysis, we present the estimates for both the D–N operator and its shape derivative with the existence of singular parts.
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