Elliptic estimates for Dirichlet-Neumann operator on a corner domain

TL;DR

This paper develops elliptic estimates for the Dirichlet-Neumann operator in corner domains, addressing boundary singularities relevant to water-wave problems, and provides singularity decompositions and estimates for the operator and its shape derivative.

Contribution

It introduces new elliptic estimates for the Dirichlet-Neumann operator in corner domains, accounting for boundary singularities, and derives singularity decompositions and estimates for the operator and its shape derivative.

Findings

Elliptic estimates for D-N operator in corner domains established.

Singularity decompositions for solutions with mixed boundary conditions derived.

Estimates for the shape derivative of D-N operator obtained.

Abstract

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.