# Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain

## TL;DR

This paper establishes weighted elliptic estimates for a mixed boundary system in a corner domain, related to the Dirichlet-Neumann operator in water waves, considering singularities and different contact angles.

## Contribution

It provides new weighted estimates for elliptic systems in corner domains, extending understanding of boundary singularities in water-wave related problems.

## Key findings

- Weighted estimates for mixed boundary systems in corner domains.
- Extension of estimates to Dirichlet and Neumann problems with different weights.
- Results applicable to water-wave problems with contact angles.

## Abstract

Based on the $H^2$ existence of the solution, we investigate weighted estimates for a mixed boundary elliptic system in a two-dimensional corner domain, when the contact angle $\om\in(0,\pi/2)$. This system is closely related to the Dirichlet-Neumann operator in the water-waves problem, and the weight we choose is decided by singularities of the mixed boundary system. Meanwhile, we also prove similar weighted estimates with a different weight for the Dirichlet boundary problem as well as the Neumann boundary problem when $\om\in(0,\pi)$.