The Generalized Dock Problem

TL;DR

This paper establishes the existence and uniqueness of solutions for a linearized water wave problem in a domain with a corner, using boundary value problem techniques and properties of the Dirichlet to Neumann map.

Contribution

It introduces a novel approach to analyze water wave problems in corner domains by linking boundary value problems with operator theory and proving key properties of the Dirichlet to Neumann map.

Findings

Existence and uniqueness of solutions in corner domains.

The Dirichlet to Neumann map is a positive self-adjoint operator.

Reduction of the water wave problem to an operator equation.

Abstract

We show existence and uniqueness for a linearized water wave problem in a two dimensional domain $G$ with corner, formed by two semi-axis $\Gamma_1$ and $\Gamma_2$ which intersect under an angle $\alpha\in (0,\pi ]$. The existence and uniqueness of the solution is proved by considering an auxiliary mixed problem with Dirichlet and Neumann boundary conditions. The latter guarantees the existence of the Dirichlet to Neumann map. The water wave boundary value problem is then shown to be equivalent to an equation like $v_{tt}+g\Lambda v=P$ with initial conditions, where $t$ stands for time, $g$ is the gravitational constant, $P$ means pressure, and $\Lambda$ is the Dirichlet to Neumann map. We then prove that $\Lambda$ is a positive self-adjoint operator.