The generalized dock problem

Abstract

We show existence and uniqueness for a linearized water wave problem in a two dimensional domain G with corner, formed by two semi-axes Γ₁ and Γ₂ which intersect under an angle α ∈ (0, π]. 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  + gΛv = P_t with initial conditions, where t stands for time, g is the gravitational constant, P means pressure and Λ is the Dirichlet to Neumann map. We then prove that Λ is a positive self-adjoint operator.

Keywords:

• water waves
• corner domain
• Dirichlet to Neumann map

AMS Subject Classifications::

• 35Q35
• 76B15

Acknowledgements

The author would like to thank prof. Gregory Eskin, for pointing out this problem and for the numerous and inspiring conversations during the writing of this article. We are indebted to the anonymous referee for comments and suggestions that improved the manuscript.