Uniform Sobolev estimates for general Lipschitz bathymetries

Establish uniform higher-regularity estimates in the shallow-water parameter for harmonic extensions and the Dirichlet–Neumann operator on general Lipschitz fluid domains, despite possible bottom-corner singularities.

Background

The paper develops uniform estimates at the energy level for linear water waves over Lipschitz bottoms, but higher-order estimates are more difficult because the harmonic extension may develop singularities at corners. For smooth bottoms, straightening diffeomorphisms and elliptic estimates can remove the singular dependence on the shallow-water parameter. The authors state that it is unclear whether an analogous procedure works for a general Lipschitz domain, and then restrict the detailed analysis to a discontinuous step geometry.