Extension of the corner-singularity strategy to general corner geometries

Extend the conformal-mapping and singular-decomposition strategy used for a sharp bottom step to more general corner geometries, including explicit shallow-water-parameter dependence and corresponding elliptic regularity estimates.

Background

For the sharp step, the paper uses an explicit Schwarz–Christoffel conformal map to decompose the harmonic potential into an H²-regular component and an explicit corner-singular component. This permits precise tracking of the dependence on the shallow-water parameter and the step height. The authors identify extending this strategy beyond the step geometry as an unresolved problem.