Elliptic Regularity Theory for the Robin Laplacian on Domains with an Exterior Cusp 

We study the Laplace operator with Robin boundary conditions on a planar domain Ω ⊂ \mathbb{R}² which has an exterior cusp in 0 ∈ ∂Ω. To this end we introduce weighted L²-based Sobolev spaces W^l_2,β,γ (Ω), that are adapted to the geometric description of Ω near the cuspidal point, and prove the existence of a weak solution in these spaces. In a second step we establish maximal regularity results.
By employing a nonlinear transformation that maps the cuspidal part of Ω onto the semi-infinite strip \mathcal{C}⁺ = \mathbb{R}⁺ × (−1, 1), we reduce the question of elliptic regularity for the Robin Laplacian on Ω to the examination of a perturbed Robin Laplacian on \mathcal{C}⁺. Regularity results for the latter are based on the analysis of the Robin Laplacian on the infinite strip \mathcal{C} = \mathbb{R} × (−1, 1) which proceeds using the Laplace transform. In particular, we establish the existence of solutions in W²_2,β,γ (Ω) and a priori estimates if the weights β and γ satisfy certain conditions.