Sharp Bounds for Ground State Eigenfunctions on Domains with Horns and Cusps

We prove a lower bound on the ground state eigenfunction of the Dirichlet Laplacian for horn-shaped and cusped domains. The novelty of the method is that for horn-shaped domains it implies a new result for dimension two, and more importantly, is valid for higher dimensions. Moreover, the same technique applies just as easily to cusps. Finally, we construct an example to show that the estimates are sharp.