We consider an anisotropic model case for a strictly convex domain of dimension with smooth boundary and we describe dispersion for the semiclassical Schrödinger equation with Dirichlet boundary condition. More specifically, we obtain the following fixed time decay rate for the linear semiclassical flow: a loss of occurs with respect to the boundaryless case due to repeated swallowtail-type singularities, and is proven optimal. Corresponding Strichartz estimates allow us to solve the cubic nonlinear Schrödinger equation on such a three-dimensional model convex domain, hence matching known results on generic compact boundaryless manifolds.
Cite this article
Oana Ivanovici, Dispersive estimates for the Schrödinger equation in a model convex domain and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 40 (2023), no. 4, pp. 959–1008DOI 10.4171/AIHPC/75