On Refined Local Smoothing Estimates for the Schrödinger Equation in Exterior Domains

Abstract

Abstract. We consider refinements of the local smoothing estimates for the Schrödinger equation in domains which are exterior to a strictly convex obstacle in R n . By restricting the solution to small, frequency dependent collars of the boundary, it is expected that taking its square integral in space-time should exhibit a larger gain in regularity when compared to the usual gain of half a derivative. By a result of Ivanovici, these refined local smoothing estimates are satisfied by solutions in the exterior of a ball. We show that when such estimates are valid, they can be combined with wave packet parametrix constructions to yield Strichartz estimates. This provides an avenue for obtaining these bounds when Neumann boundary conditions are imposed.

Strichartz and localized energy estimates for the wave equation in strictly concave domains

“…When ∂M is not assumed to have this structure, these bounds are new and will be seen as a consequence of Theorem 1.2 below in §4.1. As alluded to above, (2) is the main theorem in [Bl14]. Strictly speaking, the estimates assumed there are for the classical Schrödinger equation (1.2), and take the form (4.9) below, but as noted in [Bl14, §2.2], the estimates for the semiclassical equation which follow are the crucial element in the proof.…”

“…In [Bl14], the author furthered these ideas, proving that whenever a refined family of estimates for the semiclassical Schrödinger equation in strictly concave domains are satisfied, scale invariant Strichartz estimates follow as a result. To state them, given a small dyadic number λ − 2 3 ≤ 2 −j ≤ 1, we define (1.10)…”

“…We begin with proving the refined localized energy estimates in coordinates from Theorem 1.6 in §2, which is actually the heart of the matter. In §3, we prove Theorems 1.5 and 1.3, working in analogy to [Bl14] to obtain the subcritical Strichartz and square function bounds as a consequence of (1.20). The final section is then dedicated to proving Theorem 1.2 and seeing that the estimates in the first half of Theorem 1.1 (namely (1.11)) follow as a corollary.…”

Strichartz Estimates on Exterior Polygonal Domains

“…Scale invariant Strichartz estimates for exterior domains first appeared in Planchon-Vega [29] and Blair-Smith-Sogge [6], though not for the full range of admissible Strichartz pairs. For Strichartz estimates exterior to a smooth, convex obstacle however, scale invariant estimates have been established in the full range of estimates in Ivanovici [20], Ivanovici-Planchon [21], and Blair [3].…”

“…). 3 The reader may note that in obtaining these estimates we have not pursued optimality in F : the solution u should be one derivative more regular. We have avoided this issue owing to the breakdown of the identification of the domain D 1 with a weighted b-Sobolev space (principally relevant in our analysis of the zero-angular-mode below).…”