Strichartz estimates for the wave equation on flat cones

2011/05/26 by Matthew D. Blair, Blair, Matthew D., G. Austin Ford +3 · 2 citations
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1105.5410

openalex publication_date 2011/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the solution operator for the wave equation on the flat Euclidean cone over the circle of radius ρ> 0, the manifold ℝ₊ × ℝ / 2 πρℤ equipped with the metric \g(r,θ) = dr² + r² dθ². Using explicit representations of the solution operator in regions related to flat wave propagation and diffraction by the cone point, we prove dispersive estimates and hence scale invariant Strichartz estimates for the wave equation on flat cones. We then show that this yields corresponding inequalities on wedge domains, polygons, and Euclidean surfaces with conic singularities. This in turn yields well-posedness results for the nonlinear wave equation on such manifolds. Morawetz estimates on the cone are also treated.

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