2020/12/15 by Oana Ivanovici, Ivanovici, Oana
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Ball (mathematics) #Boundary value problem #Center (category theory) #Combinatorics #Counterexample #Dirichlet boundary condition #Discrete mathematics #Dispersion (optics) #FOS: Mathematics #Mathematical analysis #Mathematical physics #Mathematics #Physics #Quantum mechanics #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations #Wave equation
paper · pdf · open access · doi:10.48550/arxiv.2012.08366
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2020/12/15 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We consider the wave equation with Dirichlet boundary conditions in the exterior of the unit ball Bd(0,1) of ℝd. For d=3, we obtain a global in time parametrix and derive sharp dispersive estimates, matching the ℝ3 case, for all frequencies (low and high). For d≥ 4, we provide an explicit solution at large frequency 1/h, h∈ (0,1), with a smoothed Dirac data at a point at distance h-1/3 from the origin in ℝd whose decay rate exhibits h-(d-3)/3 loss with respect to the boundary less case, that occurs at observation points around the mirror image of the source with respect to the center of the ball (at the Poisson-Arago spot). Similar counterexample are obtained for the Schrödinger flow. Moreover, we generalize these counterexamples, first announced in \citeildispext, to the case of the wave and Schrödinger equations outside cylindrical domains of the form B_d_1(0,1)× ℝd_2 in ℝd with d=d_1+d_2 and d_1≥ 4, for which we construct solutions, as done \citeIaIv23 for d_1=2, d_2=1, whose decay rates exhibit a h-(d_1-3)/3 loss with respect to the boundary less case (at observation points around the mirror image of the source with respect to the origin).