Abstract
We establish Strichartz estimates for the Schrödinger equation on Riemannian manifolds (Ω, g) with boundary, for both the compact case and the case that Ω is the exterior of a smooth, non-trapping obstacle in Euclidean space. The estimates for exterior domains are scale invariant; the range of Lebesgue exponents (p, q) for which we obtain these estimates is smaller than the range known for Euclidean space, but includes the key \({L^{4}_{t}L^{\infty}_x}\) estimate, which we use to give a simple proof of well-posedness results for the energy critical Schrödinger equation in 3 dimensions. Our estimates on compact manifolds involve a loss of derivatives with respect to the scale invariant index. We use these to establish well-posedness for finite energy data of certain semilinear Schrödinger equations on general compact manifolds with boundary.
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The authors were supported by National Science Foundation grants DMS-0801211, DMS-0654415, and DMS-0555162.
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Blair, M.D., Smith, H.F. & Sogge, C.D. Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary. Math. Ann. 354, 1397–1430 (2012). https://doi.org/10.1007/s00208-011-0772-y
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DOI: https://doi.org/10.1007/s00208-011-0772-y