On the dispersive estimate for the Dirichlet Schrödinger propagator and applications to energy critical NLS (Q2922905)

One proves the dispersive estimate \( | e^{it\Delta_{D}}(r,s,t) | \leq C_{n} | t | ^{-n/2}\), \(\forall r,s > 1\), \(\forall t \not = 0\), for the kernel of the Dirichlet Laplacian in the exterior of the closed unit ball \(\Omega\) under the radial assumption, for \(n=5\) and \(n=7\). Using this result one proves next the global well-posedness of the mixed problem for the defocusing energy-critical NLS in \(\Omega\), \(i \partial _{t} u + \Delta u = | u | ^{4/(n-2)} u\) with homogeneous Dirichlet boundary conditions and initial data in the radial homogeneous Sobolev space \( \dot{H}_{0}^{1} (\Omega)\). A scattering result for the same problem is also obtained. Similar results for \(n=3\) were previously obtained in [\textit{D. Li} et al., Math. Res. Lett. 19, No. 1, 213--232 (2012; Zbl 1307.35281)].