On the energy critical Schrödinger equation in 3D non-trapping domains

Ivanovici, Oana; Planchon, Fabrice

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 5, pp. 1153-1177.

We prove that the quintic Schrödinger equation with Dirichlet boundary conditions is locally well posed for $H_0^1\left(\Omega \right)$ data on any smooth, non-trapping domain $\Omega \subset {ℝ}^3$. The key ingredient is a smoothing effect in $L_x^5\left(L_t^2\right)$ for the linear equation. We also derive scattering results for the whole range of defocusing sub quintic Schrödinger equations outside a star-shaped domain.

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