On blow-up solutions to the nonlinear Schrödinger equation in the exterior of a convex obstacle

In this paper, we consider the Schr\\"odinger equation with a-supercritical focusing nonlinearity, in the exterior of a smooth, compact, obstacle of \ᵈ with Dirichlet boundary conditions. We prove that with negative energy blow up in finite time. Assuming furthermore the nonlinearity is energy-subcritical, we also prove (under additional conditions) blow-up with the same optimal ground-state criterion than the work of Holmer and Roudenko on \ᵈ. The classical proof of Glassey, on the concavity of the variance, fails in the exterior of an obstacle of the appearance of boundary terms with an unfavorable sign in the derivative of the variance. The main idea of our proof is to introduce a modified variance which is bounded from below and strictly concave for the that we consider.