We consider a potential type perturbation of the three dimensional wave equation and we establish a dispersive estimate for the associated propagator. The main estimate is proved under the assumption that the potential satisfies where .
For optimal control problems with ordinary differential equations where the -norm of the control is minimized, often bang-bang principles hold. For systems that are governed by a hyperbolic partial differential equation, the situation is different: even if a weak form of the bang-bang principle still holds for the wave equation, it implies no restriction on the form of the optimal control. To illustrate that for the Dirichlet boundary control of the wave equation in general not even feasible...
Let be a long range metric perturbation of the Euclidean Laplacian on , . We prove local energy decay for the solutions of the wave, Klein-Gordon and Schrödinger equations associated to . The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group where has a suitable development at zero (resp. infinity).
Let £ denote the sub-Laplacian on the Heisenberg group Hm. We prove that ei√£ / (1 - £)α/2 extends to a bounded operator on Lp(Hm), for 1 ≤ p ≤ ∞, when α > (d - 1) |1/p - 1/2|.