DECAY ESTIMATES FOR VARIABLE COEFFICIENT WAVE ...

arXiv:0806.3409v2 [math.AP] 28 Aug 2009

DECAY ESTIMATES FOR VARIABLE COEFFICIENT WAVE EQUATIONS IN EXTERIOR DOMAINS JASON METCALFE AND DANIEL TATARU Abstract. In this article we consider variable coefficient, time dependent wave equations in exterior domains R × (Rn \ Ω), n ≥ 3. We prove localized energy estimates if Ω is star-shaped, and global in time Strichartz estimates if Ω is strictly convex.

1. Introduction Our goal, in this article, is to prove analogs of the well known Strichartz estimates and localized energy estimates for variable coefficient wave equations in exterior domains. We consider long-range perturbations of the flat metric, and we take the obstacle to be starshaped. The localized energy estimates are obtained under a smallness assumption for the long range perturbation. Global-in-time Strichartz estimates are then proved assuming the local-in-time Strichartz estimates, which are known to hold for strictly convex obstacles. For the constant coefficient wave equation 2 = ∂t2 − ∆ in R × Rn , n ≥ 2, we have that solutions to the Cauchy problem (1)

2u = f,

u(0) = u0 ,

∂t u(0) = u1 ,

satisfy the Strichartz estimates1 k|Dx |−ρ1 ∇ukLp1 Lq1 . k∇u(0)kL2 + k|Dx |ρ2 2ukLp′2 Lq2′ , for Strichartz admissible exponents (ρ1 , p1 , q1 ) and (ρ2 , p2 , q2 ). Here, exponents (ρ, p, q) are called Strichartz admissible if 2 ≤ p, q ≤ ∞, 2 n n 1 2 n − 1 1− , ρ= − − , ≤ 2 q p p 2 q and (ρ, p, q) 6= (1, 2, ∞) when n = 3. The work of the first author was supported in part by NSF grant DMS0800678. The work of the second author was supported in part by NSF grants DMS0354539 and DMS0301122. 1Here and throughout, we shall use ∇ to denote a space-time gradient unless otherwise specified with subscripts. 1

The Strichartz estimates follow via a T T ∗ argument and the Hardy-Littlewood-Sobolev inequality from the dispersive estimates, k|Dx |−

n+1 (1− 2q ) 2

∇u(t)kLq . t−

n−1 (1− 2q ) 2

ku1kLq′ ,

2≤q