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Counterexamples to the Strichartz inequalities for the wave equation in general domains with boundary. (English) Zbl 1254.35035
The author studies Strichartz estimates for the wave equation on a compact manifold of dimension \(d\) with boundary, with Dirichlet boundary condition. We recall that for a given sharp wave admissible pair \( (q,r) \), i.e. \[ \frac{2}{q} + \frac{d-1}{r} = \frac{d-1}{2} , \qquad 2 \leq q \leq \infty, \;2 \leq r < \infty , \] and if one sets \( \gamma := \frac{d+1}{2} \big( \frac{1}{2} - \frac{1}{r} \big) \), then the usual Strichartz estimates for the wave equation on \( {\mathbb R}^d \) read as \[ || u ||_{L^q_t L^r_x} \lesssim || u (0) ||_{\dot{H}^{\gamma}} + || \partial_t u (0) ||_{\dot{H}^{\gamma - 1}} . \] The main result of the paper, which holds in dimensions \(d = 2 , 3 ,4 \), is that if \( r > 4 \) and if there is a bicharacteristic meeting the boundary at a gliding point, then there is an unavoidable additional loss of \( \lambda := \frac{1}{6} \left( \frac{1}{4} - \frac{1}{r} \right) \) derivatives on the initial data. In other words, for this range of pairs, one needs to replace \( \gamma \) by \( \gamma + \lambda \) to obtain Strichartz estimates on a domain. Note that the assumption that there is a gliding point is always fulfilled by smooth and bounded domains of \( {\mathbb R}^d \).

MSC:
35B45 A priori estimates in context of PDEs
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
58J30 Spectral flows
58J32 Boundary value problems on manifolds
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