# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Blair, M. D., Smith, H. F., Sogge, C. D.: Strichartz estimates for the wave equation on manifolds with boundary. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 1817-1829 (2009) · Zbl 1198.58012 [2] Bouclet, J.-M., Tzvetkov, N.: On global Strichartz estimates for non-trapping metrics. J. Funct. Anal. 254, 1661-1682 (2008) · Zbl 1168.35005 [3] Burq, N., Lebeau, G., Planchon, F.: Global existence for energy critical waves in 3-D domains. J. Amer. Math. Soc. 21, 831-845 (2008) · Zbl 1204.35119 [4] Burq, N., Planchon, F.: Global existence for energy critical waves in 3-d domains : Neumann boundary conditions. Amer. J. Math. 131, 1715-1742 (2009) · Zbl 1184.35210 [5] Cazenave, T., Weissler, F. B.: The Cauchy problem for the critical nonlinear Schrödinger equation in H s . Nonlinear Anal. 14, 807-836 (1990) · Zbl 0706.35127 [6] Eskin, G.: Parametrix and propagation of singularities for the interior mixed hyperbolic prob- lem. J. Anal. Math. 32, 17-62 (1977) · Zbl 0375.35037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.