论文标题
波浪的分散和schrödinger方程在球外和反例外
Dispersion for the wave and Schrödinger equations outside a ball and counterexamples
论文作者
论文摘要
我们考虑了单位球外部$ b_ {d}（0,1）$ $ \ mathbb {r}^d $的波动方程。对于$ d = 3 $，我们获得了全局时间参数，并得出尖锐的分散估计，与$ \ mathbb {r}^{3} $ case匹配所有频率（低和高）。对于$ d \ geq 4 $，我们提供了一个明确的解决方案，以$ 1/h $，$ h \ in（0,1）$提供了明确的解决方案，从$ \ mathbb {r}^d的decay率$ h^nime the Outs $ h^（d-d-d-d}的范围$ \ mathbb {r}^d的距离为$ h^{ - 1/3} $ hime的dirac数据，以$ \ mathbb {r}相对于球的中心（位于泊松 - 阿拉莫景点）的镜像图像。对于Schr {Ö} dinger流量，获得了类似的反例。此外，我们概括了这些反例，最初是在\ cite {iLdisPext}中首次宣布的，这是波浪和schr {Ö} dinger方程的情况。形式$ b \ _ {d \ _1}（d \ _1}（0,1,1）\ times \ times \ times \ mathbb {d \ mathbb {r}^r}^r}^r}^d = n cylindrical域之外$ \ MATHBB {r}^d $ with $ d = d = d \ _1+d \ _2 $和$ d \ _1 \ geq 4 $，我们为其构建解决方案，如完成\ cite {iaiv23} for $ d \ _1 = 2 = 2 $，$ d = 2 $，$ d $，$ d \ _2 = 1 $ $ h^{ - （d \ _1-3）/3} $相对于边界较小的情况（在源镜像相对于原点的镜像图像围绕镜像）的损失）。
We consider the wave equation with Dirichlet boundary conditions in the exterior of the unit ball $B_{d}(0,1)$ of $\mathbb{R}^d$. For $d=3$, we obtain a global in time parametrix and derive sharp dispersive estimates, matching the $\mathbb{R}^{3}$ case, for all frequencies (low and high). For $d\geq 4$, we provide an explicit solution at large frequency $1/h$, $h\in (0,1)$, with a smoothed Dirac data at a point at distance $h^{-1/3}$ from the origin in $\mathbb{R}^d$ whose decay rate exhibits $h^{-(d-3)/3}$ loss with respect to the boundary less case, that occurs at observation points around the mirror image of the source with respect to the center of the ball (at the Poisson-Arago spot). Similar counterexample are obtained for the Schr{ö}dinger flow. Moreover, we generalize these counterexamples, first announced in \cite{ildispext}, to the case of the wave and Schr{ö}dinger equations outside cylindrical domains of the form $B\_{d\_1}(0,1)\times \mathbb{R}^{d\_2}$ in $\mathbb{R}^d$ with $d=d\_1+d\_2$ and $d\_1\geq 4$, for which we construct solutions, as done \cite{IaIv23} for $d\_1=2$, $d\_2=1$, whose decay rates exhibit a $h^{-(d\_1-3)/3}$ loss with respect to the boundary less case (at observation points around the mirror image of the source with respect to the origin).
