arXiv 2006.131932020-06-23math.AP

一类半线性wave equation反问题的唯一性和稳定性

Uniqueness and stability of an inverse problem for a semi-linear wave equation

Matti Lassas, Tony Liimatainen, Leyter Potenciano-Machado, Teemu Tyni

备注：: 32 pages

摘要：

考虑了$\mathbb{R}^{n+1}$，$n\geq 1$上半线性波动方程的势的恢复问题.本文给出了波动方程$\square u +a u^m=0$从Dirichlet-to-Neumann映射恢复未知势$a$的Hölder稳定性估计.我们证明了一个未知的潜在的$a(x,t)$，支持在$Ω\times[t_1,t_2]$，波动方程$\square u +a u^m=0$可以恢复在一个Hölder稳定的方式从地图$u|_{\partial Ω\times [0,T]}\mapsto \langleψ,\partial_νu|_{\partial Ω\times [0,T]}\rangle_{L^2(\partial Ω\times [0,T])}$。这个数据等价于狄利克雷-诺依曼映射与测量函数$ψ$的内积。我们还证明了类似的稳定性结果恢复$a$时，有噪声添加到边界数据。我们使用的方法是建设性的，它是基于高阶线性化。因此，我们也得到一个唯一性结果。我们还详细介绍了方程$\square u +a u^m=0$的正问题。

英文摘要：

We consider the recovery of a potential associated with a semi-linear wave equation on $\mathbb{R}^{n+1}$, $n\geq 1$. We show a Hölder stability estimate for the recovery of an unknown potential $a$ of the wave equation $\square u +a u^m=0$ from its Dirichlet-to-Neumann map. We show that an unknown potential $a(x,t)$, supported in $Ω\times[t_1,t_2]$, of the wave equation $\square u +a u^m=0$ can be recovered in a Hölder stable way from the map $u|_{\partial Ω\times [0,T]}\mapsto \langleψ,\partial_νu|_{\partial Ω\times [0,T]}\rangle_{L^2(\partial Ω\times [0,T])}$. This data is equivalent to the inner product of the Dirichlet-to-Neumann map with a measurement function $ψ$. We also prove similar stability result for the recovery of $a$ when there is noise added to the boundary data. The method we use is constructive and it is based on the higher order linearization. As a consequence, we also get a uniqueness result. We also give a detailed presentation of the forward problem for the equation $\square u +a u^m=0$.

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摘要：

英文摘要：

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