论文标题
半线性波方程的逆问题的独特性和稳定性
Uniqueness and stability of an inverse problem for a semi-linear wave equation
论文作者
论文摘要
我们考虑在$ \ mathbb {r}^{n+1} $，$ n \ geq 1 $上恢复与半线性波方程相关的电势。我们显示了Hölder稳定性估算值，用于从其dirichlet到Neumann地图中恢复Wave方程$ \ Square U +A u^m = 0 $的未知潜力$ a $。我们表明，Wave方程$ \ square U +A U +A U +A U^M = 0 $的未知潜力$ a（x，t）$可以从hölder稳定的方式中恢复为$ω\ times [t_1，t_2] $，从地图$ u | _ {\ partialω\ times [0，t] \langleψ，\partial_νu| _ {\ partialω\ times [0，t]} \ rangle_ {l^2（\ partialω\ times [0，t]）} $。该数据等于具有测量功能$ψ$的Dirichlet到Neumann地图的内部产品。当添加噪声时，我们还证明了$ a $的恢复$ A $的稳定性结果。我们使用的方法是建设性的，它基于高阶线性化。结果，我们也获得了独特的结果。我们还提供了方程$ \ square u +a u^m = 0 $的远期问题的详细介绍。
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 $\square u +a u^m=0$.
