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Stability of inverse problems for ultrahyperbolic equations. (English) Zbl 1335.35295
The authors consider inverse problems of determining a coefficient or a source term in an ultrahyperbolic equation \[ \Delta_y u(x,y)- \Delta_x u(x,y)- p(x,y') u(x,y)= F(x,y), \] where \(x= (x_1,\dots, x_n)\in \mathbb{R}^n\), \(y= (y_1,\dots, y_m)\in \mathbb{R}^m\), \(y'= (y_2,\dots, y_m)\in \mathbb{R}^{m-1}\), \(\Delta_x= \sum^n_{i=1} \partial^2_{x_i}\), \(\Delta_y= \sum^m_{j=1} \partial^2_{y_j}\), by some lateral boundary data.
Consider the bounded domain \(D\subseteq\mathbb{R}^n\) with smooth boundary \(\partial D\), \(T>0\), \(T_1>0\), \(G(t,T_1)= \{y\in\mathbb{R}^m;|y_1|< T, |y'|< T_1\}\), \(G'(T,T_1)\cap \{y_1=0\}\), \(\nu(x)= (\nu_1(x),\dots, \nu_n(x))\), the unit outward normal vector to \(\partial D,\partial_\nu u=(\nabla_x u,\nu)\), \(\nabla_x= (\partial_{x_1},\dots, \partial_{x_n})\), \(\Gamma\subseteq\partial D\), \(\partial D_+= \{x\in\partial D; ((x-x_0,\nu)\geq 0\}\), with \((\cdots)\) being the scalar product in \(\mathbb{R}^n\) or \(\mathbb{R}^m\). The authors consider the system
(1) \(\mathrm{Au}=\Delta_y u(x,y)-\Delta_x u(x,y)- p(x,y') u(x,y)= f(x,y') R(x,y)\), \((x,y)\in D\times G(T,T_1)\),
(2) \(u(x,0,y')= \partial_{y_1} u(x,0,y')= 0\), \((x,y')\in D\times G'(T, T_1)\),
(3) \(u(x,y)= 0\), \((x,y)\in\Gamma\times G(T, T_1)\),
and they use the normed spaces \((S_i,\|\cdot\|_i)\), \(1\leq i\leq 10\).
The authors consider the following hypotheses: \(M>0\) is fixed, \(f\in S_1=L^2(D\times G')\), \(p\in S_2= L^\infty(D\times G')\), \(\| p\|_2\leq M\), \(R\in H^1(-T,T; S_2)\), \(\|\partial_{y_1}R\|_3\leq M\), where \(S_3= L^2(-T, T;S_2)\), \(\| f\|_1\leq M\), \(\|\partial_{y_1} u\|_4\leq M\), where \(S_4= H^2(D\times G)\), \((\exists r_0> 0)\) \((\forall(x,y')\in D\times G')(|R(x,0,y')|\geq r_0)\), \(\max\{|x-x_0|; x\in\overline D\}<\sqrt{\beta T^2+\delta^2}\), where \(0<\beta< 1\), \(\delta> 0\) and \(x_0\not\in\overline D\), \(\partial D\cap\{|x- x_0|\geq \delta\}\subseteq\Gamma\). They denote
\[ \begin{aligned} \Omega(\delta) &= \{(x,y)\in D\times G(T, T_1);|x- x_0|^2- \beta|y|^2> \delta^2\},\\ \Omega'(\delta) &= \Omega(\delta)\cap \{y_1= 0\}.\end{aligned} \] They prove that, for any \(\delta_1> \delta\), there exist \(C> 0\) and \(\theta\in(0,1)\), depending on \(M\) and \(r_0\), such that \(\| f\|_5\leq C\|\partial_\nu \partial_{y_1} u\|^\theta_6\), where \(S_5= L^2(\Omega'(\delta_1))\) and \(S_6= L^2(\Gamma\times G)\).
The authors consider (1), (2), (3) in \(D\times G(T, 2T)\), \(u= 0\) on \(\partial D\times G(T, 2T)\), \(\|\partial^k_{y_1} u\|_7\leq M\), \(k\in \{1,2\}\), \(T>{1\over\sqrt{\beta}}\max\{|x- x_0|; x\in\overline D\}\), \(\|\partial^k_{y_1} R\|_8\leq M\), \(k\in \{1,2\}\), \(|R(x,0,y')|= 0\), \(x\in\overline D\), \(|y'|\leq 2T\), where \(S_7= H^2(D\times G(T,2T))\), \(S_8= L^2(-T, T; L^2(D\times \{|y'|< 2T\}))\), and prove that, for any \(\varepsilon> 0\), there exist constants \(C>0\) and \(\theta\in(0,1)\), depending on \(\varepsilon\), \(M\), \(x_0\), such that \[ \| f\|_9\leq C \sum^2_{k=1} \|\partial_\nu \partial^k_{y_1} u\|_{10}, \] where \(S_9= L^2(D\times\{|y'|< T-\varepsilon\})\), \(S_{10}= L^2(\partial D_+\times G(T, 2T))\).
Finally, they prove Hölder estimates which are global and local and the key tool is the Carleman estimate.

MSC:
35R30 Inverse problems for PDEs
35A25 Other special methods applied to PDEs
35B35 Stability in context of PDEs
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