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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.

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