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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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