### Stability of Lipschitz type in determination of initial heat distribution.

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Let $y\left(h\right)(t,x)$ be one solution to $${\partial}_{t}y(t,x)-\sum _{i,j=1}^{n}{\partial}_{j}\left({a}_{ij}\left(x\right){\partial}_{i}y(t,x)\right)=h(t,x),\phantom{\rule{0.166667em}{0ex}}0\<t\<T,\phantom{\rule{0.166667em}{0ex}}x\in \Omega $$ with a non-homogeneous term $h$, and ${y|}_{(0,T)\times \partial \Omega}=0$, where $\Omega \subset {\mathbb{R}}^{n}$ is a bounded domain. We discuss an inverse problem of determining $n(n+1)/2$ unknown functions ${a}_{ij}$ by $\{{\partial}_{\nu}y\left({h}_{\ell}\right){|}_{(0,T)\times {\Gamma}_{0}}$, $y\left({h}_{\ell}\right)(\theta ,\xb7){\}}_{1\le \ell \le {\ell}_{0}}$ after selecting input sources ${h}_{1},...,{h}_{{\ell}_{0}}$ suitably, where ${\Gamma}_{0}$ is an arbitrary subboundary, ${\partial}_{\nu}$ denotes the normal derivative, $0\<\theta \<T$ and ${\ell}_{0}\in \mathbb{N}$. In the case of ${\ell}_{0}={(n+1)}^{2}n/2$, we prove the Lipschitz stability in the inverse problem if we choose $({h}_{1},...,{h}_{{\ell}_{0}})$ from a set $\mathscr{H}\subset {\left\{{C}_{0}^{\infty}((0,T)\times \omega )\right\}}^{{\ell}_{0}}$ with an arbitrarily fixed subdomain $\omega \subset \Omega $. Moreover we can take ${\ell}_{0}=(n+3)n/2$ by making special choices...

Let be one solution to $${\partial}_{t}y(t,x)-\sum _{i,j=1}^{n}{\partial}_{j}\left({a}_{ij}\left(x\right){\partial}_{i}y(t,x)\right)=h(t,x),\phantom{\rule{0.166667em}{0ex}}0<t<T,\phantom{\rule{0.166667em}{0ex}}x\in \Omega $$ with a non-homogeneous term , and ${y|}_{(0,T)\times \partial \Omega}=0$, where $\Omega \subset {\mathbb{R}}^{n}$ is a bounded domain. We discuss an inverse problem of determining unknown functions by $\{{\partial}_{\nu}y\left({h}_{\ell}\right){|}_{(0,T)\times {\Gamma}_{0}}$, $y\left({h}_{\ell}\right)(\theta ,\xb7){\}}_{1\le \ell \le {\ell}_{0}}$ after selecting input sources ${h}_{1},...,{h}_{{\ell}_{0}}$ suitably, where ${\Gamma}_{0}$ is an arbitrary subboundary, ${\partial}_{\nu}$ denotes the normal derivative, $0<\theta <T$ and ${\ell}_{0}\in \mathbb{N}$. In the case of ${\ell}_{0}={(n+1)}^{2}n/2$, we prove the Lipschitz stability in the inverse problem if we choose $({h}_{1},...,{h}_{{\ell}_{0}})$ from a set $\mathscr{H}\subset {\left\{{C}_{0}^{\infty}((0,T)\times \omega )\right\}}^{{\ell}_{0}}$ with an arbitrarily fixed subdomain $\omega \subset \Omega $. Moreover we can take ${\ell}_{0}=(n+3)n/2$ by making special choices for ${h}_{\ell}$,...

In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lamé coefficients by a single measurement of solution over $(0,T)\times \omega $, where $T\>0$ is a sufficiently large time interval and a subdomain $\omega $ satisfies a non-trapping condition.

In this paper we established the Carleman estimate for the two dimensional Lamé system with the zero Dirichlet boundary conditions. Using this estimate we proved the exact controllability result for the Lamé system with with a control locally distributed over a subdomain which satisfies to a certain type of nontrapping conditions.

In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lamé coefficients by a single measurement of solution over (0,) x ω, where is a sufficiently large time interval and a subdomain satisfies a non-trapping condition.

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