Lipschitz stability in the determination of the principal part of a parabolic equation
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 525-554.

Let $y\left(h\right)\left(t,x\right)$ be one solution to

 ${\partial }_{t}y\left(t,x\right)-\sum _{i,j=1}^{n}{\partial }_{j}\left({a}_{ij}\left(x\right){\partial }_{i}y\left(t,x\right)\right)=h\left(t,x\right),\phantom{\rule{0.166667em}{0ex}}0
with a non-homogeneous term $h$, and ${y|}_{\left(0,T\right)×\partial \Omega }=0$, where $\Omega \subset {ℝ}^{n}$ is a bounded domain. We discuss an inverse problem of determining $n\left(n+1\right)/2$ unknown functions ${a}_{ij}$ by $\left\{{\partial }_{\nu }y\left({h}_{\ell }\right){|}_{\left(0,T\right)×{\Gamma }_{0}}$, $y\left({h}_{\ell }\right)\left(\theta ,·\right){\right\}}_{1\le \ell \le {\ell }_{0}}$ 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 and ${\ell }_{0}\in ℕ$. In the case of ${\ell }_{0}={\left(n+1\right)}^{2}n/2$, we prove the Lipschitz stability in the inverse problem if we choose $\left({h}_{1},...,{h}_{{\ell }_{0}}\right)$ from a set $ℋ\subset {\left\{{C}_{0}^{\infty }\left(\left(0,T\right)×\omega \right)\right\}}^{{\ell }_{0}}$ with an arbitrarily fixed subdomain $\omega \subset \Omega$. Moreover we can take ${\ell }_{0}=\left(n+3\right)n/2$ by making special choices for ${h}_{\ell }$, $1\le \ell \le {\ell }_{0}$. The proof is based on a Carleman estimate.

DOI : https://doi.org/10.1051/cocv:2008043
Classification : 35R30,  35K20
Mots clés : inverse parabolic problem, Carleman estimate, Lipschitz stability
@article{COCV_2009__15_3_525_0,
author = {Yuan, Ganghua and Yamamoto, Masahiro},
title = {Lipschitz stability in the determination of the principal part of a parabolic equation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {525--554},
publisher = {EDP-Sciences},
volume = {15},
number = {3},
year = {2009},
doi = {10.1051/cocv:2008043},
zbl = {1182.35238},
mrnumber = {2542571},
language = {en},
url = {www.numdam.org/item/COCV_2009__15_3_525_0/}
}
Yuan, Ganghua; Yamamoto, Masahiro. Lipschitz stability in the determination of the principal part of a parabolic equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 525-554. doi : 10.1051/cocv:2008043. http://www.numdam.org/item/COCV_2009__15_3_525_0/

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