Lipschitz stability in the determination of the principal part of a parabolic equation

Ganghua Yuan; Masahiro Yamamoto

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 15, Issue: 3, page 525-554
  • ISSN: 1292-8119

Abstract

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Let y(h)(t,x) be one solution to ∂ t y ( t , x ) - ∑ i , j = 1 n ∂ j ( a i j ( x ) ∂ i y ( t , x ) ) = h ( t , x ) , 0 < t < T , x ∈ Ω with a non-homogeneous term h, and y | ( 0 , T ) × ∂ Ω = 0 , where Ω ⊂ ℝ n is a bounded domain. We discuss an inverse problem of determining n(n+1)/2 unknown functions aij by { ∂ ν y ( h ℓ ) | ( 0 , T ) × Γ 0 , y ( h ℓ ) ( θ , · ) } 1 ≤ ℓ ≤ ℓ 0 after selecting input sources h 1 , . . . , h ℓ 0 suitably, where Γ 0 is an arbitrary subboundary, ∂ ν denotes the normal derivative, 0 < θ < T and ℓ 0 ∈ ℕ . In the case of ℓ 0 = ( n + 1 ) 2 n / 2 , we prove the Lipschitz stability in the inverse problem if we choose ( h 1 , . . . , h ℓ 0 ) from a set ℋ ⊂ { C 0 ∞ ( ( 0 , T ) × ω ) } ℓ 0 with an arbitrarily fixed subdomain ω ⊂ Ω . Moreover we can take ℓ 0 = ( n + 3 ) n / 2 by making special choices for h ℓ , 1 ≤ ℓ ≤ ℓ 0 . The proof is based on a Carleman estimate.

How to cite

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Yuan, Ganghua, and Yamamoto, Masahiro. "Lipschitz stability in the determination of the principal part of a parabolic equation." ESAIM: Control, Optimisation and Calculus of Variations 15.3 (2008): 525-554. <http://eudml.org/doc/90925>.

@article{Yuan2008,
abstract = { Let y(h)(t,x) be one solution to \[ \partial\_t y(t,x) - \sum\_\{i, j=1\}^\{n\}\partial\_\{j\} (a\_\{ij\}(x)\partial\_i y(t,x)) = h(t,x), \thinspace 0<t<T, \thinspace x\in \Omega \] with a non-homogeneous term h, and $y\vert_\{(0,T)\times\partial\Omega\} = 0$, where $\Omega \subset \Bbb R^n$ is a bounded domain. We discuss an inverse problem of determining n(n+1)/2 unknown functions aij by $\\{ \partial_\{\nu\}y(h_\{\ell\})\vert_\{(0,T)\times \Gamma_0\}$, $y(h_\{\ell\})(\theta,\cdot)\\}_\{1\le \ell\le \ell_0\}$ 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 $\ell_0 \in \Bbb N$. In the case of $\ell_0 = (n+1)^2n/2$, we prove the Lipschitz stability in the inverse problem if we choose $(h_1, ..., h_\{\ell_0\})$ from a set $\{\cal H\} \subset \\{ C_0^\{\infty\} ((0,T)\times \omega)\\}^\{\ell_0\}$ with an arbitrarily fixed subdomain $\omega \subset \Omega$. Moreover we can take $\ell_0 = (n+3)n/2$ by making special choices for $h_\{\ell\}$, $1 \le \ell \le \ell_0$. The proof is based on a Carleman estimate. },
author = {Yuan, Ganghua, Yamamoto, Masahiro},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Inverse parabolic problem; Carleman estimate; Lipschitz stability; parabolic equation; Lipshitz stability; inverse problem},
language = {eng},
month = {7},
number = {3},
pages = {525-554},
publisher = {EDP Sciences},
title = {Lipschitz stability in the determination of the principal part of a parabolic equation},
url = {http://eudml.org/doc/90925},
volume = {15},
year = {2008},
}

TY - JOUR
AU - Yuan, Ganghua
AU - Yamamoto, Masahiro
TI - Lipschitz stability in the determination of the principal part of a parabolic equation
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/7//
PB - EDP Sciences
VL - 15
IS - 3
SP - 525
EP - 554
AB - Let y(h)(t,x) be one solution to \[ \partial_t y(t,x) - \sum_{i, j=1}^{n}\partial_{j} (a_{ij}(x)\partial_i y(t,x)) = h(t,x), \thinspace 0<t<T, \thinspace x\in \Omega \] with a non-homogeneous term h, and $y\vert_{(0,T)\times\partial\Omega} = 0$, where $\Omega \subset \Bbb R^n$ is a bounded domain. We discuss an inverse problem of determining n(n+1)/2 unknown functions aij by $\{ \partial_{\nu}y(h_{\ell})\vert_{(0,T)\times \Gamma_0}$, $y(h_{\ell})(\theta,\cdot)\}_{1\le \ell\le \ell_0}$ 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 $\ell_0 \in \Bbb N$. In the case of $\ell_0 = (n+1)^2n/2$, we prove the Lipschitz stability in the inverse problem if we choose $(h_1, ..., h_{\ell_0})$ from a set ${\cal H} \subset \{ C_0^{\infty} ((0,T)\times \omega)\}^{\ell_0}$ with an arbitrarily fixed subdomain $\omega \subset \Omega$. Moreover we can take $\ell_0 = (n+3)n/2$ by making special choices for $h_{\ell}$, $1 \le \ell \le \ell_0$. The proof is based on a Carleman estimate.
LA - eng
KW - Inverse parabolic problem; Carleman estimate; Lipschitz stability; parabolic equation; Lipshitz stability; inverse problem
UR - http://eudml.org/doc/90925
ER -

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