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

Ganghua Yuan; Masahiro Yamamoto

ESAIM: Control, Optimisation and Calculus of Variations (2009)

  • 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 a i j 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 (2009): 525-554. <http://eudml.org/doc/245358>.

@article{Yuan2009,
abstract = {Let $y(h)(t,x)$ be one solution to\[ \partial \_t y(t,x) - \sum \_\{i, j=1\}^\{n\}\partial \_\{j\} (a\_\{ij\}(x)\partial \_i y(t,x)) = h(t,x), \,0&lt;t&lt;T, \,x\in \Omega \]with a non-homogeneous term $h$, and $y\vert _\{(0,T)\times \partial \Omega \} = 0$, 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 $\lbrace \partial _\{\nu \}y(h_\{\ell \})\vert _\{(0,T)\times \Gamma _0\}$, $y(h_\{\ell \})(\theta ,\cdot )\rbrace _\{1\le \ell \le \ell _0\}$ after selecting input sources $h_1, ..., h_\{\ell _0\}$ suitably, where $\Gamma _0$ is an arbitrary subboundary, $\partial _\{\nu \}$ denotes the normal derivative, $0 &lt; \theta &lt; T$ and $\ell _0 \in \mathbb \{N\}$. 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 $\{\mathcal \{H\}\} \subset \lbrace C_0^\{\infty \} ((0,T)\times \omega )\rbrace ^\{\ell _0\}$ 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},
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/245358},
volume = {15},
year = {2009},
}

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
PY - 2009
PB - EDP-Sciences
VL - 15
IS - 3
SP - 525
EP - 554
AB - Let $y(h)(t,x)$ be one solution to\[ \partial _t y(t,x) - \sum _{i, j=1}^{n}\partial _{j} (a_{ij}(x)\partial _i y(t,x)) = h(t,x), \,0&lt;t&lt;T, \,x\in \Omega \]with a non-homogeneous term $h$, and $y\vert _{(0,T)\times \partial \Omega } = 0$, 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 $\lbrace \partial _{\nu }y(h_{\ell })\vert _{(0,T)\times \Gamma _0}$, $y(h_{\ell })(\theta ,\cdot )\rbrace _{1\le \ell \le \ell _0}$ after selecting input sources $h_1, ..., h_{\ell _0}$ suitably, where $\Gamma _0$ is an arbitrary subboundary, $\partial _{\nu }$ denotes the normal derivative, $0 &lt; \theta &lt; T$ and $\ell _0 \in \mathbb {N}$. 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 ${\mathcal {H}} \subset \lbrace C_0^{\infty } ((0,T)\times \omega )\rbrace ^{\ell _0}$ 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/245358
ER -

References

top
  1. [1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). Zbl0314.46030MR450957
  2. [2] K.A. Ames and B. Straughan, Non-standard and Improperly Posed Problems. Academic Press, San Diego (1997). 
  3. [3] L. Baudouin and J.-P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation. Inverse Probl. 18 (2002) 1537–1554. Zbl1023.35091MR1955903
  4. [4] M. Bellassoued, Global logarithmic stability in inverse hyperbolic problem by arbitrary boundary observation. Inverse Probl. 20 (2004) 1033–1052. Zbl1061.35162MR2087978
  5. [5] M. Bellassoued and M. Yamamoto, Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation. J. Math. Pures Appl. 85 (2006) 193–224. Zbl1091.35112MR2199012
  6. [6] H. Brezis, Analyse Fonctionnelle. Masson, Paris (1983). Zbl0511.46001MR697382
  7. [7] A.L. Bukhgeim, Introduction to the Theory of Inverse Probl. VSP, Utrecht (2000). Zbl0653.35001
  8. [8] A.L. Bukhgeim and M.V. Klibanov, Global uniqueness of a class of multidimensional inverse problems. Soviet Math. Dokl. 24 (1981) 244–247. Zbl0497.35082
  9. [9] D. Chae, O.Yu. Imanuvilov and S.M. Kim, Exact controllability for semilinear parabolic equations with Neumann boundary conditions. J. Dyn. Contr. Syst. 2 (1996) 449–483. Zbl0946.93007MR1420354
  10. [10] J. Cheng and M. Yamamoto, One new strategy for a priori choice of regularizing parameters in Tikhonov’s regularization. Inverse Probl. 16 (2000) L31–L38. Zbl0957.65052MR1776470
  11. [11] P.G. Danilaev, Coefficient Inverse Problems for Parabolic Type Equations and Their Application. VSP, Utrecht (2001). 
  12. [12] A. Elayyan and V. Isakov, On uniqueness of recovery of the discontinuous conductivity coefficient of a parabolic equation. SIAM J. Math. Anal. 28 (1997) 49–59. Zbl0870.35124MR1427727
  13. [13] M.M. Eller and V. Isakov, Carleman estimates with two large parameters and applications. Contemp. Math. 268 (2000) 117–136. Zbl0973.35042MR1804792
  14. [14] C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Royal Soc. Edinburgh 125A (1995) 31–61. Zbl0818.93032MR1318622
  15. [15] A.V. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations, in Lecture Notes Series 34, Seoul National University, Seoul, South Korea (1996). Zbl0862.49004MR1406566
  16. [16] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin (2001). Zbl1042.35002MR1814364
  17. [17] R. Glowinski and J.L. Lions, Exact and approximate controllability for distributed parameter systems. Acta Numer. 3 (1994) 269–378. Zbl0838.93013MR1288099
  18. [18] L. Hörmander, Linear Partial Differential Operators. Springer-Verlag, Berlin (1963). Zbl0108.09301MR404822
  19. [19] O.Yu. Imanuvilov, Controllability of parabolic equations. Sb. Math. 186 (1995) 879–900. Zbl0845.35040MR1349016
  20. [20] O.Yu. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate. Inverse Probl. 14 (1998) 1229–1245. Zbl0992.35110MR1654631
  21. [21] O.Yu. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations. Inverse Probl. 17 (2001) 717–728. Zbl0983.35151MR1861478
  22. [22] O.Yu. Imanuvilov and M. Yamamoto, Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, in Control of Nonlinear Distributed Parameter Systems, Marcel Dekker, New York (2001) 113–137. Zbl0977.93041MR1817179
  23. [23] O.Yu. Imanuvilov and M. Yamamoto, Determination of a coefficient in an acoustic equation with a single measurement. Inverse Probl. 19 (2003) 151–171. Zbl1020.35117MR1964256
  24. [24] O.Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations. Publ. RIMS Kyoto Univ. 39 (2003) 227–274. Zbl1065.35079MR1987865
  25. [25] V. Isakov, Inverse Problems for Partial Differential Equations. Springer-Verlag, Berlin (1998), (2005). Zbl0908.35134MR1482521
  26. [26] V. Isakov and S. Kindermann, Identification of the diffusion coefficient in a one-dimensional parabolic equation. Inverse Probl. 16 (2000) 665–680. Zbl0962.35188MR1766227
  27. [27] M. Ivanchov, Inverse Problems for Equations of Parabolic Type. VNTL Publishers, Lviv, Ukraine (2003). Zbl1147.35110MR2406459
  28. [28] A. Khaĭdarov, Carleman estimates and inverse problems for second order hyperbolic equations. Math. USSR Sbornik 58 (1987) 267–277. Zbl0656.35146MR854975
  29. [29] M.V. Klibanov, Inverse problems in the “large” and Carleman bounds. Diff. Equ. 20 (1984) 755–760. Zbl0573.35083
  30. [30] M.V. Klibanov, Inverse problems and Carleman estimates. Inverse Probl. 8 (1992) 575–596. Zbl0755.35151MR1178231
  31. [31] M.V. Klibanov, Estimates of initial conditions of parabolic equations and inequalities via lateral Cauchy data. Inverse Probl. 22 (2006) 495–514. Zbl1094.35139MR2216411
  32. [32] M.V. Klibanov and A.A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. VSP, Utrecht (2004). Zbl1069.65106MR2126149
  33. [33] M.V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an accoustic equation. Appl. Anal. 85 (2006) 515–538. Zbl1274.35413MR2213073
  34. [34] M.M. Lavrent’ev, V.G. Romanov and Shishat · skiĭ, Ill-posed Problems of Mathematical Physics and Analysis. American Mathematical Society Providence, Rhode Island (1986). Zbl0593.35003
  35. [35] J.L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin (1972). Zbl0223.35039
  36. [36] L.E. Payne, Improperly Posed Problems in Partial Differential Equations. SIAM, Philadelphia (1975). Zbl0302.35003MR463736
  37. [37] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). Zbl0516.47023MR710486
  38. [38] J.C. Saut and B. Scheurer, Unique continuation for some evolution equations. J. Diff. Eq. 66 (1987) 118–139. Zbl0631.35044MR871574
  39. [39] E.J.P.G. Schmidt and N. Weck, On the boundary behavior of solutions to elliptic and parabolic equations – with applications to boundary control for parabolic equations. SIAM J. Contr. Opt. 16 (1978) 593–598. Zbl0388.93027MR497464
  40. [40] M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems. J. Math. Pures Appl. 78 (1999) 65–98. Zbl0923.35200MR1671221
  41. [41] M. Yamamoto and J. Zou, Simultaneous reconstruction of the initial temperature and heat radiative coefficient. Inverse Probl. 17 (2001) 1181–1202. Zbl0987.35166MR1861508

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