# 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

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topYuan, 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 -

## References

top- R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
- K.A. Ames and B. Straughan, Non-standard and Improperly Posed Problems. Academic Press, San Diego (1997).
- L. Baudouin and J.-P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation. Inverse Probl.18 (2002) 1537–1554.
- M. Bellassoued, Global logarithmic stability in inverse hyperbolic problem by arbitrary boundary observation. Inverse Probl.20 (2004) 1033–1052.
- 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.
- H. Brezis, Analyse Fonctionnelle. Masson, Paris (1983).
- A.L. Bukhgeim, Introduction to the Theory of Inverse Probl. VSP, Utrecht (2000).
- A.L. Bukhgeim and M.V. Klibanov, Global uniqueness of a class of multidimensional inverse problems. Soviet Math. Dokl.24 (1981) 244–247.
- 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.
- 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.
- P.G. Danilaev, Coefficient Inverse Problems for Parabolic Type Equations and Their Application. VSP, Utrecht (2001).
- 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.
- M.M. Eller and V. Isakov, Carleman estimates with two large parameters and applications. Contemp. Math.268 (2000) 117–136.
- C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Royal Soc. Edinburgh125A (1995) 31–61.
- A.V. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations, in Lecture Notes Series34, Seoul National University, Seoul, South Korea (1996).
- D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin (2001).
- R. Glowinski and J.L. Lions, Exact and approximate controllability for distributed parameter systems. Acta Numer.3 (1994) 269–378.
- L. Hörmander, Linear Partial Differential Operators. Springer-Verlag, Berlin (1963).
- O.Yu. Imanuvilov, Controllability of parabolic equations. Sb. Math.186 (1995) 879–900.
- O.Yu. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate. Inverse Probl.14 (1998) 1229–1245.
- O.Yu. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations. Inverse Probl.17 (2001) 717–728.
- 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.
- O.Yu. Imanuvilov and M. Yamamoto, Determination of a coefficient in an acoustic equation with a single measurement. Inverse Probl.19 (2003) 151–171.
- 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.
- V. Isakov, Inverse Problems for Partial Differential Equations. Springer-Verlag, Berlin (1998), (2005).
- V. Isakov and S. Kindermann, Identification of the diffusion coefficient in a one-dimensional parabolic equation. Inverse Probl.16 (2000) 665–680.
- M. Ivanchov, Inverse Problems for Equations of Parabolic Type. VNTL Publishers, Lviv, Ukraine (2003).
- A. Khaĭdarov, Carleman estimates and inverse problems for second order hyperbolic equations. Math. USSR Sbornik58 (1987) 267–277.
- M.V. Klibanov, Inverse problems in the “large” and Carleman bounds. Diff. Equ.20 (1984) 755–760.
- M.V. Klibanov, Inverse problems and Carleman estimates. Inverse Probl.8 (1992) 575–596.
- M.V. Klibanov, Estimates of initial conditions of parabolic equations and inequalities via lateral Cauchy data. Inverse Probl.22 (2006) 495–514.
- M.V. Klibanov and A.A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. VSP, Utrecht (2004).
- M.V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an accoustic equation. Appl. Anal.85 (2006) 515–538.
- M.M. Lavrent'ev, V.G. Romanov and Shishat$\xb7$skiĭ, Ill-posed Problems of Mathematical Physics and Analysis. American Mathematical Society Providence, Rhode Island (1986).
- J.L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin (1972).
- L.E. Payne, Improperly Posed Problems in Partial Differential Equations. SIAM, Philadelphia (1975).
- A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).
- J.C. Saut and B. Scheurer, Unique continuation for some evolution equations. J. Diff. Eq.66 (1987) 118–139.
- 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.
- M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems. J. Math. Pures Appl.78 (1999) 65–98.
- M. Yamamoto and J. Zou, Simultaneous reconstruction of the initial temperature and heat radiative coefficient. Inverse Probl.17 (2001) 1181–1202.

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