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Zbl 0676.35082
Isakov, Victor
On uniqueness of recovery of a discontinuous conductivity coefficient. (English)
[J] Commun. Pure Appl. Math. 41, No.7, 865-877 (1988). ISSN 0010-3640

The author considers three inverse problems related to elliptic equations. The first of them consists in identifying the coefficient a appearing in the Dirichlet problem $$ (1)\quad div(a(x)\nabla u)=0\quad in\quad \Omega \subset {\bbfR}\sp n;\quad (2)\quad u=\phi \quad on\quad \partial \Omega$$ under the assumption that a is a discontinuous function of the form $a=a\sb 0+\chi (\Omega\sp*)b,$ $\Omega\sp*$ and b being unknown. Here $\Omega\sp*$ is an open subset of $\Omega$, $\chi (\Omega\sp*)$ is the characteristic function of $\Omega\sp*$ and the functions $a\in C\sp 2({\bar \Omega})$ and $b\in C\sp 2({\bar \Omega})$ satisfy the relations: i) $0<a\sb 0(x)$ $\forall x\in {\bar \Omega}$; $ii)\quad 0<a\sb 0(x)+b(x)$ $\forall x\in \Omega\sp*$; iii) b(x)$\ne 0$ $\forall x\in \partial \Omega\sp*.$ \par The author proves the uniqueness of the unknown pair $(\Omega\sp*,b)$ when $n=3$, under the assumption that the Dirichlet-Neumann map $\phi \to \partial u(\phi)/\partial N$ is known for any $\phi \in C\sp 1(\partial \Omega)$ with Supp $\phi$ $\subset \Gamma\sb 0$, $\Gamma\sb 0$ being a neighbourhood if $\partial \Omega$. In the case $n=2$ he obtains a weaker result. \par Similar uniqueness results are deduced also in the case where equation (1) is replaced by $$ (1bis)\quad div(a(y)\nabla\sb yu(x,y))=\delta (x),\quad y\in {\bbfR}\sp n, $$ where $\delta$ (x) is the delta function with pole at x and $a\sb 0$ is constant outside a bounded domain $\Omega\sb 0$ with ${\bar \Omega}{}\sb 0\subset \Omega$, $\Omega$ being a convex domain with analytic boundary. \par The additional information needed to prove the uniqueness result for $(\Omega\sp*,b)$ is the following: u(x,$\cdot)$ is assigned on $\Gamma\sb 1$ for any $x\in \Gamma\sb 2$, $\Gamma\sb 1$ and $\Gamma\sb 2$ being two disjoint neighbourhoods in $\partial \Omega.$ \par The third identification problem is related to the anisotropic equation $$ (1ter)\quad div(A(x)\nabla u)=0\quad in\quad \Omega, $$ where the matrix A(x) admits the decomposition $$ A(x)=A\sb 0(x)+\chi (\Omega\sp*)B(x)\quad \forall x\in {\bar \Omega}. $$ Here $A\sb 0(x)$ and B(x) are known positive definite matrices $\forall x\in {\bar \Omega}$, while the open set $\Omega\sp*$ is unknown. In this case the additional information is again the Dirichlet-Neumann map as in the first problem: it ensures the uniqueness of the open set $\Omega\sp*$.
[A.Lorenzi]

MSC 2000:: *35R30 Inverse problems for PDE
35J25 Second order elliptic equations, boundary value problems
35R05 PDE with discontinuous coefficients or data
35A05 General existence and uniqueness theorems (PDE)

Keywords: identifying the coefficient; Dirichlet problem; uniqueness; Dirichlet- Neumann map; convex domain; analytic boundary; anisotropic equation

Cited in: Zbl 0787.35119

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Zentralblatt MATH Berlin [Germany]