# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
On uniqueness of recovery of a discontinuous conductivity coefficient. (English) Zbl 0676.35082

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

$\left(1\right)\phantom{\rule{1.em}{0ex}}div\left(a\left(x\right)\nabla u\right)=0\phantom{\rule{1.em}{0ex}}in\phantom{\rule{1.em}{0ex}}{\Omega }\subset {ℝ}^{n};\phantom{\rule{1.em}{0ex}}\left(2\right)\phantom{\rule{1.em}{0ex}}u=\phi \phantom{\rule{1.em}{0ex}}on\phantom{\rule{1.em}{0ex}}\partial {\Omega }$

under the assumption that a is a discontinuous function of the form $a={a}_{0}+\chi \left({{\Omega }}^{*}\right)b,$ ${{\Omega }}^{*}$ and b being unknown. Here ${{\Omega }}^{*}$ is an open subset of ${\Omega }$, $\chi \left({{\Omega }}^{*}\right)$ is the characteristic function of ${{\Omega }}^{*}$ and the functions $a\in {C}^{2}\left(\overline{{\Omega }}\right)$ and $b\in {C}^{2}\left(\overline{{\Omega }}\right)$ satisfy the relations: i) $0<{a}_{0}\left(x\right)$ $\forall x\in \overline{{\Omega }}$; $ii\right)\phantom{\rule{1.em}{0ex}}0<{a}_{0}\left(x\right)+b\left(x\right)$ $\forall x\in {{\Omega }}^{*}$; iii) b(x)$\ne 0$ $\forall x\in \partial {{\Omega }}^{*}·$

The author proves the uniqueness of the unknown pair $\left({{\Omega }}^{*},b\right)$ when $n=3$, under the assumption that the Dirichlet-Neumann map $\phi \to \partial u\left(\phi \right)/\partial N$ is known for any $\phi \in {C}^{1}\left(\partial {\Omega }\right)$ with Supp $\phi$ $\subset {{\Gamma }}_{0}$, ${{\Gamma }}_{0}$ being a neighbourhood if $\partial {\Omega }$. In the case $n=2$ he obtains a weaker result.

Similar uniqueness results are deduced also in the case where equation (1) is replaced by

$\left(1bis\right)\phantom{\rule{1.em}{0ex}}div\left(a\left(y\right){\nabla }_{y}u\left(x,y\right)\right)=\delta \left(x\right),\phantom{\rule{1.em}{0ex}}y\in {ℝ}^{n},$

where $\delta$ (x) is the delta function with pole at x and ${a}_{0}$ is constant outside a bounded domain ${{\Omega }}_{0}$ with $\overline{{\Omega }}{}_{0}\subset {\Omega }$, ${\Omega }$ being a convex domain with analytic boundary.

The additional information needed to prove the uniqueness result for $\left({{\Omega }}^{*},b\right)$ is the following: u(x,$·\right)$ is assigned on ${{\Gamma }}_{1}$ for any $x\in {{\Gamma }}_{2}$, ${{\Gamma }}_{1}$ and ${{\Gamma }}_{2}$ being two disjoint neighbourhoods in $\partial {\Omega }·$

The third identification problem is related to the anisotropic equation

$\left(1ter\right)\phantom{\rule{1.em}{0ex}}div\left(A\left(x\right)\nabla u\right)=0\phantom{\rule{1.em}{0ex}}in\phantom{\rule{1.em}{0ex}}{\Omega },$

where the matrix A(x) admits the decomposition

$A\left(x\right)={A}_{0}\left(x\right)+\chi \left({{\Omega }}^{*}\right)B\left(x\right)\phantom{\rule{1.em}{0ex}}\forall x\in \overline{{\Omega }}·$

Here ${A}_{0}\left(x\right)$ and B(x) are known positive definite matrices $\forall x\in \overline{{\Omega }}$, while the open set ${{\Omega }}^{*}$ 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 ${{\Omega }}^{*}$.

Reviewer: A.Lorenzi

##### MSC:
 35R30 Inverse problems for PDE 35J25 Second order elliptic equations, boundary value problems 35R05 PDEs with discontinuous coefficients or data 35A05 General existence and uniqueness theorems (PDE) (MSC2000)