Determining two coefficients in elliptic operators via boundary spectral data: a uniqueness result

Canuto, Bruno, and Kavian, Otared. "Determining two coefficients in elliptic operators via boundary spectral data: a uniqueness result." Bollettino dell'Unione Matematica Italiana 7-B.1 (2004): 207-230. <http://eudml.org/doc/195678>.

@article{Canuto2004,
abstract = {For a bounded and sufficiently smooth domain $\Omega$ in $\mathbb\{R\}^\{N\}$, $N\geq 2$, let $(\lambda_\{k\})_\{k=1\}^\{\infty\}$ and $(\varphi_\{k\})_\{k=1\}^\{\infty\}$ be respectively the eigenvalues and the corresponding eigenfunctions of the problem (with Neumann boundary conditions) $$ - \text\{div\} (a(x) \nabla \varphi\_\{k\})+ q(x) \varphi\_\{k\}= \lambda\_\{k\}\varrho (x) \varphi\_\{k\} \text\{ in \} \Omega, \quad a\frac\{\partial\}\{\partial \mathbf\{n\}\} \varphi\_\{k\}=0 \text\{ su \} \partial\Omega. $$ We prove that knowledge of the Dirichlet boundary spectral data $(\lambda_\{k\})_\{k=1\}^\{\infty\}$, $(\varphi_\{k|\partial\Omega\})_\{k=1\}^\{\infty\}$ determines uniquely the Neumann-to-Dirichlet (or the Steklov- Poincaré) map $\gamma$ for a related elliptic problem. Under suitable hypothesis on the coefficients $a, q, \varrho$ their identifiability is then proved. We prove also analogous results for Dirichlet boundary conditions.},
author = {Canuto, Bruno, Kavian, Otared},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {207-230},
publisher = {Unione Matematica Italiana},
title = {Determining two coefficients in elliptic operators via boundary spectral data: a uniqueness result},
url = {http://eudml.org/doc/195678},
volume = {7-B},
year = {2004},
}

TY - JOUR
AU - Canuto, Bruno
AU - Kavian, Otared
TI - Determining two coefficients in elliptic operators via boundary spectral data: a uniqueness result
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/2//
PB - Unione Matematica Italiana
VL - 7-B
IS - 1
SP - 207
EP - 230
AB - For a bounded and sufficiently smooth domain $\Omega$ in $\mathbb{R}^{N}$, $N\geq 2$, let $(\lambda_{k})_{k=1}^{\infty}$ and $(\varphi_{k})_{k=1}^{\infty}$ be respectively the eigenvalues and the corresponding eigenfunctions of the problem (with Neumann boundary conditions) $$ - \text{div} (a(x) \nabla \varphi_{k})+ q(x) \varphi_{k}= \lambda_{k}\varrho (x) \varphi_{k} \text{ in } \Omega, \quad a\frac{\partial}{\partial \mathbf{n}} \varphi_{k}=0 \text{ su } \partial\Omega. $$ We prove that knowledge of the Dirichlet boundary spectral data $(\lambda_{k})_{k=1}^{\infty}$, $(\varphi_{k|\partial\Omega})_{k=1}^{\infty}$ determines uniquely the Neumann-to-Dirichlet (or the Steklov- Poincaré) map $\gamma$ for a related elliptic problem. Under suitable hypothesis on the coefficients $a, q, \varrho$ their identifiability is then proved. We prove also analogous results for Dirichlet boundary conditions.
LA - eng
UR - http://eudml.org/doc/195678
ER -