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

Canuto, Bruno ; Kavian, Otared

Bollettino della Unione matematica italiana, Série 8, 7B (2004) no. 1, pp. 207-230.

Abstract (VO)

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.

Riassunto

Sia $\Omega$ un dominio limitato e sufficientemente regolare di $\mathbb{R}^{N}$, $N\geq 2$, e siano $(\lambda_{k})_{k=1}^{\infty}$ e $(\varphi_{k})_{k=1}^{\infty}$ rispettivamente gli autovalori e le autofunzioni corrispondenti del problema (con condizioni al bordo di Neumann) $$ - \text{div} (a(x) \nabla \varphi_{k})+ q(x) \varphi_{k}= \lambda_{k}\varrho (x) \varphi_{k} \text{ in } \Omega, \quad a\frac{\partial}{\partial \mathbf{n}} \varphi_{k}=0 \text{ su } \partial\Omega. $$ Dimostriamo che i dati spetrali al bordo di Dirichlet $(\lambda_{k})_{k=1}^{\infty}$, $(\varphi_{k|\partial\Omega})_{k=1}^{\infty}$ determinano in modo unico la mappa $\gamma$ di Neumann-Dirichlet (o la mappa di Steklov- Poincaré) per un problema ellittico relativo. Sotto opportune ipotesi sui coefficienti $a, q, \varrho$ proviamo in seguito la loro identificabilità. Dimostriamo risultati analoghi nel caso di condizioni al bordo di Dirichlet.

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