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

Canuto, Bruno ; Kavian, Otared

Bollettino dell'Unione Matematica Italiana, Tome 7-A (2004), p. 207-230 / Harvested from Biblioteca Digitale Italiana di Matematica

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.

Résumé

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.

Détail

Publié le : 2004-02-01

MR 2044267 | Zbl 1178.35152

Bibliographie

[1] Borg, G., Eine umkehrung der Sturm-Liouville eigenwertaufgabe, Acta Math., 78 (1946), 1-96. | MR 15185 | Zbl 0063.00523

[2] Brown, R. M., Global uniqueness in the impedance imaging problem for less regular conductivities, SIAM J. Math. Anal., 27 (1996), 1049-1056. | MR 1393424 | Zbl 0867.35111

[3] Brown, R. M.-Uhlmann, G., Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. in Part. Diff. Equat., 22 (1997), 1009-1027. | MR 1452176 | Zbl 0884.35167

[4] Calderón, A. P., On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matemática, Rio de Janeiro, 1980, 65-73. | MR 590275

[5] Canuto, B.-Kavian, O.: Determining coefficients in a class of heat equations via boundary measurements, SIAM J. Math. Anal., 32 (2001), 963-986. | MR 1828313 | Zbl 0981.35096

[6] Garofalo, N.-Lin, F. H., Unique Continuation for Elliptic Operators: A Geometric- Variational Approach, Comm. on Pure and Appl. Math., 40 (1987), 347-366. | MR 882069 | Zbl 0674.35007

[7] Gel'Fand, I. M.-Levitan, B. M., On determination of a differential equation from its spectral function, Izv. Akad. Nauk. SSSR Ser. Mat., 15 (1951), 309-360; Amer. Math. Soc. Transl. (Ser. 2), 1 (1955), 253-304. | Zbl 0066.33603

[8] Isakov, V., Inverse problems for partial differential equations, Applied Math. Sciences, vol. 127, Springer, New York, 1998. | MR 1482521 | Zbl 0908.35134

[9] Levinson, N., The inverse Sturm-Liouville problem, Mat. Tidsskr., B (1949), 25-30. | MR 32067 | Zbl 0045.36402

[10] Lions, J.-L., Problèmes aux limites dans les équations aux dérivées partielles; Presses de l'Université de Montréal, Montréal1965. | Zbl 0143.14003

[11] Nachman, A. I., Reconstructions from boundary measurements; Ann. Math., 128 (1988), 531-577. | Zbl 0675.35084

[12] Nachman, A. I.-Sylvester, J.-Uhlmann, G., An $n$-dimensional Borg-Levinson theorem; Comm. Math. Physics, 115 (1988), 595-605. | Zbl 0644.35095