×

Completeness of products of solutions and some inverse problems for PDE. (English) Zbl 0728.35141

Building on the completeness of products of solutions to general differential equations, the author solves inverse problems for such equations. For example, if \(\Omega \subset {\mathbb{R}}^ 2\) is a bounded domain, and \(u_ 1,u_ 2\in H^ 2(\Omega \times (0,T))\) are solutions of an initial-boundary value problem for the equation \(\partial^ 2u_ j/\partial t^ 2-\Delta u_ j+a_ ju_ j=0\) on \(\Omega\times (0,T)\), then the coincidence of \(u_ 1\) and \(u_ 2\), and \(\partial u_ 1/\partial t\) and \(\partial u_ 2/\partial t\), on the boundary implies that \(a_ 1=a_ 2\).

MSC:

35R30 Inverse problems for PDEs
35L10 Second-order hyperbolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alessandrini, G., Stable determination of conductivity by boundary measurements, Appl. Anal., 27, 153-172 (1988) · Zbl 0616.35082
[2] Friedman, A., Partial Differential Equations of Parabolic Type (1964), Prentice-Hall · Zbl 0144.34903
[3] Hörmander, L., The Analysis of Linear Partial Differential Operators: I, II (1983), Springer Pub: Springer Pub New York
[4] Isakov, V., On uniqueness of recovery of a discontinuous conductivity coefficient, Comm. Pure Appl. Math., 41, 865-877 (1988) · Zbl 0676.35082
[5] Kohn, R.; Vogelius, M., Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37, 113-123 (1984)
[6] Ladyzenskaja, O. A., The Mixed Problem for a Hyperbolic Equation (1953), GosTechlzdat: GosTechlzdat Mocsow
[7] Morrey, Ch. B., Multiple Integrals in the Calculus of Variations (1966), Springer Pub: Springer Pub New York
[8] Nachman, A., Reconstructions from boundary measurements, Ann. of Math., 128, 531-577 (1988) · Zbl 0675.35084
[9] Nachman, A.; Sylvester, J.; Uhlmann, G., An \(n\)-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115, 593-605 (1988) · Zbl 0644.35095
[10] Nirenberg, L., Uniqueness in Cauchy problems for differential equations with constant leading coefficients, Comm. Pure Appl. Math., 10, 89-105 (1957) · Zbl 0077.09402
[11] Sylvester, J.; Uhlmann, G., A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125, 153-169 (1987) · Zbl 0625.35078
[12] Sylvester, J.; Uhlmann, G., Inverse boundary value problem at the boundary-continuous dependence, Comm. Pure Appl. Math., 41, 197-221 (1988)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.