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Identifiability of the transmissivity coefficient in an elliptic boundary value problem. (English) Zbl 0777.35091

Summary: We deal with a coefficient inverse problem describing the filtration of ground water in a region \(\Omega\subset\mathbb{R}^ n\), \(n\geq 2\). Introducing a weak formulation of the problem, discretization and regularization methods can be constructed in a natural way. These methods converge to the normal solution of the problem, i.e. to a transmissivity coefficient of a minimal \(L^ 2(\Omega)\)-norm. Thus a question about \(L^ 2\)- identifiability (identifiability among functions of the class \(L^ 2(\Omega))\) of the transmissivity coefficient arises.
Our purpose is to describe subregions of \(\Omega\) where the transmissivity coefficient is really \(L^ 2\)-identifiable or even \(L^ 1\)-identifiable. Thereby we succeed in introducing physically realistic conditions on the data of the problem, e.g. piecewise smooth surfaces in \(\Omega\) are allowed where the data of the inverse problem may have discontinuities.
With some natural changes, our results about the \(L^ 1\)-identifiability extend known results about the identifiability among more smooth functions given by G. R. Richter [SIAM J. Appl. Math. 41, 210-221 (1981; Zbl 0501.35075)] C. Chicone and J. Gerlach [Inverse and ill-posed problems, Notes Rep. Math. Sci. Eng. 4, 513-521 (1987; Zbl 0641.93022)] and K. Kunisch [Automatica 24, No. 4, 531-539 (1988; Zbl 0649.93020)].

MSC:

35R30 Inverse problems for PDEs
35J20 Variational methods for second-order elliptic equations
93B30 System identification
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
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