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Stability for the reconstruction of a Riemannian metric by boundary measurements. (English) Zbl 0969.53018

In this note the author derives a stability estimate for the reconstruction of a Riemannian metric in a bounded domain \(\Omega\subset \mathbb{R}^3\) by a set of boundary measurements related to geodesics’ information from a different point of view. This problem is based on the following inverse problem. Let \(\Omega\subset \mathbb{R}^3\) with smooth boundary \(\Gamma= \partial\Omega\). Assume that \(g(x)\) is a given Riemannian metric in \(\Omega\). The author also assumes that \(\Omega\) is geodesically convex with respect to \(g\). Denote by \(L_g(x,y)\) the length of the geodesic joining \(x\) and \(y\) for \((x,y)\in \Gamma^{2}\).
An inverse problem in this setting is whether one can reconstruct the Riemannian metric by the knowledge of \(L(x,y)\) for all \((x,y)\in \Gamma^2\). Based on P. Stefanov and G. Uhlman’s ideas [J. Funct. Anal. 154, 330-358 (1988; Zbl 0915.35066)], the author derives a stability estimate for the inverse problem which is equivalent to the hodograph problem.

MSC:

53C22 Geodesics in global differential geometry
35R30 Inverse problems for PDEs

Citations:

Zbl 0915.35066
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