Institute of Computational Mathematics and Mathematical Geophysics

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Математические проблемы в геофизических исследованиях твердой Земли

The problem of determining three coefficients $c(x)$, $sigma(x)$, $q(x)$ for a hyperbolic equation is considered. They are coefficients under the Laplacian, the first derivative with respect to time and the lower term, respectively. The inverse two-dimensional problem of electrodynamics is reduced to this problem. It is assumed that coefficients $c(x)-1$, $sigma(x)$, $q(x)$ are small in a suitable norm and their supports belong to some disk $D$. The latter is equivalent to the assumption that electrodynamical parameters close to constants. The Cauchy problem for the hyperbolic equation with zero initial data and a source is considered. The source is taken of the form $delta(t),delta(xcdot u)$, where $ u$ is a unit vector and is a parameter of the problem. The electromagnetic field is measured for tree different values of the parameter $ u$ at the boundary of $D$ for some fix time interval of the length $T$ that is accounting beginning with arriving a signal from the source. It is proved that the given information determines the unknown coefficients uniquely if $T$ is greater than double diameter of $D$. A conditional stability estimate of the solution is found.

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