# Cauchy data

## Cauchy data

(relativity)
The Cauchy data for a hyperbolic partial differential equation consist of the value of the field and its time derivative on some spacelike surface.
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That is, the solution, if exists, does not depend continuously on the given Cauchy data. Any small perturbation in the given Cauchy data may cause dramatically large errors to the location, the size and the shape of the coefficient with piecewise constant.
In this paper, we consider an inverse coefficient identification problem from the Cauchy data on the exterior boundary of the solution domain.
Recovering a potential from Cauchy data in the two- dimensional case.
Moreover, the use of the mapping property leads to an explicit solution of the noncharacteristic Cauchy problem for (1) with Cauchy data belonging to a Holmgren class .
On the other hand, the inverse source problem consists in, given the Cauchy data {g, [g.sub.v]} [member of] [H.sup.1/2]([partial derivative][OMEGA]) x [H.sup.-1/2]([partial derivative][OMEGA]), finding the source term f.
In next theorem we establish an equivalence between the source reconstruction from the Cauchy data and from the reciprocity functional.
The Cauchy data uniquely determines the source f if, and only if, f is uniquely determined by R[f](v), for all v [member of] [H.sub.[lambda]]([OMEGA]).
Determination of the Source Support Boundary from Cauchy Data. Consider the subsets [omega], [??] [subset] [R.sup.2], with same barycenter, whose boundaries, [partial derivative][omega] and [partial derivative] [??], are parametrized, respectively, by R(t) and by a truncated Fourier series
(i) the given [mathematical expression not reproducible] depends only on Cauchy data;
For the determination of the barycenter, the reciprocity functional was calculated considering the Cauchy data from the solution of the direct problem, u.
Part of the Memoirs of the American Mathematical Society series, this volume uses perturbations of a Hamilton function to show that "when X is the sphere, and when the mass parameter m is outside an exceptional subset of zero measure, smooth Cauchy data of small size E give rise to almost global solutions." The book begins with an abstract and introduction and then include four chapters that elaborate on the details of this idea.
The continuation problem is ill-posed problem; its solution is unique, but it does not depend continuously on the Cauchy data [1-10].

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