Singularities of the solution to a certain Cauchy problem and an application to the Pompeiu problem
This paper consists of two parts. The first is a report on an integral transform of B. Yu. Sternin and V. E. Shatalov and its application to the holomorphic Cauchy problem with constant coefficients. The second part is an application of the first to the Pompeiu problem. The main result implies that under certain hypotheses on a domain \(\Omega_ 0\) in \(\mathbb{R}^ 2\), symmetric about the \(x_ 1\)-axis, the rotation of \(\Omega_ 0\) around the \(x_ 1\)-axis in \(\mathbb{R}^ 3\), denoted by \(\Omega\) has the Pompeiu property; i.e. \(f \equiv 0\) is the only continuous function in \(\mathbb{R}^ 3\) such that \(\int_{\sigma (\Omega)} f = 0\) for every rigid motion \(\sigma\) of \(\mathbb{R}^ 3\). In fact, a stronger property of \(\Omega\) is shown to hold: a certain family of overdetermined Cauchy problems has no solutions.
scientific article; zbMATH DE number 495439
scientific article