Propagation of singularities from singular and infinite points in certain complex-analytic Cauchy problems and an application to the Pompeiu problem
Let \(\Gamma\) be an irreducible analytic hypersurface in some domain in \(C^2\). The author investigates the local propagation of singularities of the solution of the Cauchy problem \(u= |\text{grad}(u)|= 0\) on \(\Gamma\) for the complex differential equation \[ u_{zw}+ a(z, w) u_z+ b(z, w) u_w+ c(z, w)u= g(z, w) \] with analytical coefficients. Let \(U\) be a neighborhood of some point \((z_0, w_0)\in \Gamma\) and \(\Gamma= \{f(z,w)= 0\}\) for some analytical function \(f\) on \(U\) such that \(f\) is irreducible in the ring of convergent power series at \((z_0, w_0)\). If \(g(z_0, w_0)= 0\) and if \((z_0, w_0)\) is not a singular point of \(\Gamma\), then the singularity of \(u\) propagates along the bicharacteristic hyperplane \(\{z = z_0\}\) which is tangential to \(\Gamma\). The main result of this paper in its first part is that if \((z_0, w_0)\) is a singular point, then the singularity propagates along both the bicharacteristics \(\{z= z_0\}\) and \(\{w= w_0\}\). In the second part the author investigates the special Cauchy problem \[ 4u_{zw}+ \lambda u= 1,\quad u= |\text{grad}(u)|= 0\quad\text{on }\Gamma, \] where \(\lambda\) is some constant, and \(\Gamma\) is a certain irreducible analytic hypersurface passing the point at infinity. The main result is: The singularity propagates from the infinite point along the bicharacteristic \(\{z= z_0\}\) into the finite space. It has relation with the Pompeiu problem.
scientific article; zbMATH DE number 572044
scientific article