Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients
The goal of the paper is to fill the gap between Holmgren's theorem (for linear differential operators with analytic coefficients and non-characteristic initial surface) and Hörmander's theorem (for linear differential operators with \(C^\infty\)-coefficients, where the operator is principally normal and the initial surface is pseudo-convex). A natural question is that for uniqueness results for linear differential operators \(P\) of arbitrary order with \(C^\infty\)-coefficients, where the coefficients depend analytically on \(x_a\), \(x= (x_a,x_b)\). The authors prove uniqueness results for the Cauchy problem for such operators \(P\), in this sense they fill the gap mentioned at the beginning. As usual one has to prove Carleman estimates, these are \(L^2\)-estimates with an exponential weight \(e^{-\lambda\psi}\). The assumptions of principal normality and pseudo-convexity guarantee that the operator \(P_\lambda= e^{\lambda\psi} Pe^{-\lambda\psi}\) is subelliptic. Hence Garding-type estimates can be derived. As an other important tool, the authors use the partial FBI transformation. The authors give an interesting motivation for their results from control theory. As an application, they obtain a uniqueness result for second-order strictly hyperbolic operators for any non-characteristic initial hypersurface (even time-like ones are allowed).
scientific article; zbMATH DE number 1143543