A Cauchy problem for the Cauchy–Riemann operator

Abstract

We study the Cauchy problem for a nonlinear elliptic equation with data on a piece $${\mathcal {S}}$$ of the boundary surface $$\partial {\mathcal {X}}$$ . By the Cauchy problem is meant any boundary value problem for an unknown function u in a domain $${\mathcal {X}}$$ with the property that the data on $${\mathcal {S}}$$ , if combined with the differential equations in $${\mathcal {X}}$$ , allows one to determine all derivatives of u on $${\mathcal {S}}$$ by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near $${\mathcal {S}}$$ is guaranteed by the Cauchy–Kovalevskaya theorem. We discuss a variational setting of the Cauchy problem which always possesses a generalized solution.