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.