LAUSR

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