## Abstract

We discuss the relaxation of a class of nonlinear elliptic Cauchy problems with data on a piece *S* of the boundary surface by means of a variational approach known in the optimal control literature as “equation error method”. By the Cauchy problem is meant any boundary value problem for an unknown function *y* in a domain *χ* with the property that the data on *S*, if combined with the differential equations in *χ*, allow one to determine all derivatives of *y* on *S* by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near *S* is guaranteed by the Cauchy–Kovalevskaya theorem. We also admit overdetermined elliptic systems, in which case the set of those Cauchy data on *S* for which the Cauchy problem is solvable is very “thin”. For this reason we discuss a variational setting of the Cauchy problem which always possesses a generalised solution.

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