A Variational Approach to the Cauchy Problem for Nonlinear Elliptic Systems

We discuss the Cauchy problem for a class of nonlinear elliptic equations with data on a piece \(\mathcal {S}\) of the boundary surface by means of a variational approach known in the optimal control literature as “equation error method.” To this end, we use purely nonlinear methods such as successive iterations of a Zaremba-type mixed boundary value problem. In the case of real analytic data on a non-characteristic hypersurface \(\mathcal {S}\) , the existence of a local solution for the Cauchy problem near \(\mathcal {S}\) is guaranteed by the Cauchy–Kowalewskaya theorem.