TY  - JOUR
A1  - Ly, Ibrahim
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A variational approach to the Cauchy problem for nonlinear elliptic differential equations
N2  - 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 X with the property that the data on S, if combined with the differential equations in X, 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.
Y1  - 2009
UR  - https://publishup.uni-potsdam.de/frontdoor/index/index/docId/31780
UR  - http://dx.doi.org/10.1515/jiip
SN  - 0928-0219
ER  -