Journal of Inverse and Ill-posed Problems
Editor-in-Chief: Kabanikhin, Sergey I.
6 Issues per year
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A variational approach to the Cauchy problem for nonlinear elliptic differential equations
- Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany. Email: firstname.lastname@example.org
- Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany. Email: email@example.com
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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