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A variational approach to the Cauchy problem for nonlinear elliptic differential equations

  • 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.

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Metadaten
Author:Ibrahim Ly, Nikolai Nikolaevich TarkhanovORCiDGND
URL:http://dx.doi.org/10.1515/jiip
DOI:https://doi.org/10.1515/Jiip.2009.037
ISSN:0928-0219
Document Type:Article
Language:English
Year of first Publication:2009
Year of Completion:2009
Release Date:2017/03/25
Source:Journal of inverse and ill-posed problems. - ISSN 0928-0219. - 17 (2009), 6, S. 595 - 610
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Informatik und Computational Science
Peer Review:Referiert
Institution name at the time of publication:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Informatik