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.
Author details: | Ibrahim LyGND, Nikolai Nikolaevich TarkhanovORCiDGND |
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URL: | http://dx.doi.org/10.1515/jiip |
DOI: | https://doi.org/10.1515/Jiip.2009.037 |
ISSN: | 0928-0219 |
Publication type: | Article |
Language: | English |
Year of first publication: | 2009 |
Publication year: | 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 the publication: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Informatik |