A variational approach to the Cauchy problem for nonlinear elliptic differential equations
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I. Ly
Abstract
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.
© de Gruyter 2009
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Articles in the same Issue
- Identification of a source for parabolic and hyperbolic equations with a parameter
- A sensitivity matrix based methodology for inverse problem formulation
- Well-posedness of an inverse problem of Navier–Stokes equations with the final overdetermination
- Modified Landweber iterations in a multilevel algorithm applied to inverse problems in piezoelectricity
- A variational approach to the Cauchy problem for nonlinear elliptic differential equations
- A new robust algorithm for solution of pressure/rate deconvolution problem
- International Conference and Young Scientists School Theory and Computational Methods for Inverse and Ill-posed Problems
- 5th International Conference Inverse Problems: Modeling and Simulation
