In this thesis, we study Cauchy problems for elliptic and parabolic equations. These include the stationary Stokes system and the heat equation. Data are given on a part of the boundary of a bounded domain. The aim is to reconstruct the solution from these data. These problems are ill-posed in the sense of J. Hadamard. We propose iterative regularization methods, which require solving of a sequence of well-posed boundary value problems for the same operator. Methods based on this idea were _rst proposed by V. A. Kozlov and V. G. Maz'ya for a certain class of equations which do not include the above problems. Regularizing character is proved and stopping rules are proposed. The regularizing character for the heat equation is proved in a certain weighted L2 space. In each iteration the Zaremba problem for the heat equation is solved. We also prove well-posedness of this problem in a weighted Sobolev space. This result is of independent interest and is presented as a separate paper.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:liu-140145 |
| Date | January 2003 |
| Creators | Johansson, Tomas |
| Publisher | Linköpings universitet, Kommunikations- och transportsystem, Linköpings universitet, Tekniska högskolan, Linköping |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Doctoral thesis, comprehensive summary, info:eu-repo/semantics/doctoralThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | Linköping Studies in Science and Technology. Dissertations, 0345-7524 ; 832 |
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