Mixed problems with parameter
- Let X be a smooth n-dimensional manifold and D be an open connected set in X with smooth boundary OD. Perturbing the Cauchy problem for an elliptic system Au = f in D with data on a closed set Gamma subset of partial derivativeD, we obtain a family of mixed problems depending on a small parameter epsilon > 0. Although the mixed problems are subjected to a noncoercive boundary condition on partial derivativeDF in general, each of them is uniquely solvable in an appropriate Hilbert space D-T and the corresponding family {u(epsilon)} of solutions approximates the solution of the Cauchy problem in D-T whenever the solution exists. We also prove that the existence of a solution to the Cauchy problem in D-T is equivalent to the boundedness of the family {u(epsilon)}. We thus derive a solvability condition for the Cauchy problem and an effective method of constructing the solution. Examples for Dirac operators in the Euclidean space R-n are treated. In this case, we obtain a family of mixed boundary problems for the Helmholtz equation
Verfasserangaben: | Alexander ShlapunovORCiDGND, Nikolai Nikolaevich TarkhanovORCiDGND |
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ISSN: | 1061-9208 |
Publikationstyp: | Wissenschaftlicher Artikel |
Sprache: | Englisch |
Jahr der Erstveröffentlichung: | 2005 |
Erscheinungsjahr: | 2005 |
Datum der Freischaltung: | 24.03.2017 |
Quelle: | Russian Journal of Mathematical Physics. - ISSN 1061-9208. - 12 (2005), 1, S. 97 - 119 |
Organisationseinheiten: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
Peer Review: | Referiert |