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Topics in analytic theory of partial differential equations
Stockholm University, Faculty of Science, Department of Mathematics.
Responsible organisation
2006 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The thesis consists of three papers I, II III devoted to some topics in analytic theory of partial differential equations. The main paper is II. In this paper we study a version of the classical problem on the convergence of formal solutions of systems of partial differential equations. A necessary and sufficient condition for the convergence of a given formal solution (found by any method) is proved. This convergence criterion applies to systems of partial differential equations (possibly, nonlinear) solved for the highest-order derivatives or, which is most important,"almost solved for the highest-order derivatives".

In the paper I, systems of linear partial differential equations with constant coefficients are considered. The space of formal and analytic solutions of such systems are described by elementary algebraic methods. The Hilbert and Hilbert-Samuel polynomials for systems of linear partial differential equations with constant coefficients are defined.

In the paper III, systems of linear partial differential equations with analytic coefficients are considered. Suppose the coefficients of a system are defined in a domain U in n-dimensional complex space. We study the space of germs of formal and analytic solutions of the system at a point u of the domain U. We discuss the following questions. 1) How to pose "proper" initial conditions for formal and analytic solutions for the system. 2) How depend dimensions of the space of k-jets of germs of formal and analytic solutions of the system at a point u in the domain U on the positive integer k and the point u.

Place, publisher, year, edition, pages
Stockholm: Matematiska institutionen , 2006. , 61 p.
Keyword [en]
partial differential equations
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-782ISBN: 91-7155-193-X (print)OAI: oai:DiVA.org:su-782DiVA: diva2:199102
Public defence
2006-01-20, sal 14, hus 5, Kräftriket, Stockholm, 13:00
Opponent
Supervisors
Available from: 2005-12-22 Created: 2005-12-22Bibliographically approved
List of papers
1. Hilbert and Hilbert-Samuel polynomials and partial differential equations
Open this publication in new window or tab >>Hilbert and Hilbert-Samuel polynomials and partial differential equations
2005 In: Mat. Zametki, Vol. 77, no 1, 141-151 p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:su:diva-25189 (URN)
Note
Part of urn:nbn:se:su:diva-782Available from: 2005-12-22 Created: 2005-12-22Bibliographically approved
2. On the convergence of formal solutions of a system of partial differential equations
Open this publication in new window or tab >>On the convergence of formal solutions of a system of partial differential equations
2005 In: Funks. Anal. Prilozhen, Vol. 39, no 3, 64-75 p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:su:diva-25190 (URN)
Note
Part of urn:nbn:se:su:diva-782Available from: 2005-12-22 Created: 2005-12-22Bibliographically approved
3. The Hilbert polynomial for systems of linear partial differential equations with analytic coefficients
Open this publication in new window or tab >>The Hilbert polynomial for systems of linear partial differential equations with analytic coefficients
2006 (English)In: Izvestiya. Mathematics, ISSN 1064-5632, E-ISSN 1468-4810, Vol. 70, no 1, 153-169 p.Article in journal (Refereed) Published
Abstract [en]

We consider systems of linear partial differential equations with analytic coefficients and discuss existence and uniqueness theorems for their formal and analytic solutions. Using elementary methods, we define and describe an analogue of the Hilbert polynomial for such systems.

National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-25191 (URN)10.1070/IM2006v170n01ABEH002307 (DOI)000237779300007 ()
Note

Part of urn:nbn:se:su:diva-782

Available from: 2005-12-22 Created: 2005-12-22 Last updated: 2017-12-13Bibliographically approved

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