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On separation of a degenerate differential operator in Hilbert space
2011 (English)Conference paper, Published paper (Refereed)
Abstract [en]

A coercive estimate for a solution of a degenerate second order di fferential equation is installed, and its applications to spectral problems for the corresponding dif ferential operator is demonstrated. The suffi cient conditions for existence of the solutions of one class of the nonlinear second order diff erential equations on the real axis are obtained.

Place, publisher, year, edition, pages
2011. p. 1-12, article id 1080
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:ltu:diva-68528OAI: oai:DiVA.org:ltu-68528DiVA, id: diva2:1202518
Conference
Centre de Recerca Matematica, UAB, Spain, Barcelona
Available from: 2018-04-27 Created: 2018-04-27 Last updated: 2018-04-27Bibliographically approved
In thesis
1. Maximal regularity of the solutions for some degenerate differential equations and their applications
Open this publication in new window or tab >>Maximal regularity of the solutions for some degenerate differential equations and their applications
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[sv]
Maximal regularitet av lösningarna till några degenererade differentialekvationer och deras tillämpningar
Abstract [en]

This PhD thesis deals with the study of existence and uniqueness together with coercive estimates for solutions of certain differential equations.

The thesis consists of six papers (papers A, B, C, D, E and F), two appendices and an introduction, which put these papers and appendices into a more general frame and which also serves as an overview of this interesting field of mathematics.

In the text below the functionsr = r(x), q = q(x), m = m(x) etc. are functions on (−∞,+∞), which are different but well defined in each paper. Paper A deals with the study of separation and approximation properties for the differential operator                                                                                       

                                                   

in the Hilbert space (here is the complex conjugate of ). A coercive estimate for the solution of the second order differential equation is obtained and its applications to spectral problems for the corresponding differential operator  is demonstrated. Some sufficient conditions for the existence of the solutions of a class of nonlinear second order differential equations on the real axis are obtained.

In paper B necessary and sufficient conditions for the compactness of the resolvent of the second order degenerate differential operator  in is obtained. We also discuss the two-sided estimates for the radius of fredholmness of this operator.

In paper C we consider the minimal closed differential operator

                                      

in , where are continuously differentiable functions, and is a continuous function. In this paper we show that the operator is continuously invertible when these coefficients satisfy some suitable conditions and obtain the following estimate for :

                                            ,

where is the domain of .

In papers D, E, and F various differential equations of the third order of the form

      

are studied in the space .

In paper D we investigate the case when and .

Moreover, in paper E the equation (0.1) is studied when . Finally, in paper F the equation (0.1) is investigated under certain additional conditions on .

For these equations we establish sufficient conditions for the existence and uniqueness of the solution, and also prove an estimate of the form

     

for the solution of equation (0.1).

                         

                       

                             

Place, publisher, year, edition, pages
Luleå: Luleå University of Technology, 2018
Series
Doctoral thesis / Luleå University of Technology 1 jan 1997 → …, ISSN 1402-1544
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-68293 (URN)978-91-7790-100-6 (ISBN)978-91-7790-101-3 (ISBN)
Public defence
2018-06-07, E243, Luleå, 10:00 (English)
Opponent
Supervisors
Available from: 2018-04-11 Created: 2018-04-11 Last updated: 2018-05-29Bibliographically approved

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http://www.recercat.cat/handle/2072/196881

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CiteExportLink to record
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