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Maximal regularity of the solutions for some degenerate differential equations and their applicationsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2018 (English)Doctoral thesis, comprehensive summary (Other academic)Alternative title
##### Abstract [en]

##### 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: urn:nbn:se:ltu:diva-68293ISBN: 978-91-7790-100-6 (print)ISBN: 978-91-7790-101-3 (electronic)OAI: oai:DiVA.org:ltu-68293DiVA, id: diva2:1196979
##### Public defence

2018-06-07, E243, Luleå, 10:00 (English)
##### Opponent

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##### Supervisors

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#####

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Available from: 2018-04-11 Created: 2018-04-11 Last updated: 2018-05-29Bibliographically approved
##### List of papers

Maximal regularitet av lösningarna till några degenererade differentialekvationer och deras tillämpningar (Swedish)

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).

1. Separation and the existence theorem for second order nonlineardifferential equation$(function(){PrimeFaces.cw("OverlayPanel","overlay1198454",{id:"formSmash:j_idt504:0:j_idt508",widgetVar:"overlay1198454",target:"formSmash:j_idt504:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Some new results concerning a class of third-order differential equations$(function(){PrimeFaces.cw("OverlayPanel","overlay976617",{id:"formSmash:j_idt504:1:j_idt508",widgetVar:"overlay976617",target:"formSmash:j_idt504:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Compactness of the resolvent of one second order differential operator$(function(){PrimeFaces.cw("OverlayPanel","overlay1200902",{id:"formSmash:j_idt504:2:j_idt508",widgetVar:"overlay1200902",target:"formSmash:j_idt504:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Some inequalities for second order differential operators with unbounded drift$(function(){PrimeFaces.cw("OverlayPanel","overlay1198471",{id:"formSmash:j_idt504:3:j_idt508",widgetVar:"overlay1198471",target:"formSmash:j_idt504:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Coersive solvability of the differential equation of the third order with complex valued coefficients$(function(){PrimeFaces.cw("OverlayPanel","overlay1201016",{id:"formSmash:j_idt504:4:j_idt508",widgetVar:"overlay1201016",target:"formSmash:j_idt504:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. On separation of a degenerate differential operator in Hilbert space$(function(){PrimeFaces.cw("OverlayPanel","overlay1202518",{id:"formSmash:j_idt504:5:j_idt508",widgetVar:"overlay1202518",target:"formSmash:j_idt504:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

7. On Solvability of Third-Order Singular Differential Equation$(function(){PrimeFaces.cw("OverlayPanel","overlay1199323",{id:"formSmash:j_idt504:6:j_idt508",widgetVar:"overlay1199323",target:"formSmash:j_idt504:6:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

isbn
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