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Maximal regularity of the solutions for some degenerate differential equations and their applications
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
2018 (English)Doctoral thesis, comprehensive summary (Other academic)Alternative title
Maximal regularitet av lösningarna till några degenererade differentialekvationer och deras tillämpningar (Swedish)
##### 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

$ly=-y''+r(x)y'+s(x)\bar{y}'$

in the Hilbert space $L_2:= L_2 (\mathbb{R}),\ \mathbb{R}=(-\infty, +\infty),$(here $\bar y$ is the complex conjugate of $y$). A coercive estimate for the solution of the second order differential equation $ly=f$ is obtained and its applications to spectral problems for the corresponding differential operator $l$ 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 $l$ in $L_2$ 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

$Ly = -\rho(x)(\rho(x)y')'+ r(x)y' + q(x)y$

in $L_2(\mathbb{R})$, where $\rho=\rho (x), r=r(x)$ are continuously differentiable functions, and $q=q(x)$ is a continuous function. In this paper we show that the operator $L$ is continuously invertible when these coefficients satisfy some suitable conditions and obtain the following estimate for $y \in D(L)$:

$\||-\rho(\rho y')'\||_2+\||r y'\||_2+\||q y\||_2\leq c \||L y\||_2$,

where $D(L)$ is the domain of $L$.

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

$-m_1(x)\left(m_2(x)\left(m_3(x)y'\right)'\right)'+[q(x)+ir(x)+\lambda]y=f(x) \ (0.1)$

are studied in the space $L_p(\mathbb{R})$.

In paper D we investigate the case when $m_1=m_3=m$ and $m_2=1$.

Moreover, in paper E the equation (0.1) is studied when $m_3=1$. Finally, in paper F the equation (0.1) is investigated under certain additional conditions on $m_j(x) (j=1,2,3)$.

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

$\left\|m_1(x)(m_2(x)\left(m_3(x)y')'\right)'\right\|^p_p+\left\|(q(x)+ir(x)+\lambda)y\right\|^p_p \leq c \left\|f(x)\right\|^p_p$

for the solution $y$ 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
Mathematics
##### Identifiers
ISBN: 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)
##### Supervisors
Available from: 2018-04-11 Created: 2018-04-11 Last updated: 2018-05-29Bibliographically approved
##### List of papers
1. Separation and the existence theorem for second order nonlineardifferential equation
Open this publication in new window or tab >>Separation and the existence theorem for second order nonlineardifferential equation
2012 (English)In: Electronic journal on the qualitative theory of differential equations, ISSN 1417-3875, E-ISSN 1417-3875, no 66, p. 1-12Article in journal (Refereed) Published
##### Abstract [en]

Sufficient conditions for the invertibility and separability in L2(−∞,+∞) of the degenerate second order differential operator with complex-valued coefficients are obtained, and its applications to the spectral and approximate problems are demonstrated. Usinga separability theorem, which is obtained for the linear case, the solvability of nonlinear second order differential equation is proved on the real axis.

##### Place, publisher, year, edition, pages
Hungary: University of Szeged, 2012
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-68389 (URN)000307976000001 ()2-s2.0-84866043474 (Scopus ID)
Available from: 2018-04-17 Created: 2018-04-17 Last updated: 2018-04-27Bibliographically approved
2. Some new results concerning a class of third-order differential equations
Open this publication in new window or tab >>Some new results concerning a class of third-order differential equations
2015 (English)In: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 94, no 2, p. 419-434Article in journal (Refereed) Published
##### Abstract [en]

We consider the following third-order differential equation with unbounded coefficients:Some new existence and uniqueness results are proved, and precise estimates of the norms of the solutions are given. The obtained results may be regarded as a unification and extension of all other results of this type

##### Place, publisher, year, edition, pages
Taylor & Francis, 2015
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-3756 (URN)10.1080/00036811.2014.898375 (DOI)000347395000013 ()2-s2.0-84920697521 (Scopus ID)1959cbfa-5e19-4ea4-9c4d-93b09eb1a578 (Local ID)1959cbfa-5e19-4ea4-9c4d-93b09eb1a578 (Archive number)1959cbfa-5e19-4ea4-9c4d-93b09eb1a578 (OAI)
##### Note

Validerad; 2015; Nivå 2; 20140415 (andbra)

Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2018-07-10Bibliographically approved
3. Compactness of the resolvent of one second order differential operator
Open this publication in new window or tab >>Compactness of the resolvent of one second order differential operator
2014 (English)In: 10th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences (ICNPAA 2014) / [ed] Sivasundaram, Seenith, American Institute of Physics (AIP), 2014, Vol. 1637, no 1, p. 13-17, article id 13Conference paper, Published paper (Refereed)
##### Abstract [en]

In this work a necessary and sufficient condition for the compactness of the resolvent of one second order degenerate differential operator in L-2 is obtained. We also discuss the two-sided estimates for the radius of fredholmness of this operator.

##### Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2014
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-68420 (URN)10.1063/1.4904559 (DOI)000347812200002 ()2-s2.0-85000874562 (Scopus ID)978-0-7354-1276-7 (ISBN)
##### Conference
The 10th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences (ICNPAA), Narvik, Norway, 15–18 July 2014
Available from: 2018-04-24 Created: 2018-04-24 Last updated: 2018-04-27Bibliographically approved
4. Some inequalities for second order differential operators with unbounded drift
Open this publication in new window or tab >>Some inequalities for second order differential operators with unbounded drift
2015 (English)In: Eurasian Mathematical Journal, ISSN 2077-9879, Vol. 6, no 2, p. 63-74Article in journal (Refereed) Published
##### Abstract [en]

We study coercive estimates for some second-order degenerate and damped differential operators with unbounded coefficients. We also establish the conditions for invertibility of these operators.

##### Place, publisher, year, edition, pages
L.N. Gumilyov Eurasian National University, 2015
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-68392 (URN)000374499500004 ()2-s2.0-84957683266 (Scopus ID)
Available from: 2018-04-17 Created: 2018-04-17 Last updated: 2018-04-27Bibliographically approved
5. Coersive solvability of the differential equation of the third order with complex valued coefficients
Open this publication in new window or tab >>Coersive solvability of the differential equation of the third order with complex valued coefficients
2013 (English)In: Vestnik ENU, Vol. 95, no 4, p. 355-361Article in journal (Refereed) Published
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-68491 (URN)
Available from: 2018-04-24 Created: 2018-04-24 Last updated: 2018-04-27Bibliographically approved
6. On separation of a degenerate differential operator in Hilbert space
Open this publication in new window or tab >>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.

##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-68528 (URN)
##### Conference
Centre de Recerca Matematica, UAB, Spain, Barcelona
Available from: 2018-04-27 Created: 2018-04-27 Last updated: 2018-04-27Bibliographically approved
7. On Solvability of Third-Order Singular Differential Equation
Open this publication in new window or tab >>On Solvability of Third-Order Singular Differential Equation
2017 (English)In: FAIA 2017: Functional Analysis in Interdisciplinary Applications, Springer, 2017, Vol. 216, p. 106-112Conference paper, Published paper (Refereed)
##### Abstract [en]

In this paper some new existence and uniqueness results are proved and maximal regularity estimates of solutions of third-order differential equation with unbounded coefficients are given.

Springer, 2017
##### Series
Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-68426 (URN)10.1007/978-3-319-67053-9_10 (DOI)2-s2.0-85041319582 (Scopus ID)978-3-319-67052-2 (ISBN)978-3-319-67053-9 (ISBN)
##### Conference
International Conference on Functional analysis in interdisciplinary applications, Astana, Kazakhstan, 2-5 October
Available from: 2018-04-20 Created: 2018-04-20 Last updated: 2018-04-27Bibliographically approved

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Cite
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