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Maximal regularity of the solutions for some degenerate differential equations and their applications
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
2018 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)Alternativ titel
Maximal regularitet av lösningarna till några degenererade differentialekvationer och deras tillämpningar (Svenska)
##### 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).

##### Ort, förlag, år, upplaga, sidor
Luleå: Luleå University of Technology, 2018.
##### Serie
Doctoral thesis / Luleå University of Technology 1 jan 1997 → …, ISSN 1402-1544
##### Nationell ämneskategori
Matematisk analys
Matematik
##### Identifikatorer
ISBN: 978-91-7790-100-6 (tryckt)ISBN: 978-91-7790-101-3 (digital)OAI: oai:DiVA.org:ltu-68293DiVA, id: diva2:1196979
##### Disputation
2018-06-07, E243, Luleå, 10:00 (Engelska)
##### Delarbeten
1. Separation and the existence theorem for second order nonlineardifferential equation
Öppna denna publikation i ny flik eller fönster >>Separation and the existence theorem for second order nonlineardifferential equation
2012 (Engelska)Ingår i: Electronic journal on the qualitative theory of differential equations, ISSN 1417-3875, E-ISSN 1417-3875, nr 66, s. 1-12Artikel i tidskrift (Refereegranskat) 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.

##### Ort, förlag, år, upplaga, sidor
Hungary: University of Szeged, 2012
##### Nationell ämneskategori
Matematisk analys
Matematik
##### Identifikatorer
urn:nbn:se:ltu:diva-68389 (URN)000307976000001 ()2-s2.0-84866043474 (Scopus ID)
2. Some new results concerning a class of third-order differential equations
Öppna denna publikation i ny flik eller fönster >>Some new results concerning a class of third-order differential equations
2015 (Engelska)Ingår i: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 94, nr 2, s. 419-434Artikel i tidskrift (Refereegranskat) 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

##### Ort, förlag, år, upplaga, sidor
Taylor & Francis, 2015
##### Nationell ämneskategori
Matematisk analys
Matematik
##### Identifikatorer
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 (Lokalt ID)1959cbfa-5e19-4ea4-9c4d-93b09eb1a578 (Arkivnummer)1959cbfa-5e19-4ea4-9c4d-93b09eb1a578 (OAI)
##### Anmärkning

Validerad; 2015; Nivå 2; 20140415 (andbra)

3. Compactness of the resolvent of one second order differential operator
Öppna denna publikation i ny flik eller fönster >>Compactness of the resolvent of one second order differential operator
2014 (Engelska)Ingår i: 10th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences (ICNPAA 2014) / [ed] Sivasundaram, Seenith, American Institute of Physics (AIP), 2014, Vol. 1637, nr 1, s. 13-17, artikel-id 13Konferensbidrag, Publicerat paper (Refereegranskat)
##### 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.

##### Ort, förlag, år, upplaga, sidor
American Institute of Physics (AIP), 2014
##### Nationell ämneskategori
Matematisk analys
Matematik
##### Identifikatorer
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)
##### Konferens
The 10th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences (ICNPAA), Narvik, Norway, 15–18 July 2014
4. Some inequalities for second order differential operators with unbounded drift
Öppna denna publikation i ny flik eller fönster >>Some inequalities for second order differential operators with unbounded drift
2015 (Engelska)Ingår i: Eurasian Mathematical Journal, ISSN 2077-9879, Vol. 6, nr 2, s. 63-74Artikel i tidskrift (Refereegranskat) 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.

##### Ort, förlag, år, upplaga, sidor
L.N. Gumilyov Eurasian National University, 2015
##### Nationell ämneskategori
Matematisk analys
Matematik
##### Identifikatorer
urn:nbn:se:ltu:diva-68392 (URN)000374499500004 ()2-s2.0-84957683266 (Scopus ID)
5. Coersive solvability of the differential equation of the third order with complex valued coefficients
Öppna denna publikation i ny flik eller fönster >>Coersive solvability of the differential equation of the third order with complex valued coefficients
2013 (Engelska)Ingår i: Vestnik ENU, Vol. 95, nr 4, s. 355-361Artikel i tidskrift (Refereegranskat) Published
##### Nationell ämneskategori
Matematisk analys
Matematik
##### Identifikatorer
urn:nbn:se:ltu:diva-68491 (URN)
6. On separation of a degenerate differential operator in Hilbert space
Öppna denna publikation i ny flik eller fönster >>On separation of a degenerate differential operator in Hilbert space
2011 (Engelska)Konferensbidrag, Publicerat paper (Refereegranskat)
##### 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.

##### Nationell ämneskategori
Matematisk analys
Matematik
##### Identifikatorer
urn:nbn:se:ltu:diva-68528 (URN)
##### Konferens
Centre de Recerca Matematica, UAB, Spain, Barcelona
7. On Solvability of Third-Order Singular Differential Equation
Öppna denna publikation i ny flik eller fönster >>On Solvability of Third-Order Singular Differential Equation
2017 (Engelska)Ingår i: FAIA 2017: Functional Analysis in Interdisciplinary Applications, Springer, 2017, Vol. 216, s. 106-112Konferensbidrag, Publicerat paper (Refereegranskat)
##### 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
##### Serie
Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009
##### Nationell ämneskategori
Matematisk analys
Matematik
##### Identifikatorer
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)
##### Konferens
International Conference on Functional analysis in interdisciplinary applications, Astana, Kazakhstan, 2-5 October

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