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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
Mathematics
##### Identifiers
OAI: 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

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

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
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)
##### 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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Cite
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