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Maximal regularity of the solutions for some degenerate differential equations and their applications

Akhmetkaliyeva, Raya, (författare)
Luleå tekniska universitet, Matematiska vetenskaper
Luleå tekniska universitet Institutionen för teknikvetenskap och matematik. Matematiska vetenskaper. (creator_code:org_t)
ISBN 978-91-7790-100-6
Luleå : Luleå University of Technology, 2018
Serie: Doctoral thesis / Luleå University of Technology 1 jan 1997 → …, 1402-1544
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  • Doktorsavhandling (övrigt vetenskapligt)
Abstract Ämnesord
  • 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                                                                                                                                           <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?$$%0Aly=-y''+r(x)y'+s(x)%5Cbar%7By%7D'%0A$$" />in the Hilbert space <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?L_2:=%20L_2%20(%5Cmathbb%7BR%7D),%5C%20%5Cmathbb%7BR%7D=(-%5Cinfty,%20+%5Cinfty)," />(here <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cbar%20y" /> is the complex conjugate of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?y" />). A coercive estimate for the solution of the second order differential equation <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?ly=f" /> is obtained and its applications to spectral problems for the corresponding differential operator <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?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 <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?l" /> in <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?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                                       <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%0ALy%20=%20-%5Crho(x)(%5Crho(x)y')'+%20r(x)y'%20+%20q(x)y%0A" />in <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?L_2(%5Cmathbb%7BR%7D)" />, where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Crho=%5Crho%20(x),%20r=r(x)" /> are continuously differentiable functions, and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?q=q(x)" /> is a continuous function. In this paper we show that the operator <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?L" /> is continuously invertible when these coefficients satisfy some suitable conditions and obtain the following estimate for <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?y%20%5Cin%20D(L)" />:                                            <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5C%7C%7C-%5Crho(%5Crho%20y')'%5C%7C%7C_2+%5C%7C%7Cr%20y'%5C%7C%7C_2+%5C%7C%7Cq%20y%5C%7C%7C_2%5Cleq%20c%20%5C%7C%7CL%20y%5C%7C%7C_2" />,where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?$D(L)$" /> is the domain of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?$L$" />.In papers D, E, and F various differential equations of the third order of the form       <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?-m_1(x)%5Cleft(m_2(x)%5Cleft(m_3(x)y'%5Cright)'%5Cright)'+%5Bq(x)+ir(x)+%5Clambda%5Dy=f(x)%20%20%5C%20%20%20%20%20(0.1)" />are studied in the space <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?L_p(%5Cmathbb%7BR%7D)" />.In paper D we investigate the case when <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?m_1=m_3=m" /> and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?m_2=1" />.Moreover, in paper E the equation (0.1) is studied when <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?m_3=1" />. Finally, in paper F the equation (0.1) is investigated under certain additional conditions on <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?$m_j(x)%20(j=1,2,3)$" />.For these equations we establish sufficient conditions for the existence and uniqueness of the solution, and also prove an estimate of the form      <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cleft%5C%7Cm_1(x)(m_2(x)%5Cleft(m_3(x)y')'%5Cright)'%5Cright%5C%7C%5Ep_p+%5Cleft%5C%7C(q(x)+ir(x)+%5Clambda)y%5Cright%5C%7C%5Ep_p%20%5Cleq%20c%20%5Cleft%5C%7Cf(x)%5Cright%5C%7C%5Ep_p" />for the solution <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?y" /> of equation (0.1).                                                                             


NATURVETENSKAP  -- Matematik -- Matematisk analys (hsv//swe)
NATURAL SCIENCES  -- Mathematics -- Mathematical Analysis (hsv//eng)



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