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On some singular Sturm-Liouville equations and a Hardy type inequality
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Castro, Hernán. On some singular Sturm-Liouville equations and a Hardy type inequality. Retrieved from https://doi.org/doi:10.7282/T3TX3CXN

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TitleOn some singular Sturm-Liouville equations and a Hardy type inequality
NameCastro, Hernán (author); Brezis, Haim (chair); Li, YanYan (internal member); Nussbaum, Roger (internal member); Peletier, Lambertus A (outside member); Rutgers University; Graduate School - New Brunswick
Date Created2012
Other Date2012-10 (degree)
SubjectMathematics, Sturm-Liouville equation--Numerical solutions, Hardy-Littlewood method
Extentix, 186 p. : ill.
DescriptionThe main body of this dissertation can be divided into two separate topics. The first topic deals with a Hardy type inequality for functions belonging to the Sobolev space $W^{m,1}_0(Omega)$, where $mgeq 2$ and $Omega$ is a smooth bounded domain in $RR^N$, $Ngeq 1$. We show that for such functions $uin W^{m,1}_0(Omega)$, one has [ orm{partial^kpt{frac{partial^ju(x)}{d(x)^{m-j-k}}}}_{L^1(Omega)}leq Corm{u}_{W^{m,1}(Omega)}, ] where $j,k$ are non-negative integers such that $1 leq k leq m-1$ and $1leq j+kleq m$, and $d(x)$ is a smooth positive function which coincides with $dist(x,domega)$ near $domega$. The second topic deals with the study of the singular Sturm-Liouville operator {$mathcal L_alpha u:=-(x^{2alpha}u')'$,} where $alpha>0$. We develop a linear theory for such operator by introducing suitable weighted Sobolev spaces and prove existence and uniqueness for equations of the form $mathcal L_alpha u+u=fin L^2$ under both homogeneous and non-homogeneous boundary data at the origin. In addition, the spectrum of the operator $mathcal L_alpha$ is fully described. Finally, we prove existence, non-existence and uniqueness results for positive solutions of the non-linear singular Sturm-Liouville equation $mathcal L_alpha u=lambda u+u^p, u(1)=0$, where $alpha>0$, $p>1$ and $lambdainRR$ are parameters.
NotePh.D.
NoteIncludes bibliographical references
NoteIncludes vita
Noteby Hernán Castro
Genretheses, ETD doctoral
Persistent URLhttps://doi.org/doi:10.7282/T3TX3CXN
Languageeng
CollectionGraduate School - New Brunswick Electronic Theses and Dissertations
Organization NameRutgers, The State University of New Jersey
RightsThe author owns the copyright to this work.
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