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# Second-order linear elliptic systems on the complex plane

## LAUR Repository

 dc.contributor.author Habre, Samer Said dc.date.accessioned 2018-06-19T06:36:24Z dc.date.available 2018-06-19T06:36:24Z dc.date.copyright 1991 en_US dc.identifier.uri http://hdl.handle.net/10725/8055 dc.description.abstract Let ${\cal S}$ denote the general second order linear differential operator on the complex plane. en_US It is a well-known fact that the Dirichlet boundary value problem for ${\cal S}$ on a bounded domain $\Omega$ with a smooth boundary, is not always well-posed even when ${\cal S}$ is elliptic. This phenomenon led to a homotopic classification of elliptic systems. B. Bojarski, a pioneer in this subject, showed that the family of elliptic operators forms an open set in $\doubc\sp6$ with exactly six components. It follows from his classification that the only components where the Dirichlet problem may be well-posed are the ones represented by the Laplacian or its complex conjugate. We denote by ${\cal E}(\Delta)$ the component consisting of elliptic operators that are deformable to the Laplacian. The objective of this dissertation is to establish the Fredholm alternative for the equation: ${\cal SW}(z)=g(z),{\cal S}\in{\cal E}(\Delta).$ Using the complex Hilbert transforms, the problem reduces to showing that Lw(z) = g(z) satisfies the Fredholm alternative, where L is an integral operator and $w\in L\sp{p}(\doubc)$. Assuming first that the coefficients of ${\cal S}$ are constant functions, we show that L is in fact invertible. In case the coefficients are only measurable, we construct an example of an operator with a non-trivial kernel. If the coefficients are uniformly continuous functions on $\doubc,$ we show that L may be a Fredholm operator with index equal to zero for p close to 2. The main result of this paper generalizes the latter as follows: we assume that the coefficients are functions of vanishing mean oscillation and we show that L is Fredholm with index zero for all $p\in(1,\infty).$ dc.language.iso en en_US dc.subject Differential equations, Elliptic en_US dc.title Second-order linear elliptic systems on the complex plane en_US dc.type Thesis en_US dc.author.degree PHD en_US dc.author.school SAS en_US dc.author.idnumber 199329050 en_US dc.author.department Computer Science and Mathematics en_US dc.description.embargo N/A en_US dc.description.physdesc 83 p: ill en_US dc.author.advisor Iwaniec, Tadeusz dc.description.bibliographiccitations Includes bibliographical references en_US dc.identifier.ctation Habre, S. S. (1991). Second-order linear elliptic systems on the complex plane. en_US dc.author.email shabre@lau.edu.lb en_US dc.identifier.tou http://libraries.lau.edu.lb/research/laur/terms-of-use/thesis.php en_US dc.identifier.url https://dl.acm.org/citation.cfm?id=918950 en_US dc.orcid.id https://orcid.org/0000-0002-7887-8767 en_US dc.publisher.institution Syracuse University Syracuse en_US dc.author.affiliation Lebanese American University en_US
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