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Comparison and oscillation theorems for elliptic equations and systems Noussair, Ezzat Sami

Abstract

In the first part of this thesis, strong comparison theorems of Sturm's type are established for systems of second order quasilinear elliptic partial differential equations. The technique used leads to new oscillation and nonoscillation criteria for such systems. Some criteria are deduced from a comparison theorem, and others are derived by a direct variational method. Some of our results constitute extensions of known theorems to non-self-adjoint quasilinear systems. Application of these results to first order systems leads to criteria for the existence of conjugate points. In the second part, comparison theorems are obtained for elliptic differential operators of arbitrary even order. A description of the behaviour of the smallest eigenvalue for such operators is given under domain perturbations by means of Garding's inequality. New oscillation and nonoscillation criteria are obtained by variational methods. Specialization of our theorems to elliptic equations of fourth order, and to ordinary differential equations yields various generalizations of known results.

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