dc.contributor.advisor Enache, Cristian dc.contributor.author Samuel, Sajan K. dc.date.accessioned 2019-09-09T07:59:57Z dc.date.available 2019-09-09T07:59:57Z dc.date.issued 2019-07 dc.identifier.other 29.232-2019.02 dc.identifier.uri http://hdl.handle.net/11073/16487 dc.description A Master of Science thesis in Mathematics by Sajan K. Samuel entitled, “On Some Applications of the Maximum Principles to a Variety of Elliptic and Parabolic Problems”, submitted in July 2019. Thesis advisor is Dr. Cristian Enache. Soft and hard copy available. en_US dc.description.abstract One of the most important and useful tools used in the study of partial differential equations is the maximum principle. This principle is a natural extension to higher dimensions of an elementary fact of calculus: any function, which satisfies the inequality f′′ > 0 on an interval [a,b], achieves its maximum at one of the endpoints of the interval. In this context, we say that the solution to the differential inequality f′′ > 0 satisfies a maximum principle. In this thesis we will discuss the maximum principles for partial differential equations and their applications. More precisely, we will show how one may employ the maximum principles to obtain information about uniqueness, approximation, boundedness, convexity, symmetry or asymptotic behavior of solutions, without any explicit knowledge of the solutions themselves. The thesis will be organized in two main parts. The purpose of the first part is to briefly introduce in Chapter 1 the terminology and the main tools to be used throughout this thesis. We will start by introducing the second order linear differential operators of elliptic and parabolic type. Then, we will develop the first and second maximum principles of E. Hopf for elliptic equations, respectively the maximum principles of L. Nirenberg and A. Friedman for parabolic equations. Next, in the second part, namely in Chapter 2 and 3, we will introduce various P-functions, which are nothing else than appropriate functional combinations of the solutions and their derivatives, and derive new maximum principles for such functionals. Moreover, we will show how to employ these new maximum principles to get isoperimetric inequalities, symmetry results and convexity results in the elliptic case (Chapter 2), respectively spatial and temporal asymptotic behavior of solutions, in the parabolic case (Chapter 3). en_US dc.description.sponsorship College of Arts and Sciences en_US dc.description.sponsorship Department of Mathematics and Statistics en_US dc.language.iso en_US en_US dc.relation.ispartofseries American University of Sharjah Student Work en_US dc.relation.ispartofseries Master of Science in Mathematics (MSMTH) en_US dc.subject Maximum principles en_US dc.subject Isoperimetric inequalities en_US dc.subject Overdetermined problems en_US dc.subject Symmetry en_US dc.subject Convexity en_US dc.subject Time decay estimates en_US dc.subject Spatial decay estimates en_US dc.title On Some Applications of the Maximum Principles to a Variety of Elliptic and Parabolic Problems en_US dc.type Thesis en_US
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