DescriptionThe purpose of this project is to provide an insight into the maximum principle for elliptic differential equations and present an application. The books of Protter and Weinberger and Gilbarg and Trudinger provide a good companion to the project, and provide a full treatment of the maximum principle. The project begins by introducing the maximum principle in one dimension in order to give the student a firm intuitive grasp of the subject, before proceeding to the more general elliptic case. The idea in this case is to deduce from elementary calculus that the solution to a certain generic second-order linear boundary-value problem cannot attain a maximum on the interior of the domain. Having gained a good intuitive grounding on the one dimensional maximum principle, we next move on to the n-dimensional case. It is important to note that the theorems and proofs we discuss here each have their exact one dimensional counterpart - the proofs of n-dimensional results here reduce to the proofs of the one dimensional results when n = 1. The maximum principle has a large number of applications in geometry and students are encouraged to present one of them. Number of studentsWe have a limit of five students on this project.ResourcesEvans, L.C.(2010) Partial Differential Equations, American Mathematical Society Gilbarg, D. and Trudinger, N. S. (1977) Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin Protter, M.H. and Weinberger, H.F. (1967) Maximum Principles in Differential Equations. Prentice-Hall, New Jersey. Maximum Principle, Wikipedia entry : https://en.wikipedia.org/wiki/Maximum_principle email: W Klingenberg |