# DEGREE REGULATIONS & PROGRAMMES OF STUDY 2015/2016

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# Undergraduate Course: Elliptic Partial Differential Equations (MATH10089)

 School School of Mathematics College College of Science and Engineering Credit level (Normal year taken) SCQF Level 10 (Year 4 Undergraduate) Availability Available to all students SCQF Credits 10 ECTS Credits 5 Summary The aim of the course is to give an introduction to elliptic partial differential equations and their applications in geometry, material sciences, biology and finance. The student will learn about important concepts such as the notion of a weak solution, the maximum principle as well as important theoretical results, for example the Krylov- Safonov theorem. Course description Examples of elliptic PDEs and various types of weak solutions. Weak differentiability, Sobolev spaces and classical solutions. Harmonic functions: Mean value theorem, Greens function Poisson kernel. Maximum principle for general linear equations. Divergence form equations, Caccioppoli's inequality, Morrey's theorem in two dimensions. Non-divergence form equations, strong and viscosity solutions. Aleksandrov's maximum principle. Krylov-Safonov's theorem.
 Pre-requisites Students MUST have passed: Honours Analysis (MATH10068) OR Algebra and Calculus (PHYS08041) OR Linear Algebra and Several Variable Calculus (PHYS08042) Co-requisites Prohibited Combinations Other requirements None
 Pre-requisites None
 Not being delivered
 1. Reformulate equations of divergence form through integral identities using partial integration. 2. Evaluate or estimate the maximum and minimum of solutions of elliptic equation using the maximum principle. 3. Infer regularity of solutions from that of given data. 4. Explicitly compute the (radially) symmetric solutions and Green/Poisson kernels for the Laplace operator. 5. Estimate first and second order derivatives of solutions via integral norms of solution itself. 6. Compare qualitative properties of weak and viscosity solutions.
 Recommended : E. Landis, Second Order Equations of Elliptic and Parabolic Type, AMS 1998 Qing Han, Fanghua Lin, Courant Institute Lecture notes, NYU, 2011 D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2011
 Graduate Attributes and Skills Not entered Keywords EPDE
 Course organiser Dr Martin Dindos Tel: Email: M.Dindos@ed.ac.uk Course secretary Mrs Alison Fairgrieve Tel: (0131 6)50 5045 Email: Alison.Fairgrieve@ed.ac.uk
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