Variational Methods for Elliptic Partial Differential Equations
20 credits / 30 lectures
Many phenomena in real life are described by means of partial differential equations (PDEs). Therefore, the study of PDEs or (initial) boundary value problems plays a crucial role in the mathematical modelling of real life phenomena. The purpose of these lectures is to study partial differential equations or precisely, initial bounday value problems from the theoretical view point.
After introducing the general setting of PDEs, we will present various classes of PDEs with examples of physical phenomena described by each class. This year, we will focus essentially on the class of elliptic PDEs with the Lalpace and Poisson equations as model examples. We will analyze among others, cases of non-existence results for some boundary value problems governed by the Laplace or the Poisson equations. These will be used as motivations for studying variational methods for boundary value Problems.
We will then embark on the study of fundamental tools like Sobolev spaces in which weak formulations of elliptic boundary value problems are obtained. Then we apply abstract results like the Hartman-Stampacchia and the Lax-Milgram theorems to get existence results for elliiptic boundary value problems.
We might study (time permitting) the regularity theory for some elliptic boundary value problems.
Prerequisites:Real Analysis (2RA), Topics in Analysis (MAM3000W-3TN) or any other equivalent course, Honours Analysis Core or any other equivalent course.