Partial differential equations are fundamental in the modeling of many physical phenomena and technical problems. This class will introduce you to the mathematical theory of linear elliptic, parabolic and hyperbolic partial differential equations, which are the main types of pde’s and closely related to important physical phenomena, such as diffusion, wave and transport problems. An important topic of this course is to provide the mathematical framework that is necessary to study partial differential equations. This will provide you with the theoretical background to investigate the mathematical properties of partial differential equations and to analyze numerical techniques for the solution of partial differential equations, such as finite element methods.
After completion of this course the student should be able to:
1. derive key properties of Sobolev spaces, such as weak derivatives, trace, extension and embedding theorems in a simple setting, and apply these results to analyze properties of partial differential equations.
2. prove existence, uniqueness and regularity of strongly elliptic second order PDE’s and related evolution equations, such as the heat and wave equation, using the Fredholm alternative and the Galerkin technique.
3. read relevant research papers on (numerical) methods for partial differential equations.
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Since the theory of linear partial differential equations relies heavily on Sobolev and Holder spaces we will first thoroughly study the mathematical properties of these function spaces. The second topic will be second order strongly elliptic partial differential equations. These equations have been extensively studied and provide a good starting point for the study of partial differential equations. Important topics will be techniques to determine the existence and uniqueness of solutions and their regularity. Also, eigenvalue problems and the maximum principle as well as singularities in corner domains will be discussed. The third topic will be second order parabolic and hyperbolic linear evolution equations, which model, respectively, diffusion and wave problems. The class will conclude with linear first order hyperbolic systems and a brief introduction to semigroup theory. Some of the theoretical results will be used in the analysis of finite element discretizations of elliptic partial differential equations.
The course material consists of Chapters 5, 6 and 7 of the book of Evans.
Prior knowledge necessary
Introduction to Partial Differential Equations (201700034)
Applied Functional Analysis (191506302)
Suggested prior knowledge
Measure and probability (191570401)
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