Course11110615485 lectures
Sobolev Spaces and Partial Differential Equations
Prof. S. Kesavan · IMSc
About
The modern theory of Partial Differential Equations relies heavily on functional analytic methods. With the advent of high speed computers, numerical methods, like the finite element method, have revolutionized areas like computational fluid dynamics and structural analysis. These rely on the study of weak solutions to PDEs and functional analysis plays a dominant role in this. At the core of the ideas involved lie the theory of distributions and the important function spaces, called the Sobolev spaces. These spaces form a natural framework in which we study generalized (i.e. weak) solutions of boundary value problems. This course will develop, in detail, the theory of distributions and study the important properties of Sobolev spaces. These will be applied to the study of weak solutions of elliptic boundary value problems. The theory of semigroups of operators on a Banach space will also be developped. This, together with Sobolev spaces will help us to study evolution equations. Prerequisites for this course are analysis, topology, linear algebra, functional analysis and measure theory (especially, the theory of L-p spaces).
Intended audience: MSc (Mathematics) and above
Prerequisites: MSc Real Analysis, Topology, Linear Algebra,Measure Theory, Functional Analysis
Lectures
Week 1
- Test Functions - Part 1
- Test Functions - Part 2
- Distributions
- Examples - Part 1
- Distribution Derivatives
- More operations on distributions
- Support of a distribution
Week 2
- Distributions with compact support; singular support - Part 1
- Distributions with compact support; singular support - Part 2
- Exercises - Part 1
- Convolution of functions - Part 1
- Convolution of functions - Part 2
- Convolution of functions - Part 3
- Convolution of distributions - Part 1
Week 3
- Convolution of distributions - Part 2
- Convolution of distributions - Part 3
- Exercises - Part 2
- Fundamental solutions
- The Fourier transform
- The Schwarz space - Part 1
- The Schwarz space - Part 2
Week 4
- Examples - Part 1
- Fourier inversion formula
- Tempered distributions
- Exercises - Part 3
- Sobolev spaces - Part 1
- Sobolev spaces - Part 2
- Sobolev spaces - Part 3
Week 5
- Approximation by smooth functions
- Chain rule and applications - Part 1
- Chain rule and applications - Part 2
- Extension theorems - Part 1
- Extension theorems - Part 2
- Poincare's inequlity
Week 6
- Exercises - Part 4
- Exercises - Part 5
- Imbedding theorems
- Imbedding theorems: Case p < N - Part 1
- Imbedding theorems:Case p = N - Part 2
- Imbedding theorems: Case p > N - Part 3
- Compactness theorems - Part 1
Week 7
- Compactness theorems - Part 2
- Compactness theorems - Part 3
- The spaces W^{s,p}
- Spaces W^{s,p} and Trace spaces
- Trace theory - Part 1
- Trace theory - Part 2
Week 8
- Trace theory - Part 3
- Trace theory - Part 4
- Exercises - Part 6
- Exercises - Part 7
- Abstract variational problems - Part 1
- Abstract variational problems - Part 2
- Weak solutions of elliptic boundary value problems - Part 1
Week 9
- Weak solutions of elliptic boundary value problems - Part 2
- Neumann problems
- The Biharmonic operator
- The elasticity system
- Exercises - Part 8
- Exercises - Part 9
- The Galerkin method
Week 10
- Maximum Principles - Part 1
- Maximum Principles - Part 2
- Exercises - Part 10
- Exercises - Part 11
- Eigenvalue problems - Part 1
- Eigenvalue problems - Part 2
- Eigenvalue problems - Part 3
- Exercises - Part 12
Week 11
- Exercises - Part 13
- Unbounded operators - Part 1
- Unbounded operators - Part 2
- The exponential map
- C_0 Semigroups - Part 1
- C_0 Semigroups - Part 2
- Infinitesimal generators of contraction semigroups
- Hille-Yosida theorem
Week 12
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IMSc
- Department
- Mathematics
- Teacher
- Prof. S. Kesavan
- Level
- Graduate
- Language
- English
- Subjects
- Mathematics
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
- Who can watch
- NPTEL gives no rating. Invisible Network shows it to Tweens profiles and older, judged from what the source says about it. How courses are placed