Course11110816842 lectures
Advanced Partial Differential Equations - (Part I: Distributions and Sobolev Spaces)
Prof. A. K. Nandakumaran and Prof. P. S. Datti · IISc Bangalore
About
This is intended to be a first part on Advanced Partial Differential Equations, which generally requires 40 lecture hours (One semester course). So, we plan to make this course in two parts – 20 hours each. This first part is an introduction to distributions and Sobolev spaces, which are the tools for the second part. The contents of this course are not yet part of the curriculum in many our universities. Thus, we hope that this course is beneficial to many students and teachers. We plan to offer the first part starting in August 2023 and second part may be in January (August?) 2024.
Intended audience: Mathematics, Science and Engineering
Prerequisites: Measure and Integration; Functional Analysis, Basic PDE (for example, NPTEL courses on PDE – I & II)
Lectures
Week 1 Bascis Real and Functional Analysis
Week 2 Fourier Transform and Topological Vector Spaces
Week 3 Distributions
- Introduction to Distributions
- Properties and Examples
- Convergence of distributions
- Convergence of distributions
- Calculus in the space of distributions
Week 4 Order and support of distributions
- Further discussion on Distributions
- Order and support of a distribution
- Laplace and Poisson equations - Distributions with compact support
- validity of the definition of the support
- Convolution and Fourier transform of distributions
Week 5 Convolution and Fourier transform of Distributions
- The Schwartz space and its dual
- Fourier transform of a tempered distribution, convolution
- Properties of Convolution
- Further discussion on Fourier transform and convolution
- Convolution of two distributions
Week 6 Sobolev Spaces
- convolution of distributions
- Introduction to Sobolev spaces
- Properties of Sobloev Spaces
- Extension and Density results
- General Extension result
Week 7 Extension result and compact embeddings
- Integration on a smooth surface
- A more general extension result
- Notion of the trace
- A compactness theorem
- Equivalent norms
- Sobolev lemma
Week 8 Trace theory
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IISc Bangalore
- Department
- Mathematics
- Taught by
- Prof. A. K. Nandakumaran, Prof. P. S. Datti
- 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.
- 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