# Advanced Partial Differential Equations - (Part I: Distributions and Sobolev Spaces)

- Institution: IISc Bangalore
- Department: Mathematics
- Taught by: Prof. A. K. Nandakumaran, Prof. P. S. Datti
- Level: Graduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111108168)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111108168)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

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

1. [Introduction-1](https://invisiblenetwork.ai/learn/nptel/111108168/86c83e05aa79) (YouTube)
2. [Preliminaries-1](https://invisiblenetwork.ai/learn/nptel/111108168/3d176ef740cb) (YouTube)
3. [Preliminaries-2](https://invisiblenetwork.ai/learn/nptel/111108168/07c99c5cf921) (YouTube)
4. [Priliminaries-3](https://invisiblenetwork.ai/learn/nptel/111108168/c0dcfc19fe3b) (YouTube)
5. [Priliminaries-4](https://invisiblenetwork.ai/learn/nptel/111108168/b4f17935a39c) (YouTube)
6. [Preliminaries-5](https://invisiblenetwork.ai/learn/nptel/111108168/53dca1d4e7e3) (YouTube)
7. [Prelimunaries-6](https://invisiblenetwork.ai/learn/nptel/111108168/7adaba472c25) (YouTube)
8. [Preliminaries-7](https://invisiblenetwork.ai/learn/nptel/111108168/2fd9535cc923) (YouTube)
9. [Preliminaries-8](https://invisiblenetwork.ai/learn/nptel/111108168/a730e44b41ab) (YouTube)
10. [Preliminaries-9](https://invisiblenetwork.ai/learn/nptel/111108168/532cc79ad762) (YouTube)
11. [Introduction to Distributions](https://invisiblenetwork.ai/learn/nptel/111108168/26a4d6287941) (YouTube)
12. [Properties and Examples](https://invisiblenetwork.ai/learn/nptel/111108168/d319be8dd5db) (YouTube)
13. [Convergence of distributions](https://invisiblenetwork.ai/learn/nptel/111108168/f321a9b1a184) (YouTube)
14. [Convergence of distributions](https://invisiblenetwork.ai/learn/nptel/111108168/36cb21ab54ed) (YouTube)
15. [Calculus in the space of distributions](https://invisiblenetwork.ai/learn/nptel/111108168/6694779b9182) (YouTube)
16. [Further discussion on Distributions](https://invisiblenetwork.ai/learn/nptel/111108168/02dccca5cf72) (YouTube)
17. [Order and support of a distribution](https://invisiblenetwork.ai/learn/nptel/111108168/1f2b7329bcf7) (YouTube)
18. [Laplace and Poisson equations - Distributions with compact support](https://invisiblenetwork.ai/learn/nptel/111108168/51b4285f57e1) (YouTube)
19. [validity of the definition of the support](https://invisiblenetwork.ai/learn/nptel/111108168/747d8cd695b1) (YouTube)
20. [Convolution and Fourier transform of distributions](https://invisiblenetwork.ai/learn/nptel/111108168/d69b1e30e6b4) (YouTube)
21. [The Schwartz space and its dual](https://invisiblenetwork.ai/learn/nptel/111108168/f24d16b9b2e1) (YouTube)
22. [Fourier transform of a tempered distribution, convolution](https://invisiblenetwork.ai/learn/nptel/111108168/6bfe03a46ca0) (YouTube)
23. [Properties of Convolution](https://invisiblenetwork.ai/learn/nptel/111108168/89bfe23d960e) (YouTube)
24. [Further discussion on Fourier transform and convolution](https://invisiblenetwork.ai/learn/nptel/111108168/2a07b39dd6d9) (YouTube)
25. [Convolution of two distributions](https://invisiblenetwork.ai/learn/nptel/111108168/e0650eae84c6) (YouTube)
26. [convolution of distributions](https://invisiblenetwork.ai/learn/nptel/111108168/28505bab14bf) (YouTube)
27. [Introduction to Sobolev spaces](https://invisiblenetwork.ai/learn/nptel/111108168/78f4d6272551) (YouTube)
28. [Properties of Sobloev Spaces](https://invisiblenetwork.ai/learn/nptel/111108168/d6ac0b2f00fe) (YouTube)
29. [Extension and Density results](https://invisiblenetwork.ai/learn/nptel/111108168/c6db4db616c1) (YouTube)
30. [General Extension result](https://invisiblenetwork.ai/learn/nptel/111108168/c66b44318e15) (YouTube)
31. [Integration on a smooth surface](https://invisiblenetwork.ai/learn/nptel/111108168/1431841d45f9) (YouTube)
32. [A more general extension result](https://invisiblenetwork.ai/learn/nptel/111108168/48bdd33bb68f) (YouTube)
33. [Notion of the trace](https://invisiblenetwork.ai/learn/nptel/111108168/704e2124b0dc) (YouTube)
34. [A compactness theorem](https://invisiblenetwork.ai/learn/nptel/111108168/65ec80689c36) (YouTube)
35. [Equivalent norms](https://invisiblenetwork.ai/learn/nptel/111108168/53ea2ef34a2e) (YouTube)
36. [Sobolev lemma](https://invisiblenetwork.ai/learn/nptel/111108168/655baf25595a) (YouTube)
37. [Sobolev lemma (continued)](https://invisiblenetwork.ai/learn/nptel/111108168/7814fb675777) (YouTube)
38. [Analysis near the boundary](https://invisiblenetwork.ai/learn/nptel/111108168/a5ac5c537b6b) (YouTube)
39. [Trace in the upper half space](https://invisiblenetwork.ai/learn/nptel/111108168/c1926c5b6776) (YouTube)
40. [Trace in the upper half space](https://invisiblenetwork.ai/learn/nptel/111108168/cfa2e53fdf06) (YouTube)
41. [Supplementary lecture](https://invisiblenetwork.ai/learn/nptel/111108168/885c34ab4560) (YouTube)
42. [Supplementary lecture](https://invisiblenetwork.ai/learn/nptel/111108168/dd85c7115aed) (YouTube)