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

ABOUT THE COURSE: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 EngineeringPREREQUISITES: Measure and Integration; Functional Analysis, Basic PDE (for example, NPTEL courses on PDE – I & II)

Week 1: Some topics from L^p spacesWeek 2:Concepts and results from Functional AnalysisWeek 3:Concepts and results from Functional Analysis. Space of Test FunctionsWeek 4:Space of Test Functions. Distributions.Week 5:Distributions. Weak Derivative via IntegralWeek 6:Sub-class of Distributions – Introduction to Sobolev SpacesWeek 7:Sobolev spaces – Prolongation and Density TheoremsWeek 8:Sobolev Spaces – Embedding theorems, Trace theorem