Sobolev Spaces and Partial Differential Equations

By Prof. S. Kesavan   |   IMSc
Learners enrolled: 227
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
PRE-REQUISITES         : MSc Real Analysis, Topology, Linear Algebra,Measure Theory, Functional Analysis
Course Status : Upcoming
Course Type : Elective
Duration : 12 weeks
Start Date : 26 Jul 2021
End Date : 15 Oct 2021
Exam Date : 23 Oct 2021
Enrollment Ends : 02 Aug 2021
Category :
  • Mathematics
Credit Points : 3
Level : Postgraduate

Course layout

Week 1:Test functions Distributios, calculus of distributions, support and singular support of distributions.
Week 2:Convolutions of functions, convolution of distributions, Fundamental solutions.
Week 3:Fourier transform, Fourier inversion, tempered distributions.
Week 4:Sobolev spaces, definition, approximation by smooth functions.
Week 5:Extension theorems, Poincare inequality, Imbedding theorems.
Week 6:Compactness theorems, trace theory.
Week 7:Variational problems in Hilbert spaces and Lax-Milgram lemma. Examples of weak formulations of elliptic boundary value problems.
Week 8:Regularity, Galerkin’s method, Maximum principles.
Week 9:Eigenvalue problems, introduction to the finite element method.
Week 10:Semigroups of operators. Examples, basic properties, Hille-Yosida theorem.
Week 11:Maximl dissipative operators, regularity
Week 12:Heat equation, wave equation, Schrodinger equation. Inhomogeneous equations.

Books and references

  • S. Kesavan, Topics in Functional Analysis and Applications, Third Edition, New Age International Publishers

Instructor bio

Prof. S. Kesavan

S. Kesavan retired as Professor from the Institute of Mathematical Sciences, Chennai. He obtained his doctoral degree from the Universite de Pierre et Marie Curie (Paris VI), France. His research interests are in Partial Differential Equations. He is the author of five books. He is a Fellow of the Indian Academy of Sciences and the National Academy of Sciences, India. He has served as Deputy Director of the Chennai Mathematical Institute (2007-2010) and two terms (2011-14, 2015-18) as Secretary (Grants Selection) of the Commission for Developing Countries of the International Mathematical Union. He was a member of the National Board for Higher Mathematics during 2000-2019.

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 23 October 2021 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.


Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Madras .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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