- Instructor: Saikat Mazumdar
- Email: saikat@math.iitb.ac.in, saikat.mazumdar@iitb.ac.in
- Timings: Mondays and Thursdays from 2pm, online.

This course is an
introduction to the theory of Partial Differential Equations (PDEs, for
short), focusing on second-order linear equations.

The course will run online.** Course
notes** will be uploaded regularly and the material will be
discussed online. The notes will contain exercises and you are expected to
solve them. Additional Problem-Sets will be posted.

*Grading will be based on
Homework. Exam details will be notified later. *

- Laplace's Equation and Harmonic Functions: Mean value properties, Maximum principles, Fundamental solutions, Energy methods and the Dirichlet's Principle.
- Maximum Principle for Elliptic PDEs and Existence.
- Theory of distributions and Fourier Analysis.
- Sobolev Spaces.
- Second order Linear Elliptic PDEs and L^2 Regularity theory.
- The Heat equation and Second order Linear Parabolic PDEs: Existence, Regularity and Maximum principles.
- The Wave equation and Second order Linear Hyperbolic PDEs: Existence, Propagation of disturbances.

- L. C. Evans, Partial Differential Equations; Graduate Studies in Mathematics, American Mathematical Society.

- M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations; Texts in Applied Mathematics, Springer-Verlag.

- S. Kesavan, Topics in Functional Analysis and Applications; New Age International Pvt. Ltd.

- J. Jost, Partial Differential Equations; Graduate Texts in Mathematics; Springer-Verlag.

- Gilbarg and N. Trudinger; Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag.

- Q. Han and F. Lin, Elliptic Partial Differential Equations; Courant Lecture Notes in Mathematics, American Mathematical Society.

- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations; Universitext, Springer-Verlag.

- J. Rauch, Partial Differential Equations; Graduate Texts in Mathematics, Springer-Verlag.

- G. B. Folland, Introduction to Partial Differential Equations; Prentice-Hall, or TIFR lecture notes: http://www.math.tifr.res.in/~publ/ln/tifr70.pdf.

- M. E. Taylor, Partial Differential Equations I-Basic theory; Applied Mathematical Sciences, Springer-Verlag.

- L. Hormander, The Analysis of Linear Partial Differential Operators I; Classics in Mathematics, Springer-Verlag.

- R. S. Strichartz, A guide to distribution theory and Fourier transforms, Studies in Advanced Mathematics, CRC Press.

- L. C. Evans, Partial Differential Equations, Princeton Companion to Applied Mathematics; https://math.berkeley.edu/~evans/evans_pcam.pdf.

- H. Brezis and F. Browder, Partial Differential Equations in the 20th Century; https://doi.org/10.1006/aima.1997.1713.