Differential Equations I
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
- Maximum Principle for Elliptic PDEs and Existence.
- Theory of distributions and Fourier
- 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
- 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
- G. B. Folland, Introduction to Partial Differential Equations;
Prentice-Hall, or TIFR lecture notes:
- 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.