Advanced Course on Partial Differential Equations - II

ABOUT THE COURSE:This is a continuation of our earlier NPTEL course “Advanced Course on Partial Differential Equations – I”. This course is also for 8 weeks consisting of 20 one hour lecture hours. Half of the course (roughly 10 hours) is devoted to various aspects of elliptic equations and the rest deals with evolution equations (using the semi-groups as the tool).INTENDED AUDIENCE: Mathematics, Science and Engineering.PREREQUISITES: Measure and , Integration; Functional Analysis, including Semi-group theory. Basic PDE. NPTEL Course on “Advanced Course on Partial Differential Equations – I”

Week 1: Elliptic equations: Weak (Generalized) solutions of boundary value problems. Existence and Uniqueness results

Week 2:Eigenvalues and eigenfunctions. Variational properties – Rayleigh-Ritz quotient
Week 3:Asymptotic behavior of eigenvalues. Non-homogeneous boundary values
Week 4:Regularity results. Smoothness of weak solutions
Week 5:Evolution equations. Abstract formulation. Semi-group theory. Hille-Yosida theorem (Only statements of the main results)
Week 6:Parabolic and Hyperbolic equations
Week 7:Schrodinger equation
Week 8:Perturbation results. Schrodinger equation with a potential.