NOC:Advanced Course on Partial Differential Equations - II

Lecture 1 - Introduction and Outline

Lecture 2 - Review of Sobolev spaces - 1

Lecture 3 - Review of Sobolev spaces - 2

Lecture 4 - Review of Sobolev spaces - 3

Lecture 5 - Review of Sobolev spaces - 4

Lecture 6 - Review of Sobolev spaces - 5

Lecture 7 - Elliptic equations and weak formulation

Lecture 8 - Abstract Formulation

Lecture 9 - Variational Inequality

Lecture 10 - Babuska-Brezzi Theorem

Lecture 11 - Strong vs Weak Form of PDE

Lecture 12 - General Second Order Equations

Lecture 13 - Non-uniqueness of Neumann problem

Lecture 14 - Biharmonic equation, Stokes system

Lecture 15 - Stokes system (Continued...)

Lecture 16 - Regularity of Ellptic Equations

Lecture 17 - Regularity (Continued...)

Lecture 18 - Maximum and Minimum Principles for weak formulation

Lecture 19 - Spectrum of the Laplace operator

Lecture 20 - First eigenvalue of the Laplace operator and a brief discussion of a Galerkin method

Lecture 21 - Introduction to semi-groups and unbounded operators

Lecture 22 - Spectrum of an operator

Lecture 23 - Examples of operators and operators in a Hilbert space

Lecture 24 - Perturbation of an operator, Kato theorems. Definition of a semi-group

Lecture 25 - Semi-group and its infinitesimal generator

Lecture 26 - Examples of semi-groups; spectrum and resolvent of the infinitesimal generator

Lecture 27 - An application. Saturation theorem

Lecture 28 - Hille-Yosida Theorem

Lecture 29 - Lumer-Philips Theorem. Stone's theorem

Lecture 30 - Abstract Cauchy Problem

Lecture 31 - Semi-group arising from the heat equation

Lecture 32 - Geneartor of the heat semi-group

Lecture 33 - Semi-group arising from the wave equation

Lecture 34 - Semi-group arising from the wave equation in energy space

Lecture 35 - Wave equation in energy space

Lecture 36 - Wave equation in energy space (Continued...)

Lecture 37 - Semi-group arising from the wave equation in H^1 x L^2

Lecture 38 - Schroedinger equation in free space

Lecture 39 - Schroedinger equation with a potential. Application of Kato's theorem

Lecture 40 - Equations in a bounded domain. Concluding remarks