MTH407: (Partial Differential Equations)

        [Cr:4, Lc:3, Tt:1, Lb:0]

Course Outline

• First order partial differential equations (PDEs), linear and quasi-linear equations, characteristic equations for first order PDEs. Well posed problems and Classical solutions. Weak solutions and regularity
• Second order PDEs, characteristic for linear and quasi-linear second order equations, Classification of second order PDEs, Normal forms of hyperbolic equations in two variables.
• The Cauchy initial value problem, Cauchy-Kovalevskaya theorem, Holmgren’s uniqueness theorem
• The Laplace equation: Fundamental Solution, Green’s function, Harmonic functions, Mean value theorem, The maximum principle, Harnack inequalities, Analyticity of harmonic functions, Dirichlet principle.
• The Heat equation: Fundamental solution, properties of solutions like strong maximal principle and uniqueness, infinite speed of propagation and regularity, local estimates for solution of heat equation, Energy methods
• The Wave equation: Solution by spherical means, non-homogeneous problem, finite speed of propagation, Energy methods

Additional Topics: Hamilton Jacobi equations, Elliptic equations

Reference Texts:

1. Michael Renardy and Robert C. Rogers, An Introduction to Partial Differential Equations, Springer
2. Lawrence C. Evans, Partial Differential Equations, American Mathematical Society (2014)..
3. Michael E. Taylor, Partial Differential Equations I (Basic theory), Springer
4. Gerald B. Folland, Introduction to Partial Differential Equations, Princeton University Press (2011).