**[Cr:4, Lc:3, Tt:1, Lb:0]**

- 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

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