# An Introduction to Partial Differential Equations

**Semester:** Spring Semester 2017

See also the official course catalogue and *metaphor*

### Schedule Lectures

Day | Time | Room |
---|---|---|

Wednesdays | 10-12 | HG.E.21 |

Fridays | 11-12 | HG.E.21 |

### A preliminary sketch of the course

- The method of characteristics for first order equations, linear and nonlinear, transport equation, Hamilton-Jacobi equation.
- Laplace equation, fundamental solution, harmonic functions and main properties, maximum principle. Poisson equation. Green functions. Perron method for the solution of the Dirichlet problem.
- Heat equation, fundamental solution, existence of solutions to the Cauchy problem and representation formulas, main properties, uniqueness by maximum principle, regularity.
`Time permitting`

: Wave equation, existence of the solution, D'Alembert formula, solutions by spherical means, main properties.

### Recommended bibliography

**L. Evans**,*Partial Differential Equations*, AMS 2010 (2nd edition) (Ch. 1,2,3,6).**D. Gilbarg, N.S. Trudinger**,*Elliptic Partial Differential Equations of Second Order*, Springer, 1998, (3rd edition). (Ch. 1,2,3).**E. Di Benedetto**,*Partial Differential Equations*, Birkauser, 2010 (2nd edition), (Ch. 0,2,5,6).**F. John**,*Partial Differential Equations*, Springer, 1995.**T. Rivière**,*Exploring the unknown: the work of Louis Nirenberg in Partial Differential Equations*, Notices Amer. Math. Soc. 63 (2016), no. 2, 120-125.**L. Evans**,*Partial differential equations, Princeton Companion to Applied Mathematics*.**F. Da Lio**,*Introduction to Partial Differential Equations*, Lecture Notes, FS17.