# 401-3352-09L An Introduction to Partial Differential Equations

Semester | Spring Semester 2018 |

Lecturers | C. Busch, F. Da Lio |

Periodicity | non-recurring course |

Language of instruction | English |

Abstract | This course aims at being an introduction to first and second order partial differential equations (in short PDEs). We will present the so called method of characteristics to solve quasilinear PDEs and some basic properties of classical solutions to second order linear PDEs. |

Learning objective | |

Content | A preliminary plan is the following - Laplace equation, fundamental solution, harmonic functions and main properties, maximum principle. Poisson equation. Green functions. Perron method for the solution of the Dirichlet problem. Regularity of solutions to the Poisson Equation. - Heat equation, fundamental solution, existence of solutions to the Cauchy problem and representation formulas, main properties, uniqueness by maximum principle, regularity. - The Method of characteristics for first order equations, linear and nonlinear, transport equation, Hamilton-Jacobi equation, scalar conservation laws. -Time permitting: Wave equation, existence of the solution, D'Alembert formula, solutions by spherical means, main properties, uniqueness by energy methods. |

Lecture notes | The teacher provides the students with personal notes. |

Literature | Bibliography - L.Evans Partial Differential Equations, AMS 2010 (2nd edition) - D. Gilbarg, N.S. Trudinger Elliptic Partial Differential Equations of Second Order, Springer, 1998. - E. Di Benedetto Partial Differential Equations, Birkauser, 2010 (2nd edition). - W. A. Strauss Partial Differential Equations. An Introduction, Wiley, 1992. -Q. Han, A Basic Course in Partial Differential Equations, Graduate Studies in Mathematics Volume: 120; 2011. |

Prerequisites / Notice | Differential and integral calculus for functions of several variables; elementary theory of ordinary differential equations, basic facts of measure theory. |