# Introduction to partial differential equations

Dr. Georgios Psaradakis

Lectures:

**Wednesdays, 10:15 - 11:45, A5,6 C 012**

**Thursdays, 12:00 - 13:30, A5,6 C 013**

Tutorial (from Matthew Liew): **Thursday, 15:00 - 16:30, A5,6 C 013**

**Course description:** This is a master level introductory course to Partial Differential Equations, entirely focused on their classical solutions. It includes derivation of solution formulas to the linear transport equation, the wave equation, the heat equation and Laplace's equation. Several advanced topics will be presented for the last two equations. For example, we will prove the removable singularity and unique continuation theorems for Laplace's equation, the symmetry of solutions and Pohozaev's nonexistence theorem for semilinear equations involving Laplace's operator and the mean value formula for solutions of the heat equation. Also, derivation of solution formulas by using the method of characteristics, separation of variables and Green's function techniques are presented. Classical tools such as maximum principles and energy estimates are introduced.

Language: English

References: We will go through most of the first section (titled: *Representation Formulas for Solutions*) of

**Evans, L. C. - Partial Differential Equations. 2nd edition. Grad. Stud. Math. 19. Amer. Math. Soc. 2010**.

A big part of the above section of the book is provided for free on the America Mathematical Society site here

Last year's notes of Prof. Chen can be found here

drive.google.com/file/d/0B9bnL6egCoq-S0RaRy14UC1fSWM/view

Hopefully we will add to the Laplace equation part of the course, some material from the first two chapters of

**Han Q.; Lin, F. - Elliptic Partial Differential Equations. 2nd edition. Courant Lecture Notes 1. Amer. Math. Soc. 2011**.

Two older books containing most of the material we plan to cover are

Bers, L; John, F.; Schechter, M. - *Partial Differential Equations.* Lect. Appl. Math. 3A. Amer. Math. Soc. 1964,

John, F. - *Partial Differential Equations.* 4th edition. Appl. Math. Sci. 1. Springer 1982.

Two very interesting new books on the subject are

Salsa, S. - *Partial Differential Equations in Action. From modelling to theory. *3rd edition. Unitext 99. Springer 2016,

Esposito, G. - *From Ordinary to Partial Differential Equations.* Unitext 106. Springer 2017.

You can get them in the following SpringerLinks (internet connection provided by the Univ. Mannheim is required)

link.springer.com/content/pdf/10.1007%2F978-3-319-31238-5.pdf

link.springer.com/content/pdf/10.1007%2F978-3-319-57544-5.pdf