# Lecture 1

## Parabolic maximum principle

Dear Participants,

the moment of truth has arrived: Lecture 1 has been uploaded to the wiki-page and you can download it by clicking on the PDF link

We begin this 20th edition of the Internet Seminar with one of the most useful tools used to solve several problems in the theory of partial differential equations (PDE for short): the maximum principle. In particular, in the forthcoming lectures we will use the maximum principle to prove the uniqueness of the solution to parabolic PDE's in the whole space and in bounded domains with several types of boundary conditions. In our notes we will mainly consider Dirichlet boundary conditions, but also more general boundary conditions can be considered.

From elementary calculus we know that, if a function f has positive second-order derivative on an interval [a,b], then its maximum it is achieved only on the boundary of that interval. A generalization of this argument, allows to prove the weak maximum principle.

In the first part of this lecture, we present the parabolic weak maximum principle for general uniformly elliptic operators on bounded domains. In the second part we state and prove the strong maximum principle confining ourselves to the simplest case of the heat equation.

As a tradition of the Internet Seminar (ISEM for short) each lecture
will end with a series of exercises which should be "officially" solved
by one different team per lecture.
The solutions will be posted on the web page of the ISEM so each
participants could download them. A discussion board is available on
the wiki-page Discussion board. Please, feel free to use it for discussions about the
study material and the exercises.

For this first lecture, we kindly ask the team from Salerno to produce
the "official" solutions of the exercises.

Feel free to use the wiki-page if you have any questions or comments!

Have fun and enjoy reading. Your virtual lecturers,

Abdelaziz Rhandi

Luca Lorenzi