La ecuación del calor

Abstract

[en] The objective of this work will be, first of all, to develop the basic theoretical content related to the heat equation and, after that, to state and demonstrate some applications using such content.

The work is divided into three sections, the first of which focuses on searching for solutions to the heat equation. First, we will look for solutions in all the space. To do this, we will begin by considering the case of the homogeneous heat equation and, from it, we will obtain solutions for the non-homogeneous heat equation. Later we will look for solutions in bounded domains, for this we will also see the method of separation of variables.

The second section deals with two important properties of solutions to the heat equation: uniqueness and regularity. To see these properties we will also have to look at the strong maximum principle and the mean value property for the heat equation.

In these first two sections we have mainly followed Evans' book ([1]) and in some cases we have used Folland ([2]) and Ireneo Peral's books ([3]), especially for the parts referring to the Fourier transform.

Finally, we will use everything we have learned in the first sections to prove two statements related to Liouville's theorem. The first is the Liouville theorem itself for harmonic functions, that is, if we have a harmonic function that is bounded in $\mathbb{R}^{\mathrm{n}}$, then it is constant. The second is an extension of the same theorem which says that if we have a harmonic function whose growth is limited by a power of the distance to the origin, then the function is a polynomial. For this part we will be guided by Yoichi Miyazaki's article ([4]).