Classical partial Differential equations

Specialeforsvar ved Sarah Andersen 

Titel: Classical partial Differential Equations

Abstract: In this thesis we will study three of the classical partial differential equations: the elliptic Laplace equation, the parabolic heat equation and the hyperbolic wave equation. First we look at the Laplace equation. We establish the existence of a solution by defining a fundamental solution which we prove is in fact a solution to the Laplace equation. Another important property is that the solution is unique which will be proven by using the maximum principle. The mean-value formulas are used to prove the maximum principle. Energy estimates are also used to give an alternative proof of uniqueness. Similarly to the Laplace equation we use a fundamental solution to prove existence of a solution to the Heat equation. The uniqueness of the solution will also be proven with the maximum principle which is proven using mean-value formulas. An alternative proof of uniqueness is given by energy estimates. For the Wave equation we construct a solution for n=1 and use spherical means to find solutions for higher dimensions. We will discuss the strong Huygen's principle for odd dimensions and the weak Huygen's principle for even dimensions. Energy estimates are used to prove uniqueness.

Vejleder: Jan Philip Solovej
Censor:   Michael Pedersen, DTU