Study of the Laplacian

Abstract

The Laplacian on Euclidean space is the unique second order differential operator that commutes with the isometries of the Euclidean space. As such, it features in many laws of physics and is of tremendous significance in mathematics, with connections to other important objects. Of particular interest are eigenvalues and eigenfunctions of the Laplacian on domains in Euclidean space or on manifolds. Connections between these spectral objects (we stick to domains and do not consider the situation of manifolds) and the underlying geometry are the main object of study in this report.

In the first chapter we study the Laplacian on finite graphs. This is just a symmetric, positive semi-definite matrix and is technically much simpler to study than its continuous counterpart. On the other hand, many of the features of the continuous Laplacian can already be seen in this discrete setting. In particular, we see how it helps to count the number of spanning trees of the graph (Kirchhoff's theorem) and how the well-connectedness of the graph is seen in the second eigenvalue of the Laplacian (Cheeger's inequality). We also study the discrete analogue of Courant's nodal domain theorem. The key ingredient in the proofs is the variational characterization of eigenvalues.

In the second chapter, we move to the continuous Laplacian, but in one dimension. These are the St{\"u}rm-Liouville operators. The one-dimensionality allows a more precise study of the eigenvalues (Weyl's asymptotics) and eigenfunctions (oscillation theorem). Weyl's asymptotics is a feature that is not seen in the setting of finite graphs.

In the third chapter, we study the Laplacian on bounded domains of Euclidean space, with Dirichlet or Neumann boundary conditions. The subject gets much more technical, and one needs the framework of unbounded operators and the theory of Sobolev spaces to make sense of eigenvalues and eigenfunctions. Once that is done and the variational characterization of eigenvalues is proved, the proofs become quite analogous to the situation of graphs. This is true in particular for Cheeger's inequality and Courant's nodal domain theorem. But we also prove some new theorems such as the Faber-Krahn inequality (which asserts that the ball minimizes the principal Dirichlet eigenvalue among domains with a given volume) and Weyl's asymptotics for the eigenvalues of the Laplacian.