Approximation of p-modulus in the plane with discrete grids

Abstract

This thesis contains four chapters. In the first chapter, the theory of continuous p-modulus in the plane is introduced and the background p-modulus properties are provided. Modulus is a minimization problem that gives a measure of the richness of families of curves in the plane. As the main example, we compute the modulus of a 2-by-1 rectangle using complex analytic methods. We also introduce discrete modulus on a graph and its basic properties. We end the first chapter by providing the relationship between connecting modulus and harmonic functions. This is the fact that computing the modulus of the family of walks from a to b is equivalent to minimizing the energy over all potentials with boundary values 0 at a and 1 at b. In the second chapter, we are interested in the connection between the continuous and the discrete modulus. We study the behavior of side-to-side modulus under some grid refinements and find an upper bound for the discrete modulus using the concept of Fulkerson duality between paths and cuts. These calculations show that the refinement will lower the discrete modulus. Since connecting modulus can also be computed by minimizing the Dirichlet energy of potential functions, we recall an argument of Jacqueline Lelong-Ferrand, that shows how refining a square grid in a ''geometric'' fashion, naturally decreases the 2- the energy of a potential. This monotonicity can be used to prove the convergence between continuous and discrete modulus. We first review the linear theory of discrete holomorphicity and harmonicity as provided by Skopenkov and Werness. Instead of reviewing their work in full generality, we present the outline of their arguments in the special case of square grids. Then use these results to prove the convergence between the continuous and discrete case. We believe that our method of proof generalizes to the full case of quadrangular grids that Werness studies. In the third chapter, we show how to generalize all our proofs for 2-modulus to the case of quadrangular grids with some geometric conditions on the lengths of edges and the angles between them. In the last chapter, a connection with potentials when p is not 2 is discussed in the square grid case. We study the behavior of side-to-side p-modulus under the same refinements as before and we find upper bound for the p-modulus, but only when p > 2. The rest of the chapter is dedicated to generalizing the results from Chapter 2 to the case 2 < p.