Description

Let u be a harmonic function, i.e. one that is a solution of the Laplace equation. This project aims to first study the set of points in the domain of u for which u vanishes. This set is called the nodal set of u. A very particular case of a harmonic function is that of a linear function, where the nodal set is a linear subspace. One aim of this project is to study the growth of the nodal set for a harmonic function and more generally for solutions to elliptic partial differential equations. The nodal set is related a property of u called the unique continuation property. This property states that at any point of the nodal set, the function u does not vanish of infinite order. Here, one may follow an approach by Garofalo and Lin. Some references are given below. This project requires an interest in rigorous analytic arguments.

Pre - and corequisites

Resources

• Nodal sets of harmonic functions
• Nodal sets of solutions of elliptic differential equations
• Notes
• Topology of nodal sets