Nodal Geometry and Growth of Solutions to Elliptic Partial Differential Equations

Abstract

In this thesis we will study zeros and growth properties of solutions to elliptic PDEs. In particular, we first study zeros of linear combination of Laplace-Beltrami eigenfunctions via a growth estimate for positive solutions of higher order elliptic PDEs. A large part of the thesis is then dedicated to the nodal set of Steklov eigenfunctions, which are solutions to an elliptic spectral problem. Finally, we prove a delicate gradient inequality for Laplace-Beltrami eigenfunctions. The key motive throughout the thesis is that solutions to elliptic PDEs behave like polynomials.