Nodal Geometry and Growth of Solutions to Elliptic Partial Differential Equations
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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.