Sharp $L^p$ estimates and size of nodal sets for generalized Steklov eigenfunctions

TL;DR

This paper establishes sharp $L^p$ estimates for Steklov eigenfunctions on manifolds with boundary, analyzes harmonic extension operators, and derives lower bounds on nodal set sizes, including a generalized Steklov problem with boundary potential.

Contribution

It provides new sharp $L^p$ bounds for Steklov eigenfunctions and extends results to a generalized problem with boundary potential, also analyzing nodal set sizes.

Findings

Sharp $L^p$ estimates for Steklov eigenfunctions

Bounds for harmonic extension and spectral projection operators

Lower bounds on nodal set sizes for generalized Steklov problem

Abstract

We prove sharp $L^p$ estimates for the Steklov eigenfunctions on compact manifolds with boundary in terms of their $L^2$ norms on the boundary. We prove it by establishing $L^p$ bounds for the harmonic extension operators as well as the spectral projection operators on the boundary. Moreover, we derive lower bounds on the size of nodal sets for a variation of the Steklov spectral problem. We consider a generalized version of the Steklov problem by adding a non-smooth potential on the boundary but some of our results are new even without potential.