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arXiv · 2301.00095

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

Xiaoqi Huang·Yannick Sire·Xing Wang·Cheng Zhang
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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.

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Xiaoqi Huang, Yannick Sire, Xing Wang, Cheng Zhang. 2022-12-31. Sharp $L^p$ estimates and size of nodal sets for generalized Steklov eigenfunctions. https://arxiv.org/abs/2301.00095

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