Sharp bounds for the first nonzero Steklov eigenvalues for $f$-Laplacians

Authors: GUANGYUE HUANG, BINGQING MA

Abstract: Let $M$ be an $n$-dimensional compact Riemannian manifold with a boundary. In this paper, we consider the Steklov first eigenvalue with respect to the $f$-divergence form: $$ e^{f}{\rm div}(e^{-f}A\nabla u)=0\ {\rm in}\ \ M, \ \ \ \ \ \langle A(\nabla u),\nu\rangle-\eta u=0 \ \ {\rm on}\ \partial M,$$ where $A$ is a smooth symmetric and positive definite endomorphism of $TM$, and the following three fourth order Steklov eigenvalue problems: $$ (\Delta_f)^2u=0\ \ {\rm in}\ M, \ \ \ \ \ u=\Delta_f u-q\frac{\partial u}{\partial \nu}=0\ \ {\rm on}\ \partial M; $$ $$ (\Delta_f)^2u=0\ {\rm in}\ \ M, \ \ \ \ \ u=\frac{\partial^2u}{\partial \nu^2}-\mu\frac{\partial u}{\partial \nu }=0 \ \ {\rm on}\ \partial M; $$ $$ (\Delta_f)^2u=0\ {\rm in}\ \ M, \ \ \ \ \ \frac{\partial u}{\partial \nu }=\frac{\partial(\Delta_f u)}{\partial \nu}+\xi u=0 \ \ {\rm on}\ \partial M. $$ Under the assumption that the $m$-dimensional Bakry-Emery Ricci curvature and the weighted mean curvature are bounded from below, we obtain sharp bounds for Steklov first nonzero eigenvalues. Moreover, we also study the case in which the bounds are achieved.

Keywords: $m$-dimensional Bakry-Emery Ricci curvature, $f$-Laplacian, Steklov eigenvalue

Full Text: PDF