Reilly type inequality for the first eigenvalue of the $L_{r;F}$ operator

Abstract

Given a positive function $F$ on ${\mathbb{S}}^n$ which satisfies a convexity condition, for $1\le r\le n$, we define for hypersurfaces in ${\mathbb{R}}^{n+1}$ the $r$-th anisotropic mean curvature function $H_{r;F}$, a generalization of the usual $r$-th mean curvature function. We also define $L_{r;F}$ operator, the linearized operator of the $r$-th anisotropic mean curvature, which is a generalization of the usual $L_r$ operator for hypersurfaces in the Euclidean space ${\mathbb{R}}^{n+1}$. The Reilly type inequalities for the first eigenvalue of the $L_{r;F}$ operator have been proved.

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