Eigenvalues of weighted \(p\)-Laplacian (Q2855915)

Let \(\langle M,g\rangle\) be a compact Riemannian manifold and \(\varphi\in C^2( M) \). Then the triple \(( M,g,\varphi) \) is called a Bakry-Émery manifold. The weighted Laplacian \( \Delta_\varphi\) (also known as drifting Laplacian) is defined by NEWLINENEWLINE\[NEWLINE\Delta_\varphi =\Delta -\nabla\varphi\cdot\nabla,NEWLINE\]NEWLINENEWLINE cf. [\textit{D. Bakry} and \textit{M. Émery}, Lect. Notes Math. 1123, 177--206 (1985; Zbl 0561.60080)]. Given \(\varepsilon >0\), letNEWLINE NEWLINE\[NEWLINEM_\varepsilon =\left\{( x,y) :x\in M,0\leq y\leq \varepsilon e^{-\varphi( x)}\right\} \subseteq M\times\mathbb R^+.NEWLINE\]NEWLINE NEWLINEIn [Math. Res. Lett. 19, No. 3, 627--648 (2012; Zbl 1270.53070)], \textit{Z. Lu} and \textit{J. Rowlett} proved that if \((\mu _k) _{k=0}^\infty\) are the eigenvalues of the weighted Laplacian \(\Delta_\varphi\) on \(M\) and \((\mu_k(\varepsilon))_{k=0}^\infty\) are the (Neumann) eigenvalues of the Laplacian \(\widetilde{\Delta}=\Delta +\partial_y^2\) on \(M_\varepsilon\), then \(\mu _k(\varepsilon )=\mu_k+O(\varepsilon^2)\) for every \(k\geq 0\). The \(p\)-Laplacian and the weighted \(p\)-Laplacian are defined by NEWLINE\[NEWLINE\Delta_p(u)=\mathrm{div}\left(\left| \nabla u\right| ^{p-2}\nabla u\right)NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\Delta_{p,\varphi}(u)=\Delta_p(u)-| \nabla u|^{p-2}\nabla\varphi\cdot\nabla .NEWLINE\]NEWLINE The author presents a generalization of the result by Lu-Rowlett to the first eigenvalue of the weighted \(p\)-Laplacian. Namely, the main theorem states that if \(p>1\) and if \(\mu _{p,\varphi }\) is the first eigenvalue of the weighted \(p\)-Laplacian on \(M\) and \(\mu _{p,\varepsilon }\) is the first (Neumann) eigenvalue of the \(p\)-Laplacian on \(M_\varepsilon\), then NEWLINE\[NEWLINE\mu_{p,\varepsilon }\leq\mu_{p,\varphi}\leq\mu_{p,\varepsilon}+O(\varepsilon ).NEWLINE\]