Best constant for trace Nash inequality
Let \((M,g)\) be a Riemannian smooth compact \(n\)-manifold with \(n\in \mathbb N^{\ast}\), \(n\geq 2\). Consider the following inequality, called the trace Nash inequality: \[ \left(\int_{\partial M}u^2ds_g\right)^{\frac n{n-1}}\leq \left(A\int_M \left|\nabla u\right|_g^2dv_g + B\int_{\partial M}u^2ds_g\right) \left(\int_{\partial M}\left|u\right|ds_g\right)^{\frac 2{n-1}} \tag{1} \] for all functions \(u\in C^{\infty}(M)\). Define the first best constant in the trace Nash inequality as the infimum of the \(A>0\) such that there exists \(B>0\) such that (1) holds, and denote it by \(\widetilde{A}_0(n)\). It was shown in another paper by the same author [\textit{E. Humbert}, Geom. Funct. Anal. 11, 759-772 (2001; Zbl 1005.58014)]that \[ \widetilde{A}_0(n)^{-1}=\inf\left\{\frac{\int_{\mathbb R_+^n}\left|\nabla u\right|^2dx \left(\int_{\partial \mathbb R^n_+}\left|u_0\right|ds\right)^{\frac 2{n-1}}}{\left(\int_{\partial \mathbb R^n_+} u_0^2ds\right)^{\frac n{n-1}}},\;u\in C^{\infty}_c(\mathbb R^n_+),\;u\neq 0\right\} \] where \(u_0\) is the trace of \(u\) on \(\partial \mathbb R^n_+\), and that \[ \frac {2\left(\frac{n+1}2\right)^{\frac n{n-1}}}{\sqrt{n^2-1}\lambda_{1,n-1}^{1/2}\left|{\mathcal B}^{n-1}\right|^{\frac 1{n-1}}} \leq \widetilde{A}_0(n) \leq \frac {n^{\frac n{n-1}}}{(n-1)\lambda_{1,n-1}^{1/2}\left|{\mathcal B}^{n-1}\right|^{\frac 1{n-1}}}, \] where \({\mathcal B}\) is the Euclidean unit ball in \(\mathbb R^n\), \(\left|{\mathcal B}\right|\) its volume and \(\lambda_{1,n-1}\) is the first Neumann eigenvalue of the Laplacian for radial functions in \({\mathcal B}\). The main result of the present paper states that there exists \(B>0\) such that (1) holds with \(\widetilde{A}_0(n)\) and \(B\).
first Neumann eigenvalue of the Laplacian
scientific article; zbMATH DE number 1761066