黎曼流形上加权 Sobolev-Poincaré 型迹不等式
偏微分方程分析
2022-05-17 v1
摘要
给定 ( M , g ) (M, g) ( M , g ) ∂ M \partial M ∂ M ( n + 1 ) (n+1) ( n + 1 ) ρ \rho ρ M M M σ ∈ ( 0 , 1 ) \sigma \in(0,1) σ ∈ ( 0 , 1 ) W 1 , 2 ( ρ 1 − 2 σ , M ) ↪ L p ( ∂ M ) W^{1,2}(\rho^{1-2 \sigma}, M) \hookrightarrow L^{p}(\partial M) W 1 , 2 ( ρ 1 − 2 σ , M ) ↪ L p ( ∂ M ) p = 2 n n − 2 σ p=\frac{2 n}{n-2 \sigma} p = n − 2 σ 2 n B > 0 B>0 B > 0 u ∈ W 1 , 2 ( ρ 1 − 2 σ , M ) u \in W^{1,2}(\rho^{1-2\sigma}, M) u ∈ W 1 , 2 ( ρ 1 − 2 σ , M ) ( ∫ ∂ M ∣ u ∣ p \ud s g ) 2 / p ≤ μ − 1 ∫ M ρ 1 − 2 σ ∣ ∇ g u ∣ 2 \ud v g + B ∣ ∫ ∂ M ∣ u ∣ p − 2 u \ud s g ∣ 2 / ( p − 1 ) . \Big(\int_{\partial M}|u|^{p} \,\ud s_{g}\Big)^{2/p} \leq \mu^{-1} \int_{M} \rho^{1-2 \sigma}|\nabla_{g} u|^{2} \,\ud v_{g}+B \Big|\int_{\partial M} |u|^{p-2}u \,\ud s_{g}\Big|^{2/(p-1)}. ( ∫ ∂ M ∣ u ∣ p \ud s g ) 2/ p ≤ μ − 1 ∫ M ρ 1 − 2 σ ∣ ∇ g u ∣ 2 \ud v g + B ∫ ∂ M ∣ u ∣ p − 2 u \ud s g 2/ ( p − 1 ) . μ − 1 \mu^{-1} μ − 1 μ − 1 \mu^{-1} μ − 1 n n n σ \sigma σ
引用
@article{arxiv.2205.07272,
title = {A weighted Sobolev-Poincar\'e type trace inequality on Riemannian manifolds},
author = {Zhongwei Tang and Ning Zhou},
journal= {arXiv preprint arXiv:2205.07272},
year = {2022}
}