论文标题
加权的Sobolev-Poincaré型痕量不平等现象
A weighted Sobolev-Poincaré type trace inequality on Riemannian manifolds
论文作者
论文摘要
给定$（m，g）$平滑的紧凑型$（n+1）$ - 尺寸riemannian歧管，带边界$ \ partial m $。令$ρ$为$ m $和$σ\ in（0,1）$的定义函数。在本文中，我们研究了加权的Sobolev-Poincaré类型痕迹不等式，对应于$ w^{1,2}（ρ^{1-2σ}，m），m）\ hookrightArrow l^{p}（\ partial m）$，其中$ p = \ frac = \ frac {2 n} n} n} n} n-2} $。更准确地说，根据对歧管的某些假设，我们证明存在常数$ b> 0 $，以至于对于所有$ u \ in W^{1,2}}（ρ^{1-2σ}，m）$，$ $，$ $ \ big（\ big big） s_ {g} \ big）^{2/p} \ leqμ^{ - 1} \ int_ {m}ρ^{1-2σ} | \ nabla_ {g} \，\ ud s_ {g} \ big |^{2/（p-1）}。 $$从某种意义上说，$μ^{ - 1} $不能被任何较小的常数代替。此外，与经典的Sobolev不平等不同，$μ^{ - 1} $不仅取决于$ n $和$σ$，而是取决于歧管。
Given $(M, g)$ a smooth compact $(n+1)$-dimensional Riemannian manifold with boundary $\partial M$. Let $ρ$ be a defining function of $M$ and $σ\in(0,1)$. In this paper we study a weighted Sobolev-Poincaré type trace inequality corresponding to the embedding of $W^{1,2}(ρ^{1-2 σ}, M) \hookrightarrow L^{p}(\partial M)$, where $p=\frac{2 n}{n-2 σ}$. More precisely, under some assumptions on the manifold, we prove that there exists a constant $B>0$ such that, for all $u \in W^{1,2}(ρ^{1-2σ}, M)$, $$ \Big(\int_{\partial M}|u|^{p} \,\ud s_{g}\Big)^{2/p} \leq μ^{-1} \int_{M} ρ^{1-2 σ}|\nabla_{g} u|^{2} \,\ud v_{g}+B \Big|\int_{\partial M} |u|^{p-2}u \,\ud s_{g}\Big|^{2/(p-1)}. $$ This inequality is sharp in the sense that $μ^{-1}$ cannot be replaced by any smaller constant. Moreover, unlike the classical Sobolev inequality, $μ^{-1}$ does not depend on $n$ and $σ$ only, but depends on the manifold.
