RECORD
A weighted Sobolev-Poincar\'e type trace inequality on Riemannian manifolds
Analysis of PDEs · Submitted 15 May 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim More precisely, under some assumptions on the manifold, we prove that there exists a constant $B>0$ such that, for all $u \in W^{1,2}(\rho^{1-2\sigma}, M)$, $$ \Big(\int_{\partial M}|u|^{p} \,\ud s_{g}\Big)^{2/p} \leq \mu^{-1} \int_{M} \rho^{1-2 \sigma}|\nabla_{g} u|^{2} \,\ud v_{g}+B \Big|\int_{\partial M} |u|^{p-2}u \,\ud s_{g}\Big|^{2/(p-1)}.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
Given $(M, g)$ a smooth compact $(n+1)$-dimensional Riemannian manifold with boundary $\partial M$. Let $\rho$ be a defining function of $M$ and $\sigma \in(0,1)$. In this paper we study a weighted Sobolev-Poincar\'e type trace inequality corresponding to the embedding of $W^{1,2}(\rho^{1-2 \sigma}, M) \hookrightarrow L^{p}(\partial M)$, where $p=\frac{2 n}{n-2 \sigma}$. More precisely, under some assumptions on the manifold, we prove that there exists a constant $B>0$ such that, for all $u \in W^{1,2}(\rho^{1-2\sigma}, M)$, $$ \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)}. $$ This inequality is sharp in the sense that $\mu^{-1}$ cannot be replaced by any smaller constant. Moreover, unlike the classical Sobolev inequality, $\mu^{-1}$ does not depend on $n$ and $\sigma$ only, but depends on the manifold.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Given $(M, g)$ a smooth compact $(n+1)$-dimensional Riemannian manifold with boundary $\partial M$. | background 1.00 | 0.06 | 0.01 | asserted 0.69 |
| s1 | Let $\rho$ be a defining function of $M$ and $\sigma \in(0,1)$. | background 0.98 | 0.12 | 0.02 | asserted 0.67 |
| s2 | In this paper we study a weighted Sobolev-Poincar\'e type trace inequality corresponding to the embedding of $W^{1,2}(\rho^{1-2 \sigma}, M) \hookrightarrow L^{p}(\partial M)$, where $p=\frac{2 n}{n-2 \sigma}$. | contribution 0.95 | 0.71 | 0.20 | asserted 0.84 |
| s3 | More precisely, under some assumptions on the manifold, we prove that there exists a constant $B>0$ such that, for all $u \in W^{1,2}(\rho^{1-2\sigma}, M)$, $$ \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)}. | result 0.97 | 0.93 | 0.85 | asserted 1.00 |
| s4 | $$ This inequality is sharp in the sense that $\mu^{-1}$ cannot be replaced by any smaller constant. | result 0.75 | 0.85 | 0.28 | asserted 1.00 |
| s5 | Moreover, unlike the classical Sobolev inequality, $\mu^{-1}$ does not depend on $n$ and $\sigma$ only, but depends on the manifold. | interpretation 0.60 | 0.79 | 0.13 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
Given $(M, g)$ a smooth compact $(n+1)$-dimensional Riemannian manifold with boundary $\partial M$. Let $\rho$ be a defining function of $M$ and $\sigma \in(0,1)$. In this paper we study a weighted Sobolev-Poincar\'e type trace inequality corresponding to the embedding of $W^{1,2}(\rho^{1-2 \sigma}, M) \hookrightarrow L^{p}(\partial M)$, where $p=\frac{2 n}{n-2 \sigma}$. More precisely, under some assumptions on the manifold, we prove that there exists a constant $B>0$ such that, for all $u \in W^{1,2}(\rho^{1-2\sigma}, M)$, $$ \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)}. $$ This inequality is sharp in the sense that $\mu^{-1}$ cannot be replaced by any smaller constant. Moreover, unlike the classical Sobolev inequality, $\mu^{-1}$ does not depend on $n$ and $\sigma$ only, but depends on the manifold.
Human review
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.