论文标题
在$ \ mathbb {r}^n $的域上的加权$ l^p $ -hardy不平等
On weighted $L^p$-Hardy inequality on domains in $\mathbb{R}^n$
论文作者
论文摘要
我们考虑加权$ l^p $ - hardy的不平等现象，涉及到具有非空边界的$ n $维欧几里得空间中域边界的距离。使用批判性理论，我们给出了F.〜G。〜Avkhadiev（2006）定理的以下结果的替代证明：让$ω\ subsetNeqq \ Mathbb {r}^n $，$ n \ geq 2 $，是任意的域名，是一个任意的域名，$ 1 <p <p <p <\ p <\ f \ \ f \ \ p <\ fty $ and $α+ p> p> p> n $。令$ \ mathrm {d}_Ω（x）= \ mathrm {dist}（x，x，\ partialω）$表示点$ x \ inω$至$ \ partialω$的点$ x \。然后以下强硬型不平等 $$ \ int_ {ω} \ frac {| \nablaφ|^p} {\ mathrm {d}_Ω \ int_ {ω} \ frac {|φ|^p} {\ mathrm {d}_ω^{p+α}}} \，\ Mathrm {d} x \ qquad \ forallφ\ inc^{\ infty} 下限常数$ \ left（\ frac {α+p-n} {p} \ right）^p $很清晰。
We consider weighted $L^p$-Hardy inequalities involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty boundary. Using criticality theory, we give an alternative proof of the following result of F.~G.~Avkhadiev (2006) Theorem: Let $Ω\subsetneqq \mathbb{R}^n$, $n\geq 2$, be an arbitrary domain, $1<p<\infty$ and $α+ p>n$. Let $\mathrm{d}_Ω(x) =\mathrm{dist}(x,\partial Ω)$ denote the distance of a point $x\in Ω$ to $\partial Ω$. Then the following Hardy-type inequality holds $$ \int_{Ω}\frac{|\nabla φ|^p}{\mathrm{d}_Ω^α}\,\mathrm{d}x \geq \left( \frac{α+p-n}{p}\right)^p \int_{Ω}\frac{|φ|^p}{\mathrm{d}_Ω^{p+α}}\,\mathrm{d}x \qquad \forall φ\in C^{\infty }_c(Ω),$$ and the lower bound constant $\left( \frac{α+p-n}{p}\right)^p$ is sharp.
