论文标题
对称扩散操作员的加权强度不平等和自我接触性
The weighted Hardy inequality and self-adjointness of symmetric diffusion operators
论文作者
论文摘要
令$ \ ri^d $中的一个域，带有边界$γ$$ {\！ $ c_ {kl} = c_ {lk} $是真实的，有限的，Lipschitz函数。我们假设$ c \ sim c \，d_γ^{\，δ} $是$d_γ\ to0 $，在渐近分析的意义上，$ c $是严格的正面，有限的，lipschitz的功能和$Δ\ geq0 $。我们还假设有一个$ r> 0 $和a $ b_ {δ，r}> 0 $，以便加权hardy不平等\ [\ int_ {γ_{γ_{\！\！ b_ {δ，r}^{\，2} \ int_ {γ_{\！\！r}}d_γ^{\，δ-2} \，| ψ|^2 \]对于C_C^\ Infty（γ_{\！\！r}）中的所有$ψ\均有效，其中$γ_{\！\！r} = \ {x \inΩ：d_γ（x）<r \ \} $。然后，我们证明条件$（2-δ）/2 <b_δ$足以满足$ c_c^\ infty（ω）$的基本自我相关性，而$b_Δ$ the supremum the supremum超过了所有可能的$ b_ {δ，r} $的$ r $。该结果扩展了具有平滑边界的域的所有已知结果，还提供了有关具有粗糙（例如\ \ fractal，边界）的大型域的自相关性的信息。
Let $Ω$ be a domain in $\Ri^d$ with boundary $Γ$${\!,}$ $d_Γ$ the Euclidean distance to the boundary and $H=-\divv(C\,\nabla)$ an elliptic operator with $C=(\,c_{kl}\,)>0$ where $c_{kl}=c_{lk}$ are real, bounded, Lipschitz functions. We assume that $C\sim c\,d_Γ^{\,δ}$ as $d_Γ\to0$ in the sense of asymptotic analysis where $c$ is a strictly positive, bounded, Lipschitz function and $δ\geq0$. We also assume that there is an $r>0$ and a $ b_{δ,r}>0$ such that the weighted Hardy inequality \[ \int_{Γ_{\!\!r}} d_Γ^{\,δ}\,|\nabla ψ|^2\geq b_{δ,r}^{\,2}\int_{Γ_{\!\!r}} d_Γ^{\,δ-2}\,| ψ|^2 \] is valid for all $ψ\in C_c^\infty(Γ_{\!\!r})$ where $Γ_{\!\!r}=\{x\inΩ: d_Γ(x)<r\}$. We then prove that the condition $(2-δ)/2<b_δ$ is sufficient for the essential self-adjointness of $H$ on $C_c^\infty(Ω)$ with $b_δ$ the supremum over $r$ of all possible $b_{δ,r}$ in the Hardy inequality. This result extends all known results for domains with smooth boundaries and also gives information on self-adjointness for a large family of domains with rough, e.g.\ fractal, boundaries.
