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The weighted Hardy inequality and self-adjointness of symmetric diffusion operators
Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-04-23 , DOI: 10.1016/j.jfa.2021.109067
Derek W. Robinson

Let Ω be a domain in Rd with boundary Γ, dΓ the Euclidean distance to the boundary and H=div(C) an elliptic operator with C=(ckl)>0 where ckl=clk are real, bounded, Lipschitz functions. We assume that CcdΓδI as dΓ0 in the sense of asymptotic analysis where c is a strictly positive, bounded, Lipschitz function and δ0. We also assume that there is an r>0 and a bδ,r>0 such that the weighted Hardy inequalityΓrdΓδ|ψ|2bδ,r2ΓrdΓδ2|ψ|2 is valid for all ψCc(Γr) where Γr={xΩ:dΓ(x)<r}. We then prove that the condition (2δ)/2<bδ is sufficient for the essential self-adjointness of H on Cc(Ω) with bδ the supremum over r of all possible bδ,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.



中文翻译:

对称扩散算子的加权Hardy不等式和自伴性

设Ω为 [Rd 边界为Γ dΓ 到边界的欧几里得距离和 H=-股利C 一个椭圆运算符 C=Cķ>0 在哪里 Cķ=Cķ是真实的,有界的Lipschitz函数。我们假设CCdΓδ一世 作为 dΓ0在渐近分析的意义上,其中c是严格的正有界Lipschitz函数,δ0。我们还假设存在一个[R>0 和一个 bδ[R>0 这样加权的Hardy不等式Γ[RdΓδ|ψ|2个bδ[R2个Γ[RdΓδ-2个|ψ|2个 对所有人都有效 ψCCΓ[R 在哪里 Γ[R={XΩdΓX<[R}。然后我们证明条件2个-δ/2个<bδ足以满足H on的基本自伴性CCΩbδr在所有可能之上的最高bδ[R在哈代不平等中。该结果扩展了具有平滑边界的域的所有已知结果,并且还提供了具有粗糙(例如分形)边界的大量域的自伴随性信息。

更新日期:2021-04-23
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