Lipschitz区域上Schrdinger算子Neumann问题的讨论(英文)

Notes on Neumann Problem for Schrdinger Operators in Weighted Lipschitz Domains

Ω∈R n,n≥3是一个有界Lipschitz区域.令ωα(Q)=|Q-Q0|α,其中Q0是边界Ω上的一个固定点.对带有非负奇异位势的Schrdinger方程-Δu+Vu=0,V∈B∞,研究了边值在L2(Ω,ωαdσ)中的Neumann问题,证明了当0<α<n-1时,Neumann问题存在唯一解,并且(▽u)*∈L2(Ω,ωαdσ).

Let Ω be a bounded Lipschitz domain in R^n, n ≥3. Let ωa （Q） =｜Q - Q0 ｜^a, where Q0 is a fixed point on Ω. For Schrodinger equation -△u ＋ Vu = 0 in Ω, with singular non-negative potentials V belonging to the reverse Holder class B∞, we study the Neumann problem with boundary data in the weighted space L^2 （ Ω, ωadσ） , where dσ denotes the surface measure on Ω. We show that a unique solution u can be found for the Neumann problem provided 0 〈 a 〈 n - 1. Also proven is that the non-tangential maximal function of △↓u exists in L^2 （ Ω, ω dσ）.