利普希兹区域上薛定谔方程Neumann问题的加权估计

Weighted Estimates on the Neumann Problem for L^2 for Schrodinger Equations in Lipschitz Domains

该文研究了利普希兹区域上加权空间H^p(?Ω,ω_αdσ)和L^p(?Ω,ω_αdσ)(1-ε<p≤2)上薛定谔方程-△u+Vu=0加权估计问题.记Ω是R^n(n≥3)上边界连通的有界利普希兹区域.令ω_α(Q)=|Q-Q_0|~α,这里Q_0是?Ω上的一个不动点.对于:定义在Ω上的薛定谔方程-△u+Vu=0,其中奇异非负位势V属于反H?lder类-B_n.该文研究边值落在加权空间H^p(?Ω,ω_αdσ)或L^p(?Ω,ω_αdσ)上的Neumann问题,这里dσ表示?Ω上的测度.对于特定范围的α,方程存在唯一解u,使得非切向的极大函数▽u在H^p(?Ω,ω_αdσ)或L^p(?Ω,ω_αdσ)上.此外,还建立了这些解的一致估计.

In this paper, we consider the weighted estimates in the weighted space HP（ Ω, ωαdσ） or LP（ Ω, ωαdσ） （1 - ε 〈 p ≤ 2） for Schrodinger equation -△u ＋ Vu = 0 on Lipschitz domains. Let Ω be a bounded Lipschitz domain with connected boundary in R^n, n≥ 3. Let ωα（Q） = ｜Q- Q0｜α, 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 Bn, we study the Neumann problem with boundary date lies in the weighted space HP（ Ω, ωαdσ） or LP（ Ω, ωαdσ）, where dσ denotes the surface measure on Ω. We show that for certain ranges of a, there is a unique solution u, such that the non-tangential maximal function of u is in HP（ Ω, ωαdσ） or LP（ Ω, ωαdσ）. Moreover, the uniform estimates are founded.