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

Ω∈R^n，n≥3是一个有界Lipschitz区域．令ωa（Q）=｜Q—Q0｜^a，其中Q0是边界 Ω上的一个固定点．对带有非负奇异位势的Schrodinger方程-△u＋Vu=0，V∈B∞研究了边值在L^2（ Ω，ωa dσ）中的Neumann问题，证明了当0〈a〈n-1时，Neumann问题存在唯一解，并且（△↓u）∈L^2（ Ω，ωadσ）．

Notes on Neumann Problem for Schr(o)dinger Operators in Weighted Lipschitz Domains

Let Ω be a bounded Lipschitz domain in Rn, n ≥ 3. Let ωa (Q) =| Q - Q0|a, where Q0 is a fixed point on Ω. For Schr6dinger equation -△u + Vu = 0 in Ω, with singular non-negative potentials V belonging to the reverse H(o)1der class B∞,we study the Neumann problem with boundary data in the weighted space L2(Ω,ωadσ),wheredσ 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 Vu exists in L2(Ω,ωadσ).