Potential theory for stationary Schrödinger operators: a survey of results obtained with non-probabilistic methods
Potential theory for stationary Schrödinger operators: a survey of results obtained with non-probabilistic methods
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发表时间：
1993-06
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通讯作者：
M. Bramanti
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作者：
M. Bramanti
本文研究一类一致椭圆算子：LuAu + Vu，其中主部a是散度形式，V是一个假设在“Kato类”中的函数。这个算子已经在不同的背景下进行了研究，特别是使用概率技术。本文的目的是给出有界Lipschitz域上算子L的非概率方法所得结果的统一和简化表示。这些结果涉及：Lu=0时解的连续性；Harnack不平等;格林函数和L -谐波测度的估计；Lu=0正解的边界行为，特别是“法图定理”。
In this paper we deal with a uniformly elliptic operator of the kind: Lu Au + Vu , where the principal part A is in divergence form, and V is a function assumed in a “Kato class”. This operator has been studied in different contexts, especially using probabilistic techniques. The aim of the present work is to give a unified and simplified presentation of the results obtained with non probabilistic methods for the operator L on a bounded Lipschitz domain. These results regard: continuity of the solutions of Lu=0 ; Harnack inequality; estimates on the Green's function and L -harmonic measure; boundary behavior of positive solutions of Lu=0 , in particular a “Fatou's theorem”.