Gauge Theorem for the Neumann Problem
Kai Lai Chung, Pei Hsu · Birkhäuser Boston eBooks · 1986
Let D be a bounded domain in IRd and let (Δ/2 + q)u = 0 be Schrödinger’s equation on D. The Dirichlet problem for the equation was studied first in [2] for bounded q and then in [1] and [4] for q ∈ Kd (see below for definition). The gauge function for the Dirichlet problem is defined in [2] as 1 $${\text{G(x)}}{\mkern 1mu} {\text{ = }}{\mkern 1mu} {{{\text{E}}}^{{\text{x}}}}[\exp (\int_{{\text{0}}}^{{{{\tau }_{{\text{D}}}}}} {{\text{q(}}{{{\text{B}}}_{{\text{S}}}}{\text{)}}} {\mkern 1mu} {\text{ds}})]$$ , where B = {Bt, t ≥ 0} is the standard Brownian motion on and IRd and τD is the first exit time of D. One striking property of the gauge function proved in [2] and [4] is the following.