Affiliation:
1. School of Mathematics and Physics , University of Science and Technology Beijing , Beijing , 100083 , P. R. China
Abstract
Abstract
This paper is devoted to introducing and
investigating the bounded variation capacity and the perimeter
in the abstract Wiener space X, thereby discovering some related
inequalities. Functions of bounded variation in an abstract Wiener
space X have been studied by many scholars. As the continuation of this research, we define the corresponding BV capacity
cap
H
(
⋅
)
{\operatorname{cap}_{H}(\,\cdot\,)}
(now called abstract Wiener BV capacity) and
investigate its properties. We also investigate some properties
of sets of finite γ-perimeter, with γ being a Gaussian measure.
Subsequently, the isocapacitary inequality associated with
cap
H
(
⋅
)
{\operatorname{cap}_{H}(\,\cdot\,)}
is presented and we are able to show that it is
equivalent to the Gaussian isoperimetric inequality. Finally, we
prove that every set of finite γ-perimeter in X has mean
curvature in
L
1
(
X
,
γ
)
{L^{1}(X,\gamma)}
.
Funder
National Natural Science Foundation of China