BV capacity and perimeter in abstract Wiener spaces and applications

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)} .