On integration by parts formula on open convex sets in Wiener spaces

• Published in: Journal of evolution equations Vol. 21; no. 2; pp. 1917 - 1944
• Main Authors: Addona, Davide, Menegatti, Giorgio, Miranda, Michele
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2021; Springer Nature B.V
• Subjects: Analysis; Convexity; Density; Euclidean geometry; Euclidean space; Mathematics; Mathematics and Statistics; Metric space; Infinite-dimensional analysis; Secondary: 46G05; Geometric measure theory; Integration by parts formula; Abstract Wiener spaces; Primary: 28C20; 46N10; Convex analysis
• ISSN: 1424-3199; 1424-3202
• DOI: 10.1007/s00028-020-00663-1

In Euclidean space, it is well known that any integration by parts formula for a set of finite perimeter Ω is expressed by the integration with respect to a measure P ( Ω , · ) which is equivalent to the one-codimensional Hausdorff measure restricted to the reduced boundary of Ω . The same result has been proved in an abstract Wiener space, typically an infinite-dimensional space, where the surface measure considered is the one-codimensional spherical Hausdorff–Gauss measure S ∞ - 1 restricted to the measure-theoretic boundary of Ω . In this paper, we consider an open convex set Ω and we provide an explicit formula for the density of P ( Ω , · ) with respect to S ∞ - 1 . In particular, the density can be written in terms of the Minkowski functional p of Ω with respect to an inner point of Ω . As a consequence, we obtain an integration by parts formula for open convex sets in Wiener spaces.