ABSTRACT
This chapter is devoted to deeper results about locally BV sets obtained by E. De Giorgi [18,19]. We prove that the families of all locally BV sets and all sets with locally finite perimeter coincide. In addition, we show that the variational measure https://www.w3.org/1998/Math/MathML"> D χ E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429096679/ee3e4f1b-65eb-4f80-b900-911f329ff879/content/eq4437.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of a locally BV set E lives in https://www.w3.org/1998/Math/MathML"> ∂ * E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429096679/ee3e4f1b-65eb-4f80-b900-911f329ff879/content/eq4438.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and has the polar decomposition https://www.w3.org/1998/Math/MathML"> ( H n - 1 ∟ ∂ * E ) ∟ ( - ν E ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429096679/ee3e4f1b-65eb-4f80-b900-911f329ff879/content/eq4439.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where https://www.w3.org/1998/Math/MathML"> ν E ↾ ∂ * E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429096679/ee3e4f1b-65eb-4f80-b900-911f329ff879/content/eq4440.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is the measuretheoretic exterior normal of E. Throughout this chapter Ω denotes an open subset of the ambient space https://www.w3.org/1998/Math/MathML"> R n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429096679/ee3e4f1b-65eb-4f80-b900-911f329ff879/content/eq4441.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
