ABSTRACT
In this chapter we present the analytic approach to the perimeter of measurable sets. We define BV functions and BV sets and establish their main properties. The emphasis is on BV functions — a connection with BV sets is provided by the coarea theorem. Sobolev’s and Poincare’s inequalities are proved directly within the framework of BV functions; their validity for https://www.w3.org/1998/Math/MathML"> C ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429096679/ee3e4f1b-65eb-4f80-b900-911f329ff879/content/eq2596.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> functions is not presupposed. Throughout this chapter Ω denotes an open subset of 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/eq2597.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
