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On an open question about functions of bounded variation. (English) Zbl 1047.46025
The author adresses an open question posed by J. Bourgain, H. Brezis and P. Mironescu [“Another look at Sobolev spaces”, in: J. L. Menaldi, E. Rofman, A. Sulem (eds.), Optimal control and partial differential equations. In honour of Professor Alain Bensoussan’s 60th Birthday (IOS Press, Amsterdam), 439–455 (2001)]. The question raised by Bourgain, Brezis, and Mironescu was if \(f\in \text{BV}(\Omega)\), \(\Omega\subset\mathbb{R}^n\), do we then have
\[ \lim_{i\to\infty} \int_\Omega \int_\Omega [\{| f(x)- f(y)|\}/(| x-y|)]\rho_i(x- y)\,dx\,dy= K_{1,n} \int_\Omega|\nabla f|, \]
\(K_{1,n}\) and \(\rho_i\) being, respectively, a constant depending only on \(n\), and a sequence of radial mollifiers. It may be pointed out that L. Ambrosio has already independently obtained a proof of the main result of this note.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26A45 Functions of bounded variation, generalizations
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
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