Weighted BV functions (Q2771482)

The author introduces the space \(BV (\Omega,\omega)\) of the weighted functions of bounded variation, where the weight \(\omega\) belongs to a suitable subclass of Muckenhoupt's \(A_1\) class. The main result is a characterization of weighted BV functions in terms of the summability of \(\omega\) with respect to the (non-weighted) variation measure associated to the same function. A Sobolev-Poincaré inequality for weighted BV functions is given. The local compact imbedding of \(BV (\Omega,\omega)\) in weighted \(L^1\) space and the existence of minimal surface are also proved.