Strictly continuous extension of functionals with linear growth to the space BV

The structure of optimal parameters for image restoration problems

“…where (z|y) denotes the dual product, and the subdifferential ∂Φ(y) is defined by z satisfying the same expression with γ = 0. With regard to more advanced strict convergence results, we point the reader to [25,38,35].…”

Boundary effects and weak$^*$ lower semicontinuity for signed integral functionals on $\mathrm{BV}$

“…Remark 3.1. Using the density of W 1,1 in BV with respect to area-strict convergence ( •strict convergence) 4 , together with the associated variant of Reshetnyak's continuity theo-rem ([12, Theorem 5] and, more general, [19,Theorem 1]), all our variants of quasi-sublinear growth from below have equivalent versions extending the class of test functions from W 1,1 to BV. Most importantly for our purposes, (3.2) is equivalent to the following:…”

“…Remark 5.6. If W 1,1 is dense in U with respect to area-strict convergence in BV ( •strict convergence in the notation of [12]), in particular if U = BV, u * can be chosen in W 1,1 (Ω; R n ) because F is area-strictly-continuous [19] (see also [12,Theorem 5] for a special case). In this case, χ Sn ∇u * → 0 in L 1 , and the proof of Proposition 5.5 still works even if we only assume that L N (S n ) → 0 (i.e.…”