Strictly Continuous Extension of Functionals With Linear Growth to the Space Bv

rindler, filip; shaw, giles;

In this paper, we prove that the integral functional F[u] : BV(;Rm) → R defined by F[u] :=  f (x, u(x),∇u(x)) dx +    1 0 f∞  x, uθ (x), dDsu d|Dsu| (x)  dθ d|Dsu|(x) is continuous over BV(;Rm), with respect to the topology of area-strict convergence, a topology
in which (W1,1 ∩ C∞ )(;Rm) is dense. This provides conclusive justification for the treatment of F as the natural extension of the functional u →   f (x, u(x),∇u(x)) dx, defined for u ∈ W1,1(;Rm). This result is valid for a large class of integrands satisfying |f (x, y, A)| ≤ C(1 + |y|d/(d−1) + |A|) and its proof makes use of Reshetnyak’s Continuity Theorem combined with a lifting map μ[u] : BV(;Rm) → M( × Rm;Rm×d ). To obtain the theorem in the case where f exhibits d/(d − 1) growth in the y variable, an embedding result from the theory of concentration-compactness is also employed.