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

Abstract

In this paper, we prove that the integral functional F[u]: BV(Ω;ℝ^m) → ℝ defined by

F[u] := ∫_Ω f(x,u(x),∇u(x))dx + ∫_Ω ∫_1^0 f^∞ (x, u^θ (x), (d D^s u)/(d |D^s u|) (x)

is continuous over BV(Ω;ℝ^m), with respect to the topology of area-strict convergence, a topology in which (W^(1,1) ∩ C^∞)(Ω;ℝ^m) is dense. This provides conclusive justification for the treatment of F as the natural extension of the functional

u ↦ ∫_Ω f(x,u(x),∇u(x))dx,

defined for u ∈ W^(1,1) (Ω;ℝ^m). 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(Ω;ℝ^m) → M(Ω × ℝ^m; ℝ^(m×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.

See full abstract at http://dx.doi.org/10.1093/qmath/hav022