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We study properties of the functional F loc (u, Ω) := inf where u ∈ BV(Ω; R N ), and f : R N ×n → R is continuous and satisfies 0 ≤ f (ξ) ≤ L(1 + |ξ| r ). For r ∈ [1, 2), assuming f has linear growth in certain rank-one directions, we combine a result of Braides and Coscia  with a new technique involving mollification to prove an upper bound for F loc . Then, for r ∈ [1, n n−1 ), we prove that F loc satisfies the lower bound provided f is quasiconvex, and the recession function f ∞ (defineddoi:10.1051/cocv/2014008 fatcat:yockixcodvf3jhq6n4qpibnpcy