Relaxation in BV of integrals with superlinear growth

Parth Soneji
2014 E S A I M: Control, Optimisation and Calculus of Variations  
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 [16] 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 ∞ (defined
more » ... nction f ∞ (defined as f ∞ (ξ) := lim t→∞ f (tξ)/t) is assumed to be finite in certain rank-one directions. The proof of this result involves adapting work by Kristensen [30], and Ambrosio and Dal Maso [8], and applying a non-standard blow-up technique that exploits fine properties of BV maps. It also makes use of the fact that F loc has a measure representation, which is proved in the appendix using a method of Fonseca and Malý [24] .
doi:10.1051/cocv/2014008 fatcat:yockixcodvf3jhq6n4qpibnpcy