Relaxation in BV of integrals with superlinear growth
Parth Soneji · ESAIM Control Optimisation and Calculus of Variations · 2014
We study properties of the functional where u ∈ BV(Ω;RN), and f:RN × 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 [A. Braides and A. Coscia, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 737–756] with a new technique involving mollification to prove an upper bound for Floc. Then, for , we prove that Floc satisfies the lower bound provided f is quasiconvex, and the recession function f∞ (defined as ) is assumed to be finite in certain rank-one directions. The proof of this result involves adapting work by [Kristensen, Calc. Var. Partial Differ. Eqs. 7 (1998) 249–261], and [Ambrosio and Dal Maso, J. Funct. Anal. 109 (1992) 76–97], and applying a non-standard blow-up technique that exploits fine properties of BV maps. It also makes use of the fact that Floc has a measure representation, which is proved in the appendix using a method of [Fonseca and Malý, Annal. Inst. Henri Poincaré Anal. Non Linéaire 14 (1997) 309–338].