Relaxation in BV of integrals with superlinear growth
ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 4, pp. 1078-1122.

We study properties of the functional

loc (u,Ω):=inf (u j ) lim inf j Ω f (u j ) d x (u j )W loc 1,r Ω, N u j *uinBVΩ, N ,
where u B V ( Ω ; N ) , and f : N × n 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 loc . Then, for r [ 1 , n n - 1 ) , we prove that loc satisfies the lower bound
loc (u,Ω) Ω f(u(x))dx+ Ω f D s u |D s u||D s u|,
provided f is quasiconvex, and the recession function f (defined as f ( ξ ) : = lim ¯ t f ( t ξ ) / t 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 loc 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].

DOI : 10.1051/cocv/2014008
Classification : 49J45, 26B30
Mots clés : quasiconvexity, lower semicontinuity, relaxation, BV
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Soneji, Parth. Relaxation in BV of integrals with superlinear growth. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 4, pp. 1078-1122. doi : 10.1051/cocv/2014008. http://www.numdam.org/articles/10.1051/cocv/2014008/

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