Relaxation in BV of integrals with superlinear growth

Soneji, Parth

ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 4, pp. 1078-1122.

We study properties of the functional

$${ℱ}_{\text{loc}}(u,\Omega ):=\underset{\left(u_j\right)}{inf}\left\{\underset{j\to \infty }{lim\; inf}{\int }_{\Omega }f\left(\nabla u_j\right)dx\left|\begin{array}{cc}& \left(u_j\right)\subset W_{\mathrm{loc}}^{1,r}\left(\Omega ,{ℝ}^N\right)\hfill \\ & u_j\stackrel{*}{\rightharpoonup }u\text{in}BV\left(\Omega ,{ℝ}^N\right)\hfill \end{array}\right.\right\},$$

where $u\in BV(\Omega ;{ℝ}^N)$, and $f:{ℝ}^{N\times n}\to ℝ$ is continuous and satisfies $0\le f\left(\xi \right)\le L(1+|\xi {|}^r)$ . For $r\in [1,2)$ , 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 ${ℱ}_{\text{loc}}$ . Then, for $r\in [1,\frac{n}{n-1})$ , we prove that ${ℱ}_{\text{loc}}$ satisfies the lower bound

$${ℱ}_{\text{loc}}(u,\Omega )\ge {\int }_{\Omega }f\left(\nabla u\left(x\right)\right)dx+{\int }_{\Omega }{f}^{\infty }\left(\frac{D^{s}u}{|D^{s}u|}\right)\left|D^{s}u\right|,$$

provided $f$ is quasiconvex, and the recession function $f^{\infty }$ (defined as $f^{\infty }\left(\xi \right):={\overline{lim}}_{t\to \infty }f\left(t\xi \right)/t$ 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 ${ℱ}_{\text{loc}}$ 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].

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