Relaxation in BV of integrals with superlinear growth

Ludwig Maximilians University Munich, Theresienstr. 39, 80333 Munich, Germany
soneji@math.lmu.de

Received: 22 July 2013
Revised: 27 November 2013

Abstract

We study properties of the functional $\begin{array}{ccc}{F}_{\mathrm{loc}}\mathrm{\left(}\mathit{u,\Omega }\mathrm{\right)}\mathrm{:}\mathrm{=}\underset{\mathrm{\left(}{\mathit{u}}_{\mathit{j}}\mathrm{\right)}}{\inf }\{\underset{\mathit{j}\mathrm{\to }\mathrm{\infty }}{\lim \inf }{\mathrm{\int }}_{\mathit{\Omega }}\mathit{f}\mathrm{\left(}\mathrm{\nabla }{\mathit{u}}_{\mathit{j}}\mathrm{\right)}\mathrm{d}\mathit{x}|\begin{array}{c}\\ & \\ & \end{array}\}\mathit{,}& & \end{array}$ 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 [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_loc. Then, for $\mathit{r}\mathrm{\in }\mathrm{\left[}\mathrm{1}\mathit{,}\frac{\mathit{n}}{\mathit{n}\mathrm{-}\mathrm{1}}\mathrm{\right)}$, we prove that F_loc satisfies the lower bound $F_{\mathrm{loc}}\mathrm{\left(}\mathit{u,\Omega }\mathrm{\right)}\mathrm{\ge }{\mathrm{\int }}_{\mathit{\Omega }}\mathit{f}\mathrm{\left(}\mathrm{\nabla }\mathit{u}\mathrm{\right(}\mathit{x}\mathrm{\left)}\mathrm{\right)}\mathrm{d}\mathit{x}\mathrm{+}{\mathrm{\int }}_{\mathit{\Omega }}{\mathit{f}}^{\mathrm{\infty }}(\frac{{\mathit{D}}^{\mathit{s}}\mathit{u}}{\mathrm{\left|}{\mathit{D}}^{\mathit{s}}\mathit{u}\mathrm{\right|}})\mathrm{\left|}{\mathit{D}}^{\mathit{s}}\mathit{u}\mathrm{\right|}\mathit{,}$ provided f is quasiconvex, and the recession function f^∞ (defined as ${\mathit{f}}^{\mathrm{\infty }}\mathrm{\left(}\mathit{\xi }\mathrm{\right)}\mathrm{:}\mathrm{=}{\overline{\lim }}_{\mathit{t}\mathrm{\to }\mathrm{\infty }}\mathit{f}\mathrm{\left(}\mathit{t\xi }\mathrm{\right)}\mathit{/}\mathit{t}$) 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_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].