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Relaxation in BV of integrals with superlinear growth

Published online by Cambridge University Press:  13 August 2014

Parth Soneji*
Ludwig Maximilians University Munich, Theresienstr. 39, 80333 Munich, Germany.
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We study properties of the functional

\begin{eqnarray} \mathscr{F}_{{\rm loc}}(u,\Omega):= \inf_{(u_{j})}\bigg\{ \liminf_{j\rightarrow\infty}\int_{\Omega}f(\nabla u_{j})\ud x\, \left| \!\!\begin{array}{rl} & (u_{j})\subset W_{{\rm loc}}^{1,r}\left(\Omega, \RN\right) \\ & u_{j}\tostar u\,\,\textrm{in }\BV\left(\Omega, \RN\right) \end{array} \right. \bigg\}, \end{eqnarray}Floc(u,Ω):=inf(uj)liminfjΩf(uj)dx,
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 \hbox{$r\in[1,\frac{n}{n-1})$}r[1,nn1), we prove that Floc satisfies the lower bound
\begin{equation*} \scF_{{\rm loc}}(u,\Omega) \geq \int_{\Omega} f(\nabla u (x))\ud x + \int_{\Omega}\finf \bigg(\frac{D^{s}u}{|D^{s}u|}\bigg)\,|D^{s}u|, \end{equation*}Floc(u,Ω)Ωf(u(x))dx+ΩfDsu|Dsu||Dsu|,
provided f is quasiconvex, and the recession function f (defined as \hbox{$ f^{\infty}(\xi):= \overline{\lim}_{t\rightarrow\infty}f(t\xi )/t$}f(ξ):=limtf()/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 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].

Research Article
© EDP Sciences, SMAI, 2014

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