On the representation of effective energy densities (Q2701813)

The paper under review deals with a question raised in [\textit{R. Choksi} and \textit{I. Fonseca}, Arch. Ration. Mech. Anal. 138, No. 1, 37-103 (1997; Zbl 0891.73078)] on the representation of relaxed energies given as a sum of bulk and surface integrals. More precisely, given the functional NEWLINE\[NEWLINEE(u)= \int_\Omega W(\nabla u) dx+ \int_{S_u} \phi([u],\nu) d{\mathcal H}^{n- 1}NEWLINE\]NEWLINE defined on \(\text{SBV}(\Omega;\mathbb{R}^m)\), where \(\nabla u\) is the absolutely continuous part of the gradient \(Du\), \(S_u\) is the set of jump points, \([u]\) is the jump of \(u\) across \(S_u\), and \(\nu\) is the normal to \(S_u\), we may consider its relaxation NEWLINE\[NEWLINEI(u)= \inf\Biggl\{\liminf_{n\to\infty} E(u_n):u_n\to u\Biggr\}NEWLINE\]NEWLINE or more generally the functional NEWLINE\[NEWLINEI(u,G)= \inf\Biggl\{\liminf_{n\to\infty} E(u_n): u_n\to u L^1, \nabla u_n\rightharpoonup^* G\text{ measures}\Biggr\}.NEWLINE\]NEWLINE It is known that the relaxed bulk term of \(I(u, G)\) is of the form NEWLINE\[NEWLINE\int_\Omega H(\nabla u(x), G(x)) dx,NEWLINE\]NEWLINE where the density function \(H\) is given by NEWLINE\[NEWLINEH(A, B)= \inf\Biggl\{E(u): u|_{\partial\Omega}= Ax,\;\int_Q\nabla u dx= B\Biggr\}NEWLINE\]NEWLINE being \(Q\) the unit cube of \(\mathbb{R}^N\).NEWLINENEWLINENEWLINEThe question is to know if \(H(A, B)\) can be written in the form \(F_1(B)+ F_2(A- B)\) for some functions \(F_1\) and \(F_2\). In the present paper the author shows that in general this is not possible.