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Zbl 0913.49008
Allaire, Grégire; Francfort, Gilles
Existence of minimizers for non-quasiconvex functionals arising in optimal design. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 15, No.3, 301-339 (1998). ISSN 0294-1449

The paper is concerned with the minimization on the set $D_\xi= \xi\cdot x+ H^1_0(\Omega; \bbfR^N)$ of the integral functional $F(u)= \int_\Omega f(Du)dx$, where $$f: z\in \bbfR^{nN}\to \cases \lambda+\alpha| z|^2,\quad &\text{if} \quad z\ne 0\\ 0,\quad & \text{if} \quad z = 0\endcases$$ with $0<\alpha,\lambda< +\infty$. Such functional was introduced by Kohn and Strang in the context of optimal design problems in electrostatics. Since $F$ is not sequentially weakly lower-semicontinuous on $D_\xi$, its lower-semicontinuous envelope $\overline F= \int_\Omega Qf(Du)dx$ must be taken into account, where $Qf$ is the quasiconvex envelope of $f$.\par The explicit computation of $Qf$ has been carried out by Kohn and Strang when $n=2$. Moreover, Dacorogna and Marcellini gave necessary and sufficient conditions on $\xi$ for the existence of minimizers of $F$ on $D_\xi$ again when $n= 2$.\par In the paper both the problems of the explicit computation of $Qf$, and of the determination of the conditions on $\xi$ for the existence of minimizers of $F$ on $D_\xi$ are studied for general values of the space dimension $n$.\par Fixing $n\in\bbfN$, and denoting by $z_1,\dots, z_n$ the square roots of the eigenvalues of $z^Tz$, it is proved that $$Qf(z)= \cases \lambda+ \alpha| z|^2,\quad &\text{if}\quad \sum^n_{i=1} z_i\ge \sqrt{\lambda/\alpha}\\ \alpha| z|^2- \alpha(\sum^n_{i=1} z_i)^2+ 2\sqrt{\lambda\alpha} \sum^n_{i=1} z_i,\quad &\text{if} \quad \sum^n_{i=1} z_i< \sqrt{\lambda/\alpha}\endcases.$$ Moreover, it is also proved that, if $\xi\in \bbfR^{nN}$, a sufficient condition for $F$ to have minimizers on $D_\xi$ is that either $f(\xi)= Qf(\xi)$, or $\text{rank }\xi= n$, and that a sufficient condition for $F$ to have no minimizers on $D_\xi$ is that $f(\xi)> Qf(\xi)$ and $\text{rank }\xi= 1$. It is also conjectured that, if $f(\xi)> Qf(\xi)$ and $2\le \text{rank }\xi\le n-1$, $F$ has no minimizers on $D_\xi$.\par The results of the paper are applicable to the existence of solutions of shape optimization problems in electrostatics. The proof of the existence theorem is based on the link between the considered functional and the homogenization theory for two phase composite materials. The case of the dual problem, i.e., of $F$ acting on divergence free fields, is also considered. Finally, the result about existence is partially extended to the case of nonquadratic Kohn-Strang type functionals.
[R.De Arcangelis (Napoli)]

MSC 2000:: *49J45 Optimal control problems inv. semicontinuity and convergence
49Q10 Optimization of the shape other than minimal surfaces

Keywords: integral functional; optimal design; weakly lower-semicontinuous; shape optimization problems in electrostatics; homogenization; quadratic Kohn-Strang type functionals

Cited in: Zbl 1004.49011 Zbl 0958.49008

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