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Optimal design and relaxation of variational problems. II. (English) Zbl 0621.49008
In part I of the paper [ibid. 39, 113-137 (1986; Zbl 0609.49008)] the following variational problem was considered: $(1)\quad \inf \{\int_{\Omega}G(Du)dx: u=u_ 0\quad on\quad \partial \Omega \}$ where the infimum is taken over all vector-valued functions $$u: \Omega\to {\mathbb{R}}^ N$$, and G is the function defined on matrices $$G(z)=1+| z|^ 2$$ if $$z\neq 0$$; $$G(z)=0$$ if $$z=0.$$
The relaxed problem associated to (1) is $\min \{\int_{\Omega}\Phi (Du)dx: u=u_ 0\quad on\quad \partial \Omega \}$ where $$\Phi$$ is the quasiconvexification of G. The computation of $$\Phi$$ was carried out in part I, and we get $$\Phi (z)=1+| z|^ 2$$ if $$\rho\geq 1$$; $$\Phi (z)=2\rho -2D$$ if $$\rho\leq 1$$, where $$D=[\sum_{1\leq i<j\leq N}(z_{i1}z_{j2}-z_{j1}z_{i2})^ 2]^{1/2}$$ and $$\rho =(| z|^ 2+2D)^{1/2}.$$
In the present paper, the connections between this relaxation problem and structural optimization are developed, and Section 4 is devoted to this aim. In Section 5, the calculation of the quasiconvex envelope $$\Phi$$ of G is performed algebraically by showing that the polyconvex envelope PG of G and the rank-one envelope RG of G coincide with $$\Phi$$.
The optimal design problem the authors deal with is the following: given two conductors with conductivity coefficients $$\alpha$$ and $$\beta$$ respectively, minimize $$\int_{\Omega}a(x)dx$$ with the constraint $$E(a,u_ i)\geq C_ i$$, where $$a(x)=\alpha$$ if x belongs to the first conductor; $$a(x)=\beta$$ if x belongs to the second conductor, $$E(a,u_ i)$$ is the energy $\int_{\Omega}a(x) | Du_ i|^ 2 dx- 2\int_{\partial \Omega}f_ iu_ i d\sigma,$ $$u_ i$$ are the solutions of the elliptic problems $(2)\quad div(a Du_ i)=0\quad in\quad \Omega;\quad a D_{\nu}u_ i=0\quad on\quad \partial \Omega,$ and the constants $$C_ i$$ and the functions $$f_ i$$ are given $$(i=1,...,N)$$. This problem can be attacked variationally, via the relaxation of problem (1), or by using the homogenization method which uses the G-convergence of sequences of elliptic equations of the form (2).
In Section 6, the relationship between the relaxation method and homogenization and optimal bounds is discussed.
Reviewer: G.Buttazzo

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 35J20 Variational methods for second-order elliptic equations 74E30 Composite and mixture properties 49J20 Existence theories for optimal control problems involving partial differential equations 35J25 Boundary value problems for second-order elliptic equations 74P99 Optimization problems in solid mechanics 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
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