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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

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.)
Full Text: DOI
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