Nondifferentiable optimization problems for elliptic systems. (English) Zbl 0571.49010

The authors consider two optimal design problems for a clamped plate. Let \(\Omega\) be a bounded domain of \({\mathbb{R}}^ 2\) representing a clamped plate, and let us denote by (with the summation convention) \(F(u,y)=\int_{\Omega}u^ 3(x)b_{ijhk}D_{ij}yD_{hk}ydx\) the usual deformation energy of the plate, where u(x) is the thickness of the plate, y(x) is the deflection, and \(b_{ijhk}\) are coefficients depending on the material of the plate. Let \(f\in H^{-2}(\Omega)\) be a given force, and denote by \(y_ u\) the solution of the problem \(D_{hk}(u^ 3(x)b_{ijhk}D_{ij}y)=f\) with \(y\in H^ 2_ 0(\Omega).\)
The first optimization problem considered in the paper is \[ (P_ 1)\quad \inf \{J(u):\quad u\in L^{\infty}(\Omega),\quad c_ 1\leq u\leq c_ 2,\quad \int_{\Omega}udx=c\} \] where \(c_ 1\), \(c_ 2\), c are given positive constants, and J(u) is the maximal deflection \(J(u)=\sup \{| y_ u(x)|:\) \(x\in \Omega \}\). The second optimization problem is \[ (P_ 2)\quad \sup \{\lambda (u):\quad u\in L^{\infty}(\Omega),\quad c_ 1\leq u\leq c_ 2,\quad \int_{\Omega}udx=c\} \] where the smallest eigenvalue \(\lambda\) (u) is defined by \(\lambda (u)=\inf \{F(u,y):\) \(y\in H^ 2_ 0(\Omega)\), \(\int_{\Omega}uy^ 2dx=1\}.\)
The problems above may have no solution; however, it is possible to define approximate problems \((P_ 1^{\epsilon})\), \((P_ 2^{\epsilon})\) possessing always a solution \(u_{\epsilon}\). When \(\epsilon\) \(\to 0\), by using the G-convergence theory, the authors characterize in the first case the limit \(\bar y\) of the approximate deflections \(y_{u_{\epsilon}}\), and in the second case the limit \({\bar \lambda}\) of the approximate smallest eigenvalues \(\lambda (u_{\epsilon})\). Necessary conditions of optimality for the approximating problems \((P_ 1^{\epsilon})\), \((P_ 2^{\epsilon})\) are given. Finally, some numerical results are presented.
Reviewer: G.Buttazzo


49J45 Methods involving semicontinuity and convergence; relaxation
49K20 Optimality conditions for problems involving partial differential equations
74P99 Optimization problems in solid mechanics
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
49K40 Sensitivity, stability, well-posedness
49J20 Existence theories for optimal control problems involving partial differential equations
35J45 Systems of elliptic equations, general (MSC2000)
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