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Thin insulating layers: The optimization point of view. (English) Zbl 0645.73015
Material instabilities in continuum mechanics, Proc. Symp. Edinburgh/Scotl. 1985/86, 11-19 (1988).
[For the entire collection see Zbl 0627.00023.]
The problem studied by the author is the following: Let u be the solution of the boundary value problem $$\nabla^ 2_ u+f=0$$ in $$\Omega$$, $$d(\partial u/\partial n)+u=0$$ on $$\partial \Omega$$, where $$\Omega$$ is a regular bounded open subset of $${\mathbb{R}}^ n$$, $$f\in L^ 2(\Omega)$$, and d a positive function defined on $$\partial \Omega$$. It is known that u is also the (unique) minimum of the functional $$E(u)=\int_{\Omega}| \nabla u|^ 2dx-2\int_{\Omega}fdx+\int_{\partial \Omega}(u^ 2/d)d\sigma$$ for $$u\in H^ 1(\ell).$$
If d is given, the study of the minimum of E(u) is a classic question of calculus of variation. Here, however, the inverse problem is considered. For every d, let $$u_ d$$ be the minimizer for E(u). The value of E(u) depends obviously on the choice of d, and one wants to minimize E(u) among all functions d satisfying the constraint $$\int_{\partial \Omega}d d\sigma =k$$, where k is a constant.
It is first shown that, in this case, to minimize E(u) is equivalent to find the minimum of the new functional $$\int_{\Omega}| \nabla u|^ 2dx-2\int_{\Omega}f dx+(1/k)(\int_{\partial \Omega}| u| d\sigma)^ 2$$ for $$u\in H^ 1(\Omega)$$. Thus if one denotes by $$u_{opt}$$ the corresponding minimizer, the associated function $$d_{opt}$$ is unique and admits an explicit representation.
The physical interest of the problem is that it permits to find the best distribution of an insulating layer around a conducting body.
Reviewer: P.Villaggio

##### MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 74A15 Thermodynamics in solid mechanics