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.

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