I will describe a minimization problem:

$$/inf/{E(u,/Omega)=/int_/Omega (1/2 |Du|^2-u)/dx +(/int_{/partial/Omega} |u| d/sigma)^2/}$$ over $u/in H^1(/Omega)$ and $volume(/Omega)=1$.

This minimization problem results from an optimal design for heat insulation. I will report some recent result on the existence of such a minimization problem, that involves the free boundary issues. I will also discuss a result on the stability of the unit ball solution. This is a joint work with Qinfeng Li and Hengrong Du from Purdue University.

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