A Free Boundary Problem Related to Thermal Insulation
 发布时间： 2017-11-20     17:01   【返回上一页】 发布人：Dennis Kriventsov

We consider a variational problem for domains coming from the task of finding an optimal thermal insulator. Let $/Omega /subseteq /mathbb{R}^n$ be a (nice) fixed set, and minimize

/[  F(A,u)=/int_{A}|/nabla u|^2 d/mathcal{L}^n + h /int_{/partial A}u^2 d/mathcal{H}^{n-1} + C_0 /mathcal{L}^{n}(A/setminus /Omega) /]
over all sets $A$ containing $/W$ and having smooth boundary, and all smooth functions $u/in C^1(A)$ with $u/equiv 1$ on $/Omega$. Here $h$ and $C_0$ are fixed, positive parameters. This may be thought of as a variational free boundary problem, with the unusual characteristic that along the boundary of the minimal set, $/partial A$, the harmonic function $u$ satisfies a Robin condition, not the typical Dirichlet condition. We will briefly explain how this problem arises, discuss how an appropriately relaxed version of our functional admits minimizers, and then describe some of the regularity properties of these minimizers. This is based on joint work with Luis Caffarelli.