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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


 

 
北京师范大学数学科学学院

 

偏微分方程专题报告

 

 

报告题目:A Free Boundary Problem Related to Thermal Insulation

报告人: Dennis Kriventsov (美国Courant Institute )

时间地点: 1121 15:00—16:00, 后主楼 1129

邀请人: 熊金钢

报告摘要:

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.