论文标题
与热绝缘有关的非线性椭圆形问题的形状优化
Shape optimization for a nonlinear elliptic problem related to thermal insulation
论文作者
论文摘要
在本文中，我们考虑了$$ I_ {β，p}（d;ω）= \ inf \ biggl \ {\int_Ω\ lvert {d ϕ} \ rvert^pdx+β\ int _ {\ int _ {\ partial^**东ϕ\in W^{1,p}(Ω),\;ϕ\geq 1 \;\textrm{in}\;D\biggl\}, $$ where $Ω$ is a bounded connected open set in $\mathbb{R}^n$, $D\subset \barΩ$ is a compact set and $β$ is a positive constant.我们让集合$ d $在规定的几何约束和固定厚度的$ω\ setMinus d $下变化，以便以最小化（或最大化）为$ i_ {β，p} $的最小化（或最大化）。在平面案例中，我们表明在圆盘下，磁盘最大化$ i_ {β，p} $。在$ n $维的情况下，我们将分析限制为凸起集，表明球对球是正确的，但在不同的几何约束下。
In this paper we consider a minimization problem of the type $$ I_{β,p}(D;Ω)=\inf\biggl\{\int_Ω\lvert{Dϕ}\rvert^pdx+β\int_{\partial^* Ω}\lvertϕ\rvert^pd\mathcal{H}^{n-1},\; ϕ\in W^{1,p}(Ω),\;ϕ\geq 1 \;\textrm{in}\;D\biggl\}, $$ where $Ω$ is a bounded connected open set in $\mathbb{R}^n$, $D\subset \barΩ$ is a compact set and $β$ is a positive constant. We let the set $D$ vary under prescribed geometrical constraints and $Ω\setminus D$ of fixed thickness, in order to look for the best (or worst) geometry in terms of minimization (or maximization) of $I_{β,p}$. In the planar case, we show that under perimeter constraint the disk maximize $I_{β,p}$. In the $n$-dimensional case we restrict our analysis to convex sets showing that the same is true for the ball but under different geometrical constraints.
