论文标题
通过体积限制和双重障碍物进行热传导的优化问题
An Optimization Problem in Heat Conduction With Volume Constraint and Double Obstacles
论文作者
论文摘要
我们考虑最小化$ \ int _ {\ mathbb {r}^n} | \ nabla u |^2 \，\ mathrm {d} x $带双障碍物$ ϕ \ leq uq u \ leq u \ leq 6 $ A.E.的优化问题。在$ d $和$ \ {u> 0 \} \ setMinus \ overline {d} $的卷的约束中，其中$ d \ subset \ mathbb {r}^n $是一个有界的域。通过研究一个惩罚问题，该问题可以实现少量惩罚参数值的约束数量，我们证明，每个最小化器都是$ c^{1,1} $本地，$ d $中的本地，Lipschitz在$ \ mathbb {r}^n $中持续连续，并且是free界限$ \ partial $ \ partial \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ is $ sempline ploome asseminus \ setminus {d。此外，当$ d $的边界具有平面部分时，我们表明最小化为$ c^{1，\ frac {1} {2}}} $，直到平面部分。
We consider the optimization problem of minimizing $\int_{\mathbb{R}^n}|\nabla u|^2\,\mathrm{d}x$ with double obstacles $ϕ\leq u\leqψ$ a.e. in $D$ and a constraint on the volume of $\{u>0\}\setminus\overline{D}$, where $D\subset\mathbb{R}^n$ is a bounded domain. By studying a penalization problem that achieves the constrained volume for small values of penalization parameter, we prove that every minimizer is $C^{1,1}$ locally in $D$ and Lipschitz continuous in $\mathbb{R}^n$ and that the free boundary $\partial\{u>0\}\setminus\overline{D}$ is smooth. Moreover, when the boundary of $D$ has a plane portion, we show that the minimizer is $C^{1,\frac{1}{2}}$ up to the plane portion.
