论文标题
平均距离问题与区域比率罚款
The average distance problem with perimeter-to-area ratio penalization
论文作者
论文摘要
在本文中，我们考虑函数\ begin {qore*} e_ {p，\ la}（ω）：= \int_Ω\ dist^p（x，x，\pdΩ）\ d x+\ la \ la \ frac {\ h^1（\ h^1（\pdΩ）} {\ h^2（\ h^2（ω））} {\ h^2（ω）}}。 \ end {equation*}在这里$ p \ geq 1 $，$ \ la> 0 $给出参数，未知$ω$在compact，convex，hausdorff二维集合中的$ \ r^2 $，$ \ \pdΩ$在ω）：= \ inf_ {y \ in \pdΩ} | x-y | $。整体术语$ \int_Ω\ dist^p（x，x，\pdΩ）\ d x $量化了$ω$中点的“易于”点以达到边界，而$ \ frac {\ h^1（\pdΩ）} {\ h^2（ω）} $是范围。主要目的是证明存在和$ c^{1,1} $ - $ \ e $的最小化器的规律性。
In this paper we consider the functional \begin{equation*} E_{p,\la}(Ω):=\int_Ω\dist^p(x,\pd Ω)\d x+\la \frac{\H^1(\pd Ω)}{\H^2(Ω)}. \end{equation*} Here $p\geq 1$, $\la>0$ are given parameters, the unknown $Ω$ varies among compact, convex, Hausdorff two-dimensional sets of $\R^2$, $\pd Ω$ denotes the boundary of $Ω$, and $\dist(x,\pd Ω):=\inf_{y\in\pd Ω}|x-y|$. The integral term $\int_Ω\dist^p(x,\pd Ω)\d x$ quantifies the "easiness" for points in $Ω$ to reach the boundary, while $\frac{\H^1(\pd Ω)}{\H^2(Ω)}$ is the perimeter-to-area ratio. The main aim is to prove existence and $C^{1,1}$-regularity of minimizers of $\E$.
