论文标题
Euler Elastica惩罚的平均距离问题
The average distance problem with an Euler elastica penalization
论文作者
论文摘要
我们考虑在二维域$ω$上定义的平均距离功能的最小化，并具有与$ \pdΩ$相关的Euler Elastica惩罚，即$ω$的边界。平均距离由\ begin {equination*} \int_Ω\ dist^p（x，x，\pdΩ）\ d x \ end {equation*}，其中$ p \ geq 1 $是给定的参数，$ \ dist（x，x，\pdΩ）$是$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \^$ \^$ \^$ \^$ \^$ \^$ \ \ \ \ \ \ \ \ \^$ \^$ \^$ \^$惩罚术语是边界曲线$ {\pdΩ} $的Euler Elastica（即Helfrich弯曲能量或Willmore Energy）的倍数，它与\ \ pdω$的集成平方曲率成正比，如\ pd的\ pdω$，如\ begin feben \ begin \ begin {等式*} \ la \ la \ la \ la \ la \ la \ la \ la \ la。 κ_ {\pdΩ}^2 \ d \ d \ h _ {\ llcorner \pdΩ}^1，\ end {equation*}其中$κ_ {\pdΩ} $表示$ \ pdω$和$ \ lamy和$ \ la的（签名）曲率的（签名）的曲率。域 $ω$在紧凑的凸套中有所不同，$ \ mathbb {r}^2 $，与hausdorff尺寸相当于$ 2 $ \ tcr {。}，在没有关于边界$ \ \pdΩ$的规律性的任何先验假设下，我们证明了最小的$ e_ e _ pd，我们证明的是$ e e _ p，p. la {此外，我们建立了$ c^{1,1} $ - 其最小化器的规律性。合适的竞争对手家族的原始结构在证明规律性方面起着决定性的作用。
We consider the minimization of an average distance functional defined on a two-dimensional domain $Ω$ with an Euler elastica penalization associated with $\pd Ω$, the boundary of $Ω$. The average distance is given by \begin{equation*} \int_Ω \dist^p(x,\pd Ω)\d x \end{equation*} where $p\geq 1$ is a given parameter, and $\dist(x,\pd Ω)$ is the Hausdorff distance between $\{x\}$ and $\pd Ω$. The penalty term is a multiple of the Euler elastica (i.e., the Helfrich bending energy or the Willmore energy) of the boundary curve ${\pd Ω}$, which is proportional to the integrated squared curvature defined on $\pd Ω$, as given by \begin{equation*} \la \int_{\pd Ω} κ_{\pd Ω}^2\d\H_{\llcorner \pd Ω}^1, \end{equation*} where $κ_{\pd Ω}$ denotes the (signed) curvature of $\pd Ω$ and $\la>0$ denotes a penalty constant. The domain $Ω$ is allowed to vary among compact, convex sets of $\mathbb{R}^2$ with Hausdorff dimension equal to $2$\tcr{.} Under no a priori assumptions on the regularity of the boundary $\pd Ω$, we prove the existence of minimizers of $E_{p,\la}$. Moreover, we establish the $C^{1,1}$-regularity of its minimizers. An original construction of a suitable family of competitors plays a decisive role in proving the regularity.
