论文标题
关于De Giorgi的猜想，关于Willmore功能的相位场近似值
On a conjecture of De Giorgi about the phase-field approximation of the Willmore functional
论文作者
论文摘要
在1991年，de Giorgi猜想，如果$λ> 0 $，如果$μ_\ varepsilon $代表allen-cahn Energy的密度和$ v_ \ varepsilon $代表其第一个变化，那么$ \ int $ \ int [v_ \ varepsilon^varepsilon^varepsilon^varepsilon^2 +λ] d + varepsilon $ $ - 应对于某些真实常数$ k $，$cλ\ mathrm {per}（per}（e） + k \ mathcal {w}（σ）$，其中$ \ mathrm {per}（e）$是集合$ e $，$ et $ e $，$σ= \ partial e $ e $，$ $ \ \ m m m is $ c的$ c的$ c and and n is and mortastion; Röger和Schätzle在空间尺寸中证明了该猜想的修改版本，当时$ \ int v_ \ varepsilon^2 \，dμ__\ varepsilon $被$ \ varepsilon $替换为$ \ int v_ int v_ int v_ int v_ \ v _ v _ v varepsilon^v varepsilon^2 {在本文中，我们表明，令人惊讶的是，原始的de Giorgi猜想以$ k = 0 $保存。还提供了在近似能量均匀控制下获得的极限度量的进一步特性。
In 1991 De Giorgi conjectured that, given $λ>0$, if $μ_\varepsilon$ stands for the density of the Allen-Cahn energy and $v_\varepsilon$ represents its first variation, then $\int [v_\varepsilon^2 + λ] dμ_\varepsilon$ should $Γ$-converge to $cλ\mathrm{Per}(E) + k \mathcal{W}(Σ)$ for some real constant $k$, where $\mathrm{Per}(E)$ is the perimeter of the set $E$, $Σ=\partial E$, $\mathcal{W}(Σ)$ is the Willmore functional, and $c$ is an explicit positive constant. A modified version of this conjecture was proved in space dimensions $2$ and $3$ by Röger and Schätzle, when the term $\int v_\varepsilon^2 \, dμ_\varepsilon$ is replaced by $ \int v_\varepsilon^2 {\varepsilon}^{-1} dx$, with a suitable $k>0$. In the present paper we show that, surprisingly, the original De Giorgi conjecture holds with $k=0$. Further properties on the limit measures obtained under a uniform control of the approximating energies are also provided.
