论文标题
均匀密度估计和负质量功能的$γ$ - convergence
Uniform density estimates and $Γ$-convergence for the Alt-Phillips functional of negative powers
论文作者
论文摘要
我们获得了最小化器的自由界限的密度估计值$ u \ ge 0 $的alt-phillips功能，涉及负功率电位 $ \int_Ω\ left（| \ nabla u |^2 + u^{ - γ}χ_ {\ {u> 0 \> 0 \}}} \ right）\，dx，\ quad \ quad \ quad \quadγ\ in（0,2） 这些估计值作为参数$γ\至2 $保持均匀。结果，我们确定了相应的自由边界到最小表面的均匀收敛，为$γ\至2 $。 结果基于这些能量（适当重新恢复）的$γ$ - 授权 - $$ \int_Ω| \ nabla u |^2 dx +per_Ω（\ {u = 0 \}），$$ 由Athanasopoulous，Caffarelli，Kenig和Salsa考虑。
We obtain density estimates for the free boundaries of minimizers $u \ge 0$ of the Alt-Phillips functional involving negative power potentials $$\int_Ω\left(|\nabla u|^2 + u^{-γ} χ_{\{u>0\}}\right) \, dx, \quad \quad γ\in (0,2).$$ These estimates remain uniform as the parameter $γ\to 2$. As a consequence we establish the uniform convergence of the corresponding free boundaries to a minimal surface as $γ\to 2$. The results are based on the $Γ$-convergence of these energies (properly rescaled) to the Dirichlet-perimeter functional $$\int_Ω|\nabla u|^2 dx + Per_Ω(\{ u=0\}),$$ considered by Athanasopoulous, Caffarelli, Kenig, and Salsa.
