论文标题
重新缩放一阶积分功能的质量集中
Mass concentration in rescaled first order integral functionals
论文作者
论文摘要
我们考虑一阶$ \ min \ min \ int _ {\ mathbb {r}^n} f（u，\ nabla u）$在质量约束$ \ int _ {\ mathbb {r}^n} u = m $下的一阶局部最小化问题。我们证明，最小的能量函数$ h（m）$始终是凹的，并且能量的相关重新缩放，具体取决于一个小参数$ \ varepsilon $，$γ$ - converge to $ h $ sass，定义为原子量的原子量$ \ sum_im_iΔ____________________________ih（m_imim_iΔ________ih（m）我们还考虑了拉格朗日人，具体取决于$ \ varepsilon $，以及空间不均匀的lagrangians和$ h $ sasses。我们的结果是在$ f $上的温和假设下，尤其是$α$ - 大小在任何维度上的$ n \ geq 2 $ for Exponents $α$高于关键阈值的$α$，所有凹入$ h $ smasses in Dimension $ n = 1 $。我们的结果尤其要产生Cahn-Hilliard流体的浓度到液滴中，并且与通过椭圆形能量的分支传输近似有关。
We consider first order local minimization problems of the form $\min \int_{\mathbb{R}^N}f(u,\nabla u)$ under a mass constraint $\int_{\mathbb{R}^N}u=m$. We prove that the minimal energy function $H(m)$ is always concave, and that relevant rescalings of the energy, depending on a small parameter $\varepsilon$, $Γ$-converge towards the $H$-mass, defined for atomic measures $\sum_i m_iδ_{x_i}$ as $\sum_i H(m_i)$. We also consider Lagrangians depending on $\varepsilon$, as well as space-inhomogeneous Lagrangians and $H$-masses. Our result holds under mild assumptions on $f$, and covers in particular $α$-masses in any dimension $N\geq 2$ for exponents $α$ above a critical threshold, and all concave $H$-masses in dimension $N=1$. Our result yields in particular the concentration of Cahn-Hilliard fluids into droplets, and is related to the approximation of branched transport by elliptic energies.
