# Mass concentration in rescaled first order integral functionals

created by monteil on 02 May 2021

[BibTeX]

Preprint

Inserted: 2 may 2021
Last Updated: 2 may 2021

Year: 2021

Abstract:

We consider first order local minimization problems $\min\int_{\mathbb{R}^N}f(x_0,u,\nabla u)$ over non-negative Sobolev functions $u$ satisfying a mass constraint $\int_{\mathbb{R}^N}u=m$. We prove that the minimal energy function $H(x_0,m)$ is always concave in $m$, and that relevant rescalings of the energy, depending on a small parameter $\varepsilon$, $\Gamma$-converge in the weak topology of measures towards the $H$-mass, defined for atomic measures $\sum_i m_i\delta_{x_i}$ as $\sum_i H(x_i,m_i)$. The $\Gamma$-convergence result holds under mild assumptions on the Lagrangian, and covers several situations including homogeneous $H$-masses in any dimension $N\geq 2$ for exponents 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.