Mass concentration in rescaled first order integral functionals

TL;DR

This paper studies the behavior of minimization problems with a mass constraint, showing how energies concentrate and converge to atomic measures, with applications to fluid droplets and branched transport.

Contribution

It proves the concavity of the minimal energy function and establishes $ ext{Gamma}$-convergence of rescaled energies to $H$-masses under mild assumptions.

Findings

Minimal energy function $H(m)$ is concave.

Rescaled energies $ ext{Gamma}$-converge to $H$-mass.

Results apply to $ ext{alpha}$-masses and concave $H$-masses.

Abstract

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 $\varepsilon$, $\Gamma$-converge towards the $H$-mass, defined for atomic measures $\sum_i m_i\delta_{x_i}$ 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 $\alpha$-masses in any dimension $N\geq 2$ for exponents $\alpha$ 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.