*Published Paper*

**Inserted:** 2 may 2021

**Last Updated:** 22 feb 2024

**Journal:** Journal de l'Ă‰cole Polytechnique

**Volume:** 11

**Pages:** pp. 431-472

**Year:** 2024

**Doi:** 10.5802/jep.257

**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.

**Keywords:**
Branched transport, Concentration-compactness, semicontinuity, integral functionals, \(\Gamma\)-convergence, convergence of measures, Cahn-Hilliard fluids

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