Mass concentration in rescaled first order integral functionals

We consider first order local minimization problems of the form min∫ ℝ N f(u,∇u) under a mass constraint ∫ 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 ε, Γ-converge towards the H-mass, defined for atomic measures ∑ i m i δ x i as ∑ i H(m i ). We also consider Lagrangians depending on ε, 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≥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.