Mass concentration in rescaled first order integral functionals

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 $\epsilon$, $\Gamma$-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 $\epsilon$, 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 $N\ge 2$ 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.

References in corpus (1)

• Profile decomposition for sequences of Borel measures