Mass concentration in rescaled first order integral functionals
Résumé
We consider first order local minimization problems min ∫ f(u,∇u) under a mass constraint ∫ u = m∈R. We prove that the minimal energy function H(m) is always concave on (−∞, 0) and (0, +∞), and that relevant rescalings of the energy, depending on a small parameter ε, Γ-converge in the weak topology of measures towards the H-mass, defined for atomic measures Σᵢ mᵢ δxᵢ as Σᵢ H(mᵢ). We also consider space dependent Lagrangians f(x,u,∇u), which cover the case of space dependent H-masses Σᵢ H(xᵢ,mᵢ), and also the case of a family of Lagrangians (fε) converging as ε → 0. The Γ-convergence result holds under mild assumptions on f, and covers several situations including homogeneous H-masses in any dimension N ≥ 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.
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