Mass concentration in rescaled first order integral functionals
Monteil, Antonin; Pegon, Paul (2022), Mass concentration in rescaled first order integral functionals. https://basepub.dauphine.psl.eu/handle/123456789/23176
TypeDocument de travail / Working paper
Series titleCahier de recherche CEREMADE, Université Paris Dauphine-PSL
MetadataShow full item record
Laboratoire Analyse et Mathématiques Appliquées [LAMA]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)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.
Subjects / KeywordsΓ-convergence; semicontinuity; integral functionals; convergence of measures; concentration-compactness; Cahn-Hilliard fluids; branched transport
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