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Mass concentration in rescaled first order integral functionals

hal.structure.identifierLaboratoire Analyse et Mathématiques Appliquées [LAMA]
dc.contributor.authorMonteil, Antonin
HAL ID: 748327
hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorPegon, Paul
dc.date.accessioned2022-11-22T15:43:13Z
dc.date.available2022-11-22T15:43:13Z
dc.date.issued2022
dc.identifier.urihttps://basepub.dauphine.psl.eu/handle/123456789/23176
dc.language.isoenen
dc.subjectΓ-convergenceen
dc.subjectsemicontinuityen
dc.subjectintegral functionalsen
dc.subjectconvergence of measuresen
dc.subjectconcentration-compactnessen
dc.subjectCahn-Hilliard fluidsen
dc.subjectbranched transporten
dc.subject.ddc515en
dc.titleMass concentration in rescaled first order integral functionalsen
dc.typeDocument de travail / Working paper
dc.description.abstractenWe 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.en
dc.publisher.cityParisen
dc.identifier.citationpages38en
dc.relation.ispartofseriestitleCahier de recherche CEREMADE, Université Paris Dauphine-PSLen
dc.subject.ddclabelAnalyseen
dc.identifier.citationdate2022
dc.description.ssrncandidatenon
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.date.updated2022-11-22T15:40:24Z
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