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Mass concentration in rescaled first order integral functionals

Antonin Monteil, Paul Pegon

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 $\varepsilon$, $Γ$-converge towards the $H$-mass, defined for atomic measures $\sum_i m_iδ_{x_i}$ as $\sum_i H(m_i)$. We also consider Lagrangians depending on $\varepsilon$, 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 $N\geq 2$ 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.

Mass concentration in rescaled first order integral functionals

Abstract

We consider first order local minimization problems of the form under a mass constraint . We prove that the minimal energy function is always concave, and that relevant rescalings of the energy, depending on a small parameter , -converge towards the -mass, defined for atomic measures as . We also consider Lagrangians depending on , as well as space-inhomogeneous Lagrangians and -masses. Our result holds under mild assumptions on , and covers in particular -masses in any dimension for exponents above a critical threshold, and all concave -masses in dimension . 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.
Paper Structure (29 sections, 23 theorems, 185 equations)

This paper contains 29 sections, 23 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Assumptions and main result
  4. Remarks and applications
  5. Structure of the paper
  6. Minimal cost function and H-mass
  7. Concavity and lower semicontinuity of the minimal cost function
  8. Definition and relaxation of the H-mass
  9. Slope at the origin of the minimal cost function
  10. Strict concavity of the minimal cost function in dimension N≥2
  11. Lower bound for the energy and existence of optimal profiles
  12. Profile decomposition by concentration-compactness
  13. Lower bound by concentration-compactness
  14. Existence of optimal profiles
  15. Γ-convergence of the rescaled energies towards the H-mass
  16. ...and 14 more sections

Key Result

Theorem 1

Let $f :\mathbb{R}\times \mathbb{R}^N \to [0,+\infty]$ be Borel measurable such that $f(0,0) = 0$. The function defined for every $m\in \mathbb{R}$ by vanishes at $0$ and it is either identically $+\infty$ on $(0,+\infty)$, or it is everywhere finite, continuous, concave and non-decreasing on $[0,+\infty)$. The symmetric statement on $(-\infty,0]$ holds as well.

Theorems & Definitions (52)

  • Theorem 1
  • Theorem 2
  • proof : Proof of \ref{['concaveDN']}
  • Remark 3
  • Definition 4
  • Proposition 5: bouchitteNewLowerSemicontinuity1990
  • Proposition 6
  • proof : Proof of \ref{['relaxationMass']}
  • Proposition 7
  • proof
  • ...and 42 more