Calculus of Variations and Geometric Measure Theory
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A. Monteil - P. Pegon

Mass concentration in rescaled first order integral functionals

created by monteil on 02 May 2021

[BibTeX]

Preprint

Inserted: 2 may 2021
Last Updated: 2 may 2021

Year: 2021

Abstract:

We consider first order local minimization problems \(\min\int_{\mathbb{R}^N}f(x_0,u,\nabla u)\) over non-negative Sobolev functions \(u\) satisfying a mass constraint \(\int_{\mathbb{R}^N}u=m\). We prove that the minimal energy function \(H(x_0,m)\) is always concave in \(m\), and that relevant rescalings of the energy, depending on a small parameter \(\varepsilon\), \(\Gamma\)-converge in the weak topology of measures towards the \(H\)-mass, defined for atomic measures \(\sum_i m_i\delta_{x_i}\) as \(\sum_i H(x_i,m_i)\). The \(\Gamma\)-convergence result holds under mild assumptions on the Lagrangian, and covers several situations including homogeneous \(H\)-masses in any dimension \(N\geq 2\) 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.

Keywords: relaxation, calculus of variations, functionals on measures, Concentration compactness, \(\Gamma\)-convergence, \(H\)-mass


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