Mass concentration in rescaled first order integral functionals
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866929251729539072 |
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| author | Monteil, Antonin Pegon, Paul |
| author_facet | Monteil, Antonin Pegon, Paul |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_01250 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Mass concentration in rescaled first order integral functionals Monteil, Antonin Pegon, Paul Analysis of PDEs Optimization and Control Primary: 28A33, 49J45, 46E35, Secondary: 49Q20, 76T99, 49Q22, 49J10 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. |
| title | Mass concentration in rescaled first order integral functionals |
| topic | Analysis of PDEs Optimization and Control Primary: 28A33, 49J45, 46E35, Secondary: 49Q20, 76T99, 49Q22, 49J10 |
| url | https://arxiv.org/abs/2203.01250 |