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Relaxation for partially coercive integral functionals with linear growth
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Rindler, Filip and Shaw, Giles (2020) Relaxation for partially coercive integral functionals with linear growth. SIAM Journal of Mathematical Analysis, 52 (5). pp. 4806-4860. doi:10.1137/18M1199460 ISSN 0036-1410.
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Official URL: https://doi.org/10.1137/18M1199460
Abstract
We prove an integral representation theorem for the $\mathrm{L}^1$-relaxation of the functional $\mathcal{F}\colon u\mapsto\int_\Omega f(x,u(x),\nabla u(x))\;\mathrm{d} x,\quad u\in\mathrm{W}^{1,1}(\Omega;\mathbb{R}^m)$, where $\Omega\subset\mathbb{R}^d$ ($d \geq 2$) is a bounded Lipschitz domain, to the space $\mathrm{BV}(\Omega;\mathbb{R}^m)$ under very general assumptions: we require principally that $f$ is Carathéodory, that the partial coercivity and linear growth bound $g(x,y)|A|\leq f(x,y,A)\leq Cg(x,y)(1+|A|)$, hold, where $g\colon\overline{\Omega}\times\mathbb{R}^m\to[0,\infty)$ is a continuous function satisfying a weak monotonicity condition, and that $f$ is quasi-convex in the final variable. Our result is the first that applies to integrands which are unbounded in the $u$-variable and, therefore, allows for the treatment of many problems from applications. Such functionals are out of reach of the classical blowup approach introduced by Fonseca and Müller [Arch. Ration. Mech. Anal., 123 (1993), pp. 1--49]. Our proof relies on an intricate truncation construction (in the $x$- and $u$-arguments simultaneously) made possible by the theory of liftings developed in a previous paper by the authors [Arch. Ration. Mech. Anal., 232 (2019), pp. 1227--1328], and features techniques which could be of use for other problems involving $u$-dependent integrands.
| Item Type: | Journal Article | |||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
| Library of Congress Subject Headings (LCSH): | Relaxation methods (Mathematics), Functional analysis | |||||||||
| Journal or Publication Title: | SIAM Journal of Mathematical Analysis | |||||||||
| Publisher: | Society for Industrial and Applied Mathematics | |||||||||
| ISSN: | 0036-1410 | |||||||||
| Official Date: | 10 January 2020 | |||||||||
| Dates: |
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| Volume: | 52 | |||||||||
| Number: | 5 | |||||||||
| Page Range: | pp. 4806-4860 | |||||||||
| DOI: | 10.1137/18M1199460 | |||||||||
| Status: | Peer Reviewed | |||||||||
| Publication Status: | Published | |||||||||
| Reuse Statement (publisher, data, author rights): | “First Published in SIAM Journal of Mathematical Analysis in [volume and number, or 2020, published by the Society for Industrial and Applied Mathematics (SIAM)” and the copyright notice as stated in the article itself (e.g., “Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.”) | |||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||
| Copyright Holders: | © by SIAM | |||||||||
| Date of first compliant deposit: | 5 March 2020 | |||||||||
| Date of first compliant Open Access: | 11 March 2020 | |||||||||
| Funder: | The work presented in this paper forms part of G.S.'s PhD thesis, undertaken at the Cambridge Centre for Analysis at the University of Cambridge, and supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1 for the | |||||||||
| RIOXX Funder/Project Grant: |
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