# Liftings, Young Measures, and Lower Semicontinuity

@article{Rindler2018LiftingsYM,
title={Liftings, Young Measures, and Lower Semicontinuity},
author={Filip Rindler and Giles Shaw},
journal={Archive for Rational Mechanics and Analysis},
year={2018},
volume={232},
pages={1227-1328}
}
• Published 14 August 2017
• Mathematics
• Archive for Rational Mechanics and Analysis
AbstractThis work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs (uj, Duj)j for $${(u_j)_j\subset {\rm BV}(\Omega;\mathbb{R}^m)}$$(uj)j⊂BV(Ω;Rm) under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functional $$\mathcal{F}\colon u\mapsto\int_\Omega f(x,u(x),\nabla u(x))\, {\rm d}x, \quad u \in {\rm W}^{1,1}(\Omega;\mathbb{R}^m),\quad\Omega\subset… 7 Citations Relaxation for Partially Coercive Integral Functionals with Linear Growth • Mathematics, Computer Science SIAM J. Math. Anal. • 2020 This work proves an integral representation theorem for the \mathrm{L}^1(\Omega;\mathbb{R}^m)-relaxation of the functional which applies to integrands which are unbounded in the u-variable and thus allows to treat many problems from applications. A note on the weak* and pointwise convergence of BV functions • Mathematics • 2020 We study pointwise convergence properties of weakly* converging sequences \{u_i\}_{i \in {\mathbb N}} in \mathrm{BV}({\mathbb R}^n). We show that, after passage to a suitable subsequence (not Relaxation of functionals with linear growth: Interactions of emerging measures and free discontinuities • Mathematics Advances in Calculus of Variations • 2022 Abstract For an integral functional defined on functions ( u , v ) ∈ W 1 , 1 × L 1 {(u,v)\in W^{1,1}\times L^{1}} featuring a prototypical strong interaction term between u and v, we calculate its Lower Semicontinuity in L 1 of a Class of Functionals Defined on <jats:p>We prove lower semicontinuity in <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <msup> Space-time integral currents of bounded variation Motivated by a recent model for elasto-plastic evolutions that are driven by the flow of dislocations [12], this work develops a theory of space-time integral currents with bounded variation in time. ## References SHOWING 1-10 OF 55 REFERENCES Relaxation for Partially Coercive Integral Functionals with Linear Growth • Mathematics, Computer Science SIAM J. Math. Anal. • 2020 This work proves an integral representation theorem for the \mathrm{L}^1(\Omega;\mathbb{R}^m)-relaxation of the functional which applies to integrands which are unbounded in the u-variable and thus allows to treat many problems from applications. Relaxation of signed integral functionals in BV • Mathematics • 2009 For integral functionals initially defined for$${u \in {\rm W}^{1,1}(\Omega; \mathbb{R}^m)}$$by$$\int_{\Omega} f(\nabla u) \, {\rm d}x$$we establish strict continuity and relaxation results in Relaxation of quasiconvex functional in BV(Ω, ℝp) for integrands f(x, u,∇;u) • Mathematics • 1993 AbstractIn this paper it is shown that if p(x, u,·) is a quasiconvex function with linear growth, then the relaxed functional in BV(Ω, ℝp) of$$u \to \int\limits_\Omega {f(x, u(x), \nabla u(x)) dx}
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