# 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} }

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 ScienceSIAM 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

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- 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…

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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…

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L
1
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- Abstract and Applied Analysis
- 2021

<jats:p>We prove lower semicontinuity in <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
<msup>…

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Space-time integral currents of bounded variation

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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.…

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