Generalized W1-1-Young Measures and Relaxation of Problems with Linear Growth

0487019 - ÚTIA 2019 RIV US eng J - Journal Article
Baia, M. - Krömer, Stefan - Kružík, Martin
Generalized W1-1-Young Measures and Relaxation of Problems with Linear Growth.
SIAM Journal on Mathematical Analysis. Roč. 50, č. 1 (2018), s. 1076-1119. ISSN 0036-1410. E-ISSN 1095-7154
R&D Projects: GA ČR GA14-15264S; GA ČR(CZ) GF16-34894L
Institutional support: RVO:67985556
Keywords : lower semicontinuity * quasiconvexity * Young measures
OECD category: Pure mathematics
Impact factor: 1.334, year: 2018
http://library.utia.cas.cz/separaty/2018/MTR/kruzik-0487019.pdf

In this work we completely characterize generalized Young measures generated by sequences of gradients of maps in $W^{1,1}(\Omega-{R}^M)$, where $\Omega\subset{R}^N$. This characterization extends and completes previous analysis by Kristensen and Rindler [Arch. Ration. Mech. Anal., 197 (2010), pp. 539--598 and 203 (2012), pp. 693--700] where concentrations of the sequence of gradients at the boundary of $\Omega$ were excluded. As an application of our result we study the relaxation of non-quasiconvex variational problems with linear growth at infinity, and, finally, we link our characterization to Souček spaces [J. Souček, Časopis Pro Pěstování Matematiky, 97 (1972), pp. 10--46], an extension of $W^{1,1}(\Omega-{\mathbb{R}}^M)$ where gradients are considered as measures on $\bar\Omega$.
Permanent Link: http://hdl.handle.net/11104/0282552