2019/08/08 by Adolfo Arroyo-Rabasa, Arroyo-Rabasa, Adolfo
Mathematics · #Advanced Banach Space Theory #Nonlinear Partial Differential Equations #Advanced Operator Algebra Research
paper · pdf · doi:10.48550/arxiv.1908.03186
We give two characterizations, one for the class of generalized Young\nmeasures generated by mathcal A-free measures, and one for the class\ngenerated by mathcal B-gradient measures mathcal Bu. Here, mathcal A\nand mathcal B are linear homogeneous operators of arbitrary order, which we\nassume satisfy the constant rank property. The characterization places the\nclass of generalized mathcal A-free Young measures in duality with the class\nof mathcal A-quasiconvex integrands by means of a well-known Hahn--Banach\nseparation property. A similar statement holds for generalized mathcal\nB-gradient Young measures. Concerning applications, we discuss several\nexamples that showcase the rigidity or the failure of\n\L1-compensated compactness when concentration of mass is allowed.\nThese include the failure of \L1-estimates for elliptic systems and\nthe failure of \L1-rigidity for the two-state problem. As a\nbyproduct of our techniques we also show that, for any bounded open set\n\Ω, the inclusions \
mathrmL1(
Omega)
cap
ker
mathcal A\n
hookrightarrow
mathcal M(
Omega)
cap
ker
mathcal A, \
mathcal Bν
in
mathrmC^
infty(
Omega)
hookrightarrow
mathcal B u
in
mathcal\nM(
Omega)
, are dense with respect to area-functional convergence of\nmeasures\n