2011/03/15 by Benešová, Barbora, Kružík, Martin, Pathó, Gabriel
#35B05 #49J45 #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.1103.2859
Motivated by variational problems in nonlinear elasticity depending on the deformation gradient and its inverse, we completely and explicitly describe Young measures generated by matrix-valued mappings \Yk\k∈\N ⊂ Lp(Ø;\Rn× n), Ø⊂\Rn, such that \Yk-1\k∈\N ⊂ Lp(Ø;\Rn× n) is bounded, too. Moreover, the constraint det Yk>0 can be easily included and is reflected in a condition on the support of the measure. This condition typically occurs in problems of nonlinear-elasticity theory for hyperelastic materials if Y:=∇ y for y∈ W1,p(Ø;\Rn). Then we fully characterize the set of Young measures generated by gradients of a uniformly bounded sequence in W1,∞(Ø;\Rn) where the inverted gradients are also bounded in L^∞(Ø;\Rn× n). This extends the original results due to D. Kinderlehrer and P. Pedregal.