Weak Lower Semicontinuity by Means of Anisotropic Parametrized Measures

It is well known that besides oscillations, sequences bounded only in L1 can also develop concentrations, and if the latter occurs, we can at most hope for weak∗ convergence in the sense of measures. Here we derive a new tool to handle mutual interferences of an oscillating and concentrating sequence with another weakly converging sequence. We introduce a couple of explicit examples showing a variety of possible kinds of behavior and outline some applications in Sobolev spaces.

[1]  Stefan Krömer,et al.  On the role of lower bounds in characterizations of weak lower semicontinuity of multiple integrals , 2010 .

[2]  M. Kružík,et al.  On the measures of DiPerna and Majda , 1997 .

[3]  Tomáš Roubíček,et al.  Relaxation in Optimization Theory and Variational Calculus , 1997 .

[4]  C. B. Morrey.  Multiple Integrals in the Calculus of Variations , 1966 .

[5]  P. Pedregal.  Multi-scale Young measures , 2005 .

[6]  Guy Bouchitté,et al.  Non-Uniform Integrability and Generalized Young Measures , 1997 .

[7]  Margarida Baía,et al.  Generalized W1, 1-Young Measures and Relaxation of Problems with Linear Growth , 2018, SIAM J. Math. Anal..

[8]  Lawrence C. Evans,et al.  Weak convergence methods for nonlinear partial differential equations , 1990 .

[9]  Nicola Fusco,et al.  Semicontinuity problems in the calculus of variations , 1984 .

[10]  A. Majda,et al.  Oscillations and concentrations in weak solutions of the incompressible fluid equations , 1987 .

[11]  Didier Henrion,et al.  Optimal control problems with oscillations, concentrations and discontinuities , 2018, Autom..

[12]  Stefan Kr.  OSCILLATIONS AND CONCENTRATIONS IN SEQUENCES OF GRADIENTS UP TO THE BOUNDARY , 2011 .

[13]  M. Kružík,et al.  Generalized W1, 1-Young Measures and Relaxation of Problems with Linear Growth , 2016, SIAM J. Math. Anal..

[14]  P. Pedregal.  Parametrized measures and variational principles , 1997 .

[15]  J. Ball.  A version of the fundamental theorem for young measures , 1989 .

[16]  Pablo Pedregal,et al.  Gradient Young measures generated by sequences in Sobolev spaces , 1994 .

[17]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[18]  Irene Fonseca,et al.  Analysis of Concentration and Oscillation Effects Generated by Gradients , 1998 .

[19]  Pablo Pedregal,et al.  Characterizations of young measures generated by gradients , 1991 .

[20]  Paolo Marcellini,et al.  Semicontinuity problems in the calculus of variations , 1980 .

[21]  G. Tomassetti,et al.  A Variational Justification of Linear Elasticity with Residual Stress , 2009 .

[22]  C. Licht,et al.  A model of elastic adhesive bonded joints through oscillation–concentration measures , 2007 .

[23]  Tomáš Roubíček,et al.  Optimization problems with concentration and oscillation effects: relaxation theory and numerical approximation , 1999 .

[24]  Agnieszka Kaïamajska.  On Young measures controlling discontinuous functions , 2006 .

[25]  Martin Kružík,et al.  Oscillations and concentrations in sequences of gradients , 2008 .

[26]  John M. Ball,et al.  Lower semicontinuity of multiple integrals and the Biting Lemma , 1990, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[27]  D. Henrion,et al.  Semi-definite relaxations for optimal control problems with oscillation and concentration effects , 2014, 1412.2278.

[28]  M. Schonbek,et al.  Convergence of solutions to nonlinear dispersive equations , 1982 .

[29]  G. Tomassetti,et al.  From non-linear elasticity to linear elasticity with initial stress via Γ-convergence , 2011 .

[30]  Martin Kruzík,et al.  Weak Lower Semicontinuity of Integral Functionals and Applications , 2017, SIAM Rev..

[31]  L. Ambrosio,et al.  Multiscale Young Measures in almost Periodic Homogenization and Applications , 2009 .

[32]  J. Warga.  Optimal control of differential and functional equations , 1972 .

[33]  B. Dacorogna.  Direct methods in the calculus of variations , 1989 .