Liftings, Young Measures, and Lower Semicontinuity View Full Text


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Article Info

DATE

2019-06

AUTHORS

Filip Rindler, Giles Shaw

ABSTRACT

This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs (uj, Duj)j for (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 F:u↦∫Ωf(x,u(x),∇u(x))dx,u∈W1,1(Ω;Rm),Ω⊂Rdopen,to the space BV(Ω;Rm). Lower semicontinuity results of this type were first obtained by Fonseca and Müller (Arch Ration Mech Anal 123:1–49, 1993) and later improved by a number of authors, but our theorem is valid under more natural, essentially optimal, hypotheses than those currently present in the literature, requiring principally that f be Carathéodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising F in the x and u variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow-up procedure. More... »

PAGES

1227-1328

References to SciGraph publications

  • 1993-03. Relaxation of quasiconvex functional in BV(Ω, ℝp) for integrands f(x, u,∇;u) in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 1979-12. Integral representation on BV(ω) of Γ-limits of variational integrals in MANUSCRIPTA MATHEMATICA
  • 1987. Some Results and Conjectures in the Gradient Theory of Phase Transitions in METASTABILITY AND INCOMPLETELY POSED PROBLEMS
  • 1987-06. The gradient theory of phase transitions and the minimal interface criterion in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 1991-09. Variational integrals on mappings of bounded variation and their lower semicontinuity in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2016-07. On Rank One Convex Functions that are Homogeneous of Degree One in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 1998-11. A Global Method for Relaxation in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 1999-04. Lower semicontinuity in spaces of weakly differentiable functions in MATHEMATISCHE ANNALEN
  • 1989-09. Phase transitions of elastic solid materials in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 1984-06. Semicontinuity problems in the calculus of variations in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 1985-02. Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals in MANUSCRIPTA MATHEMATICA
  • 2010-01. Relaxation of signed integral functionals in BV in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
  • 2010-08. Characterization of Generalized Gradient Young Measures Generated by Sequences in W1,1 and BV in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 1986-12. Harmonic maps with defects in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1998. Cartesian Currents in the Calculus of Variations II, Variational Integrals in NONE
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    http://scigraph.springernature.com/pub.10.1007/s00205-018-01343-8

    DOI

    http://dx.doi.org/10.1007/s00205-018-01343-8

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    38 schema:description This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs (uj, Duj)j for (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 F:u↦∫Ωf(x,u(x),∇u(x))dx,u∈W1,1(Ω;Rm),Ω⊂Rdopen,to the space BV(Ω;Rm). Lower semicontinuity results of this type were first obtained by Fonseca and Müller (Arch Ration Mech Anal 123:1–49, 1993) and later improved by a number of authors, but our theorem is valid under more natural, essentially optimal, hypotheses than those currently present in the literature, requiring principally that f be Carathéodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising F in the x and u variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow-up procedure.
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