On the measures of DiPerna and Majda

Martin Kružík; Tomáš Roubíček

Mathematica Bohemica (1997)

  • Volume: 122, Issue: 4, page 383-399
  • ISSN: 0862-7959

Abstract

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DiPerna and Majda generalized Young measures so that it is possible to describe “in the limit” oscillation as well as concentration effects of bounded sequences in L p -spaces. Here the complete description of all such measures is stated, showing that the “energy” put at “infinity” by concentration effects can be described in the limit basically by an arbitrary positive Radon measure. Moreover, it is shown that concentration effects are intimately related to rays (in a suitable locally convex geometry) in the set of all DiPerna-Majda measures. Finally, a complete characterization of extreme points and extreme rays is established.

How to cite

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Kružík, Martin, and Roubíček, Tomáš. "On the measures of DiPerna and Majda." Mathematica Bohemica 122.4 (1997): 383-399. <http://eudml.org/doc/248149>.

@article{Kružík1997,
abstract = {DiPerna and Majda generalized Young measures so that it is possible to describe “in the limit” oscillation as well as concentration effects of bounded sequences in $L^p$-spaces. Here the complete description of all such measures is stated, showing that the “energy” put at “infinity” by concentration effects can be described in the limit basically by an arbitrary positive Radon measure. Moreover, it is shown that concentration effects are intimately related to rays (in a suitable locally convex geometry) in the set of all DiPerna-Majda measures. Finally, a complete characterization of extreme points and extreme rays is established.},
author = {Kružík, Martin, Roubíček, Tomáš},
journal = {Mathematica Bohemica},
keywords = {bounded sequences in Lebesgue spaces; oscillations; Young measures; DiPerna and Majda measures; rays; extreme points; extreme rays; concentrations; bounded sequences in Lebesgue spaces; oscillations; concentrations; Young measures; DiPerna and Majda measures; rays; extreme points; extreme rays},
language = {eng},
number = {4},
pages = {383-399},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the measures of DiPerna and Majda},
url = {http://eudml.org/doc/248149},
volume = {122},
year = {1997},
}

TY - JOUR
AU - Kružík, Martin
AU - Roubíček, Tomáš
TI - On the measures of DiPerna and Majda
JO - Mathematica Bohemica
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 122
IS - 4
SP - 383
EP - 399
AB - DiPerna and Majda generalized Young measures so that it is possible to describe “in the limit” oscillation as well as concentration effects of bounded sequences in $L^p$-spaces. Here the complete description of all such measures is stated, showing that the “energy” put at “infinity” by concentration effects can be described in the limit basically by an arbitrary positive Radon measure. Moreover, it is shown that concentration effects are intimately related to rays (in a suitable locally convex geometry) in the set of all DiPerna-Majda measures. Finally, a complete characterization of extreme points and extreme rays is established.
LA - eng
KW - bounded sequences in Lebesgue spaces; oscillations; Young measures; DiPerna and Majda measures; rays; extreme points; extreme rays; concentrations; bounded sequences in Lebesgue spaces; oscillations; concentrations; Young measures; DiPerna and Majda measures; rays; extreme points; extreme rays
UR - http://eudml.org/doc/248149
ER -

References

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Citations in EuDML Documents

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  1. Tomáš Roubíček, Nonconcentrating generalized Young functionals
  2. Martin Kružík, Johannes Zimmer, Evolutionary problems in non-reflexive spaces
  3. Martin Kružík, On convergence of gradient-dependent integrands
  4. Martin Kružík, DiPerna-Majda measures and uniform integrability
  5. Martin Kružík, Quasiconvexity at the boundary and concentration effects generated by gradients
  6. Martin Kružík, Agnieszka Kałamajska, Oscillations and concentrations in sequences of gradients
  7. Agnieszka Kałamajska, Martin Kružík, Oscillations and concentrations in sequences of gradients
  8. Irene Fonseca, Martin Kružík, Oscillations and concentrations generated by 𝒜 -free mappings and weak lower semicontinuity of integral functionals

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