Necessary conditions for weak lower semicontinuity on domains with infinite measure

Stefan Krömer

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 2, page 457-471
  • ISSN: 1292-8119

Abstract

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We derive sharp necessary conditions for weak sequential lower semicontinuity of integral functionals on Sobolev spaces, with an integrand which only depends on the gradient of a scalar field over a domain in N . An emphasis is put on domains with infinite measure, and the integrand is allowed to assume the value + .

How to cite

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Krömer, Stefan. "Necessary conditions for weak lower semicontinuity on domains with infinite measure." ESAIM: Control, Optimisation and Calculus of Variations 16.2 (2010): 457-471. <http://eudml.org/doc/250704>.

@article{Krömer2010,
abstract = { We derive sharp necessary conditions for weak sequential lower semicontinuity of integral functionals on Sobolev spaces, with an integrand which only depends on the gradient of a scalar field over a domain in $\{\mathbb R\}^N$. An emphasis is put on domains with infinite measure, and the integrand is allowed to assume the value $+\infty$. },
author = {Krömer, Stefan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Scalar integral functionals; weak lower semicontinuity; necessary conditions; scalar integral functionals; domains with infinite measure},
language = {eng},
month = {4},
number = {2},
pages = {457-471},
publisher = {EDP Sciences},
title = {Necessary conditions for weak lower semicontinuity on domains with infinite measure},
url = {http://eudml.org/doc/250704},
volume = {16},
year = {2010},
}

TY - JOUR
AU - Krömer, Stefan
TI - Necessary conditions for weak lower semicontinuity on domains with infinite measure
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/4//
PB - EDP Sciences
VL - 16
IS - 2
SP - 457
EP - 471
AB - We derive sharp necessary conditions for weak sequential lower semicontinuity of integral functionals on Sobolev spaces, with an integrand which only depends on the gradient of a scalar field over a domain in ${\mathbb R}^N$. An emphasis is put on domains with infinite measure, and the integrand is allowed to assume the value $+\infty$.
LA - eng
KW - Scalar integral functionals; weak lower semicontinuity; necessary conditions; scalar integral functionals; domains with infinite measure
UR - http://eudml.org/doc/250704
ER -

References

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  1. B. Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences78. Springer, Berlin etc. (1989).  Zbl0703.49001
  2. I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces, Springer Monographs in Mathematics. Springer, New York (2007).  Zbl1153.49001
  3. E. Giusti, Direct methods in the calculus of variations. World Scientific, Singapore (2003).  Zbl1028.49001
  4. W. Gustin, On the interior of the convex hull of an Euclidean set. Bull. Am. Math. Soc.53 (1947) 299–301.  Zbl0034.10502
  5. V.G. Maz'ya, Sobolev spaces. Springer-Verlag, Berlin etc. (1985).  
  6. Yu.S. Nikol'skij, Integral estimates for differentiable functions on unbounded domains. Proc. Steklov Inst. Math.170 (1987) 267–283. Translation from Tr. Mat. Inst. Steklova170 (1984) 233–247 (Russian).  

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