Local minimizers of integral functionals are global minimizers

Author:
E. Giner

Journal:
Proc. Amer. Math. Soc. **123** (1995), 755-757

MSC:
Primary 49J10; Secondary 28B05, 49K10

MathSciNet review:
1254839

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Abstract: We show that local minimizers of integral functionals associated with a measurable integrand are actually global minimizers. Here is a measured space with an atomless -finite positive measure, *E* is a separable Banach space, and the integral functional is defined on or, more generally, on some decomposable set of measurable mappings *x* from into *E*.

**[1]**C. Castaing and M. Valadier,*Convex analysis and measurable multifunctions*, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR**0467310****[2]**Fumio Hiai,*Representation of additive functionals on vector-valued normed Köthe spaces*, Kodai Math. J.**2**(1979), no. 3, 300–313. MR**553237****[3]**Joram Lindenstrauss,*A short proof of Liapounoff’s convexity theorem*, J. Math. Mech.**15**(1966), 971–972. MR**0207941****[4]**R. Tyrrell Rockafellar,*Integral functionals, normal integrands and measurable selections*, Nonlinear operators and the calculus of variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975) Springer, Berlin, 1976, pp. 157–207. Lecture Notes in Math., Vol. 543. MR**0512209**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1995-1254839-1

Keywords:
Integral functional,
measurable integrand,
global optimization

Article copyright:
© Copyright 1995
American Mathematical Society