Local minimizers of integral functionals are global minimizers
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- by E. Giner PDF
- Proc. Amer. Math. Soc. 123 (1995), 755-757 Request permission
Abstract:
We show that local minimizers of integral functionals associated with a measurable integrand $f:\Omega \times E \to \mathbb {R} \cup \{ \pm \infty \}$ are actually global minimizers. Here $(\Omega , \mathcal {S},\mu )$ is a measured space with an atomless $\sigma$-finite positive measure, E is a separable Banach space, and the integral functional ${I_f}(x) = \smallint _\Omega ^ \ast f(\omega ,x(\omega ))d\mu$ is defined on ${L_p}(\Omega ,E)$ or, more generally, on some decomposable set of measurable mappings x from $\Omega$ into E.References
- C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310, DOI 10.1007/BFb0087685
- Fumio Hiai, Representation of additive functionals on vector-valued normed Köthe spaces, Kodai Math. J. 2 (1979), no. 3, 300–313. MR 553237
- Joram Lindenstrauss, A short proof of Liapounoff’s convexity theorem, J. Math. Mech. 15 (1966), 971–972. MR 0207941
- R. Tyrrell Rockafellar, Integral functionals, normal integrands and measurable selections, Nonlinear operators and the calculus of variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975) Lecture Notes in Math., Vol. 543, Springer, Berlin, 1976, pp. 157–207. MR 0512209
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 755-757
- MSC: Primary 49J10; Secondary 28B05, 49K10
- DOI: https://doi.org/10.1090/S0002-9939-1995-1254839-1
- MathSciNet review: 1254839