A sufficient condition for the lower semicontinuity of parametric integrals

Silverman, Edward

doi:10.1090/S0002-9947-1972-0296785-5

A sufficient condition for the lower semicontinuity of parametric integrals
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by Edward Silverman PDF

Trans. Amer. Math. Soc. 167 (1972), 465-469 Request permission

Abstract:

We use simple convex functions and standard techniques in area theory to treat Morrey’s extension of McShane’s lower semicontinuity theorem for parametric integrals. This enables us to eliminate some technical hypotheses, simplify the proof and obtain a more general result.

References

E. J. McShane, On the semi-continuity of double integrals in the calculus of variations, Ann. of Math. (2) 33 (1932), no. 3, 460–484. MR 1503070, DOI 10.2307/1968529
Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511, DOI 10.1007/978-3-540-69952-1
T. Rado and P. V. Reichelderfer, Continuous transformations in analysis. With an introduction to algebraic topology, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXXV, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955. MR 0079620
Edward Silverman, A problem of least area, Pacific J. Math. 14 (1964), 309–331. MR 170996, DOI 10.2140/pjm.1964.14.309
Edward Silverman, Simple areas, Pacific J. Math. 15 (1965), 299–303. MR 177321, DOI 10.2140/pjm.1965.15.299