Lower semicontinuity of parametric integrals

Silverman, Edward

doi:10.1090/S0002-9947-1973-0310744-6

Lower semicontinuity of parametric integrals
HTML articles powered by AMS MathViewer

by Edward Silverman

Trans. Amer. Math. Soc. 175 (1973), 499-508

DOI: https://doi.org/10.1090/S0002-9947-1973-0310744-6

PDF | Request permission

Abstract:

It has been known for a long time that the usual two-dimensional parametric integrals in three-space are lower semicontinuous with respect to uniform convergence. In an earlier paper we saw that an easy argument extends this result to all parametric integrals generated by simply-convex integrands, with no restrictions on the dimension of the surfaces or the containing space. By using these techniques again, and generalizing to surfaces a result concerning convergent sequences of closed curves we show that a parametric integral generated by a parametric integrand which is convex in the Jacobians is lower semicontinuous with respect to uniform convergence provided all of the functions lie in a bounded subset of the Sobolev space $H_s^1$ where $s + 1$ exceeds the dimension of the parametric integral.

References

Topologie

Lamberto Cesari, Condizioni necessarie per la semicontinuità degli integrali sopra una superficie in forma parametrica, Ann. Mat. Pura Appl. (4) 29 (1949), 199–224 (Italian). MR 36294, DOI 10.1007/BF02413927
Lamberto Cesari, An existence theorem of calculus of variations for integrals on parametric surfaces, Amer. J. Math. 74 (1952), 265–295. MR 50187, DOI 10.2307/2371993

Surface area

17

John M. Danskin Jr., On the existence of minimizing surfaces in parametric double integral problems of the calculus of variations, Riv. Mat. Univ. Parma 3 (1952), 43–63. MR 50186
Herbert Federer, Essential multiplicity and Lebesgue area, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 611–616. MR 27837, DOI 10.1073/pnas.34.12.611
Casper Goffman and William P. Ziemer, Higher dimensional mappings for which the area formula holds, Ann. of Math. (2) 92 (1970), 482–488. MR 271283, DOI 10.2307/1970629
L. H. Loomis and H. Whitney, An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc. 55 (1949), 961–962. MR 0031538, DOI 10.1090/S0002-9904-1949-09320-5
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., The parametric variational problem for double integrals, Comm. Pure Appl. Math. 14 (1961), 569–575. MR 131790, DOI 10.1002/cpa.3160140328
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
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
A. G. Sigalov, Two-dimensional problems of the calculus of variations, Uspehi Matem. Nauk (N.S.) 6 (1951), no. 2(42), 16–101 (Russian). MR 0043400
Edward Silverman, A problem of least area, Pacific J. Math. 14 (1964), 309–331. MR 170996
Edward Silverman, Simple areas, Pacific J. Math. 15 (1965), 299–303. MR 177321
Edward Silverman, A sufficient condition for the lower semicontinuity of parametric integrals, Trans. Amer. Math. Soc. 167 (1972), 465–469. MR 296785, DOI 10.1090/S0002-9947-1972-0296785-5