Bouchitté, Guy; Buttazzo, Giuseppe; Fragalà, Ilaria Mean curvature of a measure and related variational problems. (English) Zbl 1015.49015 Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 25, No. 1-2, 179-196 (1997). Summary: We introduce the notion of generalized mean curvature of a measure. We then focus attention on functionals depending on curvatures, investigating their weak lower semicontinuity. A crucial role in this study is played by the dimension of the tangent spaces to a measure. Cited in 8 Documents MSC: 49J45 Methods involving semicontinuity and convergence; relaxation 49Q20 Variational problems in a geometric measure-theoretic setting Keywords:integral functional; weak convergence; tangential derivatives; varifolds; tangent space; generalized mean curvature of a measure; weak lower semicontinuity; dimension × Cite Format Result Cite Review PDF Full Text: Numdam EuDML References: [1] G. Bouchitté - G. Buttazzo - P. Seppecher , Energies with respect to a measure and applications to low dimensional structures , Calc. Var. 5 ( 1997 ), 37 - 54 . MR 1424348 | Zbl 0934.49011 · Zbl 0934.49011 · doi:10.1007/s005260050058 [2] G. Bouchitté - G. Buttazzo - P. Seppecher , Shape optimization solutions via Monge-Kantorovich equation , C. R. Acad. Sci. Paris I-324 ( 1997 ), 1185 - 1191 . MR 1451945 | Zbl 0884.49023 · Zbl 0884.49023 · doi:10.1016/S0764-4442(97)87909-8 [3] L. Euler , Additamentum I de curvis elasticis, methodus inveniendi lineas curvas maximi minimve proprietate gaudentes , Opera omnia , Lausanne I-24 ( 1744 ), 231 - 297 . [4] H. Federer , Geometric measure theory , Springer Verlag , Berlin , 1969 . MR 257325 | Zbl 0176.00801 · Zbl 0176.00801 [5] I. Fragalà - C. Mantegazza , On some notions of tangent space to a measure , preprint del Dipartimento di Matematica dell’ Universita di Pisa ( 1997 ). MR 1686704 [6] A.E.H. Love , A treatise on the mathematical theory of elasticity , Dover , New York , 1944 . MR 10851 | Zbl 0063.03651 · Zbl 0063.03651 [7] P. Mattila , Geometry of sets and measures in Euclidean Spaces , Cambridge Univ. Press , London and New York , 1995 . MR 1333890 | Zbl 0819.28004 · Zbl 0819.28004 [8] D. Preiss , Geometry of measures on Rn: distribution, rectifiability and densities , Ann. Math. 125 ( 1987 ), 573 - 643 . MR 890162 | Zbl 0627.28008 · Zbl 0627.28008 · doi:10.2307/1971410 [9] Y.G. Reshetnyak , Weak convergence of completely additive vector measures on a set , Sibirsk. Mat. Zh. 9 ( 1968 ), 1386 - 1394 . MR 240274 | Zbl 0169.18301 · Zbl 0169.18301 [10] L. Simon , Lectures on geometric measure theory , Proc. C. M. A. 3 , Australian Natl. U. Canberra , 1983 . MR 756417 | Zbl 0546.49019 · Zbl 0546.49019 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.