Abstract
We present a survey of results for different kinds of variational problems where weak geometric structures intervene as a common feature. A unified method is adopted, which has been developed in several papers during recent years; it is based on the use of measures, fitted out with suitable tangential properties and functional spaces. Some of the results are new, and their proof will be given in a forthcoming paper.
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References
W. K. Allard, First variation of a varifold,Annals of Math., 95 (1972), 417–491.
H. Attouch and G. Buttazzo, Homogenization of reinforced periodic one-codimensional structures, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (4) (1987), 465–484.
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press (2000).
A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam (1978).
G. Bouchitté, G. Bellettini and I. Fragalà, BV functions with respect to a measure and relaxation of metric integral functionals, J. Convex Anal., 6 (2) (1999), 349–366.
G. Bouchitté and G. Buttazzo, Characterization of optimal shapes and masses through Monge-Kantorovich equation, J. Eur. Math. Soc. 3 (2001), 139–168.
G. Bouchitté, G. Buttazzo and I. Fragalà, Mean curvature of a measure and related variational problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., XXV (4) (1997), 179–196.
G. Bouchitté, G. Buttazzo and I. Fragalà, Convergence of Sobolev spaces on varying manifolds, to appear on J. Geom. Anal.
G. Bouchitté, G. Buttazzo and I. Fragalà, Bounds on the effective coefficients of homogenized low dimensional structures, Preprint, (2001).
G. Bouchitté, G. Buttazzo and P. Seppecher, Energies with respect to a measure and applications to low dimensional structures, Calc. Var. Partial Differential Equations 5 (1997), 37–54.
G. Bouchitté, G. Buttazzo and P. Seppecher, Optimization solutions via MongeKantorovich equation, C. R. Acad. Sci. Paris, Série I 324 (1997), 1185–1191.
G. Bouchitté and I. Fragalà, Homogenization of thin structures by two-scale method with respect to measures, SIAM J. Math. Anal., 32 (6) (2001), 1198–1226.
G. Bouchitté and I. Fragalà, Homogenization of elastic thin structures: a measure-fattening approach, Preprint, (2000).
G. Bouchitté and I. Fragalà, Second order energies on thin structures: variational theory and non-local effects, Paper in preparation.
G. Bouchitté and M. Valadier, Integral representation of convex functionals on a space of measures, J. Funct. Anal. 80 (1988), 398–420.
G. Buttazzo, Semicontinuity, Relaxation, and Integral Representation in the Calculus of Variations, Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow (1989).
P. G. Ciarlet and H. Le Dret, Justification de la condition aux limites d’encastrement d’une plaque par une méthode asymptotique,C. R. Acad. Sci. Paris. Sér. I Math., 307 (20) (1988), 1015–1018.
D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures. Applied Mathematical Sciences 136, Springer Verlag, Berlin-Heidelberg-New York (1999).
B. Dacorogna, Direct Methods in the Calculus of Variations, Applied Mathematical Sciences 78, Springer-Verlag, Berlin-Heidelberg-New York (1988).
G. Dal Maso, An Introduction to Γ-convergence, Birkhäuser, Boston (1993).
H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin-Heidelberg-New York (1969).
H. Federer and W. Fleming, Normal and integral currents, Annals of Math., 72 (1960), 458–520.
I. Fragalà, Lower semi continuity of μ-quasiconvex integrals, Preprint, (2000).
I. Fragalà and C. Mantegazza, On some notions of tangent space to a measure, Proc. Roy. Soc Edinburgh, 129A (1999), 331–342.
M. Giaquinta, G. Modica and J. Soucek, Cartesian Currents in the Calculus of Variations, Springer-Verlag, Berlin-Heidelberg-New-York (1998).
V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin-Heidelberg-New York (1994).
H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three dimensional elasticity, J. Math. Pures Appl., 74 (1995), 549–578.
D. Percivale, The variational methods for tensile structures, Preprint, (1991).
D. Preiss, Geometry of measures on \( {\mathbb{R}^n} \) : distribution, rectifiability and densities, Annals of Math., 125 (1987), 573–643.
L. Simon, Lectures on Geometric Measure Theory, Proc. Centre for Math. Anal., Australian Nat. Univ., 3 (1983).
M. Valadier, Multiapplications mesurables à valeurs convexes compactes, J. Math. Pures Appl., 50 (1971), 265–297.
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Bouchitté, G., Fragalà, I. (2002). Variational Theory of Weak Geometric Structures: The Measure Method and Its Applications. In: dal Maso, G., Tomarelli, F. (eds) Variational Methods for Discontinuous Structures. Progress in Nonlinear Differential Equations and Their Applications, vol 51. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8193-7_3
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DOI: https://doi.org/10.1007/978-3-0348-8193-7_3
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9470-8
Online ISBN: 978-3-0348-8193-7
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