We obtain an integral representation for certain functionals arising in the context of optimal design and damage evolution problems under non-standard growth conditions and perimeter penalisation. Under our hypotheses, the integral representation includes a term which is absolutely continuous with respect to the Lebesgue measure and a perimeter term, but no additional singular term.
We also study some dimension reduction problems providing results for the optimal design of thin films.
E.Acerbi, G.Bouchitté and I.Fonseca, Relaxation of convex functionals: The gap problem, Ann. Inst. H. Poincaré Anal. Non Linéaire20(3) (2003), 359–390. doi:10.1016/S0294-1449(02)00017-3.
2.
G.Allaire and V.Lods, Minimizers for a double-well problem with affine boundary conditions, Proc. Royal Soc. Edinburgh129 A, no. 3 (1999), 439–466. doi:10.1017/S0308210500021454.
3.
L.Ambrosio and G.Buttazzo, An optimal design problem with perimeter penalization, Calc. Var. Partial Differential Equations1 (1993), 55–69. doi:10.1007/BF02163264.
4.
L.Ambrosio, N.Fusco and D.Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, 2000.
5.
J.-F.Babadjian, F.Iurlano and F.Rindler, Concentration versus oscillation effects in brittle damage, Comm. Pure Appl. Math. doi:10.1002/cpa.21953.
6.
J.M.Ball and F.Murat, -quasiconvexity and variational problems for multiple integrals, Journal of Functional Analysis58 (1984), 225–253. doi:10.1016/0022-1236(84)90041-7.
7.
A.C.Barroso and E.Zappale, Relaxation for optimal design problems with nonstandard growth, Appl. Math. Optim.80(2) (2019), 515–546. doi:10.1007/s00245-017-9473-6.
8.
G.Bouchitté, I.Fonseca, G.Leoni and L.Mascarenhas, A global method for relaxation in and in , Arch. Rational Mech. Anal.165 (2002), 187–242. doi:10.1007/s00205-002-0220-y.
9.
G.Bouchitté, I.Fonseca and J.Malý, The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent, Proc. Royal Soc. Edinburgh128A (1998), 463–479. doi:10.1017/S0308210500021600.
10.
G.Bouchitté, I.Fonseca and M.L.Mascarenhas, Bending moment in membrane theory, J. Elasticity73(1–3) (2004), 75–99. doi:10.1023/B:ELAS.0000029996.20973.92.
11.
A.Braides, I.Fonseca and G.Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J.49(4) (2000), 1367–1404. doi:10.1512/iumj.2000.49.1822.
12.
G.Carita and E.Zappale, 3D-2D dimensional reduction for a nonlinear optimal design problem with perimeter penalization, C. R. Math. Acad. Sci. Paris350(23–24) (2012), 1011–1016. doi:10.1016/j.crma.2012.11.005.
13.
A.Coscia and D.Mucci, Integral representation and Γ-convergence of variational integrals with -growth, ESAIM Control Optim. Calc. Var.7 (2002), 495–519. doi:10.1051/cocv:2002065.
14.
D.Cruz-Uribe and A.Fiorenza, Variable Lebesgue Spaces, Foundations and Harmonic Analysis, Birkhäuser, 2013.
15.
B.Dacorogna, Direct Methods in the Calculus of Variations, 2nd edn, Applied Mathematical Sciences, Vol. 78, Springer, 2008.
16.
G.Dal Maso, An Introduction to Γ-Convergence, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 1983.
17.
G.Dal Maso and F.Iurlano, Fracture models as Γ-limits of damage models, Commun. Pure Appl. Anal.12(4) (2013), 1657–1686.
18.
L.Diening, P.Harjulehto, P.Hästö and M.Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Vol. 2017, Springer, 2011.
19.
I.Fonseca and G.Francfort, 3D-2D asymptotic analysis of an optimal design problem for thin films, Journal für die Reine und Angewandte Mathematik505 (1998), 173–202. doi:10.1515/crll.1998.505.173.
20.
I.Fonseca and G.Leoni, Modern Methods in the Calculus of Variations: Spaces, Springer Monographs in Mathematics, Springer, New York, to appear.
21.
I.Fonseca and J.Malý, Relaxation of multiple integrals below the growth exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire14 (1997), 309–338. doi:10.1016/S0294-1449(97)80139-4.
22.
G.A.Francfort and J.-J.Marigo, Stable damage evolution in a brittle continuous medium, European J. Mech. A Solids12(2) (1993), 149–189.
23.
E.Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, Vol. 80, Birkhäuser, 1984.
24.
P.Harjulehto and P.Hästö, Orlicz Spaces and Generalized Orlicz Spaces, Lecture Notes in Mathematics, Vol. 2236, Springer, 2019.
25.
A.D.Ioffe, On lower semicontinuity of integral functionals I, SIAM Journal of Control and Optimization15 (1977), 521–538. doi:10.1137/0315035.
26.
F.Iurlano, Fracture and plastic models as Γ-limits of damage models under different regimes, Adv. Calc. Var.6(2) (2013), 165–189. doi:10.1515/acv-2011-0011.
27.
R.V.Kohn and F.H.Lin, Partial regularity for optimal design problems involving both bulk and surface energies, Chinese Ann. Math. Ser. B20(2) (1999), 137–158. doi:10.1142/S0252959999000175.
28.
R.V.Kohn and G.Strang, Optimal design and relaxation of variational problems I, Commun. Pure Appl. Math.39 (1986), 113–137. doi:10.1002/cpa.3160390107.
29.
R.V.Kohn and G.Strang, Optimal design and relaxation of variational problems II, Commun. Pure Appl. Math.39 (1986), 139–182. doi:10.1002/cpa.3160390202.
30.
R.V.Kohn and G.Strang, Optimal design and relaxation of variational problems III, Commun. Pure Appl. Math.39 (1986), 353–377. doi:10.1002/cpa.3160390305.
31.
J.Kristensen, A necessary and sufficient condition for lower semicontinuity, Nonlinear Analysis: Theory, Methods & Applications120 (2015), 43–56. doi:10.1016/j.na.2015.02.018.
32.
H.Le Dret and A.Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. (9)74(6), (1995), 549–578.
33.
A.Mielke and T.Roubíček, Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci.16(2) (2006), 177–209. doi:10.1142/S021820250600111X.
34.
G.Mingione and D.Mucci, Integral functionals and the gap problem: Sharp bounds for relaxation and energy concentration, SIAM J. Math. Anal.36(5) (2005), 1540–1579. doi:10.1137/S0036141003424113.
35.
D.Mucci, Relaxation of variational functionals with piecewise constant growth conditions, J. Convex Anal.10(2) (2003), 295–324.
36.
F.Murat and L.Tartar, Calcul des variations et homogénéisation, Homogenization methods: theory and applications in physics, Bréau-Sans-Nappe, 1983, Vol. 57, Eyrolles, Paris, 1985, pp. 319–369.
37.
P.Pedregal, Jensen’s inequality in the calculus of variations, Differential and Integral Equations1 (1994), 57–72.
38.
K.Pham, J.-J.Marigo and C.Maurini, The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models, J. Mech. Phys. Solids59(6) (2011), 1163–1190. doi:10.1016/j.jmps.2011.03.010.
39.
T.Schmidt, Strict interior approximation of sets of finite perimeter and functions of bounded variation, Proc. Amer. Math. Soc.143(5) (2015), 2069–2084. doi:10.1090/S0002-9939-2014-12381-1.
40.
V.V.Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, Journal of Mathematical Sciences173(5) (2011), 463–570. doi:10.1007/s10958-011-0260-7.
