Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-28T12:50:06.044Z Has data issue: false hasContentIssue false

The interaction between bulk energy and surface energy in multiple integrals

Published online by Cambridge University Press:  14 November 2011

Andrea Braides
Affiliation:
S.I.S.S.A., via Beirut 2/4,1-34014 Trieste, Italy
Alessandra Coscia
Affiliation:
S.I.S.S.A., via Beirut 2/4,1-34014 Trieste, Italy

Abstract

This paper is devoted to the study of integral functional denned on the space SBV(Ω ℝk) of vector-valued special functions with bounded variation on the open set Ω⊂ℝn, of the form

We suppose only that f is finite at one point, and that g is positively 1-homogeneous and locally bounded on the sets ℝkvm, where {v1,…, vR} ⊂ Sn−1 is a basis of ℝn. We prove that the lower semicontinuous envelope of F in the L1(Ω;ℝk)-topology is finite and with linear growth on the whole BV(Ω;ℝk), and that it admits the integral representation

A formula for ϕ is given, which takes into account the interaction between the bulk energy density f and the surface energy density g.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Acerbi, E. and Fusco, N.. Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86(1984), 125145.Google Scholar
2Alberti, G.. Rank one property for derivatives of functions with bounded variation. Proc. Roy. Soc. Edinburgh Sect. A (to appear).Google Scholar
3Ambrosio, L.. A compactness theorem for a special class of functions of bounded variation. Boll. Un. Mat. Ital. B 3 (1989), 857881.Google Scholar
4Ambrosio, L.. Existence theory for a new class of variational problems. Arch. Rational Mech. Anal. 111 (1990), 291322.Google Scholar
5Ambrosio, L.. On the lower semicontinuity of quasi-convex integrals in SBV(Ω; ርk) (preprint, University of Pisa, 1993).Google Scholar
6Ambrosio, L. and Braides, A.. Functionals defined on partitions of sets of finite perimeter, I: integral representation and Г-convergence. J. Math. Pures Appl. 69 (1990), 285305.Google Scholar
7Ambrosio, L. and Braides, A.. Functionals defined on partitions of sets of finite perimeter, II: Semicontinuity, relaxation and homogenization. J. Math. Pures Appl. 69 (1990), 307333.Google Scholar
8Ambrosio, L. and Dal Maso, G.. On the relaxation in BV(Ω; ℝmm) of quasi-convex integrals. J. Funct. Anal. 109(1992), 7697.Google Scholar
9Ambrosio, L. and Pallara, D.. Integral representation of relaxed functionals on BV(ℝn; ℝk) and polyhedral approximation. Indiana Univ. Math. J. (to appear).Google Scholar
10Bouchitte, G. and Buttazzo, G.. Relaxation for a class of nonconvex functionals defined on measures. Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear).Google Scholar
11Bouchitte, G., Braides, A. and Buttazzo, G.. Relaxation results for some free discontinuity problems (preprint, SISSA, Trieste, 1992).Google Scholar
12Braides, A. and Coscia, A.. A singular perturbation approach to problems in fracture mechanics. Math. Mod. Meth. Appl. Sci. 3 (1993) (to appear).Google Scholar
13Buttazzo, G.. Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Research Notes in Mathematics 207 (Harlow: Longman, 1989).Google Scholar
14Buttazzo, G. and Dal Maso, G.. Integral representation and relaxation of local functionals. Nonlinear Anal. 6(1985), 515532.Google Scholar
15Congedo, G. and Tamanini, I.. On the existence of solutions to a problem in multidimensional segmentation. Ann. Inst. H. Poincaré. Anal. Non Linéaire 2 (1991), 175195.Google Scholar
16Dacorogna, B.. Weak Continuity and Weak Lower Semicontinuity of Multiple Integrals, Lecture Notes in Mathematics 992 (Berlin: Springer, 1989).Google Scholar
17Dacorogna, B.. Direct Methods in the Calculus of Variations (Berlin: Springer, 1989).Google Scholar
18Dal Maso, G.. An Introduction to Г-convergence (Boston: Birkhäuser, 1993).Google Scholar
19De Giorgi, E. and Ambrosio, L.. Un nuovo tipo di funzionale del calcolo delle variazioni. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82 (1988), 199210.Google Scholar
20De Giorgi, E., Colombini, F. and Piccinini, L. C.. Frontiere orientate di misura minima e questioni collegate (Quaderno Scuola Normale di Pisa, Pisa, 1972).Google Scholar
21De, E. Giorgi and Letta, G.. Une notion generate de convergence faible pour des fonctions croissantes d'ensemble. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1977), 6199.Google Scholar
22Federer, H.. Geometric Measure Theory (New York: Springer, 1969).Google Scholar
23Fonseca, I.. Lower semicontinuity of surface energies. Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), 99115.Google Scholar
24Fonseca, I. and Müller, S.. Relaxation of quasiconvex functionals in BV(Ω; ℝp) for integrands f(x, u ∇ u) (preprint, Carnegie Mellon University, Pittsburgh, 1991).Google Scholar
25Giusti, E.. Minimal Surfaces and Functions of Bounded Variation. (Basel: Birkhäuser, 1983).Google Scholar
26Morrey, C. B.. Multiple Integrals in the Calculus of Variations (Berlin: Springer, 1966).Google Scholar
27Morrey, C. B.. Quasiconvexity and the semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 2553.Google Scholar
28Müller, S.. On quasiconvex functions which are homogeneous of degree 1. Indiana Univ. Math. J. 41 (1992), 295301.Google Scholar
29Šverák, V.. Quasiconvex functions with subquadratic growth, Proc. Roy. Soc. London Ser. A 433 (1991), 733735.Google Scholar
30Šverak, V.. Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), 185189.Google Scholar
31Vol'pert, A. I.. Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics (Dordrecht: Martinus Nijhoff, 1985).Google Scholar
32Ziemer, W. P.. Weakly Differentiable Functions (Berlin: Springer, 1989).Google Scholar
33Zhang, K.. A construction of quasiconvex functions with linear growth at infinity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 19 (1992), 313326.Google Scholar