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Lower semicontinuity in BV of quasiconvex integrals withsubquadratic growth

Published online by Cambridge University Press:  14 March 2013

Parth Soneji
Parth Soneji*
Affiliation:
Mathematical Institute, University of Oxford, 24-29 St Giles’ Oxford, OX1 3LB, UK. [email protected]; [email protected]
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Abstract

A lower semicontinuity result in BV is obtained for quasiconvexintegrals with subquadratic growth. The key steps in this proof involve obtainingboundedness properties for an extension operator, and a precise blow-up technique thatuses fine properties of Sobolev maps. A similar result is obtained by Kristensen in[Calc. Var. Partial Differ. Equ. 7 (1998) 249–261], wherethere are weaker asssumptions on convergence but the integral needs to satisfy a strongergrowth condition.

Keywords

Lower semicontinuityquasiconvex integralsfunctions of bounded variation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 2 , April 2013 , pp. 555 - 573
Copyright
© EDP Sciences, SMAI, 2013

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References

Acerbi, E. and Dal Maso, G., New lower semicontinuity results for polyconvex integrals. Calc. Var. Partial Differ. Equ. 2 (1994) 329–371. Google Scholar
Ambrosio, L. and Dal Maso, G., On the relaxation in BV(Ω;Rm) of quasi-convex integrals. J. Funct. Anal. 109 (1992) 76–97. Google Scholar
L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000).
Ball, J.M. and Murat, F., W 1, p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225–253. Google Scholar
Bouchitté, G., Fonseca, I. and Malý, J., The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent. Proc. of R. Soc. Edinburgh Sect. A 128 (1998) 463–479. Google Scholar
Carbone, L. and De Arcangelis, R., Further results on Γ-convergence and lower semicontinuity of integral functionals depending on vector-valued functions. Ric. Mat. 39 (1990) 99–129. Google Scholar
M. Carozza, J. Kristensen and A. Passarelli di Napoli, Lower semicontinuity in a borderline case. Preprint (2008).
Černý, R., Relaxation of an area-like functional for the function \hbox{$\frac x{\vert x\vert }$}x|x| . Calc. Var. Partial Differ. Equ. 28 (2007) 203–216. Google Scholar
B. Dacorogna, Direct methods in the calculus of variations. Appl. Math. Sci. 78 (1989).
Dacorogna, B., Fonseca, I., Malý, J. and Trivisa, K., Manifold constrained variational problems. Calc. Var. Partial Differ. Equ. 9 (1999) 185–206. Google Scholar
Fonseca, I. and Malý, , J. Relaxation of multiple integrals below the growth exponent. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14 (1997) 309–338. Google Scholar
Fonseca, I. and Malý, J., From Jacobian to Hessian: distributional form and relaxation. Riv. Mat. Univ. Parma 4 (2005) 45–74. Google Scholar
Fonseca, I. and Marcellini, P., Relaxation of multiple integrals in subcritical Sobolev spaces. J. Geom. Anal. 7 (1997) 57–81. Google Scholar
Fonseca, I. and Müller, S., Quasi-convex integrands and lower semicontinuity in L 1. SIAM J. Math. Anal. 23 (1992) 1081–1098. Google Scholar
Fonseca, I. and Müller, S., Relaxation of quasiconvex functionals in BV(Ω, Rp) for integrands f(x, u, ∇u). Arch. Ration. Mech. Anal. 123 (1993) 1–49. Google Scholar
Fonseca, , I. Leoni, G. and Müller, , S. 𝒜 quasiconvexity: weak-star convergence and the gap. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21 (2004) 209–236. Google Scholar
Greco, L., Iwaniec, T. and Moscariello, G., Limits of the improved integrability of the volume forms. Indiana Univ. Math. J. 44 (1995) 305–339. Google Scholar
T. Iwaniec and G. Martin, Geometric function theory and non-linear analysis. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2001).
Kristensen, , J. Lower semicontinuity of quasi-convex integrals in BV(Ω;Rm). Calc. Var. Partial Differ. Equ. 7 (1998) 249–261. Google Scholar
Malý, J., Weak lower semicontinuity of polyconvex integrals. Proc. of R. Soc. Edinburgh Sect. A 123 (1993) 681–691. Google Scholar
J. Malý, Weak lower semicontinuity of polyconvex and quasiconvex integrals. Preprint (1993).
Malý, J., Lower semicontinuity of quasiconvex integrals. Manusc. Math. 85 (1994) 419–428. Google Scholar
Marcellini, , P. On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3 (1986) 391–409. Google Scholar
Meyers, N.G., Quasi-convexity and lower semi-continuity of multiple variational integrals of any order. Trans. Amer. Math. Soc. 119 (1965) 125–149. Google Scholar
Müller, S., On quasiconvex functions which are homogeneous of degree 1. Indiana Univ. Math. J. 41 (1992) 295–301. Google Scholar
Rindler, F., Lower semicontinuity and Young measures in BV without Alberti’s rank-one theorem. Adv. Calc. Var. 5 (2012) 127–159. Google Scholar
W. Rudin, Real and complex analysis, 3rd edition, McGraw-Hill Book Co., New York (1987).
Serrin, J., A new definition of the integral for nonparametric problems in the calculus of variations. Acta Math. 102 (1959) 23–32. Google Scholar
Serrin, J., On the definition and properties of certain variational integrals. Trans. Amer. Math. Soc. 101 (1961) 139–167. Google Scholar
Šverák, V., Quasiconvex functions with subquadratic growth. Proc. of R. Soc. London A 433 (1991) 723–725. Google Scholar
Zhang, K., A construction of quasiconvex functions with linear growth at infinity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992) 313–326. Google Scholar
W.P. Ziemer, Weakly differentiable functions, Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics 120 (1989).