Lower semicontinuity in BV of quasiconvex integrals with subquadratic growth
ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 2, pp. 555-573.

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

DOI: 10.1051/cocv/2012021
Classification: 49J45
Keywords: lower semicontinuity, quasiconvex integrals, functions of bounded variation
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     title = {Lower semicontinuity in {BV} of quasiconvex integrals with subquadratic growth},
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Soneji, Parth. Lower semicontinuity in BV of quasiconvex integrals with subquadratic growth. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 2, pp. 555-573. doi : 10.1051/cocv/2012021. http://www.numdam.org/articles/10.1051/cocv/2012021/

[1] E. Acerbi and G. Dal Maso, New lower semicontinuity results for polyconvex integrals. Calc. Var. Partial Differ. Equ. 2 (1994) 329-371. | MR | Zbl

[2] L. Ambrosio and G. Dal Maso, On the relaxation in BV(Ω;Rm) of quasi-convex integrals. J. Funct. Anal. 109 (1992) 76-97. | MR | Zbl

[3] 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). | MR | Zbl

[4] J.M. Ball and F. Murat, W1, p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225-253. | MR | Zbl

[5] G. Bouchitté, I. Fonseca and J. Malý, 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. | MR | Zbl

[6] L. Carbone and R. De Arcangelis, Further results on Γ-convergence and lower semicontinuity of integral functionals depending on vector-valued functions. Ric. Mat. 39 (1990) 99-129. | MR | Zbl

[7] M. Carozza, J. Kristensen and A. Passarelli Di Napoli, Lower semicontinuity in a borderline case. Preprint (2008).

[8] R. Černý, Relaxation of an area-like functional for the function | MR | Zbl

[9] B. Dacorogna, Direct methods in the calculus of variations. Appl. Math. Sci. 78 (1989). | MR | Zbl

[10] B. Dacorogna, I. Fonseca, J. Malý and K. Trivisa, Manifold constrained variational problems. Calc. Var. Partial Differ. Equ. 9 (1999) 185-206. | MR | Zbl

[11] I. Fonseca and J. Malý, Relaxation of multiple integrals below the growth exponent. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14 (1997) 309-338. | Numdam | MR | Zbl

[12] I. Fonseca and J. Malý, From Jacobian to Hessian: distributional form and relaxation. Riv. Mat. Univ. Parma 4 (2005) 45-74. | MR | Zbl

[13] I. Fonseca and P. Marcellini, Relaxation of multiple integrals in subcritical Sobolev spaces. J. Geom. Anal. 7 (1997) 57-81. | MR | Zbl

[14] I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in L1. SIAM J. Math. Anal. 23 (1992) 1081-1098. | MR | Zbl

[15] I. Fonseca and S. Müller, Relaxation of quasiconvex functionals in BV(Ω, Rp) for integrands f(x, u, ∇u). Arch. Ration. Mech. Anal. 123 (1993) 1-49. | MR | Zbl

[16] I. Fonseca, G. Leoni and S. Müller, 𝒜 quasiconvexity: weak-star convergence and the gap. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21 (2004) 209-236. | Numdam | Zbl

[17] L. Greco, T. Iwaniec and G. Moscariello, Limits of the improved integrability of the volume forms. Indiana Univ. Math. J. 44 (1995) 305-339. | MR | Zbl

[18] T. Iwaniec and G. Martin, Geometric function theory and non-linear analysis. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2001). | MR | Zbl

[19] J. Kristensen, Lower semicontinuity of quasi-convex integrals in BV(Ω;Rm). Calc. Var. Partial Differ. Equ. 7 (1998) 249-261. | MR | Zbl

[20] J. Malý, Weak lower semicontinuity of polyconvex integrals. Proc. of R. Soc. Edinburgh Sect. A 123 (1993) 681-691. | MR | Zbl

[21] J. Malý, Weak lower semicontinuity of polyconvex and quasiconvex integrals. Preprint (1993). | MR

[22] J. Malý, Lower semicontinuity of quasiconvex integrals. Manusc. Math. 85 (1994) 419-428. | MR | Zbl

[23] P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3 (1986) 391-409. | Numdam | MR | Zbl

[24] N.G. Meyers, Quasi-convexity and lower semi-continuity of multiple variational integrals of any order. Trans. Amer. Math. Soc. 119 (1965) 125-149. | MR | Zbl

[25] S. Müller, On quasiconvex functions which are homogeneous of degree 1. Indiana Univ. Math. J. 41 (1992) 295-301. | MR | Zbl

[26] F. Rindler, Lower semicontinuity and Young measures in BV without Alberti's rank-one theorem. Adv. Calc. Var. 5 (2012) 127-159. | MR | Zbl

[27] W. Rudin, Real and complex analysis, 3rd edition, McGraw-Hill Book Co., New York (1987). | MR | Zbl

[28] J. Serrin, A new definition of the integral for nonparametric problems in the calculus of variations. Acta Math. 102 (1959) 23-32. | MR | Zbl

[29] J. Serrin, On the definition and properties of certain variational integrals. Trans. Amer. Math. Soc. 101 (1961) 139-167. | MR | Zbl

[30] V. Šverák, Quasiconvex functions with subquadratic growth. Proc. of R. Soc. London A 433 (1991) 723-725. | MR | Zbl

[31] K. Zhang, A construction of quasiconvex functions with linear growth at infinity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992) 313-326. | Numdam | MR | Zbl

[32] W.P. Ziemer, Weakly differentiable functions, Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics 120 (1989). | MR | Zbl

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