Relaxation in BV of integrals with superlinear growth

Soneji, Parth

ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 4, pp. 1078-1122

We study properties of the functional

$${ℱ}_{\text{loc}}(u,\Omega ):=\underset{\left(u_j\right)}{inf}\left\{\underset{j\to \infty }{lim\; inf}{\int }_{\Omega }f\left(\nabla u_j\right)dx\left|\begin{array}{cc}& \left(u_j\right)\subset W_{\mathrm{loc}}^{1,r}\left(\Omega ,{ℝ}^N\right)\hfill \\ & u_j\stackrel{*}{\rightharpoonup }u\text{in}BV\left(\Omega ,{ℝ}^N\right)\hfill \end{array}\right.\right\},$$

where $u\in BV(\Omega ;{ℝ}^N)$, and $f:{ℝ}^{N\times n}\to ℝ$ is continuous and satisfies $0\le f\left(\xi \right)\le L(1+|\xi {|}^r)$ . For $r\in [1,2)$ , assuming $f$ has linear growth in certain rank-one directions, we combine a result of [A. Braides and A. Coscia, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 737-756] with a new technique involving mollification to prove an upper bound for ${ℱ}_{\text{loc}}$ . Then, for $r\in [1,\frac{n}{n-1})$ , we prove that ${ℱ}_{\text{loc}}$ satisfies the lower bound

$${ℱ}_{\text{loc}}(u,\Omega )\ge {\int }_{\Omega }f\left(\nabla u\left(x\right)\right)dx+{\int }_{\Omega }{f}^{\infty }\left(\frac{D^{s}u}{|D^{s}u|}\right)\left|D^{s}u\right|,$$

provided $f$ is quasiconvex, and the recession function $f^{\infty }$ (defined as $f^{\infty }\left(\xi \right):={\overline{lim}}_{t\to \infty }f\left(t\xi \right)/t$ is assumed to be finite in certain rank-one directions. The proof of this result involves adapting work by [Kristensen, Calc. Var. Partial Differ. Eqs. 7 (1998) 249-261], and [Ambrosio and Dal Maso, J. Funct. Anal. 109 (1992) 76-97], and applying a non-standard blow-up technique that exploits fine properties of BV maps. It also makes use of the fact that ${ℱ}_{\text{loc}}$ has a measure representation, which is proved in the appendix using a method of [Fonseca and Malý, Annal. Inst. Henri Poincaré Anal. Non Linéaire 14 (1997) 309-338].

Classification : 49J45, 26B30
Keywords: quasiconvexity, lower semicontinuity, relaxation, BV

[1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86 (1984) 125-145. | Zbl | MR

[2] G. Alberti, Rank one property for derivatives of functions with bounded variation. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 239-274. | Zbl | MR

[3] G. Alberti and L. Ambrosio, A geometrical approach to monotone functions in Rn, Math. Z. 230 (1999) 259-316. | Zbl | MR

[4] M. Amar and V. De Cicco, Quasi-polyhedral approximation of BV-functions. Ric. Mat. 54 (2005) 485-490 (2006). | Zbl | MR

[5] L. Ambrosio, A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. B 3 (1989) 857-881. | Zbl | MR

[6] L. Ambrosio, Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal. 111 (1990) 291-322. | Zbl | MR

[7] L. Ambrosio, On the lower semicontinuity of quasiconvex integrals in SBV(Ω,Rk). Nonlinear Anal. 23 (1994) 405-425. | Zbl | MR

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

[9] L. Ambrosio, N. Fusco and J. Hutchinson, Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford-Shah functional. Calc. Var. Partial Differ. Eq. 16 (2003) 187-215. | Zbl | MR

[10] L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxf. Math. Monogr. The Clarendon Press Oxford University Press, New York (2000). | Zbl | MR

[11] L. Ambrosio, S. Mortola, and V. Tortorelli, Functionals with linear growth defined on vector valued BV functions. J. Math. Pures Appl. 70 (1991) 269-323. | Zbl | MR

[12] L. Ambrosio and D. Pallara, Integral representations of relaxed functionals on BV(Rn,Rk) and polyhedral approximation. Indiana Univ. Math. J. 42 (1993) 295-321. | Zbl | MR

[13] P. Aviles and Y. Giga, Variational integrals on mappings of bounded variation and their lower semicontinuity. Arch. Ration. Mech. Anal. 115 (1991) 201-255. | Zbl | MR

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

[15] G. Bouchitté, I. Fonseca, and J. Malý, The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent. Proc. Roy. Soc. Edinburgh Sect. A 128 (1998) 463-479. | Zbl | MR

[16] A. Braides and A. Coscia, The interaction between bulk energy and surface energy in multiple integrals. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 737-756. | Zbl | MR

[17] G. Buttazzo, Semicontinuity, relaxation and integral representation in the calculus of variations. Pitman Res. Notes in Math. Ser., vol. 207. Longman Scientific & Technical, Harlow (1989). | Zbl | MR

[18] 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. | Zbl | MR

[19] G. Dal Maso, Integral representation on BV(Ω) of Γ-limits of variational integrals. Manuscr. Math. 30 (1979/80) 387-416. | Zbl | MR

[20] E. De Giorgi and L. Ambrosio, New functionals in the calculus of variations. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82 (1988) 199-210 (1989). | Zbl | MR

[21] E. De Giorgi, F. Colombini, and L. Piccinini, Frontiere orientate di misura minima e questioni collegate. Scuola Normale Superiore, Pisa (1972). | Zbl | MR

[22] L. Evans and R. Gariepy, Measure theory and fine properties of functions. Stud.Adv. Math. CRC Press, Boca Raton, FL (1992). | Zbl | MR

[23] I. Fonseca, Lower semicontinuity of surface energies. Proc. Roy. Soc. Edinburgh Sect. A 120 (1992) 99-115. | Zbl | MR

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

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

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

[27] 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. | Zbl | MR

[28] I. Fonseca and P. Rybka, Relaxation of multiple integrals in the space BV(Ω,Rp). Proc. Roy. Soc. Edinburgh Sect. A 121 (1992) 321-348. | Zbl | MR

[29] C. Goffman and J. Serrin, Sublinear functions of measures and variational integrals. Duke Math. J. 31 (1964) 159-178. | Zbl | MR

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

[31] C. Larsen, Quasiconvexification in W1,1 and optimal jump microstructure in BV relaxation. SIAM J. Math. Anal. 29 (1998) 823-848. | Zbl | MR

[32] H. Lebesgue, Intégrale, longueur, aire. Ann. Mat. Pura Appl. 7 (1902) 231-359. | JFM

[33] J. Malý, Weak lower semicontinuity of polyconvex and quasiconvex integrals. Manuscr. Math. 85 (1994) 419-428. | Zbl

[34] P. Marcellini, Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscr. Math. 51 (1985) 1-28. | Zbl | MR

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

[36] P. Mattila, Geometry of sets and measures in Euclidean spaces. Cambridge Stud. Adv. Math., vol. 44. Cambridge University Press, Cambridge (1995), Fractals and rectifiability. | Zbl | MR

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

[38] C. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952) 25-53. | Zbl | MR

[39] C. Morrey, Multiple integrals in the calculus of variations. Classics Math. (1966). | Zbl | MR

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

[41] J. Reshetnyak, General theorems on semicontinuity and convergence with functionals. Sibirsk. Mat. Ž. 8 (1967) 1051-1069. | Zbl | MR

[42] F. Rindler, Lower semicontinuity and Young measures in BV(Ω;Rm) without Alberti's Rank-One Theorem. Adv. Calc. Var. 5 (2012) 127-159. | Zbl | MR

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

[44] T. Schmidt, Regularity of relaxed minimizers of quasiconvex variational integrals with (p,q)-growth. Arch. Ration. Mech. Anal. 193 (2009) 311-337. | Zbl | MR

[45] T. Schmidt, A simple partial regularity proof for minimizers of variational integrals. NoDEA Nonlinear Differ. Eq. Appl. 16 (2009) 109-129. | Zbl | MR

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

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

[48] P. Soneji, Lower semicontinuity in BV of quasiconvex integrals with subquadratic growth. ESAIM: COCV 19 (2013) 555-573. | Zbl | MR | Numdam

[49] W. Ziemer, Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts Math., vol. 120. Springer-Verlag, New York (1989). | Zbl | MR