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Published online by Cambridge University Press:
**13 August 2014**

We study properties of the functional

\begin{eqnarray} \mathscr{F}_{{\rm loc}}(u,\Omega):=
\inf_{(u_{j})}\bigg\{ \liminf_{j\rightarrow\infty}\int_{\Omega}f(\nabla u_{j})\ud
x\, \left| \!\!\begin{array}{rl} & (u_{j})\subset W_{{\rm
loc}}^{1,r}\left(\Omega, \RN\right) \\ & u_{j}\tostar u\,\,\textrm{in
}\BV\left(\Omega, \RN\right) \end{array} \right. \bigg\}, \end{eqnarray}$\begin{array}{ccc}{F}_{\mathrm{loc}}\mathrm{\left(}\mathit{u,\Omega}\mathrm{\right)}\mathrm{:}\mathrm{=}\underset{\mathrm{\left(}{\mathit{u}}_{\mathit{j}}\mathrm{\right)}}{\mathrm{inf}}\left\{\right.\underset{\mathit{j}\mathrm{\to}\mathrm{\infty}}{\mathrm{lim}\mathrm{inf}}{\mathrm{\int}}_{\mathit{\Omega}}\mathit{f}\mathrm{\left(}\mathrm{\nabla}{\mathit{u}}_{\mathit{j}}\mathrm{\right)}\mathrm{d}\mathit{x}\left|\begin{array}{c}\\ & \\ & \end{array}\right.\left\}\right.\mathit{,}& & \end{array}$

where u ∈
BV(Ω;RN), and
f:RN ×
n → R is continuous and satisfies
0 ≤ f(ξ) ≤
L(1 + | ξ |
r). For r ∈ [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
Floc. Then, for
\hbox{$r\in[1,\frac{n}{n-1})$}$\mathit{r}\mathrm{\in}\mathrm{\left[}\mathrm{1}\mathit{,}\frac{\mathit{n}}{\mathit{n}\mathrm{-}\mathrm{1}}\mathrm{\right)}$, we prove that
Floc satisfies
the lower bound \begin{equation*} \scF_{{\rm loc}}(u,\Omega) \geq \int_{\Omega}
f(\nabla u (x))\ud x + \int_{\Omega}\finf
\bigg(\frac{D^{s}u}{|D^{s}u|}\bigg)\,|D^{s}u|, \end{equation*}${F}_{\mathrm{loc}}\mathrm{\left(}\mathit{u,\Omega}\mathrm{\right)}\mathrm{\ge}{\mathrm{\int}}_{\mathit{\Omega}}\mathit{f}\mathrm{\left(}\mathrm{\nabla}\mathit{u}\mathrm{\right(}\mathit{x}\mathrm{\left)}\mathrm{\right)}\mathrm{d}\mathit{x}\mathrm{+}{\mathrm{\int}}_{\mathit{\Omega}}{\mathit{f}}^{\mathrm{\infty}}\left(\right.\frac{{\mathit{D}}^{\mathit{s}}\mathit{u}}{\mathrm{\left|}{\mathit{D}}^{\mathit{s}}\mathit{u}\mathrm{\right|}}\left)\right.\mathrm{\left|}{\mathit{D}}^{\mathit{s}}\mathit{u}\mathrm{\right|}\mathit{,}$

provided f is quasiconvex, and the recession function
f∞ (defined as
\hbox{$ f^{\infty}(\xi):= \overline{\lim}_{t\rightarrow\infty}f(t\xi )/t$}${\mathit{f}}^{\mathrm{\infty}}\mathrm{\left(}\mathit{\xi}\mathrm{\right)}\mathrm{:}\mathrm{=}{\overline{)\mathrm{lim}}}_{\mathit{t}\mathrm{\to}\mathrm{\infty}}\mathit{f}\mathrm{\left(}\mathit{t\xi}\mathrm{\right)}\mathit{/}\mathit{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 Floc 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].- Type
- Research Article
- Information
- ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 4 , October 2014 , pp. 1078 - 1122
- Copyright
- © EDP Sciences, SMAI, 2014

Acerbi, E. and Fusco, N., Semicontinuity problems in the
calculus of variations. *Arch. Ration. Mech.
Anal.*
86 (1984)
125–145. Google Scholar

Alberti, G., Rank one property for
derivatives of functions with bounded variation. *Proc.
Roy. Soc. Edinburgh Sect. A*
123 (1993)
239–274.
Google Scholar

Alberti, G. and Ambrosio, L., A geometrical approach to
monotone functions in **R**^{n},
*Math. Z.*
230 (1999)
259–316. Google Scholar

Amar, M. and De Cicco, V., Quasi-polyhedral approximation
of BV-functions. *Ric.
Mat.*
54 (2005) 485–490
(2006). Google Scholar

Ambrosio, L., A compactness theorem for a
new class of functions of bounded variation. *Boll. Un.
Mat. Ital. B*
3 (1989) 857–881.
Google Scholar

Ambrosio, L., Existence theory for a new
class of variational problems. *Arch. Ration. Mech.
Anal.*
111 (1990)
291–322. Google Scholar

Ambrosio, L., On the lower semicontinuity of
quasiconvex integrals in SBV(*Ω,* **R**^{k}).
*Nonlinear Anal.*
23 (1994)
405–425. Google Scholar

Ambrosio, L. and Maso, G.
Dal, On the relaxation in BV(*Ω*;**R**^{m})
of quasi-convex integrals. *J. Funct. Anal.*
109 (1992) 76–97. Google Scholar

Ambrosio, L., Fusco, N. and Hutchinson, J., 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. Google Scholar

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).

Ambrosio, L., Mortola, S., and Tortorelli, V., Functionals with linear
growth defined on vector valued BV functions. *J. Math.
Pures Appl.*
70 (1991)
269–323. Google Scholar

Ambrosio, L. and Pallara, D., Integral representations of
relaxed functionals on BV(**R**^{n}*,* **R**^{k})
and polyhedral approximation. *Indiana Univ. Math.
J.*
42 (1993)
295–321. Google Scholar

Aviles, P. and Giga, Y., Variational integrals on mappings
of bounded variation and their lower semicontinuity.
*Arch. Ration. Mech. Anal.*
115 (1991)
201–255. Google Scholar

Ball, J. 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. Roy. Soc. Edinburgh Sect. A*
128 (1998)
463–479. Google Scholar

Braides, A. and Coscia, A., The interaction between bulk
energy and surface energy in multiple integrals. *Proc.
Roy. Soc. Edinburgh Sect. A*
124 (1994)
737–756. Google Scholar

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).

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

Dal Maso, G., Integral representation on
BV(*Ω*) of
*Γ*-limits of
variational integrals. *Manuscr. Math.*
30 (1979/80) 387–416. Google Scholar

De Giorgi, E. and
Ambrosio, L., New functionals in the
calculus of variations. *Atti Accad. Naz. Lincei Rend.
Cl. Sci. Fis. Mat. Natur.*
82 (1988) 199–210
(1989). Google Scholar

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

L. Evans and R. Gariepy, Measure theory and fine
properties of functions. *Stud.Adv. Math.* CRC Press, Boca Raton, FL
(1992).

Fonseca, I., Lower semicontinuity of surface
energies. *Proc. Roy. Soc. Edinburgh Sect.
A*
120 (1992)
99–115. Google Scholar

Fonseca, I. and Malý, J., Relaxation of multiple integrals
below the growth exponent. *Annal. Inst. Henri Poincaré
Anal. Non Linéaire*
14 (1997)
309–338. 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,* **R**^{p})
for integrands *f*(*x,u,*∇*u*).
*Arch. Ration. Mech. Anal.*
123 (1993) 1–49.
Google Scholar

Fonseca, I. and Rybka, P., Relaxation of multiple integrals
in the space BV(*Ω,* **R**^{p}).
*Proc. Roy. Soc. Edinburgh Sect. A*
121 (1992)
321–348. Google Scholar

Goffman, C. and Serrin, J., Sublinear functions of measures
and variational integrals. *Duke Math.
J.*
31 (1964)
159–178. Google Scholar

Kristensen, J., Lower semicontinuity of
quasi-convex integrals in BV(*Ω*;**R**^{m}).
*Calc. Var. Partial Differ. Eqs.*
7 (1998) 249–261. Google Scholar

Larsen, C., Quasiconvexification in
*W* ^{1,1} and optimal jump
microstructure in BV relaxation. *SIAM J. Math.
Anal.*
29 (1998)
823–848. Google Scholar

Malý, J., Weak lower semicontinuity of
polyconvex and quasiconvex integrals. *Manuscr.
Math.*
85 (1994)
419–428. Google Scholar

Marcellini, P., Approximation of quasiconvex
functions, and lower semicontinuity of multiple integrals.
*Manuscr. Math.*
51 (1985) 1–28.
Google Scholar

Marcellini, P., On the definition and the
lower semicontinuity of certain quasiconvex integrals.
*Annal. Inst. Henri Poincaré Anal. Non Linéaire*
3 (1986) 391–409.
Google Scholar

P. Mattila, Geometry of sets and measures in
Euclidean spaces. *Cambridge Stud. Adv. Math.*, vol. 44. Cambridge
University Press, Cambridge (1995), Fractals and rectifiability.

Meyers, N., Quasi-convexity and lower
semi-continuity of multiple variational integrals of any order.
*Trans. Amer. Math. Soc.*
119 (1965)
125–149. Google Scholar

Morrey, C., Quasi-convexity and the lower
semicontinuity of multiple integrals. *Pacific J.
Math.*
2 (1952) 25–53.
Google Scholar

C. Morrey, Multiple integrals in the calculus of
variations. *Classics Math.* (1966).

Müller, S., On quasiconvex functions which
are homogeneous of degree 1. *Indiana Univ. Math.
J.*
41 (1992)
295–301. Google Scholar

Reshetnyak, J., General theorems on
semicontinuity and convergence with functionals.
*Sibirsk. Mat. Ž.*
8 (1967)
1051–1069. Google Scholar

Rindler, F., Lower semicontinuity and Young
measures in BV(*Ω*;**R**^{m})
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).

Schmidt, T., Regularity of relaxed
minimizers of quasiconvex variational integrals with (*p,q*)-growth. *Arch.
Ration. Mech. Anal.*
193 (2009)
311–337. Google Scholar

Schmidt, T., A simple partial regularity
proof for minimizers of variational integrals. *NoDEA
Nonlinear Differ. Eq. Appl.*
16 (2009)
109–129. Google Scholar

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

Soneji, P., Lower semicontinuity in
BV of quasiconvex
integrals with subquadratic growth. *ESAIM:
COCV*
19 (2013)
555–573. Google Scholar

W. Ziemer, Weakly differentiable functions.
Sobolev spaces and functions of bounded variation. *Graduate Texts Math.*,
vol. 120. Springer-Verlag, New York (1989).

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