Affiliation:
1. DISIM, Università degli Studi dell’Aquila , Via Vetoio snc , Coppito , 67100 L’Aquila , Italy
2. Dipartimento di Matematica, Università degli Studi di Padova , Via Trieste , 63-35121 Padova , Italy
Abstract
Abstract
We consider the functional
ℱ
(
u
)
≔
∫
Ω
f
(
x
,
D
u
(
x
)
)
d
x
,
{\mathcal{ {\mathcal F} }}\left(u):= \mathop{\int }\limits_{\Omega }f\left(x,Du\left(x)){\rm{d}}x,
where
f
(
x
,
z
)
f\left(x,z)
satisfies a
(
p
,
q
)
\left(p,q)
-growth condition with respect to
z
z
and can be approximated by means of a suitable sequence of functions. We consider
B
R
⋐
Ω
{B}_{R}\hspace{0.33em}\Subset \hspace{0.33em}\Omega
and the spaces
X
=
W
1
,
p
(
B
R
,
R
N
)
and
Y
=
W
1
,
p
(
B
R
,
R
N
)
∩
W
loc
1
,
q
(
B
R
,
R
N
)
.
X={W}^{1,p}\left({B}_{R},{{\mathbb{R}}}^{N})\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}Y={W}^{1,p}\left({B}_{R},{{\mathbb{R}}}^{N})\cap {W}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{1,q}\left({B}_{R},{{\mathbb{R}}}^{N}).
We prove that the lower semicontinuous envelope of
ℱ
∣
Y
{\mathcal{ {\mathcal F} }}{| }_{Y}
coincides with
ℱ
{\mathcal{ {\mathcal F} }}
or, in other words, that the Lavrentiev term is equal to zero for any admissible function
u
∈
W
1
,
p
(
B
R
,
R
N
)
u\in {W}^{1,p}\left({B}_{R},{{\mathbb{R}}}^{N})
. We perform the approximations by means of functions preserving the values on the boundary of
B
R
{B}_{R}
.
Reference43 articles.
1. E. Acerbi, G. Bouchitté, and I. Fonseca, Relaxation of convex functionals: the gap problem, Ann. Inst. H. Poincaré Anal. Anal. Non Lineaire 20 (2003), no. 3, 359–390.
2. G. Alberti and P. Majer, Gap phenomenon for some autonomous functionals, J. Convex Analysis 1 (1994), 31–45.
3. G. Alberti and F. Serra Cassano, Non-occurrence of gap for one-dimensional autonomous functionals, in: Calculus of variations, homogenization and continuum mechanics (Marseille, 1993), Ser. Adv. Math. Appl. Sci. 18 (1994), 1–17. World Scientific Publishing, River Edge, NJ.
4. A. K. Balci, L. Diening, and M. Surnachev, New examples on Lavrentiev gap using fractals, Calc. Var. Partial Differential Equations 59 (2020), 180.
5. J. M. Ball and V. J. Mizel, One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Rational Mech. Anal. 90 (1985), no. 4, 325–388.
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