Journal of Inequalities in Pure and
Applied Mathematics
REGULARITY FOR VECTOR VALUED MINIMIZERS OF SOME
ANISOTROPIC INTEGRAL FUNCTIONALS
volume 7, issue 3, article 88,
2006.
FRANCESCO LEONETTI AND PIER VINCENZO PETRICCA
Universita’ di L’Aquila
Dipartimento di Matematica Pura ed Applicata
Via Vetoio, Coppito
67100 L’Aquila
Italy.
EMail: leonetti@univaq.it
Via Sant’Amasio 18
03039 Sora
Italy.
Received 31 March, 2006;
accepted 06 April, 2006.
Communicated by: A. Fiorenza
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
R
We deal with anisotropic integral functionals Ω f(x, Du(x))dx defined on vector
valued mappings u : Ω ⊂ Rn → RN . We show that a suitable "monotonicity"
inequality, on the density f, guarantees global pointwise bounds for minimizers
u.
2000 Mathematics Subject Classification: 49N60, 35J60.
Key words: Anisotropic, Integral, Functional, Regularity, Minimizer.
Regularity for Vector Valued
Minimizers of Some Anisotropic
Integral Functionals
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
Francesco Leonetti and
Pier Vincenzo Petricca
3
7
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1.
Introduction
We consider the integral functional
Z
f (x, Du(x))dx
(1.1)
F(u) =
Ω
where u : Ω ⊂ Rn → RN and Ω is a bounded open set. When N = 1 we
are dealing with scalar functions u : Ω → R; on the contrary, vector valued
mappings u : Ω → RN appear when N ≥ 2. Local and global pointwise
bounds for scalar minimizers of (1.1) have been proved in [2], [7], [5], [4]. A
model functional for these results is
! p2
Z
n
X
dx,
(1.2)
aij (x)Dj u(x)Di u(x)
Ω
i,j=1
where coefficients aij are measurable, bounded and elliptic. Previous results
for scalar minimizers are no longer true in the vector valued case N ≥ 2 as De
Giorgi’s counterexample shows, [3]. Some years later, attention has been paid
to anisotropic functionals whose model is
Z
(1.3)
(|D1 u(x)|p1 + |D2 u(x)|p2 + · · · + |Dn u(x)|pn ) dx,
Ω
where each component Di u of the gradient Du = (D1 u, D2 u, . . . , Dn u) may
have a (possibly) different exponent pi : this seems useful when dealing with
some reinforced materials, [9]; see also [6, Example 1.7.1, page 169]. In
the framework of anisotropic functionals, global pointwise bounds have been
Regularity for Vector Valued
Minimizers of Some Anisotropic
Integral Functionals
Francesco Leonetti and
Pier Vincenzo Petricca
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proved for scalar minimizers in [1] and [8]. If no additional conditions are assumed, these bounds are false in the vectorial case, as the above mentioned
counterexample shows, [3]. The aim of this paper is to present a “monotonicity” assumption ensuring boundedness of vector valued minimizers. In order
to do that, we recall that u : Ω ⊂ Rn → RN thus Du(x) is a matrix with N
rows and n columns; the density f (x, A) in (1.1) is assumed to be measurable
with respect to x, continuous with respect to A and f : Ω × RN ×n → [0, +∞).
Every matrix A = {Aαi } ∈ RN ×n will have N rows A1 , . . . , AN and n columns
A1 , . . . , An . In this paper we will show that the following “monotonicity” inequality guarantees global pointwise bounds for vector valued minimizers of
(1.1):
(1.4)
Regularity for Vector Valued
Minimizers of Some Anisotropic
Integral Functionals
Francesco Leonetti and
Pier Vincenzo Petricca
n pi
X
f (x, Ã) + µ
Ãi − Ai ≤ f (x, A) + M (x)
i=1
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N ×n
β
for every pair of matrices Ã, A ∈ R
such that there exists a row β with à =
0 and for every remaining row α 6= β we have Ãα = Aα . In (1.4) µ, p1 , . . . , pn
are positive constants with pi > 1 and M : Ω → [0, +∞) with M ∈ Lr (Ω),
r ≥ 1.PIf we keep in mind that A = Du(x), then the left hand side of (1.4)
shows ni=1 |Ãi − Di u(x)|pi , thus each component Di u of the gradient Du may
have a possibly different exponent pi , so we are in the anisotropic framework:
u ∈ W 1,1 (Ω, RN ) with Di u ∈ Lpi (Ω, RN ). In this case the harmonic mean
P
−1
p = n1 ni=1 p1i
comes into play. In Section 2 we will prove the following
Theorem 1.1. We consider the functional (1.1) under the “monotonicity” in-
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equality (1.4) with
p∗
p
(1.5)
1
1−
r
>1
where p∗ is the Sobolev exponent of p < n. We consider u = (u1 , . . . , uN ) ∈
W 1,1 (Ω, RN ), with Di u ∈ Lpi (Ω, RN ) ∀i ∈ {1, . . . , n}, such that
F(u) < +∞
Regularity for Vector Valued
Minimizers of Some Anisotropic
Integral Functionals
and
Francesco Leonetti and
Pier Vincenzo Petricca
F(u) ≤ F(v)
(1.6)
for every v ∈ u + W01,1 (Ω, RN ) with Di v ∈ Lpi (Ω, RN ) ∀i ∈ {1, . . . , n}. Then,
for every component uβ , we have
Contents
inf uβ − c∗ ≤ uβ (x) ≤ sup uβ + c∗
(1.7)
∂Ω
∂Ω
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for almost every x ∈ Ω, where
c∗ = c
kM kLr (Ω)
µ
p1
h
∗
1− 1 p −1
|Ω| ( r ) p
i
1
p∗
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∗h
∗
1− 1 p
1− 1 p −1
2( r ) p ( r ) p
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i−1
,
c = c(n, p1 , . . . , pn ) > 0 and |Ω| is the Lebesgue measure of Ω.
A model density f for the “monotonicity” inequality (1.4) is given in the
following.
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Lemma 1.2. For every i = 1, . . . , n, let us consider pi ∈ [2, +∞) and ai ∈
(0, +∞); we take m : Ω → [0, +∞) and h : R → R with −∞ < inf R h. Let us
consider f : Ω × Rn×n → R defined as follows:
(1.8)
f (x, A) =
n
X
ai |Ai |pi + m(x)h(det A).
i=1
Then the “monotonicity” inequality (1.4) holds true with µ = minj aj and
M (x) = m(x)[h(0) − inf R h]. Moreover, if h ≥ 0, then f ≥ 0 too.
Regularity for Vector Valued
Minimizers of Some Anisotropic
Integral Functionals
Francesco Leonetti and
Pier Vincenzo Petricca
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2.
Proofs
In order to prove Theorem 1.1, we need the following
Lemma 2.1. Let us consider the functional (1.1) under the “monotonicity”
assumption (1.4). Then, for every v = (v 1 , . . . , v N ) ∈ W 1,1 (Ω, RN ) with
Di v ∈ Lpi (Ω, RN ) ∀i ∈ {1, . . . , n}, for any β ∈ {1, . . . , N }, for all t ∈ R,
it results that
n Z
X
(2.1) F(Iβ,t (v)) + µ
|Di (Iβ,t (v(x))) − Di v(x)|pi dx
i=1
Ω
Regularity for Vector Valued
Minimizers of Some Anisotropic
Integral Functionals
Z
≤ F(v) +
M (x)dx
{v β >t}
where Iβ,t : RN → RN is defined as follows:
∀y = (y 1 , . . . , y N ) ∈ RN ,
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1
N
(y), . . . , Iβ,t
(y))
Iβ,t (y) = (Iβ,t
with
(
(2.2)
α
Iβ,t
(y) =
y
α
Francesco Leonetti and
Pier Vincenzo Petricca
if α 6= β
y β ∧ t = min {y β , t} if α = β.
Proof. For every v ∈ W 1,1 (Ω, RN ), with Di v ∈ Lpi (Ω, RN ) ∀i ∈ {1, . . . , n}, it
results that Iβ,t (v) ∈ W 1,1 (Ω, RN ); moreover
(
Di v α
if α 6= β
α
(2.3)
Di (Iβ,t (v)) =
1{vβ ≤t} Di v β if α = β,
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where 1B is the characteristic function of the set B, that is, 1B (x) = 1 if x ∈ B
and 1B (x) = 0 if x ∈
/ B. Therefore Di (Iβ,t (v)) ∈ Lpi (Ω, RN ) ∀i ∈ {1, . . . , n}.
β
β
α
On {x ∈ Ω : v (x) > t} we have D(Iβ,t
(v)) = 0 and, for α 6= β, D(Iβ,t
(v)) =
α
Dv ; so we can apply (1.4) with à = D(Iβ,t (v)) and A = Dv; we obtain
(2.4) f (x, D(Iβ,t (v(x)))) + µ
n
X
|Di (Iβ,t (v(x))) − Di v(x)|pi
i=1
≤ f (x, Dv(x)) + M (x)
for x ∈ {v β > t}. On {x ∈ Ω : v β (x) ≤ t} D(Iβ,t (v)) = Dv, thus
(2.5) f (x, D(Iβ,t (v(x)))) + µ
n
X
pi
|Di (Iβ,t (v(x))) − Di v(x)| = f (x, Dv(x))
Regularity for Vector Valued
Minimizers of Some Anisotropic
Integral Functionals
Francesco Leonetti and
Pier Vincenzo Petricca
i=1
for x ∈ {v β ≤ t}. From (2.4) and (2.5) we have
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f (x, D(Iβ,t (v(x)))) + µ
n
X
|Di (Iβ,t (v(x))) − Di v(x)|pi
i=1
≤ f (x, Dv(x)) + M (x)1{vβ >t} (x)
1
for x ∈ Ω. If x → f (x, Dv(x)) ∈ L (Ω), then x → f (x, D(Iβ,t (v(x)))) ∈
L1 (Ω) too and, integrating the last inequality with respect to x, we get (2.1).
When x → f (x, Dv(x)) ∈
/ L1 (Ω), we have F(v) = +∞ and (2.1) holds true.
This ends the proof of Lemma 2.1.
Now we are ready to prove Theorem 1.1.
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Proof. Let us fix β ∈ {1, . . . , N }. If sup∂Ω uβ = +∞ then the right hand side
of (1.7) is satisfied. Thus we assume sup∂Ω uβ < t0 ≤ t < +∞ and we note that
under this assumption Iβ,t (u) ∈ u+W01,1 (Ω, RN ) and Di (Iβ,t (u)) ∈ Lpi (Ω, RN )
∀i ∈ {1, . . . , n} since
uβ ∧ t = min {uβ , t} = uβ − [max {uβ − t, 0}] = uβ − [(uβ − t) ∨ 0],
where (uβ − t) ∨ 0 ∈ W01,1 (Ω) and Di ((uβ − t) ∨ 0) = Di uβ 1{uβ >t} ∈ Lpi (Ω)
∀i ∈ {1, . . . , n}. From (1.6) and (2.1) it results that
Regularity for Vector Valued
Minimizers of Some Anisotropic
Integral Functionals
F(u) ≤ F(Iβ,t (u))
Z
n Z
X
pi
≤ F(u) − µ
|Di (Iβ,t (u(x))) − Di u(x)| dx +
i=1
M (x)dx,
{uβ >t}
Ω
that is
(2.6)
µ
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n Z
X
i=1
Francesco Leonetti and
Pier Vincenzo Petricca
|Di (Iβ,t (u(x))) − Di u(x)|pi dx ≤
Z
M (x)dx.
{uβ >t}
Ω
β
If we define φ = (u − t) ∨ 0, then we can write (2.6) as follows:
Z
n Z
X
pi
(2.7)
µ
|Di φ(x)| dx ≤
M (x)dx.
i=1
Ω
{uβ >t}
R
If r < +∞, we apply Hölder’s inequality to {uβ >t} M (x)dx and we obtain
Z
1− r1
β
M (x)dx ≤ kM kLr (Ω) |{u > t}|
.
{uβ >t}
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If r = +∞, then
Z
M (x)dx ≤ kM kL∞ (Ω) |{uβ > t}| = kM kLr (Ω) |{uβ > t}|
1− r1
.
{uβ >t}
In both cases, from (2.7) it results that
n Z
X
i=1
kM kLr (Ω) β
|Di φ(x)| dx ≤
|{u > t}|
µ
Ω
pi
in particular, ∀i ∈ {1, . . . , n}
Z
kM kLr (Ω) β
|{u > t}|
|Di φ(x)|pi dx ≤
µ
Ω
1− r1
Regularity for Vector Valued
Minimizers of Some Anisotropic
Integral Functionals
1− r1
Francesco Leonetti and
Pier Vincenzo Petricca
Title Page
from which
Z
pi
|Di φ(x)| dx
p1
i
Ω
kM kLr (Ω) β
≤
|{u > t}|
µ
1− r1
Contents
p1i
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and finally
"
(2.8)
n Z
Y
i=1
Ω
|Di φ(x)|pi dx
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p1 # n1
i
II
I
≤
kM kLr (Ω) β
|{u > t}|
µ
1− r1
p1
.
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We apply the anisotropic imbedding theorem [10] and we use (2.8):
p1∗
Z
β
p∗
(2.9)
0≤
u (x) − t dx
{uβ >t}
= kφkLp∗ (Ω)
" n Z
p1 # n1
Y
i
≤c
|Di φ(x)|pi dx
i=1
Ω
kM kLr (Ω) β
≤c
|{u > t}|
µ
1− r1
p1
Regularity for Vector Valued
Minimizers of Some Anisotropic
Integral Functionals
,
where c = c(n, p1 , . . . , pn ) > 0. If kM kLr (Ω) = 0, then from (2.9) it results that
uβ ≤ t almost everywhere in Ω and we are done. If kM kLr (Ω) > 0, then for
T > t we have
Z
∗
p∗
β
(2.10)
(T − t) |{u > T }| =
(T − t)p dx
{uβ >T }
Z
β
p∗
≤
u (x) − t dx
{uβ >T }
Z
β
p∗
≤
u (x) − t dx
{uβ >t}
(2.11)
β
|{u > T }| ≤ c
p∗
kM kLr (Ω)
µ
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and from (2.9) and (2.10) we get
Francesco Leonetti and
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pp∗
1
β
> t}|
∗ |{u
p
(T − t)
1− r1
p∗
p
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for every T, t with T > t ≥ t0 . We set χ(t) = |{uβ > t}| and we use [7,
Lemma 4.1, p. 93], that we provide below for the convenience of the reader.
Lemma 2.2. Let χ : [t0 , +∞) → [0, +∞) be decreasing. We assume that there
exist k, a ∈ (0, +∞) and b ∈ (1, +∞) such that
T > t ≥ t0 =⇒ χ(T ) ≤
(2.12)
k
(χ(t))b .
(T − t)a
Then it results that
"
χ(t0 + d) = 0
(2.13)
d = k(χ(t0 ))b−1 2
where
ab
(b−1)
Regularity for Vector Valued
Minimizers of Some Anisotropic
Integral Functionals
# a1
.
Francesco Leonetti and
Pier Vincenzo Petricca
We use the previous Lemma 2.2 and we have
|{uβ > t0 + d}| = 0
(2.14)
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that is
Contents
β
u ≤ t0 + d
(2.15)
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J
almost everywhere in Ω, where
d=c
kM kLr (Ω)
µ
p1
h
β
{u > t0 }
1− r1
p∗
−1
p
i
1
p∗
2
1− r1
h
p∗
p
1− r1
p∗
−1
p
i−1
.
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β
In order to get the right hand side of (1.7), we control |{u > t0 }| by means
of |Ω| and we take a sequence {(t0 )m }m with (t0 )m → sup∂Ω uβ . Let us show
how we obtain the left hand side of (1.7): we apply the right hand side of (1.7)
to −u. This ends the proof of Theorem 1.1.
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Now we are going to prove Lemma 1.2.
Proof. We assume that Ã, A ∈ Rn×n with Ãβ = 0 and Ãα = Aα for α 6= β.
Then
X
X
(2.16)
|Aαi |2 = |Aβi |2 +
|Aαi |2
α
α6=β
=
|Aβi
=
X
− Ãβi |2 +
X
|Ãαi |2
α6=β
|Aαi
−
Ãαi |2
α
+
X
Regularity for Vector Valued
Minimizers of Some Anisotropic
Integral Functionals
|Ãαi |2
α
Francesco Leonetti and
Pier Vincenzo Petricca
so
|Ai |2 = |Ai − Ãi |2 + |Ãi |2 .
(2.17)
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Since pi ≥ 2, the previous equality gives
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|Ai |pi ≥ |Ai − Ãi |pi + |Ãi |pi .
(2.18)
Moreover
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I
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(2.19) h(det A) ≥ inf h = h(0) − [h(0) − inf h] = h(det Ã) − [h(0) − inf h].
R
R
R
Now we are able to estimate f (x, A) and f (x, Ã) by means of (2.18) and (2.19)
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as follows:
(2.20)
f (x, Ã) + (min aj )
j
≤
≤
n
X
i=1
n
X
n
X
|Ai − Ãi |pi
i=1
pi
ai |Ãi | + m(x)h(det Ã) +
n
X
ai |Ai − Ãi |pi
i=1
ai |Ai |pi + m(x)h(det A) + m(x)[h(0) − inf h]
R
i=1
= f (x, A) + m(x)[h(0) − inf h]
R
thus the “monotonicity” inequality (1.4) holds true with µ = minj aj and M (x) =
m(x)[h(0) − inf R h]. This ends the proof of Lemma 1.2.
Regularity for Vector Valued
Minimizers of Some Anisotropic
Integral Functionals
Francesco Leonetti and
Pier Vincenzo Petricca
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References
[1] L. BOCCARDO, P. MARCELLINI AND C. SBORDONE, L∞ -regularity
for a variational problems with sharp non standard growth conditions, Boll.
Un. Mat. Ital., 4-A (1990), 219–226.
[2] E. DE GIORGI, Sulla differenziabilitá e l’analiticitá delle estremali degli
integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat.,
3 (1957), 25–43.
[3] E. DE GIORGI, Un esempio di estremali discontinue per un problema
variazionale di tipo ellittico, Boll. Un. Mat. Ital., 4 (1968), 135–137.
Regularity for Vector Valued
Minimizers of Some Anisotropic
Integral Functionals
[4] M. GIAQUINTA AND E. GIUSTI, On the regularity of the minima of variational integrals, Acta. Math., 148 (1982), 31–46.
Francesco Leonetti and
Pier Vincenzo Petricca
[5] O. LADYZHENSKAYA AND N. URAL’TSEVA, Linear and Quasilinear
Elliptic Equations, Academic Press, New York, 1968.
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[6] J.L. LIONS, Quelques Methodes de Resolution des Problemes aux Limites
Non Lineaires, Dunod, Gauthier - Villars, Paris, 1969.
[7] G. STAMPACCHIA, Equations Elliptiques du Second Ordre a Coefficientes Discontinus, Semin. de Math. Superieures, Univ. de Montreal, 16,
1966.
[8] B. STROFFOLINI, Global boundedness of solutions of anisotropic variational problems, Boll. Un. Mat. Ital., 5-A (1991), 345–352.
[9] Q. TANG, Regularity of minimizers of non-isotropic integrals of the calculus of variations, Ann. Mat. Pura Appl., 164 (1993), 77–87.
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[10] M. TROISI, Teoremi di inclusione per spazi di Sobolev non isotropi,
Ricerche Mat., 18 (1969), 3–24.
Regularity for Vector Valued
Minimizers of Some Anisotropic
Integral Functionals
Francesco Leonetti and
Pier Vincenzo Petricca
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