Journal of Applied Mathematics and Physics, 2014, 2, 194-203
Published Online April 2014 in SciRes. http://www.scirp.org/journal/jamp
http://dx.doi.org/10.4236/jamp.2014.25024
How to cite this paper: Granucci, T. (2014) Remarks on the Harnak Inequality for Local-Minima of Scalar Integral Function-
als with General Growth Conditions. Journal of Applied Mathematics and Physics, 2, 194-203.
http://dx.doi.org/10.4236/jamp.2014.25024
Remarks on the Harnak Inequality for
Local-Minima of Scalar Integral Functionals
with General Growth Conditions
Tiziano Granucci
Scuola Superiore, Istituto P. Calamandrei, Firenze, Italy
Received September 2013
Abstract
In this paper we proof a Harnack inequality and a regularity theorem for local-minima of scalar
intagral functionals with general growth conditions.
Keywords
Harnack Inequality, Regularity, Hölder Continuity
1. Introduction
In this paper we proof a Harnack inequality for local-minima of scalar intagral functionals of the calculus of
variation of that type
(1.1)
where Ω is a bounded open subset of N , Φ:[0,+∞)→[0,+∞) is a N-function and Φ globally satisfies the Δ′-
condition in [0,+∞), N f:    is a Carathéodory function and there exist

L,L0, L₁₂ ₂
and
∈∈ℝℝ for a. e. xΩ and for every (s,z)×N . The risearch of regularity results for elliptic and parabolic equations
start from the basic and most important results of E. De Giorgi [5] and J. Nash [27]. In 1990s, beginning from
the papers of G. Astarita and G. Marrucci [3] and J. P. Gosez [13] has been developed a remarkable production
of regularity results for functionals with general growths. In [7], [8] and [25], M. Fuchs, G. Mingione, G.
Seregin and F. Siepe have studied functionals of the type
(1.2)
showing results of partial and global regularity for the minimizer of such functional in the scalar and vectorial
case. Moreover in [8] M. Fuchs and G. Mingione, have already studied functionals of this type
(1.3)








J,,, ufxuxuxdx

 






,,|| || zfxszL z

 ₂








|,1|
J
uuxlnuxdx  




||,.
J
uudx

 

T. Granucci
195
In papers [7,8,25] the regularity of the minimizer of the functionals (1.2) and (1.3) has been obtained starting
from the weak Eulero-∈ Lagrange equations using the hypothesis: ΦC². We remember that in [7,8,25] there are
important estimations on the L^{∞} norm of the gradient of the minima both in the scalar case and in the vec-
torial one. In [24] E. Mascolo and G. Papi have determined an inequality of Harnack for the minimizer of the
functional (1.3) under the condition ∈ ΦΔ
2∇
∩2∈ . We observe that Φ Δ2∇ ∩2 implies
(1.4)
with real positive constants c,c,c,c₁₂₃₄
and 1 < p ≤ m. Therefore the functional (1.3) satisfies non-standard
growth conditions. Classical regularity theorem for functionals with standard growth conditions (p = m) has
been proved in [9] and [10] (for a didactic explanation refer to [2,11,12]). In [26], G. Moscariello and L. Nania
has obtain a results of hölder continuity for the local-minima of functional of the type (1.1) under the hypothesis
that (1.4) holds with 1 < p ≤ m < ((Np)/(N-p)). In [17], G. M. Lieberman proved an Harnack inequality for the
local- ∈ minima of the functional (1.1) with ΦC² suth that verifies the following relation
with 0 < c5 < c6. We are interested in functionals with quasi-linear growths and we will proof a regularity result
which extend the ones obtained in [17,24,26] to a wider N-functional class. In particular we get that the local-
minima of the following functionals:
(1.5)
are holder continuous functions. In [14] and [15] we start to study the regularity of the local-minima introducing
a maximal LΦ-L∞ inequality and estimating the measure of the level set A(k,R). Moreover in [15] and [16] we
have shown that the following hypothesis can be used in order to give a new estimation of the measure of the li-
vel set A(k,R):
H-1) Φ globally satisfies the Δ′-condition in [0,+∞);
H-2) there exists a constant H c0
₂
(1.6)
H-3) there exists a constant H
c0
₃
(1.7)
Under these hypotheses we can show the following result.
Theorem 1: If uW¹LΦ(Ω) is a quasi-minima of the functional (1.1) and if Φ confirm the hypotheses H-1,
H-2 and H-3; then u is locally hölder continuous.
In these pages we show that the hypotheses H-2 and H-3 are purely technical and they can be eliminated. We
can subsequently weaken besides H-1.
We will suppose that the following hypothesis hold.
G-1) Let ϖ:   be an increasing function such that
(1.8)
for every t  and for every ɛ(0,1), where cG > 0 is a real constant. Moreover we suppose that
We say that Φ∈G if (1.8) holds. The hypothesis G-1 implicates a type of quasi-sub-homogeneity condition
on the N-function Φ.
Remark 1: We observe that if Φ∈Δ₂∩∇₂ then by Lemma 3 (i) we have
Then the functions Φ∈Δ₂∩∇₂ verify the hypothesis G-1.
Remark 2: We observe that if Φ∈Δ′ on (0,+∞) then Φ verify the hypothesis G-1; in fact


0
pm
tc tctcfort   ₂₃₄




'' '
56
/ 0 ct ttcfort

 




|,1 |1
p
Juulnudx withp


 


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

1/ 0,1;
H
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
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m
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
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G
tc t 



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0
lim 0.
x
s 




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


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

1/ .
rr r
ttt    






. tc t  

T. Granucci
196
Our principal results will be, a weak inequality of Harnack [Theorem 5] and the corollary of regularity that it
follows of it [Corollary 2]. The proof of the Harnack inequality uses the techniques introduced in [6,17] and [24].
The only present novelty in the demonstrative technique is the use of an ɛ-Young inequality. This simple trick
allows to recover the results introduced in [15-17,24,26] in a simple way and without using the properties of the
functions Δ2∩2
 (see Lemma of [15,24] and [26]). We finally observe that the hypotheses Δ2∩2  it is not, in
general, equivalent to H-1; therefore the hypothesis G-1 seems to be slightly more general of those introduced in
[15-17,24,26].
Definition 1: Let p be a real valued function defined on [0,+∞) and having the following properties: p(0) = 0,
p(t) > 0 if t > 0, p is nondecreasing and right continuous on (0,+∞). Then the real valued function Φ defined on
[0,+∞) by
(1.9)
is called an N-function.
The function Φ: [0,+∞)→[0,+∞) defined by (1.9) satisfies the following properties:
Definition 2: Let p be a real valued function defined on [0,+∞) and having the following properties: p(0) = 0,
p(t) > 0 if t > 0, p is nondecreasing and right continuous on (0,+∞). We define
and
(1.10)
The N-functions Φ and Ψ given by (1.9) and (1.10) are said to be complementary.
Particularly for us it will be important the following Lemma.
Lemma 1: Let Φ be an N-function, let Ψ be the complemantary N-function of Φ then we have
(1.11)
∀s, t∈ℝ⁺. Moreover for every ɛ > 0 we get
(1.12)
Definition 3: A N-function Φ is of class

₂ globally in (0,+∞) if exists k > 1 such that
(1.13)
Definition 4: A N-function Φ is of class

₂^{m} globally in (0,+∞), with m>1, if for every λ>1
(1.14)
The N-functions m
₂ are characterized by the following result
Lemma 2: Let Φ be a N-function and let Φ'₋ be its left derivative. For m > 1 the following properties are
equivalent:
1) Φ(λt) ≤ λmΦ(t), for every t ≥ 0, for every λ>1;
2) tΦ'₋(t) ≤ mΦ(t), for every t ≥ 0;
3) the function Φ(t)/tr is non-increasing on (0,+∞).




0,[] t
tpsds 


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
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0,; is continuous on

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0,; isstrictlyincreasingon 


0,; is convex on






0
lim/0 lim/;
x
x
tt andtt







0, //. if stthensstt  





pt s
qs supt






0[],
.
t
tqsds 




s
ts t  








1/1/ ,. ststst  


  ¡






2 0,. tkt t

 

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


0, .
m
ttt 

 

T. Granucci
197
The N-functions Φ∈∇₂r are characterized by the following result
Lemma 3: Let Φ be a N-function and let Φ'₋ be its left derivative. For r>1 the following properties are equiv-
alent:
1) Φ(λt) ≥ λrΦ(t), for every t ≥ 0, for every λ > 1;
2) tΦ'₋(t) ≥ rΦ(t), for every t ≥ 0;
3) the function Φ(t)/λr is non-decreasing on (0,+∞).
Definition 5: We say that a N-function Φ belongs to the class Φ∈∇₂r if any of the three condition (i)', (ii)' or
(iii)' is satisfied.
Definition 6: We say that the N-function Φ satisfies the Δ′-condition if there exist positive constants—c and
t₀—such that
(1.15)
for every t,s ≥ t₀.
Definition 7: We say that the N-function Φ globally satisfies the Δ′-condition in [0,+∞) if (1.12) holds for
every t, s ≥ 0.
We remember that if Φ∈C² then Φ∈Δ′ if tΦ''(t)/Φ'(t) is a non-increasing function, for further details refer to
Theorems 5.1 and 5.2 and to the Lemma 5.2 of [19].
Lemma 4: If the N-function Φ satisfies the Δ′-condition then it also satisfies the ₂-condition
The N-functions
satisfy the Δ′-condition. Moreover ₁ and

₂ satisfy the Δ′-condition globally in [0,+∞) and belong to the
class ∇2 globally in [0,+∞). The function

₃ does not satisfy Δ′-condition for all t,s ≥ 0 and Φ3∇2. Ossevia-
mo inoltre che la funzione ₄₂₂ but

₄ does not satisfy the Δ′-condition. For further details refer to
[1,19,28]. Now we can introduce Orlicz spaces and Orlicz Sobolev Spaces, LΦ and W¹LΦ; in these definitions
and throughout the article we assume that Φ is a N-function of class m

₂ for some m > 1 and that Ω⊂ℝN is a
bounded open set with Lipschitz boundary.
Definition 8: If u is a LN-measurable function on Ω and: ∫ΩΦ(|u|)dx <+ ∞ then u∈LΦ(Ω). Moreover
(1.16)
where ∂iu, for I = 1,...,N, are the weak derivatives of u.
Theorem 2: LΦ(Ω) e W¹LΦ (Ω) are Banach spaces with the following norms
(1.17)
and
(1.18)
For greater details we refer to [1,19,28]. If u∈Wloc¹LΦ (Ω), k is a real number and R
Q, we set
Remark 3: For almost each k∈ℝ we get |A(k,R)| = |QR|-|B(k,R)|.
Definition 9: If uWloc¹LΦ (Ω), we say that u∈ODGΦ⁺(Ω,H, R) if for every couple of concentric balls
Qϱ⊂QR⋐QR₀⋐Ω, with R<R₀, and for every k∈ℝ we have
(1.19)






ts cts  ₄


1;
p
ttwith p

 ₁






1 1;
p
ttlntwith p

 ₂






11; ttlntt

  ₃








²/11 . tt lnt

 ₄








¹: 1,...,
i
WLuLuLforiN

  






,
0:|/1| uinfkukdx
 

  ‖‖
1, ,,1,....,,
.
iNi
uu u

 

‖‖‖‖‖ ‖







,: ,
R
R
AkRxQuxkukQ  








,: .
R
R
BkRx QuxkukQ 








() (,,)
|| /
Ak AkR
udxHukRdx  
ñ
ñ

T. Granucci
198
Definition 10: If u∈Wloc¹LΦ (Ω), we say that u
∈ODGΦ⁻ (Ω,H,R₀) if for every couple of concentric balls
Qϱ⊂QR⋐QR₀⋐Ω, with R < R₀, and for every k∈ℝ we have
(1.20)
Definition 11: If u∈Wloc¹LΦ (Ω), we say that u∈ODG_{Φ}(Ω,H, R₀) if u∈ODG_{Φ}^{±}(Ω,H,R₀), that is
Theorem 3: If u∈ODGΦ⁺(Ω,H,R₀) then u is locally bounded above on Ω. Furthermore, for each x₀∈Ω and R
≤ min(R₀,d(x₀,∂Ω),1) there exists an universal constant c5 = c5(N,m,H) such that
Proof: The proof follows using the demonstration methods presented in [24].
Corollary 1: If u∈ODGΦ(Ω,H,R₀) then u is locally bounded on Ω. Furthermore, for each x₀∈Ω and R ≤
min(R₀,d(x₀,∂Ω),1) there exists an universal constant c6 = c6(N,m,H) such that
Proof: The proof comes after Theorem 3 remembering that if u∈DGΦ⁻(Ω,H,R₀) then -u∈DGΦ⁺(Ω,H,R₀).
Moreover the following lemma is valid:
Lemma 5: If u∈DGΦ⁺(Ω,H,R₀) then u is locally bounded above on Ω. Furthermore, for each x₀∈Ω, R ≤
min(R₀,d(x₀,∂Ω),1) and for every p > 1 there exists an universal constant c7 = c7(p,N,m,H) such that
(1.21)
for each R
QQ
 and 0 < ϱ < R.
Proof: The proof comes after Theorem 3 using the demonstration methods presented in [24].
Definition 12: Let u∈Wloc¹LΦ(Ω) then it is a local minima of (1.1) if for every ϕ∈W₀¹LΦ(Ω) we have
Moreover we get:
Theorem 4 (Caccioppoli inequalities): If

₂ and u∈Wloc¹LΦ(Ω) is a local minima of (1.1) then
u∈ODGΦ(Ω,H,R₀).
Using the previous results we obtain the following theorems:
Theorem 5 (Weak Harnack inequality): Let Φ be a N-function. Let u be a positive function satisfying (1,17).
If Φ∈G; then there exists p > 1 and a constant c > 0 such that
(1.22)
Theorem 6 (Main Theorem-Harnack inequality): Let Φ be a N-function. Let u be a positive local minimizer of
(1.1). If Φ∈G; then there exists a constant c > 0 such that, for σ∈(0,1) we have
(1.23)
Proof (Proof of the Main Theorem): Using the (1.21) and (1.22) we have (1.23).
Corollary 2: Let Φ be a N-function. If Φ∈G and u∈W¹LΦ(Ω) is a local minimizer of the functional (1.1);
then u is locally hölder continuous.
Proof: Using (1.20) and the technique introduced in [6,11,12] we get the proof.
We finish observing that with small changes our demonstrative technique can also be applied to the quasi-mi-
nima of the functional (1.1). Besides we can also apply this demonstration using equivalent N-functions. Unfor-
tunately,




ttln1t ₂ does not verify H1; for this Φ₂∈Δ′ on [t₀,+∞) with t₀>0 but Φ₂∉△′ globally on
[0,+∞). We should think to solve this problem using the concept of equivalent N-function; the demonstrative
technique allows it, but we do not know if it exists a N-function

₃ equivalent to ₂ which globally verifies
Δ′ globally on [0,+∞). It is still an unsolved problem.I thank the colleague Dott. Elisa Albano who translated the
article into English supporting and encouraging me so much.








() (,,)
|| /
Bk BkR
udx HkuRdx 
ñ
ñ






000
,,,,,, . ODGHRODGH RODGH R


 
 



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




/2 7
/| |
QRR QR
ess supuxcQu dx


 










/2 7
|| ||/.||
QRR QR
esssupu xcQudx  






1/ 1/
7(
||) ||/
N
pp
QQR
esssupucRu dx  
ñ
ñ








,,. JusuppJusupp

 

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





1/ 1/
/2
1/ .
p
Np
QR QR
essinfucRu dx  




.
QR QR
esssupucessinfu




T. Granucci
199
2. Proof of the Weak Harnack Inequality
2.1. Lemmata
Let define
then we have the following Caccioppoli inequalities
(2.1)
and
(2.2)
where 0 < σ < τ < 1 and k∈ℝ.
Let us start remembering the following lemma:
Lemma 6: Let g(t), h(t) be a non-negative and increasing functions on [0,+∞) then g(t)h(s) ≤ g(t)h(t) + g(s)h(s)
for every s, t∈[0,+∞).
Proof: If s ≤ t then g(t)h(s) ≤ g(t)h(t) ≤ g(t)h(t) + g(s)h(s). If t ≤ s then g(t)h(s) ≤ g(s)h(s) ≤ g(t)h(t) + g(s)h(s).
Let us remember for the sake of completeness the following lemma:
Lemma 7: Let ₂ and u∈W¹LΦ(Ω). Suppose that u is positive in QR and satisfies (2.2) then there exists
a positive constants δ₀ such that if for some θ > 0 we have |B(θ,u,R)| ≤ δ₀|QR|, then
(2.3)
Proof: The proof follows using the demonstration methods presented in [24]. Refer to Lemma 4.1 of [24].
Our demonstration of the weak inequality of Harnack founds him on the following Lemma 8. We have shown
the Lemma 8 using an opportune ɛ-Young inequality.
Lemma 8: Let be Φ a N-function and Φ∈G. Let u∈W¹LΦ(Ω). Suppose that u satisfies (2.2). For every
δ∈(0,1) and T > (1/2), there exists a positive constant μ(δ,T) such that if u is positive on Q2TR and there exists θ >
0 such that |B(θ,u,R)| ≤ δ|QR|, we have
(2.4)
Proof: Let δ∈(0,1). We first prove that if u is positive in QR and there exists θ > 0 such that |B(θ,u,R)| < δ|QR|,
there exists a constant λ(θ) such that
(2.5)
We consider the function wR define by wR(y) = 0 if vR(y) ≥ k, wR(y) = k − vR if k > vR(y) > h, wR(y) = k − h if
vR(y) ≤ h
where vR(y) = ((u(Ry))/R), y∈Q₁. Let us consider ki = (θ/(2iR)) with I ≤ ν, since wR = 0 in


iR
Q\Bk,v ,1₁
and
by Sobolev inequality we have
and
(2.6)
where Δi = B(ki,vR,1)\B(ki-1,vR,1). Using the Young inequality ab ≤ Ψ(a) + Φ(b), where Φ is the complementary
function of Φ, we have








/,
R
vyuRyRyQ

 ₁









),(,, ),(
|| /
AkvRRAk vRR
vdxHvk dx

 










),(,,),(
|| /
BkvRRBk vRR
vdxHkv dx

  




/2
/2 .
QR
inf u  




,.
QTR
inf uT

 




/2 QR
inf u

 




\,,11
iR R
QBkv Q   ₁
 


/1
1/ () )1/
1
,,1 ,,1(]
NN
NNN
i iiRiRQR
kk BkvBkvwdx


 


₁
  
1/
1
,,1,,1|'
N
iiiRSNiRiR R
kkBkvCBkvww dx

 






/('/
iRRi RR
ww dxmwmw dx 



T. Granucci
200
and
then
(2.7)
Since
from the inequality
(see inequality (6), page 230 of [1]) we have
then
Moreover, since Φ globally satisfies the Δ′-condition in [0,+∞), it follows
since
using Caccioppoli’s inequality (2.2) we have
Summing the last inequality on i from 0 to ν we have
and
Fix ɛ = (1/(1 + ν)1/2), then
and
(2.9)
From (2.9) we have
Since ϖ(s)↓0 for s↓0 then we can choose ν such that
where δ₀ is the constant in Lemma 7, then there exists λ(δ₀) such that












/'//'(||/
iRRi RR
mwmwdxm wmwdx 














'|.|//
iRRi RR
wwdxmwmwdx 


 








)
'/ ('//
R
RR RRR
wmw w mwww  






/) ttt  








/||,
iRRiR R
wwdxm wwdx 



1/
1
,,1,,1/|| .
N
iiiRSNiRiRR
k kBkvC Bkvmwwdx 


 


 
1/
1 1
,,1,,1/| |
N
i iiRSNiRi iiGiR
kk BkvCBkvmkkmccwdx



 




₁




|| ||
iR iR
wdx vdx

 





11/
|
,,1 /.
N
iRSNiSN
BkvCmC mcQ



 ₂₂
   
11/
1,,1 /1
N
i RSNSN
BkvCmQC mcQ





  ₁₂ ₂
 





11/
,,1/ 1.
N
i RSNSN
BkvCmQC mcQ





 ₁₂₂
 





11/ 1/ 21/ 2
,,1/11/1
N
i RSNSN
BkvCmQC mcQ



  ₁₂ ₁
  




11/ 11/1/ 21/2
,,1 1/11/1
N
N
iR SN
BkvC mcQ 

 ₂₁
 


/1
/1 1/2( 1) 2)/(
(,,11/1 1/1.
NN
NN
iR SN
BkvC mcQ 


 ₂₁
 




()
() /1 /1 1/21/21/
0
1/11/1/ 2(1
NN
N
NNN
SN
Cm c

 

 
  ₂




/2
.
QR
inf u  

T. Granucci
201
Let now T > (1/2) and assume |B(θ,u,R)| ≤ δ|QR| and u positive in Q2R. Since
we have
then there exists a constant depending on δ and T such that (2.4) holds.
Using the technique introduces in [11] we get the following lemma.
Lemma 9: Let u∈DGΦ⁻ with k₀ = 0 and let u be positive in Q₂. Let δ∈(0,1) and t > 0. If
then
where c = c(δ) being as in Lemma 8 with δ = 2-N-1.
Proof: For s = 0 the claim is true by Lemma 8. Now we use the inductive process. We assume the claim true
for some s and we prove it for s + 1. Let us define Ai = {x∈Q₁:u(x) > cit}; by hipothesis, if A₀ = {x∈Q₁:u(x) >
t} then
We have two alternative.
1) We assume |A₀| > 2-s|Q₁|, the by inductive hypothesis
2) Otherwise 2-s-1|Q₁| < |A₀| < 2-s|Q₁|. Let us assume g = χ_{A₀} and apply the Calderon-Zygmund argument
to g in Q₁ with parameter (1/2) then we find a sequence of dyadic cubes {Qj} such that


1
1/2 ;
j
j
g
in QQ 
if Qj is one of the 2N subcubes of Pi arising during the Calderon-Zygmund process, then
From (2) and (3) we get

000 1/2; ii i
ii
i
A
AP APP 

moreover
We apply Lemma 8 and we obtain
Let us consider
then PiA₁ and
by inductive hypotesis









2
,,211/2
R
TR
A
uTRQT Q   ⁿ







2
,,21 1/2
TR
BuTRTQ

  ⁿ




:2
s
x
Qux tQ

 ₁₁


1/ 2
s
Q
infuc t 


1
0
1/ 2.
s
AQ

 ₁


1
1/2
s
s
Q
infuc tct






1
|1/ 21/2;|
N
jj
QQgdx







||1/1 /2.
iPi
P gdx 




1
00| |
1/ 21/ 2|.
N
ij ji
A
PAQ QP

 


Pi
inf uct 




:
A
xQuxct

 ₁₁


1
0
21/2
s
QA A

 ₁₁


1
1/ 2
s
Q
infuc t



T. Granucci
202
2.2. Proof of the Weak Harnack Inequality
Now we can proof the inequality (1.19) using the technique introduced by Di Benedetto-Trudinger in [6].
Proof (Proof of Theorem 5); Given any t > 0 choose an integer s such that
i.e.
then by Lemma 9 we get
therefore
Let us define
then
where α = ln((1/2)/ln(c). Since Φ'(t)t ≤ mΦ(t) for p > max{1,(m/α)} we have
Integrating by parts, we have
hence
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



:2
s
tR R
x
QuxtQ 

 




|//1/2;
tR
slnQ ln  


/2
s
QR
essinfuc t








/1)/2(
|/| .
ln cln
tR
ux tQ  


/2 QR
ess infu  


/
tR
tQ

 









1/1/ 11/1/1
], [],[0
1/1/ //
pp pp
QRtR R
udxptt dtpQmpQttdt



 
 
 
  
 







1/ 11/
,[]
/1/1/
pp
ttdt mp


















1/ 1/
/2
|| {}1/ .
pp
R
QR QR
Qudxcessinfu 

T. Granucci
203
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