Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 835736, 11 pages
doi:10.1155/2008/835736
Research Article
Local Regularity Results for Minima of Anisotropic
Functionals and Solutions of Anisotropic Equations
Gao Hongya,1, 2 Qiao Jinjing,1 Wang Yong,3 and Chu Yuming4
1
College of Mathematics and Computer Science, Hebei University, Baoding 071002, China
Study Center of Mathematics of Hebei Province, Shijiazhuang 050016, China
3
Department of Mathematics, Chengde Teachers College for Nationalities, Chengde 067000, China
4
Faculty of Science, Huzhou Teachers College, Huzhou 313000, China
2
Correspondence should be addressed to Gao Hongya, hongya-gao@sohu.com
Received 13 July 2007; Accepted 21 November 2007
Recommended by Alberto Cabada
This paper gives some local regularity results for minima of anisotropic functionals I u; Ω Ω fx,
N
1,qi
u, Dudx, u ∈ Wloc Ω and for solutions of anisotropic equations −divAx, u, Du − i1 ∂f/
1,q
∂xi , u ∈ Wloc i Ω which can be regarded as generalizations of the classical results.
Copyright q 2008 Gao Hongya et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
Let Ω be an open bounded subset of RN , N ≥ 2. Let qi > 1, i 1, . . . , N. Denote
q max qi ,
p min qi ,
1≤i≤N
1≤i≤N
q:
N
1
1
1
.
q N i1 qi
Throughout this paper, we will make use of the anisotropic Sobolev space
∂v
1, q
q
qi
Wloc i Ω v ∈ Lloc Ω :
∈ Lloc
Ω, ∀i 1, . . . , N .
∂xi
1.1
1.2
Let x0 ∈ Ω and t > 0, we denote by Bt the ball of radius t centered at x0 . For functions u and
k > 0, let Ak {x ∈ Ω : |ux| > k}, Ak,t Ak ∩ Bt . Moreover, if p > 1, then p is always the real
number p/p − 1, and if s < N, s∗ is always the real number satisfying 1/s∗ 1/s − 1/N.
This paper mainly considers the functions u minimizing the anisotropic functionals
1, q
Iu; Ω fx, u, Dudx, u ∈ Wloc i Ω
1.3
Ω
2
Journal of Inequalities and Applications
and weak solutions of the anisotropic equations
−divAx, u , Du −
N
∂fi
∂xi
i1
,
1, q
u ∈ Wloc i Ω.
1.4
We refer to the classical books by Ladyženskaya and Ural’ceva 1, Morrey 2, Gilbarg and
Trudinger 3, and Giaquinta 4 for some details of isotropic cases.
For isotropic cases, global Ls -summability was proved in the 1960s by Stampacchia 5
for solutions of linear elliptic equations. This result was extended by Boccardo and Giachetti to
the nonlinear case in 6. For anisotropic cases, Giachetti and Porzio recently proved in 7 the
local Ls -summability for minima of anisotropic functionals and weak solutions of anisotropic
nonlinear elliptic equations. Precisely, the authors considered the minima of functionals whose
prototype is 1.3, f is a Carathéodory function satisfying the growth conditions
N N q
q
a ξi i ≤ fx, s, ξ ≤ b ξi i ϕ1 x,
i1
1.5
i1
where the function ϕ1 ∈ Lrloc Ω with 1 < r < N/q. The authors also considered the local
1, q
solutions u ∈ Wloc i Ω of the anisotropic equations 1.4, where A : Ω × R × RN → RN is a
Carathéodory function satisfying the following structural conditions:
Ax, u, ξ·ξ ≥ m0
N ξi qi ,
i1
N
Aj x, u, ξ ≤ m1 hx |ξi |qi
1.6
1−1/qj
,
j 1, . . . , N,
i1
where ml , l 0, 1 are positive constants, the function h is in L1loc Ω and the functions fi belong,
q respectively, to the spaces Lloci Ω. Under the above conditions, the authors obtained some
local regularity results.
The aim of the present paper is to prove the local regularity property for minima of the
anisotropic functionals of type 1.3 with the more general growth conditions than 1.5, that
is, we assume the integrand f satisfies the following growth conditions:
N N ξi qi − buα − ϕ0 x ≤ fx, u, ξ ≤ a ξi qi b|u|α ϕ1 x,
i1
1.7
i1
where
1
Ω,
ϕ0 ∈ Lrloc
p ≤ α < p∗ ,
2
ϕ1 ∈ Lrloc
Ω,
q < q ∗,
q < N,
r1 , r2 > 1,
a ≥ 1, b ≥ 0,
1 < min r1 , r2 <
N
.
q
1.8
We also consider weak solutions of the type 1.4 with more general growth conditions than
1.6, that is, we assume the operator A satisfies the following coercivity and growth conditions:
N q
Ax, u, ξ· ξ ≥ b0 ξi i − b1 |u|α1 − ϕ2 x,
1.9
i1
N Ax, u, ξ ≤ b2 ξi qi −1 b3 |u|α2 kx,
i1
1.10
Gao Hongya et al.
3
where b0 ≥ 1, bi > 0, i 1, 2, 3, q < q ∗ , q < N, p ≤ α1 < p∗ , p − 1 ≤ α2 ≤ Np − 1/N − p,
0
i
N1
ϕ2 ∈ Lrloc
Ω with r0 > 1, k ∈ Lrloc
Ω, fi ∈ Lrloc
Ω, i 1, . . . , N.
Remark 1.1. Notice that we have confined ourselves to the case q < N because when such
1,q
inequality is violated, every function in Wloc i Ω is trivially in Lsloc Ω for every fixed s < ∞
by 7, Lemma 3.2.
Remark 1.2. Since we have assumed in 1.7, 1.9, and 1.10 that the integrand f and the
operator A satisfy some growth conditions depending on u, in the proof of the local regularity
results, we have to estimate the integral of some power of |u| by means of |Du|. To do this, we
will make use of the Sobolev inequality that has been used in 8.
2. Preliminary lemmas
In order to prove the local Ls -integrability of the local unbounded minima of the anisotropic
functionals and weak solutions of anisotropic equations, we need a useful lemma from 7.
1,q
Lemma 2.1. Let u ∈ Wloc i Ω, φ0 ∈ Lrloc Ω, where q, q, and r satisfy
1<r<
N
,
q
q < q ∗,
2.1
q < N.
Assume that the following integral estimates hold:
N ∂u qi
dx ≤ c0
∂x Ak,τ i1
i
φ0 dx t − τ
−γ
Ak,t
N
|u|qi dx
2.2
Ak,t i1
for every k ∈ N and R0 ≤ τ < t ≤ R1 , where c0 is a positive constant that depends only on N, qi , r, R0 ,
R1 and |Ω| and γ is a real positive constant. Then u ∈ Lsloc Ω, where
s
q ∗q
.
q − q ∗ 1 − 1/r
2.3
One will also need a lemma from [8].
Lemma 2.2. Let ft be a nonnegative bounded function defined for 0 ≤ T0 ≤ t ≤ T1 . Suppose that for
T0 ≤ t < s ≤ T1 ,
ft ≤ As − t−γ B θfs,
2.4
where A, B, γ, θ are nonnegative constants, and θ < 1. Then there exists a constant c, depending only
on γ and θ such that for every , R, T0 ≤ < R ≤ T1 , one has
f ≤ c AR − −γ B .
2.5
4
Journal of Inequalities and Applications
3. Minima of anisotropic functionals
In this section, we prove a local regularity result for minima of anisotropic functionals.
Definition 3.1. By a local minimum of the anisotropic functional I in 1.3, we mean a function
1,q
u ∈ Wloc i Ω, such that for every function ψ ∈ W 1,qi Ω with supp ψ ⊂⊂ Ω, it holds that
Iu; supp ψ ≤ Iu ψ; supp ψ.
3.1
Theorem 3.2. Assume that the functional I satisfies the conditions 1.7. If u is a local minimum of I,
then it belongs to Lsloc Ω, where
q ∗q
.
q − q 1 − 1/ min r1 , r2
s
3.2
∗
Proof. Owing to Lemma 2.1, it is sufficient to prove that u satisfies the integral estimates 2.2
with γ q and φ0 ϕ0 ϕ1 . Let BR1 ⊂⊂ Ω and 0 ≤ R0 ≤ τ < t ≤ R1 be arbitrarily but fixed. It is
no loss of generality to assume that R1 − R0 < 1. For k > 0, let
Ak x ∈ Ω : ux > k ,
A−k x ∈ Ω : ux < −k .
3.3
It is obvious that Ak Ak ∪ A−k . Denote Ak,t Ak ∩ Bt and A−k,t A−k ∩ Bt . Let w maxu − k, 0.
Choose ψ −ηw in 3.1, where η is a cut-off function such that
supp η ⊂ Bt ,
0 ≤ η ≤ 1, η 1 in B τ ,
|Dη| ≤ 2t − τ−1 .
3.4
We obtain from the minimality of u that
fx, u, Dudx ≤
Bt
fx, u ψ, Du Dψdx
Bt
Ak,t
f x, u − ηw, Du − Dηw dx 3.5
Bt ∩{u≤k}
fx, u, Dudx.
This implies that
Ak,t
fx, u, Dudx ≤
Ak,t
f x, u − ηw, Du − Dηw dx.
3.6
By 1.7, we obtain
N ∂u qi
∂x dx
Ak,t i1
i
N ∂u ∂ηw qi
α
≤b
u dx ϕ0 dx a
−
dx b
u − ηw dx ϕ1 dx.
∂xi Ak,t
Ak,t
Ak,t i1 ∂xi
Ak,t
Ak,t
α
3.7
Gao Hongya et al.
5
We first estimate the 3rd term on the right-hand side of 3.7. Using the elementary inequality
a bq ≤ 2q−1 aq bq ,
a, b ≥ 0, q ≥ 1,
3.8
we obtain
N N ∂u ∂ηw qi
∂u ∂ηw qi
dx a
dx
−
−
∂xi ∂xi Ak,t i1 ∂xi
Ak,t \Ak,τ i1 ∂xi
a
≤ 2q−1 a
Ak,t \Ak,τ i1
≤2
q−1
a
Ak,t \Ak,τ
≤ 2q−1 a
∂u qi ∂η qi q
w i dx
1 − ηqi ∂x ∂xi i
N Ak,t \Ak,τ
N N
2q−1
∂u qi
a
dx 2
wqi dx
qi
∂x t
−
τ
i
Ak,t \Ak,τ i1
i1
3.9
N N
2q−1
∂u qi
a
dx 2
uqi dx
q
∂x t
−
τ
i
Ak,t \Ak,τ i1
i1
since wqi ≤ uqi in Ak,t and t − τ < 1. The summation of the 1st and the 4th terms on the righthand side of 3.7 can be estimated as
b
Ak,t
uα dx b
Ak,t
u − ηwα dx ≤ 2b
Ak,t
uα dx.
3.10
Substituting 3.9 and 3.10 into 3.7 yields
N ∂u qi
∂x dx
Ak,t i1
≤
i
Ak,t
ϕ0 ϕ1 dx 2b
Ak,t
uα dx 2q−1 a
Ak,t \Ak,τ
N N
2q−1
∂u qi
a
dx 2
uqi dx.
q
∂x t
−
τ
i
Ak,t \Ak,τ i1
i1
3.11
We know from 8 that if u
∈ W 1,p Bt and | supp u
| ≤ 1/2|Bt |, we then have the Sobolev
inequality
p∗
u
dx
p/p∗
≤ c1 N, p
Bt
|Du
|p dx.
3.12
Bt
Let
u
⎧
⎨u,
x ∈ Ak,t ,
⎩0,
x ∈ Ω \ Ak,t .
3.13
6
Journal of Inequalities and Applications
By assumption, p ≤ α < p∗ , which implies
u dx u
dx ≤ u
α
Ak,t
α
Bt
≤ c1 u
≤ c1 u
c1 u
∗
α−p 1−α/p
p∗ Bt
u
dx
p/p∗
Bt
∗
α−p 1−α/p
p∗ Bt
p∗
|Du
|p dx
Bt
∗
α−p 1−α/p
p∗ Bt
max 1, 2
p/2−1
qi
N ∂
u dx
∂x Bt i1
3.14
i
∗
α−p 1−α/p
p∗ Bt
max 1, 2p/2−1
N ∂u qi
dx,
Ak,t i1 ∂xi
provided that |supp u
|Bt | ≤ 1/2|Bt |. We can choose T so small that for t ≤ T we get
c1 u
∗
α−p 1−α/p
p∗ Bt
max 1, 2p/2−1 ≤
1
.
4b
3.15
It is obvious that
∗
k p Ak ≤ u
p∗
p∗ , Ω ,
3.16
and therefore, there exists a constant k0 , such that for k ≥ k0 , we have
1 A ≤ BT/2 .
k
2
3.17
For such values of k we then have |supp u
| < 1/2|BT/2 | and therefore, if T/2 ≤ t ≤ T,
1
u dx ≤
4b
AK,t
α
N
∂u qi
|
| dx.
AK,t i1 ∂xi
3.18
Thus, from 3.11 and, we get
N ∂u qi
∂x dx
Ak,t i1
i
≤2
Ak,t
ϕ0 ϕ1 dx 2q a
Suppose now T/2 ≤
Ak,t \Ak,τ
N N 2q
∂u qi
qi
dx 2 a
u dx.
∂x t − τq Ak,t \Ak,τ i1 i
i1
≤ τ < t ≤ R ≤ T , we get
N ∂u qi
∂x dx
Ak, i1
i
≤2
Ak,R
3.19
ϕ0 ϕ1 dx 2q a
Ak,t \Ak,
N N 2q
∂u qi
dx 2 a
uqi dx.
q
∂x t
−
τ
i
A
i1
k,R i1
3.20
Gao Hongya et al.
7
Adding to both sides 2q a times the left-hand side, we get eventually
N ∂u qi
∂x dx
Ak, i1
≤
i
2
q
2 a1
Ak,R
ϕ0 ϕ1 dx 2q a
q
2 a1
N N 2q
∂u qi
uqi dx,
dx 2 a · 1
2q a 1 t − τq Ak,R i1
Ak,t i1 ∂xi
3.21
we can now apply Lemma 2.2 to conclude that
N ∂u qi
2
dx ≤ c
∂x 2q a 1
Ak,τ i1
22q a
1
ϕ0 ϕ1 dx q
·
2 a 1 t − τq
Ak,t
i
where c depends only on q and a.
Since −u minimizes the functional
Ω Fv;
Ω
N
Ak,t i1
|u|qi dx ,
v, Dvdx,
fx,
3.22
3.23
v, p fx, −v, −p satisfies the same growth conditions 1.7, inequality 3.22
where fx,
holds with u replaced by −u. We then conclude that
N ∂u qi
2
dx ≤ c
∂x 2q a 1
A−k,τ i1
A−k,t
i
22q a
1
ϕ0 ϕ1 dx q
·
2 a 1 t − τq
N
A−k,t i1
|u|qi dx .
3.24
|u|qi dx .
3.25
Adding 3.22 and 3.24 yields
N ∂u qi
2
dx ≤ c
∂x 2q a 1
Ak,τ i1
i
Ak,t
1
22q a
·
ϕ0 ϕ1 dx q
2 a 1 t − τq
N
Ak,t i1
This shows that u satisfies estimates 2.2 with γ q and φ0 ϕ0 ϕ1 . Theorem 3.2 follows
from Lemma 2.1.
4. Local solutions of anisotropic equations
In this section, we prove a local regularity result for weak solutions of anisotropic equations.
1,q
Let u ∈ Wloc i Ω be a local solution of the anisotropic equation 1.4, where A : Ω×R×Rn → Rn
is a Carathéodory function satisfying the structural conditions 1.9 and 1.10.
1,q
Definition 4.1. By a weak solution of 1.4 we mean a function u ∈ Wloc i Ω, such that for every
function ψ ∈ W 1,qi Ω with supp ψ ⊂⊂ Ω it holds
Ax, u, Du·Dψ dx supp ψ
where f f1 , f2 , . . . , fN .
f·Dψ dx,
supp ψ
4.1
8
Journal of Inequalities and Applications
0
Theorem 4.2. Under the previous assumptions 1.9 and 1.10, if one assumes that ϕ2 ∈ Lrloc
Ω,
ri
rN1
fi ∈ Lloc Ω, i 1, 2, . . . , N, k ∈ Lloc Ω, and ri , i 0, . . . , N 1 satisfy
1 < r min
1≤i≤N
ri
rN1
, r0 ,
p
qi
<
N
,
q
4.2
then u ∈ Lsloc Ω, where
q ∗q
.
q − q ∗ 1 − 1/r
s
4.3
Proof. By virtue of Lemma 2.1, it is sufficient to prove that u satisfies the integral estimates 2.2
qi
with γ q and φ0 ϕ2 |k|p N
i1 fi . Let BR1 ⊂⊂ Ω and 0 ≤ R0 ≤ τ < t ≤ R1 be arbitrarily but
fixed. Assume again that R1 −R0 < 1. Let w max{u−k, 0}. Choose ψ ηw as a test function in
4.1, where the cut-off function η satisfies the conditions 3.4. We obtain from Definition 4.1
that
Ax, u, Du·Dηwdx f· Dηwdx.
4.4
Ak,t
Ak,t
We now estimate the integrals in 4.4. Applying the assumption 1.9, we deduce from 4.4
that
b0
N ∂u qi
dx ≤ b1
∂x Ak,τ i1
Ak,t
i
2
t−τ
α1
u dx Ak,t \Ak,τ
Ak,t
ϕ2 dx Ak,t
f·Du dx 2
t−τ
Ak,t
f w dx
Ax, u, Duw dx.
4.5
The 3rd term on the right-hand side of the above inequality can be estimated as
Ak,t
f· Du dx Ak,t
N
∂u
fi ·
dx ≤ ε
∂x
i
i1
Ak,t
N N
∂u qi
dx C ε, qi
∂x i1
i
Ak,t
i1
q fi i dx.
4.6
By Young’s inequality, the 4th term on the right-hand side of inequality 4.5 can be estimated
as
2
t−τ
2q
|f|w dx ≤
t − τq
Ak,t
Ak,t
N
i1
qi
u − k dx N i1
Ak,t
q
fi i dx.
4.7
By 1.10, the last term on the right-hand side of 4.5 can be estimated as
2
t−τ
Ak,t \Ak,τ
2
|Ax, u, Du|w dx ≤
t−τ
Ak,t \Ak,τ
N ∂u qi −1
α2
b3 |u| k w dx I1 I2 I3 .
b2 ∂xi i1
4.8
Gao Hongya et al.
9
By Young’s inequality, we derive that
I1 ≤ b 2
Ak,t \Ak,τ
N N
q
∂u qi
dx b2 2
u − kqi dx.
q
∂x t
−
τ
i
A
\A
i1
k,t
k,τ i1
4.9
Hölder’s inequality and Young’s inequality yield
I2 ≤ b3 ε
Ak,t \Ak,τ
|u|α2 p dx ≤ b3 ε
α2 p
Ak,t \Ak,τ
|u|
Cε, p2p
t − τp
Cε, p2p
dx Nt − τq
Ak,t \Ak,τ
u − kp dx
N
Ak,t \Ak,τ i1
4.10
u − k dx,
qi
where ε is a positive constant to be determined later. Further,
I3 ≤
2q
|k| dx Nt − τq
p
Ak,t \Ak,τ
N
u − kqi dx.
4.11
Ak,t \Ak,τ i1
Combining 4.6–4.11 with 4.5 yields
b0
N ∂u qi
∂x dx
Ak,τ i1
i
≤
Ak,t
N
ϕ2 |k|p Cε, qi 1 |fi |qi dx b1
ε
Ak,t
i1
i
N ∂u q
dx b2
∂x Ak,t i1
2q
t − τq
Ak,t
i
N ∂u q
∂x dx
Ak,t \Ak,τ i1
i
b2 Cε, p 2
|u|α1 dx b3 ε
N
Ak,t i1
|u|α2 p dx
4.12
i
i
u − kq dx.
Since p ≤ α1 < p∗ , then as in the proof of Theorem 3.2, we know that there exist a sufficiently
small T and a sufficiently large k0 , such that for all T/2 ≤ t ≤ T and k ≥ k0 , we have
1
|u| dx ≤
2b
1
Ak,t
α1
N ∂u qi
∂x dx.
Ak,t i1
i
4.13
Similarly, since p − 1 ≤ α2 ≤ Np − 1/n − p, then p ≤ α2 p ≤ p∗ , therefore
Ak,t
|u|α2 p dx ≤ C
N ∂u qi
∂x dx.
Ak,t i1
i
4.14
10
Journal of Inequalities and Applications
Thus, from 4.12–4.14 we can derive that
b0
N ∂u qi
dx ≤
∂x Ak,τ i1
Ak,t
i
N
q ϕ2 |k|p Cε, qi 1 fi i dx
i1
1
Cb3 1ε
2
b2 Cε, p 2
N ∂u qi
dx b2
∂x Ak,t i1
2q
t − τq
Ak,t \Ak,τ i1
i
N ∂u qi
∂x dx
i
4.15
N
u − kqi dx.
Ak,t i1
Adding to both sides
b2
N ∂u qi
∂x dx,
Ak,τ i1
4.16
i
we get eventually
Ak,τ
N N
∂u qi
q
1
p
dx ≤
i
ϕ
dx
|k|
|
C
ε,
q
1
|f
2
i
i
∂x b0 b2 Ak,t
i
i1
i1
N N ∂u qi
∂u qi
1 1
dx b2
dx
Cb3 1 ε
2
b0 b2 Ak,t i1 ∂xi b0 b2 Ak,t i1 ∂xi b2 Cε, p 2
1
2q
b0 b2 t − τq
Ak,t
N
u − kqi dx.
i1
4.17
Choosing ε small enough, such that
θ
1/2 Cb3 1 ε b2
< 1,
b0 b2
4.18
4.17 implies that
Ak,τ
N N
∂u qi
p
qi
dx ≤ C
ϕ
dxθ
|k|
|f
|
2
i
∂x i1
i
Ak,t
Suppose now that T/2 ≤
i1
i
i1
i
Ak,t
N
i1
u − kqi dx.
4.19
≤ τ < t ≤ R ≤ T , we get
N ∂u qi
dx ≤ C
∂x Ak, i1
Ak,t
N ∂u qi
dx C
∂x t−τq
Ak,t
N
ϕ2 |k|p |fi |qi dx
θ
Ak,R
i1
qi
N
C
∂u dx u − kqi dx.
q
∂x R
−
i
A
i1
k,t i1
N 4.20
Gao Hongya et al.
11
Applying Lemma 2.2, we conclude that
Ak,τ
N ∂u qi
dx ≤ cC
∂x i1
i
Ak,t
ϕ2 |k|p N fi qi dx i1
cC
t − τq
Ak,t
N
u − kqi dx.
4.21
i1
Since −u is a weak solution of
u, Du −
−div Ax,
N
∂fi
i1
∂xi
4.22
,
s, ξ Ax, −s, −ξ satisfies the same conditions 1.9 and 1.10, inequality 4.21
where Ax,
holds with u replaced by −u. We then conclude that
N ∂u qi
dx ≤ cC
∂x A−k,τ i1
i
A−k,t
N q
p
ϕ2 |k| fi i dx i1
cC
t − τq
N
A−k,t i1
u − kqi dx.
4.23
u − kqi dx.
4.24
Adding 4.21 with 4.23 yields
N ∂u qi
dx ≤ cC
∂x Ak,τ i1
i
N q
ϕ2 |k|p fi i dx Ak,t
i1
Thus, u satisfies 2.2 with φ0 ϕ2 |k|p Lemma 2.1.
cC
t − τq
N
Ak,t i1
N qi
and α q. Theorem 4.2 follows from
i1 fi
Acknowledgments
Gao Hongya is supported by Special Fund of Mathematics Research of Natural Science Foundation of Hebei Province no. 07M003 and Doctoral Foundation of the Department of Education of Hebei Province B2004103. Chu Yuming is supported by NSF of Zhejiang Province
No. Y607128 and NSFC no. 10771195. This research was done when the first author was visiting Institute of Mathematics, Nankai University. He wished to thank the Institute for support
and hospitality.
References
1 O. A. Ladyženskaya and N. N. Ural’ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New
York, NY, USA, 1968.
2 C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, Berlin, Germany, 1968.
3 D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin,
Germany, 1977.
4 M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of
Mathematics Studies, no. 105, Princeton University Press, Princeton, NJ, USA, 1983.
5 G. Stampacchia, Èquations Elliptiques du Second Ordre á Coefficients Discontinus, Séminaire de
Mathématiques Supérieures, Les Presses de l’Université de Montréal, Montreal, Quebec, Canada, 1966.
6 L. Boccardo and D. Giachetti, “Alcune osservazioni sulla regolaritá delle soluzioni di problemi nonlineari e applicazioni,” Ricerche di Matematica, vol. 34, pp. 309–323, 1985.
7 D. Giachetti and M. M. Porzio, “Local regularity results for minima of functionals of the calculus of
variation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 39, no. 4, pp. 463–482, 2000.
8 M. Giaquinta and E. Giusti, “On the regularity of the minima of variational integrals,” Acta Mathematica,
vol. 148, no. 1, pp. 31–46, 1982.