Volume 2008, Article ID 835736, 11 pagesdoi:10.1155/2008/835736 Research Article Local Regularity Results for Minima of Anisotropic Functionals and Solutions of Anisotropic Equations 1 C
Trang 1Volume 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
1 College of Mathematics and Computer Science, Hebei University, Baoding 071002, China
2 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
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
Ωfx,
u, Dudx, u ∈ W 1,q i
locΩ and for solutions of anisotropic equations −divAx, u, Du −N
i1 ∂f/
∂x i, u ∈ W 1,q i
loc 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
LetΩ be an open bounded subset of RN , N ≥ 2 Let q i > 1, i 1, , N Denote
q max
1≤i≤Nq i , p min
1≤i≤Nq i , q : 1
N
N
i1
1
Throughout this paper, we will make use of the anisotropic Sobolev space
W 1, q i
loc Ω
v ∈ L qlocΩ : ∂v
∂x i ∈ L q i
locΩ, ∀i 1, , N
Let x0 ∈ Ω and t > 0, we denote by B t the ball of radius t centered at x0 For functions u and
k > 0, let A k {x ∈ Ω : |ux| > k}, A k,t A k ∩ B t Moreover, if p > 1, then pis 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
Iu; Ω
Ωfx, u, Dudx, u ∈ W 1, q i
Trang 2and weak solutions of the anisotropic equations
−divAx, u , Du −N
i1
∂f i
∂x i , u ∈ W 1, q i
We refer to the classical books by Ladyˇzenskaya and Ural’ceva 1, Morrey 2, Gilbarg and Trudinger3, and Giaquinta 4 for some details of isotropic cases
For isotropic cases, global L s-summability was proved in the 1960s by Stampacchia5 for solutions of linear elliptic equations This result was extended by Boccardo and Giachetti to the nonlinear case in6 For anisotropic cases, Giachetti and Porzio recently proved in 7 the
local L s-summability for minima of anisotropic functionals and weak solutions of anisotropic nonlinear elliptic equations Precisely, the authors considered the minima of functionals whose prototype is1.3, f is a Carath´eodory function satisfying the growth conditions
a
N
i1
ξ iq i ≤ fx, s, ξ ≤ bN
i1
ξ iq i ϕ1x, 1.5
where the function ϕ1 ∈ L r
locΩ with 1 < r < N/q The authors also considered the local solutions u ∈ W 1, q i
loc Ω of the anisotropic equations 1.4, where A : Ω × R × RN → RN is a Carath´eodory function satisfying the following structural conditions:
Ax, u, ξ·ξ ≥ m0
N
i1
ξ iq i ,
Aj x, u, ξ ≤ m1
hx
N
i1
|ξ i|q i
1−1/qj , j 1, , N,
1.6
where m l , l 0, 1 are positive constants, the function h is in L1locΩ and the functions f ibelong,
respectively, to the spaces L q i
loc 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 type1.3 with the more general growth conditions than 1.5, that
is, we assume the integrand f satisfies the following growth conditions:
N
i1
ξ iq i − buα − ϕ0x ≤ fx, u, ξ ≤ aN
i1
ξ iq i b|u| α ϕ1x, 1.7 where
ϕ0 ∈ L r1
locΩ, ϕ1∈ L r2
locΩ, r1, r2> 1, a ≥ 1, b ≥ 0,
p ≤ α < p∗, q < q∗, q < N, 1 < min
r1, r2
< N
q .
1.8
We also consider weak solutions of the type1.4 with more general growth conditions than
1.6, that is, we assume the operator A satisfies the following coercivity and growth condi-tions:
Ax, u, ξ· ξ ≥ b0
N
i1
ξ iq i − b1|u| α1− ϕ2x, 1.9
Ax,u,ξ ≤ b2
N
i1
ξ iq i−1 b3|u| α2 kx, 1.10
Trang 3where b0 ≥ 1, b i > 0, i 1, 2, 3, q < q∗, q < N, p ≤ α1 < p∗, p − 1 ≤ α2 ≤ Np − 1/N − p,
ϕ2 ∈ L r0
locΩ with r0> 1, k ∈ L r N1
loc Ω, f i ∈ L r i
locΩ, i 1, , N.
Remark 1.1 Notice that we have confined ourselves to the case q < N because when such
inequality is violated, every function in W 1,q i
locΩ is trivially in L s
locΩ for every fixed s < ∞
by7, Lemma 3.2
Remark 1.2 Since we have assumed in 1.7, 1.9, and 1.10 that the integrand f and the
operatorA 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 in8
2 Preliminary lemmas
In order to prove the local L s-integrability of the local unbounded minima of the anisotropic functionals and weak solutions of anisotropic equations, we need a useful lemma from7
locΩ, φ0∈ L r
locΩ, where q, q, and r satisfy
1 < r < N
q , q < q
Assume that the following integral estimates hold:
A k,τ
N
i1
∂x ∂u iq i dx ≤ c0
A k,t φ0dx t − τ −γ
A k,t
N
i1
|u| q i dx
2.2
for every k ∈ N and R0 ≤ τ < t ≤ R1, where c0is a positive constant that depends only on N, q i , r, R0 , R1 and |Ω| and γ is a real positive constant Then u ∈ L s
locΩ, where
∗
q
One will also need a lemma from [ 8 ].
T0 ≤ t < s ≤ T1,
where A, B, γ, θ are nonnegative constants, and θ < 1 Then there exists a constant c, depending only
Trang 43 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
u ∈ W 1,q i
locΩ, such that for every function ψ ∈ W 1,q i Ω with supp ψ ⊂⊂ Ω, it holds that
then it belongs to L slocΩ, where
q − q∗
Proof Owing toLemma 2.1, it is sufficient to prove that u satisfies the integral estimates 2.2
with γ q and φ0 ϕ0 ϕ1 Let B R1 ⊂⊂ Ω and 0 ≤ R0≤ τ < t ≤ R1be 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
It is obvious that A k A
k ∪ A−
k Denote Ak,t A
k ∩ B t and A−k,t A−
k ∩ B t Let w maxu − k, 0 Choose ψ −ηw in 3.1, where η is a cut-off function such that
supp η ⊂ B t , 0≤ η ≤ 1, η 1 in B τ , |Dη| ≤ 2t − τ−1. 3.4
We obtain from the minimality of u that
B t fx, u, Dudx ≤
B t fx, u ψ, Du Dψdx
Ak,t
f
dx
B t ∩{u≤k} fx, u, Dudx.
3.5
This implies that
Ak,t
fx, u, Dudx ≤
Ak,t
f
By1.7, we obtain
Ak,t
N
i1
∂x ∂u iq i dx
≤ b
Ak,t
u α dx
Ak,t
ϕ0dx a
Ak,t
N
i1
∂x ∂u i− ∂ηw
∂x i
q i dx b
Ak,t
u − ηw α dx
Ak,t
ϕ1dx.
3.7
Trang 5We first estimate the 3rd term on the right-hand side of3.7 Using the elementary inequality
a b q≤ 2q−1
a q b q
we obtain
a
Ak,t
N
i1
∂x ∂u i −∂ηw
∂x i
q i dx a
Ak,t \A
k,τ
N
i1
∂x ∂u i −∂ηw
∂x i
q i dx
≤ 2q−1 a
Ak,t \A
k,τ
N
i1
1 − η q i
∂x ∂u iq i
∂x ∂η iq i w q i
dx
≤ 2q−1 a
Ak,t \A
k,τ
N
i1
∂x ∂u iq i dx 2
2q−1 a
t − τ q i
Ak,t \A
k,τ
N
i1
w q i dx
≤ 2q−1 a
Ak,t \A
k,τ
N
i1
∂x ∂u iq i dx 2
2q−1 a
t − τ q
Ak,t \A
k,τ
N
i1
u q i dx
3.9
since w q i ≤ u q i in Ak,t and t − τ < 1 The summation of the 1st and the 4th terms on the
right-hand side of3.7 can be estimated as
b
Ak,t
u α dx b
Ak,t
u − ηw α dx ≤ 2b
Ak,t
Substituting3.9 and 3.10 into 3.7 yields
Ak,t
N
i1
∂x ∂u iq i dx
≤
Ak,t
ϕ0 ϕ1
dx 2b
Ak,t
u α dx 2 q−1 a
Ak,t \A
k,τ
N
i1
∂x ∂u iq i dx 22q−1 a
t − τ q
Ak,t \A
k,τ
N
i1
u q i dx.
3.11
We know from 8 that if u ∈ W 1,p B t and | supp u| ≤ 1/2|B t|, we then have the Sobolev inequality
B t
u p∗dx
p/p∗
≤ c1N, p
B t
Let
u
⎧
⎨
⎩
u, x ∈ Ak,t ,
Trang 6By assumption, p ≤ α < p∗, which implies
Ak,t u α dx
B t
u α α−p
p∗ B t1−α/p ∗
B t
u p∗dx
p/p∗
≤ c1 α−p
p∗ B t1−α/p ∗
B t
|Du| p dx
≤ c1 α−p
p∗ B t1−α/p ∗
max
1, 2 p/2−1
B t
N
i1
∂x ∂ u iq i dx
c1 α−p
p∗ B t1−α/p ∗
max
1, 2 p/2−1
Ak,t
N
i1
∂x ∂u iq i dx,
3.14
provided that|supp u| B t | ≤ 1/2|B t | We can choose T so small that for t ≤ T we get
c1 α−p p∗ B t1−α/p ∗
max
1, 2 p/2−1
≤ 1
It is obvious that
k p∗A
k
p∗
and therefore, there exists a constant k0, such that for k ≥ k0, we have
A
k ≤ 1
For such values of k we then have |supp u| < 1/2|B T/2 | and therefore, if T/2 ≤ t ≤ T,
AK,t
u α dx ≤ 1
4b
AK,t
N
i1
|∂u
Thus, from3.11 and, we get
Ak,t
N
i1
∂x ∂u iq i dx
≤ 2
Ak,t
ϕ0 ϕ1
dx 2 q a
Ak,t \A
k,τ
N
i1
∂x ∂u iq i dx 22q a
t − τ q
Ak,t \A
k,τ
N
i1
uqi dx.
3.19
A
N
i1
∂x ∂u iq i dx
≤ 2
A
ϕ0 ϕ1
dx 2 q a
A \A
N
i1
∂x ∂u iq i dx 22q a
t − τ q
A
N
i1
uq i dx.
3.20
Trang 7Adding to both sides 2q a times the left-hand side, we get eventually
A
N
i1
∂x ∂u iq i dx
2q a 1
Ak,R
ϕ0 ϕ1
q a
2q a 1
Ak,t
N
i1
∂x ∂u iq i dx 2
2q a
2q a 1· 1
t − τ q
Ak,R
N
i1
uq i dx,
3.21
we can now applyLemma 2.2to conclude that
Ak,τ
N
i1
∂x ∂u iq i dx ≤ c
2
2q a 1
Ak,t
ϕ0 ϕ1
2q a
2q a 1· 1
t − τ q
Ak,t
N
i1
|u| q i dx
, 3.22
where c depends only on q and a.
Since−u minimizes the functional
Fv; Ω
Ω
where fx, v, p fx, −v, −p satisfies the same growth conditions 1.7, inequality 3.22
holds with u replaced by −u We then conclude that
A−k,τ
N
i1
∂x ∂u iq i dx ≤ c
2
2q a 1
A−k,t
ϕ0 ϕ1
dx 22q a
2q a 1· 1
t − τ q
A−k,t
N
i1
|u| q i dx
. 3.24
Adding3.22 and 3.24 yields
A k,τ
N
i1
∂x ∂u iq i dx ≤ c
2
2q a 1
A k,t
ϕ0 ϕ1
2q a
2q a 1· 1
t − τ q
A k,t
N
i1
|u| q i dx
. 3.25
This shows that u satisfies estimates 2.2 with γ q and φ0 ϕ0 ϕ1 Theorem 3.2follows fromLemma 2.1
4 Local solutions of anisotropic equations
In this section, we prove a local regularity result for weak solutions of anisotropic equations
Let u ∈ W 1,q i
locΩ be a local solution of the anisotropic equation 1.4, where A : Ω×R×Rn→ Rn
is a Carath´eodory function satisfying the structural conditions1.9 and 1.10
Definition 4.1 By a weak solution of1.4 we mean a function u ∈ W 1,q i
locΩ, such that for every
function ψ ∈ W 1,q i Ω with supp ψ ⊂⊂ Ω it holds
supp ψ
Ax, u, Du·Dψ dx
supp ψ
where f f1, f2, , fN
Trang 8Theorem 4.2 Under the previous assumptions 1.9 and 1.10, if one assumes that ϕ2 ∈ L r0
locΩ,
f i ∈ L r i
locΩ, i 1, 2, , N, k ∈ L r N1
loc Ω, and r i , i 0, , N 1 satisfy
1 < r min
1≤i≤N
r i
qi , r0,
r N1
p
< N
then u ∈ L s
locΩ, where
Proof By virtue ofLemma 2.1, it is sufficient to prove that u satisfies the integral estimates 2.2
with γ q and φ0 ϕ2|k| pN
i1f iqi
Let B R1 ⊂⊂ Ω and 0 ≤ R0≤ τ < t ≤ R1be 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 fromDefinition 4.1 that
Ak,t
Ax, u, Du·Dηwdx
Ak,t
We now estimate the integrals in4.4 Applying the assumption 1.9, we deduce from 4.4 that
b0
Ak,τ
N
i1
∂x ∂u iq i dx ≤ b1
Ak,t
uα1
dx
Ak,t
ϕ2dx
Ak,t
t − τ
Ak,t
fw dx
2
t − τ
Ak,t \A
k,τ
The 3rd term on the right-hand side of the above inequality can be estimated as
Ak,t
f· Du dx
Ak,t
N
i1
f i· ∂u
∂x i dx ≤ ε
Ak,t
N
i1
∂x ∂u iq i dx
N
i1
C
ε, q i
Ak,t
f iqi
dx. 4.6
By Young’s inequality, the 4th term on the right-hand side of inequality4.5 can be estimated as
2
t − τ
Ak,t
|f|w dx ≤ 2q
t − τ q
Ak,t
N
i1
u − k q i
dx
N
i1
Ak,t
f iqi
By1.10, the last term on the right-hand side of 4.5 can be estimated as
2
t − τ
Ak,t \A
k,τ
|Ax, u, Du|w dx ≤ 2
t − τ
Ak,t \A
k,τ
b2
N
i1
∂x ∂u iq i−1 b3|u| α2 k
w dx I1 I2 I3.
4.8
Trang 9By Young’s inequality, we derive that
I1 ≤ b2
Ak,t \A
k,τ
N
i1
∂x ∂u iq i dx b22
q
t − τ q
Ak,t \A
k,τ
N
i1
u − k q i dx. 4.9
H ¨older’s inequality and Young’s inequality yield
I2 ≤ b3ε
Ak,t \A
k,τ
|u| α2pdx Cε, p2
p
t − τ p
Ak,t \A
k,τ
u − k p dx
≤ b3ε
Ak,t \A
k,τ
|u| α2pdx Cε, p2
p
Nt − τ q
Ak,t \A
k,τ
N
i1
u − k q i dx,
4.10
where ε is a positive constant to be determined later Further,
I3≤
Ak,t \A
k,τ
|k| pdx 2
q
Nt − τ q
Ak,t \A
k,τ
N
i1
u − k q i dx. 4.11 Combining4.6–4.11 with 4.5 yields
b0
Ak,τ
N
i1
∂x ∂u iq i dx
≤
Ak,t
ϕ2 |k| pN
i1
Cε, q i 1|f i|qi
dx b1
Ak,t
|u| α1dx b3ε
Ak,t
|u| α2pdx
ε
Ak,t
N
i1
∂x ∂u iq i dx b2
Ak,t \A
k,τ
N
i1
∂x ∂u iq i dx
b2 Cε, p 2 2q
t − τ q
Ak,t
N
i1
u − k q i
dx.
4.12
Since p ≤ α1 < p∗, then as in the proof ofTheorem 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
Ak,t
|u| α1dx ≤ 1
2b1
Ak,t
N
i1
∂x ∂u iq i dx. 4.13
Similarly, since p − 1 ≤ α2≤ Np − 1/n − p, then p ≤ α2p ≤ p∗, therefore
A
|u| α2pdx ≤ C
A
N
i1
∂x ∂u iq i dx. 4.14
Trang 10Thus, from4.12–4.14 we can derive that
b0
Ak,τ
N
i1
∂x ∂u iq i dx ≤
Ak,t
ϕ2 |k| pN
i1
Cε, q i 1f iqi
dx
1
2 Cb3 1ε
Ak,t
N
i1
∂x ∂u iqi dx b2
Ak,t \A
k,τ
N
i1
∂x ∂u iq i dx
b2 Cε, p 2 2q
t − τ q
Ak,t
N
i1
u − k q i dx.
4.15
Adding to both sides
b2
Ak,τ
N
i1
∂x ∂u iq i dx, 4.16
we get eventually
Ak,τ
N
i1
∂x ∂u iq i dx ≤ 1
b0 b2
Ak,t
ϕ2 |k| pN
i1
C
ε, q i
1|f i|qi
dx
1
2 Cb3 1ε
1
b0 b2
Ak,t
N
i1
∂x ∂u iq i dx b2
b0 b2
Ak,t
N
i1
∂x ∂u iq i dx
b2 Cε, p 2 1
b0 b2
2q
t − τ q
Ak,t
N
i1
u − k q i dx.
4.17
Choosing ε small enough, such that
θ 1/2
Cb3 1ε b2
4.17 implies that
Ak,τ
N
i1
∂x ∂u iq i dx ≤ C
Ak,t
ϕ2 |k| pN
i1
|f i|qi
dxθ
Ak,t
N
i1
∂x ∂u iq i dx C
t−τ q
Ak,t
N
i1
u − k q i dx.
4.19
A
N
i1
∂x ∂u iq i dx ≤ C
Ak,t
ϕ2 |k| pN
i1
|f i|qi
dx
θ
A
N
i1
∂x ∂u iq i dx C q
A
N
i1
u − k q i dx.
4.20
...In order to prove the local L s-integrability of the local unbounded minima of the anisotropic functionals and weak solutions of anisotropic equations, we need a... and θ < Then there exists a constant c, depending only
Trang 43 Minima of anisotropic functionals< /b>
In... functionals< /b>
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,