Global integrability related to anisotropic operators

J. Math. Anal. Appl. 442 (2016) 244–258

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Global integrability related to anisotropic operators Gao Hongya ∗ , Liang Shuang, Cui Yi College of Mathematics and Information Science, Hebei University, Baoding, 071002, China

a r t i c l e

i n f o

Article history: Received 19 January 2016 Available online 27 April 2016 Submitted by B. Kaltenbacher Keywords: Global integrability Anisotropic elliptic equation Anisotropic integral functional

a b s t r a c t We consider boundary value problems of the form ⎧  n n  ⎨ Di (ai (x, Du(x))) = Di f i (x), i=1 i=1 ⎩ u(x) = u∗ (x),

in Ω, on ∂Ω,

where ai : Ω × Rn → R, i = 1, 2, · · · , n, are Carathéodory functions satisfying n 



|ai (x, z)|pi ≤ c1

i=1

n 

|zi |pi + g1 (x),

i=1

and c2

n 

|zi |pi − g2 (x) ≤

i=1

n 

ai (x, z)zi

i=1

for some positive constants c1 , c2 and some functions g1 , g2 . We also consider 1,(p ) minimizers u ∈ u∗ + W0 i (Ω) of the integral functional  I(u) =

f (x, Du(x))dx, Ω

where the integrand f (x, z) : Ω × Rn → [0, +∞) satisfies

c3

n  i=1

⎛ ⎝

n 

⎞ pi −2 pi

pj ⎠

|zj |

|zi |2 − g3 (x) ≤ f (x, z)

j=1

≤ c4

n  i=1

⎛ ⎝

n 

⎞ pi −2 pi

pj ⎠

|zj |

|zi |2 + g4 (x)

j=1

for some positive constants c3 , c4 and some functions g3 , g4 . We show, by a different method from the classical ones, that higher integrability of the boundary datum u∗ * Corresponding author. E-mail addresses: [email protected] (H. Gao), [email protected] (S. Liang), [email protected] (Y. Cui). http://dx.doi.org/10.1016/j.jmaa.2016.04.057 0022-247X/© 2016 Elsevier Inc. All rights reserved.

H. Gao et al. / J. Math. Anal. Appl. 442 (2016) 244–258

245

forces u to have higher integrability as well. Similar results are also obtained for obstacle problems. © 2016 Elsevier Inc. All rights reserved.

1. Introduction and statement of results Let Ω be a bounded open subset of Rn , n ≥ 2. For p1 , · · · , pn ∈ (1, +∞), let p¯ : pi

1 p¯

=

1 n

n  i=1

1 pi

and

= be the harmonic mean of p1 , · · · , pn and the Hölder conjugate of pi , respectively. In this paper we np¯ 1,(pi ) assume p¯ < n and we introduce the Sobolev exponent p¯∗ = n− (Ω) p¯ . The anisotropic Sobolev space W is defined as usual by pi pi −1



W 1,(pi ) (Ω) = v ∈ W 1,1 (Ω) : Di v ∈ Lpi (Ω) for every i = 1, · · · , n , and W0 i (Ω) is denoted to be the closure of C0∞ (Ω) in the norm of W 1,(pi ) (Ω). We refer to [9,17] for the ∂v theory of these spaces. The word anisotropic means that the exponent pi of the derivative Di v = ∂x might i be different from the exponent pj of the derivative Dj v when i = j. We also work in weak Lq spaces, known also as Marcinkiewicz spaces: if q > 1, then the space Lqweak (Ω) consists of measurable functions g on Ω such that 1,(p )

1

sup t|{x ∈ Ω : |g(x)| > t}| q < ∞.

(1.1)

t>0

This condition is equivalently stated as |||g|||q =

sup E⊂Ω,|E|>0



1

|g|dx < ∞.

1

|E| q

(1.2)

E

It is a rather easy exercise to prove that Lqweak (Ω) is a Banach space under ||| · |||q and, moreover, if the supremum in (1.1) is denoted by Aq (g), then Aq (g) ≤ |||g|||q ≤ q  Aq (g).

(1.3)

We recall that Lq (Ω) is a proper subspace of Lqweak (Ω), and if g ∈ Lqweak (Ω) for some q > 1, then g ∈ Lq−ε (Ω) for every 0 < ε ≤ q − 1. For a detailed analysis of Lqweak spaces we refer to [15]. With the above notations and symbols in hands, we now consider weak solutions to boundary value problems of the form ⎧ n n   ⎪ ⎨ Di (ai (x, Du(x))) = Di f i (x), x ∈ Ω, i=1 i=1 (1.4) ⎪ ⎩ u(x) = u∗ (x), x ∈ ∂Ω, where ai : Ω × Rn → R with x → ai (x, z) measurable and z → ai (x, z) continuous, and satisfying n 



|ai (x, z)|pi ≤ c1

i=1

n 

|zi |pi + g1 (x),

(1.5)

i=1

and c2

n  i=1

|zi |pi − g2 (x) ≤

n  i=1

ai (x, z)zi

(1.6)

H. Gao et al. / J. Math. Anal. Appl. 442 (2016) 244–258

246

for some positive constants c1 , c2 . As far as the boundary datum u∗ , the nonhomogeneous terms f i , and the functions g1 , g2 in (1.5) and (1.6) are concerned, we assume that there exist r > np¯ and qi > pi , i = 1, · · · , n, such that 

u∗ ∈ W 1,(qi ) (Ω), f i ∈ Lrpi (Ω), i = 1, 2, · · · , n, g1 , g2 ∈ Lr (Ω).

(1.7)

The anisotropic framework seems to be useful when dealing with some reinforced materials, see [16]. We recall that problems as (1.4) have been studied by several authors. In [12], Leonetti and Siepe studied (1.4) n with i=1 Di f i (x) = 0, and obtained a global integrability result, which shows that higher integrability of the boundary datum u∗ forces solutions u to have higher integrability. The main tool used in [12] is a technical lemma, see [12, Lemma 3.2] at page 2872 and [14, Lemma 4.1] at page 19. Gao, Liu and Tian [5] gave some remarks on the above mentioned paper by showing that obstacle problems can also be considered and the right-hand side of the equation in (1.4) can be nonhomogeneous. In the meantime, the proof was simplified. See [1,13] for some related results. Recently, Kovalevsky [8] gave a series of results on the improved integrability and boundedness of solutions to several anisotropic problems. In particular, he considered nonhomogeneous Dirichlet problem for anisotropic elliptic second-order equations. Moreover, he studied anisotropic variational inequalities with unilateral and bilateral obstacles as well as gradient constraints. Finally, variational problems for integral functionals with anisotropic integrands are considered. In the present paper, we continue to consider problem (1.4). We provide an alternative approach, that is, we give a sufficient condition ensuring weak integrability of functions, see Lemma 2.2 in Section 2, then we derive a global integrability result. The first result in the present paper is the following theorem. Theorem 1.1. Let r > verify   n

n p¯ .

1,(pi )

Under the previous assumptions (1.5), (1.6) and (1.7), let u ∈ u∗ + W0

ai (x, Du(x))Di ϕ(x)dx =

Ω i=1

  n

1,(pi )

fi (x)Di ϕ(x)dx, ∀ϕ ∈ W0

(Ω).

(Ω)

(1.8)

Ω i=1

Then 1 u ∈ u∗ + LTweak (Ω)

where p¯p¯∗ > p¯∗ , p¯ − b1 p¯∗

T1 =

(1.9)

and b1 is any number such that   pi p¯ and b1 < 1 − . 0 < b1 ≤ min 1 − 1≤i≤n qi n

(1.10)

Remark 1.1. The result obtained in Theorem 1.1 is the same as Corollary 3.8 in [8] (in case of f = 0) and is better than [12, Theorem 2.1]. Meanwhile, we use an alternative method to derive our result. 1,(pi )

We also consider minimizers u ∈ u∗ + W0

(Ω) of the integral functional

 I(u) =

f (x, Du(x))dx, Ω

(1.11)

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247

where the integrand f (x, z) : Ω × Rn → [0, +∞) is assumed to satisfy the growth conditions

c3

n 

⎛ ⎝

i=1

n 

⎞ pip−2 i

pj ⎠

|zj |

|zi | − g3 (x) ≤ f (x, z) ≤ c4 2

j=1

n 

⎛ ⎝

i=1

n 

⎞ pip−2 i

pj ⎠

|zj |

|zi |2 + g4 (x)

(1.12)

j=1

for some positive constants c3 , c4 . A prototype satisfying (1.12) is the integrand of the integral functional 

⎛⎛ ⎜⎝ ⎝

n 

⎛

⎞ p1p−2 1

|D1 u|2 + · · · + ⎝

|Dj u|pj ⎠

j=1

Ω

n 

⎞ pnp −2 n

|Dj u|pj ⎠

⎞ ⎟ |Dn u|2 ⎠ dx.

j=1

For the boundary datum u∗ and the functions g3 , g4 in (1.12) we assume that there exists 1 < σ < that u∗ ∈ W 1,(σpi ) (Ω) and g3 , g4 ∈ Lσ (Ω). 1,(pi )

For minimizers u ∈ u∗ +W0 we have the following result.

n p¯

such

(1.13)

(Ω) of the integral functional (1.11) under the conditions (1.12) and (1.13),

1,(pi )

Theorem 1.2. Under the previous assumptions (1.12) and (1.13), let u ∈ u∗ + W0 integral functionals of (1.11), that is, 

 f (x, Du(x))dx ≤ Ω

1,(pi )

∀w ∈ u∗ + W0

f (x, Dw(x))dx,

(Ω) minimize the

(Ω).

(1.14)

Ω

Then, 2 u ∈ u∗ + LTweak (Ω),

(1.15)

where T2 =

n¯ pσ . n − p¯σ

(1.16)

Remark 1.2. In [8], Kovalevsky considered the integral functional (1.11) under the condition c˜3

n 

|zi |pi − g3 (x) ≤ f (x, z) ≤ c˜4

i=1

n 

|zi |pi + g4 (x).

i=1

We compare (1.17) with (1.12). Note that

|zi |2 = (|zi |pi )

2 pi

⎛ ⎞ p2 i n  ≤⎝ |zj |pj ⎠ , j=1

thus n  i=1

⎛ ⎞ pip−2 i n n   ⎝ |zj |pj ⎠ |zi |2 ≤ n |zj |pj . j=1

j=1

(1.17)

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248

Let j∗ be such that |zj∗ |pj∗ = max |zj |pj , j

then ⎛ ⎝

n 

⎞− p2

i

− p2

|zj |pj ⎠

≥ (n|zj∗ |pj∗ )

i

− p2

=n

− p2

(|zj∗ |pj∗ )

i

i

− min2 p

≥n

j

j

− p2

(|zj∗ |pj∗ )

i

,

j=1

so that n 

⎛ ⎝

i=1

⎛ ≥⎝ ⎛ ≥⎝

i

|zj |

n 

2

pj ⎠

|zj |

j=1

⎞ − min2 p

|zj |pj ⎠ n

j=1

⎞

n 

|zi | = ⎝

pj ⎠

j=1

n 

⎛

⎞ pip−2

n 

j

j

n 

− min2 p

|zj |pj ⎠ n

j

j

⎛ ⎝

i=1 − p2

(|zj∗ |pj∗ )

i

⎛ − p2

(|zj∗ |pj∗ )

j∗

n 

⎞− p2

i

pj ⎠

|zj |

|zi |2

j=1

|zi |2

i=1

⎞

n 

|zj∗ |2 = ⎝

j=1

n 

⎞ − min2 p

|zj |pj ⎠ n

j

j

.

j=1

This means that − min2 p

n

j

j

n 

n 

|zi |pi ≤

i=1

⎛ ⎞ pip−2 i n n   pj ⎠ 2 ⎝ |zj | |zi | ≤ n |zi |pi ,

i=1

j=1

− min2 p

so that (1.12) implies (1.17) with c˜3 = n c3 =

1 ˜3 nc

and c4 = n

2 min pj j

j

j

i=1

c3 and c˜4 = nc4 . Conversely, (1.17) implies (1.12) with

c˜4 .

We also consider obstacle problems. Let   (pi ) 1,(pi ) 1,(pi ) Kψ,u (Ω) = v ∈ W (Ω) : v ≥ ψ, a.e. Ω, and v − u ∈ W (Ω) , ∗ 0 ∗ where for the obstacle function ψ, we assume that ψ ∈ W 1,(qi ) (Ω) with qi > pi for every i = 1, · · · , n.

(1.18)

For a recent development related to anisotropic obstacle problems, we refer the reader to [1,8]. (pi ) (Ω) The next two theorems show that higher integrability of θ = max{ψ, u∗ } forces solutions u ∈ Kψ,u ∗ to obstacle problems to be more integrable. (p )

i Theorem 1.3. Let r > np¯ . Under the assumptions (1.5), (1.6), (1.7) and (1.18), let u ∈ Kψ,u (Ω) be a ∗ solution to the following inequality

  n

≤

Ω i=1   n Ω i=1

ai (x, Du(x)) · (Di u(x) − Di v(x))dx (1.19) f (x) · (Di u(x) − Di v(x))dx, i

∀v ∈

(pi ) Kψ,u (Ω). ∗

H. Gao et al. / J. Math. Anal. Appl. 442 (2016) 244–258

249

Then 1 u ∈ θ + LTweak (Ω)

where T1 satisfies (1.9), and b1 is any number such that (1.10) holds true. (p )

i Theorem 1.4. Let 1 < σ < np¯ . Under the assumptions (1.12), (1.13) and ψ ∈ W 1,(σpi ) (Ω), let u ∈ Kψ,u (Ω) ∗ be a solution to the following inequality



 f (x, Du(x))dx ≤ Ω

f (x, Dw(x))dx,

(p )

i ∀w ∈ Kψ,u (Ω). ∗

(1.20)

Ω

Then 2 u ∈ θ + LTweak (Ω)

where T2 satisfies (1.16). When we are in the isotropic case, that is, pi = p for every i = 1, 2, · · · , n, we denote W 1,(pi ) (Ω), 1,(p) (p) pi and Kψ,u (Ω) by W 1,(p) (Ω), W0 (Ω) and Kψ,u∗ (Ω), respectively. When n > qi = q > p = pi ∗ for every i = 1, 2, · · · , n, then we take 1,(p ) W0 i (Ω)

 b1 = min

1≤i≤n

pi 1− qi

 =1−

p p <1− . q n

In this case, T1 =

nq . n−q

Thus we have the following corollary of Theorem 1.1. n p

Corollary 1.1. Let 1 < p < q < n, r >   n

1,(p)

and u ∈ u∗ + W0

ai (x, Du(x))Di ϕ(x)dx =

Ω i=1

  n

(Ω) verify 1,(p)

f i (x)Di ϕ(x)dx, ∀ϕ ∈ W0

Ω i=1

Under the growth condition n 

p

|ai (x, z)|

i=1

≤ c1

n 

|zi |p + g1 (x)

i=1

and the coercivity condition c2

n 

|zi |p − g2 (x) ≤

i=1

n 

ai (x, z)zi ,

i=1

where c1 , c2 are positive constants, and 

u∗ ∈ W 1,(q) (Ω), f i ∈ Lrp (Ω), i = 1, · · · , n, g1 , g2 ∈ Lr (Ω),

(Ω).

250

H. Gao et al. / J. Math. Anal. Appl. 442 (2016) 244–258

one has nq

n−q u ∈ u∗ + Lweak (Ω).

Theorems 1.2, 1.3 and 1.4 may have corollaries in the isotropic case, we omit the details. This paper is organized as follows. In section 2, we give two preliminary lemmas that will be repeatedly used in the proof of the main theorems. In section 3, we are devoted to prove the main results. We refer the reader to [2–4,6,7,10,11] for some other results related to anisotropic elliptic equations and anisotropic integral functionals. 2. Two preliminary lemmas First of all, let us recall the anisotropic Sobolev embedding theorem, which will be used in the sequel, see [17]. Lemma 2.1. Let Ω be a bounded open subset of Rn ; let p1 , · · · , pn be in [1, +∞); let v : Ω → R be in ∗ 1,(p ) W0 i (Ω); if p¯ < n then v ∈ Lp¯ (Ω) with 

v p¯∗ ≤ c∗

n 

 n1

Di v pi

i=1

and

v p¯∗ ≤ c∗

n 

Di v pi ,

i=1

where  c∗ = max

1≤i≤n

∗ pi

1 + p¯

−1 pi

 .

We now give a sufficient condition ensuring weak integrability of functions, which plays an important role in showing the results listed in the previous section. Lemma 2.2. Let v ∈ L1 (Ω) satisfy Lb |{|v| > 2L}| ≤ A|{|v| > L}|a

(2.1) b

1−a for all L > 0, where b, A > 0 and 0 ≤ a < 1 are some constants. Then v ∈ Lweak (Ω).

Proof. By (1.1), it is sufficient to prove that there exists a constant M > 0 such that L|{|v| > L}|

1−a b

holds true for any L > 0. For any integer k, (2.1) yields

≤M <∞

(2.2)

H. Gao et al. / J. Math. Anal. Appl. 442 (2016) 244–258

|{|v| > L}|   b  b L 2 |{|v| > L}| = L 2  b  a  L  2 ≤ A  |v| > L 2    b kj=1 aj−1  ak   k L  1 b k jaj−1 aj−1  j=1 j=1 ≤2 A  |v| > 2k  . L

251

(2.3)

Since the right hand side of (2.3) monotonically non-decreasingly tends to 2

b (1−a)2

b   1−a 1 1 A 1−a L

as k → +∞, then we get b

b

1

L 1−a |{|v| > L}| ≤ 2 (1−a)2 A 1−a < ∞, which is equivalent to (2.2). 2 3. Proofs of the theorems In the following we will write C to denote positive constants, possibly different depending on the data. Proof of Theorem 1.1. Let us consider ϕ = TL (u − u∗ − TL (u − u∗ )), where TL is the truncation operator at level L > 0, that is, 

L TL (s) = min 1, |s|

 s.

It is easy to see that 1,(pi )

ϕ ∈ W0

(Ω)

and Dϕ = (Du − Du∗ )1{L<|u−u∗ |<2L} , where 1A (x) = 1 if x ∈ A and 1A (x) = 0 otherwise. (1.8) with ϕ acting as a test function yields  {L<|u−u∗ |<2L}



= {L<|u−u∗ |<2L}

n 

ai (x, Du)(Di u − Di u∗ )dx

i=1 n 

(3.1) f (Di u − Di u∗ )dx. i

i=1

(3.1) together with the coercivity condition (1.6) yields

H. Gao et al. / J. Math. Anal. Appl. 442 (2016) 244–258

252



n  i=1

|Di u − Di u∗ |pi dx

{L<|u−u∗ |<2L}

⎛ n ⎜ ≤ 2pm −1 ⎝ i=1

≤

 |Di u|pi dx +

⎛ n ⎜ ⎝

pm −1

2

c2

i=1

i=1

2pm −1 ≤ c2

+

i=1

:=

2

c2

{L<|u−u∗ |<2L}



|Di u∗ |pi dx

{L<|u−u∗ |<2L}

(3.2)

 f i (Di u − Di u∗ )dx ⎞



⎟ g2 dx⎠

ai (x, Du)Di u∗ dx +

{L<|u−u∗ |<2L}

{L<|u−u∗ |<2L}



n  i=1

pm −1

⎟ g2 dx⎠

ai (x, Du)Di udx +



+ 2pm −1

⎞



{L<|u−u∗ |<2L}

n  i=1

{L<|u−u∗ |<2L}

{L<|u−u∗ |<2L}

⎛ n ⎜ ⎝

⎟ |Di u∗ |pi dx⎠



n 

+ 2pm −1

i=1

{L<|u−u∗ |<2L}

⎞



n 

|Di u∗ |pi dx

{L<|u−u∗ |<2L}

(I1 + I2 + I3 ) + 2pm −1 I4 ,

where pm = max {pi }. By Young inequality and the growth condition (1.5) we obtain that 1≤i≤n

|I1 | ≤ C(ε1 )



n  i=1



n 



|f i |pi dx + ε1

i=1

{L<|u−u∗ |<2L}

|Di u − Di u∗ |pi dx

{L<|u−u∗ |<2L}

and 

n 

|I2 | ≤ ε2 {L<|u−u∗ |<2L}

i=1



n 

≤ c1 ε2 {L<|u−u∗ |<2L}

{L<|u−u∗ |<2L}

g1 dx

|Di u∗ |pi dx

i=1 n 

c1 ε2 {L<|u−u∗ |<2L}

+ (2

i=1

{L<|u−u∗ |<2L}



pm −1

|Di u∗ |pi dx

 |Di u|pi dx + ε2 n 

+ C(ε2 )

≤2

n 

{L<|u−u∗ |<2L}

i=1



pm −1





|ai (x, Du)|pi dx + C(ε2 )

 |Di u − Di u∗ | dx + ε2 pi

i=1



c1 ε2 + C(ε2 )) {L<|u−u∗ |<2L}

{L<|u−u∗ |<2L} n  i=1

|Di u∗ |pi dx,

g1 dx

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253

where ε1 and ε2 are positive constants that will be determined later. Substituting the above two estimates pm −1 into (3.2), and take ε1 and ε2 small enough such that 2 c2 (ε1 + 2pm −1 c1 ε2 ) < 1, then we arrive at 

n  i=1

|Di u − Di u∗ |pi dx

{L<|u−u∗ |<2L}

⎡

≤C

n  i=1



⎢ ⎣

⎤





⎥ |Di u∗ |pi dx⎦

|f i |pi dx + {L<|u−u∗ |<2L}

(3.3)

{L<|u−u∗ |<2L}



(g1 + g2 )dx,

+C {L<|u−u∗ |<2L}

where C is a constant depending only on pm , c1 , c2 . Condition (1.7) implies 

n  i=1

≤

{L<|u−u∗ |<2L}

n  i=1

=



|f i |pi dx

n 

⎛ ⎝

⎞ r1



rpi

|f i |

dx⎠ |{L < |u − u∗ | < 2L}|1− r

1

(3.4)

Ω p

1

f i rpi  |{L < |u − u∗ | < 2L}|1− r i

i=1 1

= A1 |{L < |u − u∗ | < 2L}|1− r  n  |Di u∗ |pi dx i=1

≤

{L<|u−u∗ |<2L}

n 

|Di u∗ | pqi |{|L < |u − u∗ | < 2L}| pi

i

i=1

=

n 

 1  qi pi

(3.5)

Di u∗ pqii |{|L

p

1− qi

< |u − u∗ | < 2L}|

i

i=1

≤ A2 |{|L < |u − u∗ | < 2L}| (1 + |Ω|) b1

! " p max 1− qi −b1 i

i

= A2 B1 |{|L < |u − u∗ | < 2L}|b1 , and  (g1 + g2 )dx {L<|u−u∗ |<2L}

(3.6)

1

≤ g1 + g2 r |{|L < |u − u∗ | < 2L}|1− r 1

= A3 |{|L < |u − u∗ | < 2L}|1− r , where A1 =

n 

p

f i rpi  , A2 =

n 

i

i=1

i=1

Di u∗ pqii , B1 = (1 + |Ω|)

! " p max 1− qi −b1 i

i

, A3 = g1 + g2 r ,

H. Gao et al. / J. Math. Anal. Appl. 442 (2016) 244–258

254

and b1 is any number satisfying  0 < b1 ≤ min

1≤i≤n

Since r >

n p¯ ,

pi 1− qi

 .

then

b1 < 1 −

1 p¯ <1− , n r

this implies 1

|{|L < |u − u∗ | < 2L}|1− r

= |{|L < |u − u∗ | < 2L}|1− r −b1 |{|L < |u − u∗ | < 2L}|b1 1

≤ (1 + |Ω|)1− r −b1 |{|L < |u − u∗ | < 2L}|b1 1

= B2 |{|L < |u − u∗ | < 2L}|b1 , where B2 = (1 + |Ω|)1− r −b1 . 1

This result together with (3.3)–(3.6) yields n  i=1

1 pi

 |Di u − Di u∗ |pi dx ≤ C(A1 B2 + A2 B1 + A3 B2 )|{|L < |u − u∗ | < 2L}|b1 .

(3.7)

{L<|u−u∗ |<2L}

Now we lower the left hand side of (3.7) by considering only the ith piece of the sum; then we take the power; eventually we take the product over all i and we get ⎛ n  ⎜ ⎝ i=1

≤

n 



⎞ p1

i

⎟ |Di u − Di u∗ | dx⎠ pi

{L<|u−u∗ |<2L}

# $1 C(A1 B2 + A2 B1 + A3 B2 )|{|L < |u − u∗ | < 2L}|b1 pi

i=1 n

= (C(A1 B2 + A2 B1 + A3 B2 )) p¯ |{|L < |u − u∗ | < 2L}| n

≤ (C(A1 B2 + A2 B1 + A3 B2 )) p¯ |{||u − u∗ | > L}|

nb1 p ¯

nb1 p ¯

.

Since |ϕ| = L on the set {|u − u∗ | > 2L}, then Lemma 2.1 together with (3.8) implies

(3.8)

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255

1

L|{|u − u∗ | > 2L}| p¯∗ ⎛ ⎞ p¯1∗  ∗ ⎜ ⎟ =⎝ |ϕ|p¯ dx⎠ {|u−u∗ |>2L}

⎛ ≤⎝

⎞ p¯1∗



∗

|ϕ|p¯ dx⎠

Ω

⎡ ⎢ ≤ c∗ ⎣

n 

⎛ ⎝

i=1

⎡



(3.9)

Ω

⎛



n ⎢ ⎜ = c∗ ⎢ ⎣ ⎝ i=1

⎞ p1 ⎤ n1 i ⎥ pi ⎠ |Di ϕ| dx ⎦ ⎞ p1 ⎤ n1 i ⎟ ⎥ pi |Di u − Di u∗ | dx⎠ ⎥ ⎦

{L<|u−u∗ |<2L} b1

1

≤ c∗ (C(A1 B2 + A2 B1 + A3 B2 )) p¯ |{||u − u∗ | > L}| p¯ . (3.9) is equivalent to

∗

∗

Lp¯ |{|u − u∗ | > 2L}| ≤ cp∗¯ (C(A1 B2 + A2 B1 + A3 B2 )) 1 Then Lemma 2.2 implies u − u∗ ∈ LTweak (Ω) with T1 =

p¯∗ b p ¯∗ 1− 1p¯

=

p ¯∗ p ¯

|{||u − u∗ | > L}|

p¯p¯∗ p−b ¯ 1 p¯∗

b1 p ¯∗ p ¯

provided that

b1 p¯∗ p¯

equivalent to b1 < 1 − np¯ . This ends the proof of Theorem 1.1. 2 Proof of Theorem 1.2. We take ⎧ u∗ + L, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u∗ − L, w = u − TL ((u − u∗ ) − TL (u − u∗ )) = u, ⎪ ⎪ ⎪ u + L, ⎪ ⎪ ⎩ u − L,

if if if if if

L < u − u∗ < 2L, −2L < u − u∗ < −L, −L ≤ u − u∗ ≤ L, u − u∗ ≤ −2L, u − u∗ ≥ 2L.

It is obvious that 1,(pi )

w ∈ u∗ + W0

(Ω).

Thus w can be used as a test function in (1.14). From

Dw = (Du∗ )1{L<|u−u∗ |<2L} + (Du)1{|u−u∗ |≤L} + (Du)1{|u−u∗ |≥2L} , we derive that

.

(3.9) < 1, which is

H. Gao et al. / J. Math. Anal. Appl. 442 (2016) 244–258

256

⎛



⎜ ⎝



⎞



+

⎟ ⎠ f (x, Du(x))dx

+

{L<|u−u∗ |<2L}

{|u−u∗ |≤L}

{|u−u∗ |≥2L}







⎛ ⎜ ≤⎝

+

{L<|u−u∗ |<2L}



⎟ ⎠ f (x, Dw(x))dx

+

{|u−u∗ |≤L}

{|u−u∗ |≥2L}



f (x, Du∗ )dx +

= {L<|u−u∗ |<2L}

⎞ (3.10)



f (x, Du)dx +

{|u−u∗ |≤L}

f (x, Du)dx.

{|u−u∗ |≥2L}

By assumption, all integrals in (3.10) are finite and we drop the integrals over {|u −u∗ | ≤ L} and {|u −u∗ | ≥ 2L} from both sides of (3.10) arriving at 

 f (x, Du(x))dx ≤ {L<|u−u∗ |<2L}

f (x, Du∗ (x))dx. {L<|u−u∗ |<2L}

Remark 1.2 tells us that (1.17) holds true. Left hand side of (1.17), minimality of u and right hand side of (1.17) merge into 

n  i=1

≤c

|Di u − Di u∗ |pi dx

{L<|u−u∗ |<2L}

i=1

c ≤ c˜3 c ≤ c˜3 ≤

+

|Di u| dx + c pi

{L<|u−u∗ |<2L}



{L<|u−u∗ |<2L}

i=1

c c˜3

n  i=1

{L<|u−u∗ |<2L}

i=1



g3 (x) + c



g3 (x)dx + c

c c˜3



g3 (x)dx + c 

{L<|u−u∗ |<2L} n 



|Di u∗ |pi dx

{L<|u−u∗ |<2L}

g4 (x)dx {L<|u−u∗ |<2L}



n 

|Di u∗ |pi dx

{L<|u−u∗ |<2L}





|Di u∗ |pi dx + c˜

{L<|u−u∗ |<2L}

|Di u∗ |pi dx

i=1

{L<|u−u∗ |<2L}

i=1

{L<|u−u∗ |<2L}



n  i=1

{L<|u−u∗ |<2L}

c f (x, Du∗ (x))dx + c˜3 |Di u∗ |pi dx +

n 

|Di u∗ |pi dx

{L<|u−u∗ |<2L}

c f (x, Du(x))dx + c˜3



n 



n  i=1

{L<|u−u∗ |<2L}



c˜ c4 c˜3

= c˜



n 

(g3 + g4 )dx.

{L<|u−u∗ |<2L}

We use Hölder’s inequality with exponent σ:  {L<|u−u∗ |<2L}

⎛ |Di u∗ |pi dx ≤ ⎝

 Ω

⎞ σ1 |Di u∗ |σpi dx⎠ |{L < |u − u∗ | < 2L}|1− σ 1

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257

and ⎛



(g3 + g4 )dx ≤ ⎝

{L<|u−u∗ |<2L}



⎞ σ1 (g3 + g4 )σ dx⎠ |{L < |u − u∗ | < 2L}|1− σ , 1

Ω

then n  i=1



1

|Di u − Di u∗ |pi dx ≤ c˜˜|{L < |u − u∗ | < 2L}|1− σ .

{L<|u−u∗ |<2L}

This inequality is similar to (3.7) with b1 replaced by 1 − σ1 . As in the proof of Theorem 1.1, we arrive at the desired result. This ends the proof of Theorem 1.2. 2 (p )

i Proof of Theorem 1.3. Recall θ = max{ψ, u∗ }. Let u be a solution to obstacle problem, that is, u ∈ Kψ,u (Ω) ∗ satisfies (1.19). For L ∈ (0, +∞), we define ϕ = TL (u − θ − TL (u − θ)). We now show that

⎧ θ + L, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ θ − L, v = u − ϕ = u, ⎪ ⎪ ⎪ u + L, ⎪ ⎪ ⎩ u − L,

if if if if if

L < u − θ < 2L, −2L < u − θ < −L, −L ≤ u − θ ≤ L, u − θ ≤ −2L, u − θ ≥ 2L

(p )

i lies in the space Kψ,u (Ω). For the first case L < u − θ < 2L, one has v = θ + L ≥ θ ≥ ψ; for the second ∗ case −2L < u − θ < −L, we obviously have v = θ − L > u ≥ ψ; for the third case −L ≤ u − θ ≤ L, we have v = u ≥ ψ; for the fourth case u − θ ≤ −2L, we have v = u + L > u ≥ ψ; for the fifth case 1,(p ) 1,(p ) u − θ ≥ 2L, we have v = u − L ≥ θ + L > θ ≥ ψ; since u ∈ u∗ + W0 i (Ω) and ϕ ∈ W0 i (Ω), then 1,(p ) (pi ) v = u − ϕ ∈ u∗ + W0 i . This implies v = u on ∂Ω, and therefore v ∈ Kψ,u (Ω). 2 ∗

Take v(x) as the test function in (1.19). The next proof is similar to the proof of Theorem 1.1 with θ in place of u∗ , we omit the details. (p )

i Proof of Theorem 1.4. For u ∈ Kψ,u (Ω) a solution to (1.20), as in the proof of Theorem 1.3, one can show ∗ that

(p )

i w = u − TL (u − θ − TL (u − θ)) ∈ Kψ,u (Ω), ∗

thus w can act as a test function in (1.20). The next proof is similar to that of Theorem 1.2 with θ in place of u∗ , we omit the details. 2 Acknowledgments The first author was supported by NSFC (11371050) and Natural Science Foundation of Hebei Province (A2015201149). All authors would like to thank the referee for very helpful comments and suggestions. References [1] H.Y. Gao, Regularity for solutions to anisotropic obstacle problems, Sci. China Math. 57 (2014) 111–122. [2] H.Y. Gao, Y.M. Chu, Quasiregular Mappings and A-Harmonic Equation, Science Press, Beijing, 2013.

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