Existence of local strong solutions for motions of electrorheological fluids in three dimensions

Computers and Mathematics with Applications 53 (2007) 595–604 www.elsevier.com/locate/camwa

Existence of local strong solutions for motions of electrorheological fluids in three dimensions F. Ettwein ∗ , M. R˚uzˇ iˇcka Section of Applied Mathematics, University of Freiburg, Eckerstr. 1, 79104 Freiburg, Germany Received 13 September 2005; accepted 27 February 2006

Abstract We prove for three-dimensional domains the existence of local strong solutions to systems of nonlinear partial differential equations with p(·)-structure, p∞ ≤ p(·) ≤ p0 , and Dirichlet boundary conditions for p∞ > 95 without restriction on the upper bound p0 . In particular this result is applicable to the motion of electrorheological fluids. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Electrorheological fluids; Strong solutions; Nonlinear partial differential equation

1. Introduction Electrorheological fluids are special viscous liquids that change their viscosity rapidly when an electric field is applied. The motion of such fluids is governed by a system of nonlinear partial differential equations which read (cf. [1,2]).1 div (E + P) = 0, curl E = 0,

(1a)

∂v − div S + ρ0 [∇v] v + ∇φ = ρ0 f + [∇E] P, ∂t div v = 0,

(1b)

ρ0

where E is the electric field, P the polarisation, ρ0 the density, v the velocity, S the extra stress tensor, φ the pressure and f the mechanical force. The above system has to be supplemented with constitutive relations for P and S and with boundary and initial conditions. According to the model in [1] the extra stress S is given by

∗ Corresponding author.

E-mail address: [email protected] (F.Ettwein). 1 Here and in the following we use the notation [∇u] w = ∂u i w ∂x j

j

i=1,2,3

little Latin indices run from 1 to 3. c 2007 Elsevier Ltd. All rights reserved. 0898-1221/$ - see front matter doi:10.1016/j.camwa.2006.02.032

, where the summation convention over repeated indices is used and

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F. Ettwein, M. R˚uzˇiˇcka / Computers and Mathematics with Applications 53 (2007) 595–604

S(D, E) = α21

!   p(|E|2 )−2   p(|E|2 )−1  2 2 2 2 2 1 + |D| − 1 E ⊗ E + α31 + α33 |E| 1 + |D| D

  p(|E|2 )−2 2 2 + α51 1 + |D| (DE ⊗ E + E ⊗ DE) .

(2)

Here αi j are material constants, p is a given material function with 1 < p∞ ≤ p(|E|2 ) ≤ p0 and D is the symmetric velocity gradient, i.e. 2D(v) = ∇v + (∇v)t . These equations with p(·)-structure are one example of systems with non-standard growth conditions. In the context of calculus of variations, structures with variable exponents were first considered in [3,4]. In recent years there has been some substantial progress in the existence and regularity theory of problems with p(·)-structure (cf. [5–10]), in the theory of Lebesgue and Sobolev spaces with variable exponents (cf. [11–15]), the theory of singular integrals in these spaces (cf. [7–10]) and the development of models with p(·)-structure, e.g. in mathematical physics and image processing. If we choose in (1) a linear relation for P, i.e. P = χ E E, where χ E is the dielectric susceptibility, the system decouples in the mechanical and electric variables. Since Maxwell’s equations are well understood (cf. [16,2]) we will only treat the system (1b). Moreover, we restrict ourselves to steady situations, Dirichlet boundary conditions and homogeneous electrorheological fluids.2 Hence we study the existence of strong solutions to the system −div S(D(v), E) + [∇v] v + ∇φ = f + χ E [∇E] E,

in Ω ,

div v = 0 in Ω , v = 0 on ∂Ω .

(3)

It is shown in [2] that S has under suitable assumptions on the coefficients αi j and on the material function p ∈ W 1,∞ the following properties: strong monotonicity 

∂kl Si j (D, E)Bi j Bkl ≥ c 1 + |D|

2

 p(|E|2 )−2 2

|B|2 ,

(4a)

coercivity 

2

S(D, E) : D ≥ c 1 + |D|

 p(|E|2 )−2 2

|D|2 ,

(4b)

growth conditions  p(|E|2 )−2  2 2 , |∂kl S(D, E)| ≤ C 1 + |D|  p(|E|2 )−1     2 |∂n S(D, E)| ≤ C 1 + |D|2 1 + ln 1 + |D|2 3 for all B, D ∈ X := {D ∈ R3×3 sym , tr D = 0} and E ∈ R . Here we have used the abbreviations ∂kl Si j := ∂ Si j ∂ En .

∂n Si j := The constants c, C depend on with the following properties (cf. [17–19,2]) 1 + |D|2 ∂kl Siλj (D, E)Bi j Bkl ≥ c 1 + λ|D|2 

1 + |D|2 S (D, E) : D ≥ c 1 + λ|D|2 λ



|E|2

∂ Si j ∂ Dkl

and

and k pk1,∞ . It can be easily seen that S can be approximated by Sλ

 p(|E|22 )−2

 p(|E|22 )−2

(4c)

|B|2 ,

|D|2 ,

2 Thus we divide the equations through the constant density ρ and obtain new variables S, E and φ. 0

(5a)

(5b)

F. Ettwein, M. R˚uzˇiˇcka / Computers and Mathematics with Applications 53 (2007) 595–604



λ

|∂kl S (D, E)| ≤ C |∂n Sλ (D, E)| ≤ C



1 + |D|2 1 + λ|D|2

1 + |D|2 1 + λ|D|2

 p(|E|22 )−2

,

597

(5c)

 p(|E|22 )−2    1  1 + |D|2 2 2 1 + |D| × 1 + ln 1 + λ|D|2

(5d)

for all B, D ∈ X and E ∈ R3 . The constants c, C are independent of λ. We remark that due to the inequality 1+y 1 ≤ 1+λy ≤ λ1 , which holds for all λ ∈ (0, 1] and all y ≥ 0, we get that for each λ there exist constants cλ , Cλ which

depend on λ, p0 and p∞ such that Sλ has the properties (4) with p ≡ 2. For a detailed discussion of the underlying model and structure, the relevant functions spaces, especially Lebesgue and Sobolev spaces with variable exponents and notations we refer the reader to [2]. In the following we consider instead of the concrete model (2) a more general situation, namely

Assumption 1. S can be locally uniformly approximated by Sλ with the properties (5). Besides the above example it can be easily checked that stress tensors S which are derived from a potential, , fulfil the Assumption 1 (cf. [20,21]). i.e. Si j (D) := ∂ Φ∂ D(|D|) ij In [2] it is shown amongst other things that the system (3) possesses weak solutions if 95 < p∞ ≤ p(|E|2 ) and local strong solutions if 59 < p∞ ≤ p(|E|2 ) ≤ p0 < 6. In the present paper we remove the artificial upper bound p0 < 6. Acerbi and Mingione proved in their paper [5] partial regularity, if p(|E|2 ) > 59 . In a series of papers, Kaplick´y, M´alek and Star´a showed the existence of a local strong solution in two dimensions for constant p > 65 . This solution 1,α is actually Cloc (cf. [19,22,20]). In the unsteady case with constant p satisfying 94 ≤ p < 3, the existence of a strong solution up to the boundary was proved by M´alek, Neˇcas and R˚uzˇ iˇcka in [23]. 0

Theorem. Let Ω ⊂ R3 be a bounded domain, let S fulfil Assumption 1, f ∈ L q (Ω ) with3 q = min(2, p∞ ) and p∞ > 9/5 and E ∈ W 1,∞ (Ω ). Then there exists a local strong solution of the problem (3), i.e. v ∈ 2, p(·) ˜ Wloc (Ω ) ∩ V3 p(·),loc ∩ V p(·) with p(·) ˜ := min( p(|E(·)|2 ), 2) solves Z Z Z S(D(v), E) : D(ϕ)dx + (f + χ E [∇E] E) · ϕdx, ∀ϕ ∈ V p(·) . [∇v] v · ϕdx = Ω

Ω

Ω

Further, for all U ⊂⊂ Ω we have Z   p(|E|2 )−2 2 2 1 + |D(v)| |D(∇v)|2 dx < ∞. U 2,

3 p(·) ˜

˜ (Ω ). Remark 1. From the above theorem we conclude that v ∈ Wlocp(·)+1

In order to prove this theorem we construct approximate solutions vλ as solutions of problem (3) with S substituted by Sλ . For this approximate problem it holds that: Proposition 2. Let Ω ⊂ R3 be a bounded domain, f ∈ L 2 (Ω ). Further, let E ∈ W 1,∞ (Ω ). Then there exists a local 2,2 strong solution vλ ∈ Wloc (Ω ) ∩ V2 of the problem   −div Sλ (D(vλ ), E) + ∇vλ vλ + ∇φ = f + χ E [∇E] E in Ω , (6a) div vλ = 0 λ

v =0

in Ω ,

on ∂Ω .

We get the existence of a weak solution vλ

(6b) (6c)

Proof. ∈ V2 with the help of Brezis’ theorem on pseudomonotone operators 2,2 (cf. [24,25,2]). By the standard difference quotient technique one sees that vλ ∈ Wloc (cf. [2,17,18]) holds.  3 Here and in the following we use the convention that for every number q ∈ (1, ∞) we define the number q 0 := q/(q − 1).

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F. Ettwein, M. R˚uzˇiˇcka / Computers and Mathematics with Applications 53 (2007) 595–604

2. Proof of the theorem 2,2 Let vλ ∈ Wloc (Ω )∩V2 be the approximate solution given by the above proposition. By testing the weak formulation R   λ of (6a) with v we conclude with the help of (5b) and using that Ω ∇vλ vλ vλ dx = 0

Z  Ω

1 + |D(vλ )|2 1 + λ|D(vλ )|2

 p(|E|22 )−2

|D(vλ )|2 dx ≤ C(f, E),

uniformly in λ (cf. [2,18]). Thus we get from Lemma 44 with α = Z ˜ |D(vλ )| p(·) dx ≤ C(f, E),

(7) 1 2

(8)

Ω

where p(x) ˜ = min( p(|E(x)|2 ), 2). Now we want to multiply (6a) by a function which is locally equal to −∆vλ . In order to avoid problems with the pressure we choose5 ψ := curl (ξ 2α curl vλ ) = −ξ 2α ∆vλ + h, where ξ is a usual cut-off function, α > 1 big enough and h := ∇ξ × curl vλ ξ 2α−1 . Since h is linear in ∇vλ and ∇ξ , and the estimates |h| ≤ Cξ 2α−1 |∇vλ |,

|∇h| ≤ Cξ 2α−1 |∇ 2 v| + Cξ 2α−2 |∇vλ |

are true with C = C(∇ξ, ∇ 2 ξ ), we have at our disposal a test function ψ which is weakly divergence-free and possesses pointwise estimates for the correction term. We therefore have Z Z Z       λ  λ  λ 2α div Sλ (D(vλ ), E) · ∆vλ ξ 2α − h dx − ∇v v · ∆v ξ − h dx + f · ∆vλ ξ 2α − h dx Ω

Ω

+χE

Z Ω

  [∇E] E · ∆vλ ξ 2α − h dx =:

4 X

Ω

Ii = 0.

(9)

i=1

Using a density argument in order to perform the partial integration we obtain Z   I1 = div Sλ (D(vλ ), E) · ξ 2α ∆vλ − h dx Ω  λ  λ Z Z ∂v ∂v ∂ξ ∂ λ Si j (D(vλ ), E)Di j ξ 2α dx + 2α Siλj (D(vλ ), E)Di j ξ 2α−1 dx = ∂ x ∂ x ∂ x ∂ xk k k k Ω Ω   Z Z ∂ξ ∂ξ dx + Sλ (D(vλ ), E) : D(h)dx −α Siλj (D(vλ ), E) ∆viλ ξ 2α−1 + ∆v λj ξ 2α−1 ∂ x ∂ x j i Ω Ω =: J1 + J2 + J3 + J4 . With the chain rule, the term J1 can be written as  λ  λ  λ Z Z ∂v ∂v ∂ En ∂v λ λ 2α λ λ J1 = ∂lm Si j (D(v ), E)Dlm Di j ξ dx + ∂n Si j (D(v ), E) ξ 2α dx Di j ∂ xk ∂ xk ∂ xk ∂ xk Ω Ω =: J1,1 + J1,2 . Using the monotonicity condition (5a) we obtain Z  J1,1 ≥ c =:

1 + |D(vλ )|2 1 + λ|D(vλ )|2

Ω λ cI p (D(∇vλ )ξ α ).

 p(|E|22 )−2

|D(∇vλ )|2 ξ 2α dx

4 Please note that all technical lemmata related to the properties of the operator S are collected in the Appendix. 5 This test function, which is divergence-free, was used in [26] to study Bingham-type fluids and earlier in the two-dimensional situation by [19, 22,20].

F. Ettwein, M. R˚uzˇiˇcka / Computers and Mathematics with Applications 53 (2007) 595–604

599

This term will give us all the information and all other terms have to be estimated. With the Sobolev embedding theorem, Korn’s inequality, estimate (7) and the inequalities (21), (23) and (24) (cf. Lemmas 5, 7 and 9) we get the following lower bound for I pλ :  Z   

1

 3( p(|E|22 )−2)

1 + |D(vλ )|2

3

λ 6 6α

|D(v )| ξ

1 + λ|D(vλ )|2

Ω

Z + Ω

Z  dx  +

λ 3 p(x) ˜ 6α

ξ

|∇v | Ω

1 3

dx

˜ ˜ |∇ 2 vλ | p(x) ξ α p(x) dx + k∇ 2 vλ ξ α k2p˜− ≤ C I pλ (D(∇vλ )ξ α ) + C(f, E),

(10)

where p(x) ˜ = min(2, p(|E(x)|2 )) and p˜ − := infsupp ξ p. ˜ The constants C, C(E, f) are independent of λ. From the growth of ∂n Sλ (cf. (5c)) we obtain that the term J1,2 can be estimated by  p(|E|22 )−2   1 + |D(vλ )|2 1 + |D(vλ )|2 C ln |D(vλ )||D(∇vλ )|ξ 2α dx + C λ 2 1 + λ|D(vλ )|2 Ω 1 + λ|D(v )|  1 3 3( p(|E|2 )−2)  Z  2 1 + |D(vλ )|2   λ 6 6α ≤ δ |D(v )| ξ dx  + δ I pλ (D(∇vλ )ξ α ) + C(δ, f, E), λ )|2 1 + λ|D(v Ω Z 

where we used similar calculations as in the proof of Lemma 5. Note that the first two terms can be absorbed with the help of (10). We notice that the terms J2 − J4 can be estimated with the help of the growth condition for Sλ , the Cauchy–Schwarz inequality and estimate (7) as follows: Z  |J2 + J3 + J4 | ≤ C Ω

1 + |D(vλ )|2 1 + λ|D(vλ )|2

 p(|E|22 )−2

|D(vλ )||D(∇vλ )|ξ 2α−1 dx + C(f, E)

≤ δ I pλ (D(∇vλ )ξ α ) + C(f, E). We estimate the convective term through |I2 | ≤ δk∇ 2 vλ ξ α k2p˜− + c(∇ξ )

Z

|∇vλ | p˜− |vλ | p˜− ξ α p˜− dx 0

Ω

0

0



2 0 p˜ −

Z

|∇vλ |2 |vλ |ξ 2α−1 dx.

+C

(11)

Ω

Now we have to distinguish two cases, namely p− ≤ 2 and p− ≥ 2. In the case p− ≤ 2 the second term in (11) can be written as Z

0 λ p−

|∇v | Ω

0 0 λ p− αp−

|v |

ξ

 dx

2 0 p−

Z =

0 −z) λ z+( p−

|∇v | Ω

0 0 λ p− αp−

|v |

ξ

 dx

2 0 p−

.

(12)

In (10) we have at our disposal the L 3 p− -norm of ∇vλ raised to the power p− ( p− ≤ 2 ⇒ p(x) ˜ = p(x) ≥ p− ) ∗ and further we know that ∇vλ ∈ L p− and vλ ∈ L ( p− ) are bounded uniformly in λ (cf. (8)). Thus we can handle the 0 − z ∈ (0, p 0 ) if the following right-hand side in (12) by H¨older’s inequality and the additive splitting of p 0 in z, p− − relations are satisfied: 2 0 = p− , p− z p0 − z p0 + + ∗ ≤ 1. 3 p− p− p−

z

The first condition comes from the fact that we want to absorb the term with the L 3 p− norm on the left-hand side and the powers must fit. The second condition comes from H¨older’s inequality. That leads for three-dimensional domains

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F. Ettwein, M. R˚uzˇiˇcka / Computers and Mathematics with Applications 53 (2007) 595–604

to the condition (z = p− >

0 p− p− 2

=

2 p− 2( p− −1) )

9 , 5

(13)

which coincides with the condition that ensures the existence of weak solution with the theory of monotone operators. However, we also need to apply Young’s inequality to absorb one term on the left-hand side and thus we have to modify the above argument slightly and introduce a parameter s > 1. We get with H¨older’s inequality p− 0 , p / p 0 /(1− p /(2s)), 3( p −1)/(3− p ), 3s( p− −1) ), where 1 < s < (6s/ p− − − − − − 6s−(s+1) p 9−4 p− , and with Young’s inequality Z

0

Ω



|∇vλ | p− |vλ | p− ξ αp− dx 0

0

2 0 p−

Z

|∇vλ | p− p− /(2s) |∇vλ | p− (1− p− /(2s)) |vλ | p− ξ αp− dx 0

= Ω

Z

0

λ 3 p− 6αs

ξ

|∇v |

≤ Ω

Z

∗ λ p−

|v |

×

 1 Z 3s

≤δ

|∇v |

Ω

0



2 0 p−

2/ p− −1/s dx

Ω

 2(3− p− ) 3 p−

dx

λ 3 p− 6α

|∇v |

λ p−

dx

Ω

Z

0

ξ

1 3

dx

C(Ω ) + Cδ .

We easily estimate the last term in (11) in a similar way: Z Z |∇vλ |2 |vλ |ξ 2α−1 dx = |∇vλ | p− +(2− p− ) |vλ |ξ 2α−1 dx Ω

(14)

Ω

if 2 − p− 1 p− 3 + + ∗ ≤ 1 ⇔ p− ≥ . 3 p− p− p− 2 To handle the powers of the cut-off function we need p− > 23 . Altogether the requirement which has to be satisfied is p− > 95 . In the case p− ≥ 2 we use H¨older’s inequality with (6, 2, 3): Z Z λ λ λ 2α−1 |∇vλ ||∇vλ ||vλ |2 ξ 2α dx + |∇v ||∇v ||v |ξ dx Ω

Ω

Z

λ 6 12α

≤ Ω

|∇v | ξ

Z + Ω

≤ δ

Z Ω

 1 Z 6

dx

supp ξ

|∇vλ |6 ξ 6(2α−1) dx |∇vλ |6 ξ 6α dx

λ 2

1 3

 1 Z 2

|∇v | dx

 1 Z 6

supp ξ

|∇vλ |2 dx

λ 6

supp ξ

1 3

|v | dx

 1 Z 2

supp ξ

|vλ |3 dx

1 3

+ Cδ ,

(15)

where we also used Young’s inequality with (2, 2), Sobolev’s embedding theorem and estimate (7). The term I3 is easy and we have by H¨older’s and Young’s inequality |I3 | ≤ δk∇ 2 vλ ξ 2α k2p˜− + Cδ kfk2p˜ 0 + Ckfk p˜−0 k∇vλ k p˜− . −

(16)

I4 can be treated analogously to I3 . Altogether we therefore get from (9)–(16) the uniform estimate with respect to λ: 2

α p

λ p− k∇ 2 vλ ξ α k2Ω , p− kΩ−,3 p˜− + I pλ (D(∇vλ )ξ α ) ≤ c(f, E, ∇ξ, p+ ). ˜ + k∇v ξ

(17)

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601

Thus D(vλ ) converges pointwise to D(v) by compact embedding. With Fatou’s lemma we derive from the inequalities (7), (10) and (17) that Z  Z   p(|E|2 )−2  3( p(|E|2 )−2) 2 2 1 + |D(v)|2 1 + |D(v)|2 |D(v)|2 dx + |D(v)|6 ξ 6α dx ≤ C(f, E). Ω

(18)

Ω

From Lemma 4 with α = 23 we therefore get Z 2 |D(v)|3 p(|E| ) ξ 6α dx ≤ C(f, E). Ω

Further, (17) yields that for all U ⊂⊂ Ω Z   p(|E|2 )−2 2 1 + |D(v)|2 |D(∇v)|2 dx ≤ C.

(19)

U



 p(x)−2

1+|u|2 1+λ|u|2 u λ := D(vλ )

Indeed, set f (λ, x, u) :=

2

and F(λ, x, u, z) := f (λ, x, u)|z|2 . Let U ⊂⊂ Ω . For all  > 0 there exists

a set K ⊂ U such that ⇒ u := D(v) uniformly in K , u, z := ∇u are continuous on K and |U − K | ≤ . This is possible with the help of the theorems of Egorov and Lusin. Because 2hz 1 , z 2 i ≤ |z 1 |2 + |z 2 |2 , one sees that |z 1 |2 − |z 2 |2 ≥ 2hz 2 , z 1 − z 2 i. Z Z C≥ F(λ, x, u λ , z λ )dx ≥ F(λ, x, u λ , z λ )dx U ZK = F(λ, x, u λ , z) + (F(λ, x, u λ , z λ ) − F(λ, x, u λ , z)) dx {z } | K ≥hFz (λ,x,u λ ,z),z λ −zi

Z ≥ K

F(λ, x, u λ , z)dx +

Z K

hFz (λ, x, u λ , z), z λ − zi.

The second integral on the right-hand side converges to 0 due to the weak convergence of z λ * z in L p˜− and the above uniform convergence of u λ . With the help of Fatou’s lemma, the claim (19) is proven for  → 0. Thus all statements of the theorem are proven, if we apply Vitali’s theorem and use a diagonal sequence argument (cf. [27]). α (Ω ) and if p β Remark 3. We remark that our solution is locally continuous, i.e. v ∈ Cloc ∞ > 3 actually C (Ω ) with suitable α, β.

Appendix. Auxiliary results Here we present some technical results, mostly estimates related to I pλ . Lemma 4. For y, p, α ≥ 0 it holds that y 2αp ≤ 1 + c(1 + y 2 )α( p−2) y 4α . If y, α ≥ 0, 0 ≤ p ≤ 2 and 0 ≤ λ ≤ 1 then  α( p−2) 1 + y2 y 2αp ≤ 1 + c y 4α . 1 + λy 2 Proof. The only non-obvious case we have to look at is 0 ≤ p ≤ 2 and y ≥ 1: (1 + y 2 )α( p−2) y 4α ≥ (2y 2 )α( p−2) y 4α = 2α( p−2) y 2αp . The second inequality follows easily from the first and

1+y 2 1+λy 2

≤ 1+y 2 which holds for all λ ∈ [0, 1] and all y ≥ 0.



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F. Ettwein, M. R˚uzˇiˇcka / Computers and Mathematics with Applications 53 (2007) 595–604

Lemma 5. Let ξ be a usual cut-off function, λ ∈ [0, 1] and p ∈ W 1,∞ with p ≥ p∞ > 1, then

2







 p(|E|2 )−2 p(|E|2 )−2

1 + |D(u)|2  4

1 + |D(u)|2  4

 λ



 |D(u)|ξ |D(u)| I (D(∇u)ξ ) + ≤ C

 + C,

 p

1 + λ|D(u)|2

1 + λ|D(u)|2



2

r

holds with 1 ≤ r ≤ 2∗ =

2n n−2

for n ≥ 3; in particular, r ≤ 6 for n = 3.

Proof. We calculate   p(|E|2 )−2  p(|E|42 )−2     2 2  4 1 + |D(u)|2 1 + |D(u)|   1 + |D(u)| 0 |D(u)| |D(u)| ≤ C( p , ∇E) ln ∇  1 + λ|D(u)|2 1 + λ|D(u)|2 1 + λ|D(u)|2 

1 + |D(u)|2 1 + λ|D(u)|2



1 + |D(u)|2 1 + λ|D(u)|2

+C

+C  ≤C

1 + |D(u)|2 1 + λ|D(u)|2 

+C

for arbitrary s > 1 where we used the estimate ln





≤

|D(u)| 1−λ |D(∇u)| 2 1 + λ|D(u)| 1 + |D(u)|2

 p(|E|42 )−2 |D(∇u)|

 p(|E|42 )−2 s

1 + |D(u)|2 1 + λ|D(u)|2 1+y 2 1+λy 2

 p(|E|42 )−2

|D(u)|s + Cs

 p(|E|42 )−2 (20)

|D(∇u)|,



1+y 2 1+λy 2

 p−2 4

! y

+ C for every  > 0 and

p ≥ p∞ > 1. The constant C may depend on p 0 , E and ∇E. Since W01,2 (Ω ) is embedded into L r (Ω ), for every 2n r ≤ n−2 if Ω ⊂ Rn is bounded, we get using also Poincar´e’s inequality







p(|E|2 )−2   p(|E|42 )−2

1 + |D(u)|2  4

2

 1 + |D(u)| 

|D(u)|ξ ≤ c ∇  |D(u)|ξ 

2 2



1 + λ|D(u)| 1 + λ|D(u)|



r

  2 2

p(|E| )−2 

2  4

 1 + |D(u)|  ≤ c ∇  |D(u)| ξ



1 + λ|D(u)|2

2



p(|E|2 )−2 

1 + |D(u)|2

4

+c |D(u)|∇ξ

1 + λ|D(u)|2



  2 2

p(|E| )−2 

2  4

 1 + |D(u)|  ≤ c ∇  |D(u)| ξ



1 + λ|D(u)|2

2



p(|E|2 )−2 

1 + |D(u)|2

4

+ c(∇ξ ) |D(u)|

.

1 + λ|D(u)|2

2

603

F. Ettwein, M. R˚uzˇiˇcka / Computers and Mathematics with Applications 53 (2007) 595–604

Therefore we get using (20) and interpolation of L 2s between L 2 and L r , 2 < 2s < r , Young’s inequality and |ξ | ≤ 1,

2

 





p(|E|2 )−2 p(|E|2 )−2

1 + |D(u)|2  4

1 + |D(u)|2  4

 

≤ C 1 + I (D(∇u)ξ ) + |D(u)|ξ |D(u)|



.   p

1 + λ|D(u)|2

1 + λ|D(u)|2



2

r

Remark 6. Using estimate (7) and Lemma 4 we get from Lemma 5, with r = 6,  1 3 1  3( p(|E|22 )−2) Z Z  λ 2 3 1 + |D(v )|   λ 3 p(x) ˜ 6α λ 6 6α |∇v | ξ dx |D(v )| ξ dx  +  λ 2 Ω 1 + λ|D(v )| Ω ≤ C I pλ (D(∇vλ )ξ α ) + C(f, E).

(21)

Lemma 7. Let w ∈ L ∞ (Ω ) be a positive weight, i.e. w > 0 a.e. Then the following inequality is true: Z Z p(·) ˜ p(·)−2 ˜ ˜ ˜ |D(∇u)| p(·) ξ α p(·) dx ≤ C( p0 , p∞ ) w 2 + w 2 |D(∇u)|2 ξ 2α dx. Ω

Ω

Note that if w ≥ 1 or w ≥  > 0 a.e., then Z Z p(·) p(·)−2 ˜ ˜ |D(∇u)| p(·) ξ α p(·) dx ≤ C( p0 , p∞ , ) w 2 + w 2 |D(∇u)|2 ξ 2α dx. Ω

(22)

Ω

Proof. If p(x) ˜ < 2 we use Young’s inequality with ( 2− 2p(x) , ˜ |D(∇u(x))| p(x) = w(x)

˜ 2− p(x) 4

p(x) ˜

w

≤ C( p0 , p∞ )w(x)

p(x)−2 ˜ 4 p(x) ˜ 2

p(x) ˜

2 p(x) )

˜ |D(∇u)| p(x)

+ w(x)

p(x)−2 ˜ 2

|D(∇u)(x)|2 .

If p(x) ˜ = 2, obviously |D(∇u)(x)|2 ≤ w(x) + |D(∇u)(x)|2 . If we now integrate over Ω the first inequality follows. The second assertion follows from the first and ω ≥ .    1+|D(vλ )|2 1+x 2 2 Remark 8. If we take w = 1+λ|D(v in (22) and use estimate (7), 1+λx λ )|2 2 ≤ 1 + x and Lemma 10 we get from inequality (22) Z ˜ ˜ |∇ 2 vλ | p(x) ξ α p(x) dx ≤ C I pλ (D(∇vλ )ξ α ) + C(f, E). (23) Ω

Lemma 9. Let q < 2 and u ∈ W 2,q , then k∇ 2 uξ α kq2 ≤ Ck(1 + |D(u)|2 ) 2 kq 1

2−q

Iq (D(∇u)ξ α ).

(24)

2 Proof. By H¨older’s inequality with ( 2−q , q2 ) we directly get

Z

q αq

Ω

|D(∇u)| ξ

2

Z

q

dx

(1 + |D(u)| ) 2

= Ω

≤ k(1 + |D(u)| ) 2

1 2

2−q 4 q

2−q kq

Z Ω

1

2−q

= k(1 + |D(u)|2 ) 2 kq Now we use Lemma 10 to get the desired result.

(1 + |D(u)| ) 2



q−2 4 q

(1 + |D(u)|2 )

Iq (D(∇u)ξ α ).

q αq

|D(∇u)| ξ

q−2 2

2 q

dx

|D(∇u)|2 ξ 2α dx

604

F. Ettwein, M. R˚uzˇiˇcka / Computers and Mathematics with Applications 53 (2007) 595–604

Lemma 10. The terms |D(∇u)| and |∇ 2 u| are equivalent, i.e. there exist constants c, C, such that c|∇ 2 u| ≤ |D(∇u)| ≤ C|∇ 2 u|. Proof. The assertion follows from the algebraic identity 1 1 ∂i j u k = D jk (∂i u) − ∂ik u j + Dik (∂ j u) − ∂ jk u i = D jk (∂i u) + Dik (∂ j u) − Di j (∂k u). 2 2



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