A strong convergence of the weak gradient to A-harmonic type operators with L1 data

J. Math. Anal. Appl. 430 (2015) 381–389

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A strong convergence of the weak gradient to A-harmonic type operators with L1 data ✩ Shenzhou Zheng Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

a r t i c l e

i n f o

Article history: Received 29 December 2014 Available online 5 May 2015 Submitted by J. Xiao Keywords: A-harmonic type operator Lipschitz truncation technique Strong convergence

a b s t r a c t 1,q We prove uk → u strongly in Wloc (Ω) with 1 ≤ q < p by Lipschitz truncation argument if u ∈ W 1,p (Ω) is a weak solution of A-harmonic type equations −divA(x, Du) = f (x) with f ∈ L1 (Ω), and uk is a sequence of their weak solutions with uk  u weakly in W 1,p (Ω) and fk  f weakly in L1 (Ω). As an application, we obtain a compactness property for p-harmonic maps defined from L∞ -metric Riemannian manifold. © 2015 Elsevier Inc. All rights reserved.

1. Introduction As we know, the compactness property is a powerful technique in the modern theory of partial differential equations and geometric analysis. Generally speaking, it asserts that a loss of energy accounts for the failure of strong convergence for weakly convergent during the process of convergence. In this paper, we are devoted 1,q to consider a strong convergence in the Sobolev spaces Wloc (Ω) with any 1 ≤ q < p if uk is a sequence of weak solutions of (1.1) with uk  u in W 1,p (Ω) and fk  f in L1 (Ω). As an application, we observe a compactness property for p-harmonic maps with L∞ -metric Riemannian manifold. First, let’s recall our involved PDEs and some notations. Let Ω ⊂ Rn (n ≥ 2) be a smooth bounded domain and any given vector-valued functions f (x) ∈ L1 (Ω, Rd ). We consider the weak solutions u ∈ W 1,p (Ω, Rd ) with any p > 1 to the following elliptic system −divA(x, Du) = f (x),

x ∈ Ω;

(1.1)

where A : Ω × Rnd → Rnd is a Carathéodory vector fields satisfying the following growth and monotonicity conditions with A(x, 0) = 0 and μ ≥ 0: ✩

This paper is supported by the National Natural Science Foundation of China (Grant No. 11371050). E-mail address: [email protected].

http://dx.doi.org/10.1016/j.jmaa.2015.05.001 0022-247X/© 2015 Elsevier Inc. All rights reserved.

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• H1. (growth) There exists a constant L > 0 such that |A(x, w)| ≤ L(μ2 + |w|2 )

p−1 2

.

(1.2)

• H2. (monotonicity) There exists a constant 0 < ν ≤ L such that (A(x, w1 ) − A(x, w2 )) · (w1 − w2 ) ≥ ν(μ2 + |w1 |2 + |w2 |2 )

p−2 2

|w1 − w2 |2 ,

∀x ∈ Ω, ∀w1 , w2 ∈ Rnd . (1.3)

Of course, the prototype of a vector field A(·,·) satisfying (1.2) and (1.3) can be given by A(x, w) = p−1 B(x)(μ2 + |w|2 ) 2 w with ν|ξ|2 ≤ ξ t B(x)ξ ≤ L|ξ|2 for any ξ ∈ RN n , which have been abundant investigated and applied in different context to various purposes. Note that by taking its very special situation while μ = 0 and B(x) ≡ Id, we find it to be a p-Laplacian. In [10], Hardt, Lin, and Mou proved its strong convergence in W 1,q for 1 ≤ q < p to W 1,p weakly convergence sequences of p-harmonic maps by a few useful inequalities on the growth with p-power. Recently Tuhola-Kujanpää and Varpanenb [15] studied the first eigenfunction of p-Laplace operator with respect to positive and finite Borel measures that satisfy an Adams-type embedding condition. Here, we observe that it allows a discontinuous dependence on x in that the map x → A(x, ·) under the above assumptions of H1 and H2. This is suitable to the p-harmonic maps from the manifolds of L∞ -Riemannian metrics and piecewise Lipschitz continuous metrics, see [11,12]. However, the approaches in [10] are inadaptable to our PDEs due to the non-specific expression of the operator A(·,·). It is an important observation that since f ∈ L1 (Ω) one considers Eq. (1.1) in the weak sense by picking so-called test functions only in the classes of Lipschitz functions. So, recent development of the so-called Lipschitz truncation argument in Sobolev space W 1,p offered a useful technique as a test function, see [1,2,7]. Indeed, in the procedure of main proof one needs a Lipschitz truncation tool by way of cutting-off in W 1,p. Now, we are in the position to give our main conclusion as follows. Theorem 1.1. Let 1 < p < ∞. For k = 1, 2, · · · and fk ∈ L1 (Ω, Rd ), assume that uk ∈ W 1,p (Ω, Rd ) with K = supk uk W 1,p + supk fk L1 < ∞, is any weak solution of the following systems −div A(x, Duk ) = fk , where A(x, Duk ) satisfies the structural assumptions H1–H2. If uk  u weakly in W 1,p (Ω) and fk  f weakly in L1 (Ω), then uk → u strongly in W 1,q (Ω) for any 1 ≤ q < p. In order to prove the main result, we adapt the Lipschitz truncation argument traced back to Acerbi– Fusco’s work [2]. It shows how to approximate a Sobolev function u ∈ W 1,p by a λ-Lipschitz function that coincide with u excepting a set of small measure. This is a useful technique to various areas of analysis, in particular, to the existence theory and regularity of partial differentiable equations (see [1,8,5]). The paper is organized as follows. In Section 2, we establish the strong convergence in the Sobolev spaces 1,q Wloc (Ω) with any 1 ≤ q < p to the sequence uk ∈ W 1,p (Ω, Rd ) of weak solutions to Eq. (1.1) by Lipschitz truncation argument. In Section 3, we prove a weak compactness for p-harmonic maps with L∞ -metric on the basis of Theorem 1.1. 2. Proof of main theorem Lipschitz truncations of Sobolev functions were already successfully used in different analytic purposes. Its main idea is construct a Lipschitz test function to show that for conveniently introduced approximations uk ∈ W 1,p (Ω). The novelty of our application of the Lipschitz approximations of Sobolev functions consists

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of discovering the mechanism of obtaining a strong convergence of gradients under the lower exponent for any weakly convergent sequences of weak solutions to A-harmonic type maps in uk ∈ W 1,p (Ω). One finds its roots in the Lipschitz truncation techniques introduced in many fundamental works [1,2,6,7]. Lemma 2.1 (Lipschitz truncation argument). There exists a positive constant c depending only on n such that whenever {wk } is a bounded sequence in W01,p (Ω, Rd ), then for any λ > 0 there exists a sequence {wkλ } of maps wkλ ∈ W01,∞ (Ω, Rd ) such that

wkλ W 1,∞ (Ω) ≤ cλ.

(2.1)

Moreover, denoting Aλk = {x ∈ Ω : wkλ (x) = wk (x)}, then |Aλk | ≤

c

Dwk pLp (Ω) . λp

(2.2)

Consequently

Dwkλ pLp (Ω) ≤ c sup Dwk pLp (Ω) < ∞.

(2.3)

k

Moreover, we have Aλk ⊂ Rkλ = Fkλ ∪ Gλk ∪ Hkλ , where Fkλ , Gλk and Hkλ satisfy |Fkλ | ≤

c

Dwk pLp (Ω) , λp

|Gλk | ≤

c

Dwk pLp (Ω) , λ2p

|Hkλ | = 0.

(2.4)

In fact, the sets Fkλ and Gλk are actually given by Fkλ =: {x ∈ Ω : λ ≤ M (Dwk )(x) ≤ λ2 },

Gλk =: {x ∈ Ω : M (Dwk )(x) > λ2 },

(2.5)

where M (Du) is the maximal function of Du in Ω. ´ Thanks to keep in mind the notations in (2.5) and a priori estimates we see that F λ (|Duk |p +|fk | +μp )dx k satisfy the uniform small bound by recalling an argument of “Slicing-selection lemma” proved in [8,4]. Lemma 2.2. For a given ε > 0, there are a subsequence {wk }k∈N of {wl }l∈N and large enough λ ≥ independent of k such that ˆ

p

(|Duk |p + |fk | + μp )dx ≤ ε p−1 , Fkλ

where Fkλ =: {x ∈ B : λ ≤ M (Dwk )(x) ≤ λ2 }.

1 ε

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In addition, in proving the main results of strong convergence we still needs the following lemma induced from Brezis’ monograph [3]. Lemma 2.3. Assume that E is a uniformly convex Banach space. Let {xk } be a sequence in E such that xk  x weakly in σ(E, E ∗ ) and lim supk→∞ xk ≤ x . Then xk → x strongly in σ(E, E ∗ ). Proof of Theorem 1.1. Let’s denote g k := (A(x, Duk ) − A(x, Du) · (Duk − Du). Since uk  u weakly in W 1,p (Ω) and fk  f weakly in L1 (Ω), by a weakly lower semi-continuity of the norms · Lp (Ω) (1 ≤ p < ∞) it implies

Du Lp (Ω) ≤ lim inf Duk Lp (Ω) ,

f L1 (Ω) ≤ lim inf fk L1 (Ω) .

k→∞

k→∞

Therefore ˆ

ˆ gk dx ≤ C(L)

ˆ (|Duk |p + |Du|p + |fk | + |f |)dx ≤ C sup

(|Duk |p + |fk |)dx ≤ C(n, L, K),

k Ω

Ω

Ω

where the last inequality is due to K = supk Duk Lp (Ω) + supk fk L1 (Ω) < ∞. Let wk := uk − u ∈ W01,p (Ω, RN ) for all k ∈ N. Applying Lemma 2.1 of so-called Lipschitz truncation argument, then we conclude that for any λ > 0 there exists a sequence wkλ ∈ W01,∞ (Ω, RN ) such that

wkλ W 1,∞ (Ω) ≤ c(n)λ, and |Aλk | ≤

c(n)

Dwk pLp (Ω,RN ) ≤ c(n, K)λp λp

with Aλk = {x ∈ Ω : wkλ (x) = wk (x)}. Now, for any 0 < θ < 1 we write ˆ

ˆ gkθ dx Ω

ˆ gkθ dx

=

+

Aλ k

gkθ dx := I + II .

Ω\Aλ k

In the sequel, we estimate I and II in the right hand side of the above formula. For I, from Hölder inequality we get I ≤ |Aλk |1−θ

ˆ Ω

provided that we take λ ≥ 1ε .

θ gk dx

≤

 C 1−θ λp

C θ = C(n, K, L, θ)εp(1−θ) ,

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For II, by Hölder inequality we have ˆ



II =

θ

(A(x, Duk ) − A(x, Du)) · (Duk − Du)

Ω\Aλ k

≤ |Ω \ Aλk |1−θ

≤ |Ω|1−θ

 ˆ

dx

(A(x, Duk ) − A(x, Du)) · wkλ dx

θ

Ω\Aλ k

 ˆ

(A(x, Duk ) − A(x, Du)) · wkλ dx

θ .

(2.6)

Ω\Aλ k

Observe that ˆ (A(x, Duk ) − A(x, Du)) · wkλ dx Ω\Aλ k

ˆ  ˆ      λ ≤  (A(x, Duk ) − A(x, Du)) · wk dx +  (A(x, Duk ) − A(x, Du)) · wkλ dx Ω

Aλ k

ˆ  ˆ      λ =  (fk − f )wk dx +  (A(x, Duk ) − A(x, Du)) · wkλ dx. Ω

(2.7)

Aλ k

´ Note that Ω (fk − f )wkλ dx → 0 due to fk  f in L1 (Ω), we only consider the estimate of the second part of the right hand side in the above inequality. Thanks to Aλk ⊂ Rkλ = Fkλ ∪ Gλk ∪ Hkλ with |Hkλ | = 0, then ˆ     (A(x, Duk ) − A(x, Du)) · wkλ dx Aλ k

ˆ  ˆ      ≤  (A(x, Duk ) − A(x, Du)) · wkλ dx +  (A(x, Duk ) − A(x, Du)) · wkλ dx Gλ k

Fkλ

:= J1 + J2 .

(2.8)

For J1 , on the basis of the growth condition (1.2) and |Gλk | ≤ deduces

c p λ2p Dwk L (Ω) ,

from Hölder inequality it

ˆ 1− p1  ˆ  p1 J1 ≤ c (|Duk |p + |Du|p + μp )dx |Dwkλ |p dx Gλ k

Gλ k

ˆ 1− p1 1 ≤c (|Duk |p + |Du|p + μp )dx λ|Gλk | p Ω

≤ cK p−1 For J2 , in terms of |Fkλ | ≤ it implies

Dwk Lp (Ω) cK p ≤ ≤ cεK p . λ λ

c p λp Dwk L (Ω)

and

´ Fkλ

(2.9) p

(|Duk |p + |Du|p + μp )dx ≤ ε p−1 , from Hölder inequality

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ˆ 1− p1  ˆ  p1 J2 ≤ c (|Duk |p + |Du|p + μp )dx |Dwkλ |p dx Fkλ

Fkλ

ˆ 1− p1 1 ≤c (|Duk |p + |Du|p + μp )dx λ|Fkλ | p Fkλ

≤ cK

ˆ

(|Duk |p + |Du|p + μp )dx

1− p1

≤ cεK.

(2.10)

Fkλ

Therefore, combining the above estimates of I, II , J1 and J2 we have ˆ gkθ dx ≤ c(εp(1−θ) + ε)|Ω|1−θ ,

lim sup k→∞

∀0 < θ < 1,

Ω

provided that λ ≥ 1ε ; which implies lim sup

ˆ  θ p−2 (μ2 + |Duk |2 + |Du|2 ) 2 |Duk − Du|2 dx

k→∞ Ω

≤

1 lim sup ν k→∞

ˆ  θ (A(x, Duk ) − A(x, Du))(Duk − Du)| dx Ω

≤ c(ε

p(1−θ)

+ ε)|Ω|1−θ .

(2.11)

In the case of p ≥ 2, note that  θ p−2 |Duk − Du|pθ ≤ (μ2 + |Duk |2 + |Du|2 ) 2 |Duk − Du|2 , which yields ˆ |Duk − Du|pθ dx ≤ c(εp(1−θ) + ε)|Ω|1−θ .

lim sup k→∞ Ω

In the case of 1 < p < 2, by Hölder inequality we get ˆ |Duk − Du|pθ dx ≤ Ω

ˆ 

(μ2 + |Duk |2 + |Du|2 )

p−2 2

θ  12 |Duk − Du|2 dx

Ω

θ  12 ˆ  2−p (μ2 + |Duk |2 + |Du|2 ) 2 |Duk − Du|2(p−1) dx · Ω

ˆ  12  ˆ  θ  12 p−2 p p p θ ≤c (μ + |Duk | + |Duk | ) dx (μ2 + |Duk |2 + |Du|2 ) 2 |Duk − Du|2 dx Ω

≤ c|Ω|

1−θ 2

ˆ

Ω

(μp + |Duk |p + |Duk |p )dx

Ω θ 2

≤ cK |Ω|

1−θ 2

 θ2  ˆ  θ  12 (A(x, Duk ) − A(x, Du))(Duk − Du) dx Ω

(ε

p(1−θ)

1 2

+ ε) .

Thanks to the arbitrary of ε, on the basis of cases of p ≥ 2 and 1 < p < 2 we conclude

(2.12)

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ˆ |Duk − Du|pθ dx = 0.

lim sup k→∞ Ω

According to Lemma 2.2, for all 1 ≤ q < p there holds ˆ |Duk − Du|q dx = 0.

lim

k→∞ Ω

2

This completes the proof of Theorem 1.1.

3. A compactness of p-harmonic maps with L∞ -metric Let Ω be a smooth bounded domain endowed with L∞ -metric in Rn , and N be a closed Riemann submanifold of Rd . For 1 < p < ∞, we study a p-harmonic maps (Ω; g) → (N ; h), where the Riemannian metric g is just measurable matrix in Ω such that λ|dx|2 ≤ gαβ dxα dxβ ≤ λ−1 |dx|2 with some constants 0 < λ < 1. Here, we also use the Einstein convention for summation. For the metric gαβ dxα dxβ , let g αβ = (gαβ )−1 and g = detgαβ . As we know, p-harmonic maps from Ω to a Riemannian manifold N are critical points of the p-energy functional ˆ |∇u|pg

Ep (u) =

 |g| dx

(3.1)

Ω

over W 1,p (Ω, N ), where |∇u|2g =

n

i

i,α,β=1

i

∂u ∂u g αβ ∂x . More precisely, we have α ∂xβ

Definition 3.1. A map u ∈ W 1,p (Ω, N ) is called a weakly p-harmonic map if u satisfies −Δp,g u = Ag (u)(Du, Du)|∇u|p−2 , g where Δp,g u =

∂ 1 |g| ∂xα



|∇u|p−2 g αβ

(3.2)

 ∂u  n ∂u ∂u |g| ∂xβ and Ag (u)(Du, Du) = α,β=1 g αβ A(u)( ∂x ) : Tz Ω × α ∂xβ

Tz Ω −→ Tz Ω⊥ is the second fundamental form on z ∈ N . Once p-harmonic maps from manifolds with C ∞ -Riemannian metrics g, this can be actually transferred to a p-harmonic maps by freezing the coefficients so that there holds a strong convergence of gradients in W 1,q (Ω) with 1 < q < p due to Hardt–Lin–Mou’s work [10]. Here, since g is L∞ -Riemannian metric on Ω, we can also prove its compactness of p-harmonic maps due to the conclusion of Theorem 1.1. As being showed in [9,10], for blow-up sequence there is a sequence there is a strongly convergence conclusion similar to Theorem 1.1 above. Then one may verify that the limit function satisfies the blow-up equation, too. We state it in more precise way. Theorem 3.2. Let Ω denote an open bounded set with L∞ -Riemannian metrics in Rn . Suppose that the sequence {uk } in W 1,p (Ω, Rd ) are the weak solutions of p-harmonic maps (3.2) such that  uk  u weakly in ffl uk −¯ uk 1,p W (Ω, N ) with εk := ∇uk → 0 as k → 0. Denoting vk = εk with u ¯k = Ω uk |g| dx; then vk → v strongly in W 1,q whenever 1 < q < p, and v is a weak solution in W 1,p (Ω, Rd ) to A-harmonic equation as follows ˆ

n

Ω α,β=1

|∇v|p−2 g αβ

∂v ∂ζ  |g| dx = 0, ∂xβ ∂xα

∀ζ ∈ C0∞ (Ω).

(3.3)

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Proof. Since g is L∞ -Riemannian metric on Ω, one may rewrite Eq. (3.2) as −divA(x, Du) = f (x, u, Du),

x ∈ Ω;

(3.4)

where A(x, Du) satisfies the hypotheses of H1 and H2 with 0 < ν(λ) ≤ L(λ), and |f (x, u, Du)| ≤ C(λ)|Du|p ∈ L1 (Ω).

(3.5)

Therefore, vk → v strongly in W 1,q whenever 1 < q < p due to Theorem 1.1, and observe that the “blow-up” sequences {vk } satisfies ˆ  

n

|∇vk |p−2 g αβ

Ω α,β=1

ˆ  = εk1−p 

n

 ∂vk ∂ζ   |g| dx ∂xβ ∂xα

|∇uk |p−2 g αβ

Ω α,β=1

ˆ ≤ εk1−p

 ∂uk ∂ζ   |g| dx ∂xβ ∂xα

 |Ag (u)(Du, Du)|∇u|p−2 | · |ζ| |g| dx g

Ω

ˆ

≤ C(λ, n, d)εk1−p ζ L∞ ˆ = C(λ, n, d)εk ζ L∞

|∇uk |p



|g| dx

Ω

|∇vk |p



|g| dx → 0, as εk → 0.

Ω

This proves Theorem 3.2. 2 Remark 3.3. It’s well known that the weak convergence for p-harmonic maps with energy minimizers is necessarily strong convergence in W 1,2 by Luckhaus [13]. After this, Toro and Wang [14] showed a weak limit of p-harmonic maps is p-harmonic while the target manifolds N is a homogeneous space or the sequence consists of stationary maps; and Hardt, Lin, and Mou [10] got the weak compact of the sequence of p-harmonic maps by showing its strong convergence in W 1,q for 1 ≤ q < p. Later, Wang [16] proved the weak compactness of n-harmonic maps to general Riemannian manifolds and Zheng [17] obtained the weak compact property of biharmonic maps by way of PS sequence. Owing to our own interest, in this note we are actually devoted to the compactness of the sequences of p-harmonic maps with discontinuous metric Riemannian manifold. References [1] E. Acerbi, N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Ration. Mech. Anal. 86 (1984) 125–145. [2] E. Acerbi, N. Fusco, An approximation lemma for W 1,p functions, in: Material Instabilities in Continuum Mechanics, Oxford Univ. Press, 1988. [3] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [4] L. Diening, J. Malek, M. Steinhauer, On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications, ESAIM Control Optim. Calc. Var. 14 (2008) 211–232. [5] L. Diening, S. Schwarzacher, Solenoidal Lipschitz truncation for parabolic PDEs, Math. Models Methods Appl. Sci. 23 (14) (2013) 2671–2700. [6] G. Dolzmann, N. Hungerbühler, S. Müller, Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right hand side, J. Reine Angew. Math. 520 (2000) 1–35. [7] F. Duzaar, G. Mingione, Harmonic type approximation lemmas, J. Math. Anal. Appl. 352 (1) (2009) 301–335. [8] J. Frehse, J. Málek, M. Steinhauer, On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method, SIAM J. Math. Anal. 34 (2003) 1064–1083.

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