Calculus of Variations
Calc. Var. 22, 283–301 (2005) DOI: 10.1007/s00526-004-0273-x
Georg Dolzmann · Jan Kristensen
Higher integrability of minimizing Young measures Received: 5 April 2003 / Accepted: 2 February 2004 / c Springer-Verlag 2004 Published online: 16 July 2004 – Abstract. We prove local higher integrability with large exponents for minimizers and Young measure minimizers of variational integrals of the form F (x, ∇u(x)) dx Ω
where F is a Carath´eodory integrand that resembles the p-Dirichlet integrand at infinity. The result yields existence of minimizing sequences with higher equi-integrability properties locally in Ω. Mathematics Subject Classification (2000): 26B15,74N15
1. Introduction In this paper, we consider variational problems of the following type: Minimize F (x, ∇u(x)) dx subject to u ∈ A,
(1.1)
Ω
where for a prescribed u0 ∈ W 1,p (Ω, RN ) the class of admissible maps is given by A = u0 + W01,p (Ω, RN ). Here Ω is a bounded and open subset of Rn , RN ×n is the space of all real N × n matrices, and F : Ω × RN ×n → R is a Carath´eodory integrand, i.e., x → F (x, ξ) is measurable on Ω for all ξ ∈ RN ×n , and ξ → F (x, ξ) is continuous for almost all x ∈ Ω. Assume that F satisfies the growth condition c1 |ξ|p − c2 ≤ F (x, ξ) ≤ c3 (1 + |ξ|p ),
(1.2)
where c1 , c2 , c3 > 0 and 1 < p < ∞. Without further assumptions on the integrand F the minimization problem (1.1) might not admit a minimizer in A. However, it G. Dolzmann: Mathematics Department, University of Maryland, College Park, MD 207424015, USA J. Kristensen: Mathematical Institute, University of Oxford, 24–29 St. Giles’, Oxford OX1 3LB, United Kingdom GD was supported by NSF through grant DMS0104118, and both authors were partially supported by EPSRC grant GR/R25002.
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always admits at least oneYoung measure minimizer: because of (1.2) any minimizing sequence {uj } for (1.1) has a subsequence {ujk }, such that {∇ujk } generates a Young measure ν. (We refer to Sect. 2 for an explanation of notation and terminology.) The Young measure ν is minimizing for (1.1). In this paper we investigate the integrability properties of such measures locally in Ω. (When the Young measure corresponds to a classical minimizer u ∈ W 1,p (Ω, RN ) our results yield local higher integrability for large exponents of the derivative ∇u.) Even though the approach to higher integrability for large exponents used here seems particularly suited for variational problems of the form (1.1)-(1.2) it can also be applied in certain non-variational settings; we comment on that below. Our method is inspired by the approach used by Iwaniec in [23]. It is important to note that in the general multi-dimensional case (i.e. n, N > 1) minimizers to the variational problem (1.1)-(1.2) are not locally integrable to arbitrarily large powers. Some striking examples to this effect can be found in [36]. It is therefore necessary to impose an additional condition on the behaviour of the integrand F = F (x, ξ) for large values of |ξ|. The main result of this paper yields local higher integrability with large exponents forYoung measure minimizers provided that F resembles the p-Dirichlet integrand in a C 0 -sense at infinity. At the same time we are able to quantify the improved higher integrability. To the best of our knowledge, even when N = 1 and the minimizing Young measure corresponds to a classical minimizer (i.e., an admissible function u : Ω → R) this result is new. Theorem 1. Let n, N ∈ N, p ∈ (1, ∞), 0 ≤ δ < , Ω ⊂ Rn be an open and bounded subset, and F : Ω × RN ×n → R a locally bounded Carath´eodory integrand satisfying |F (x, ξ) − |ξ|p | ≤ δ. (1.3) lim sup sup |ξ|p x∈Ω |ξ|→∞ Then there exist positive constants α = α(n, p) and β = β(n, p), depending on n, p only, with the following property. If q ∈ (p, ∞) satisfies δ (1.4) ≤ α10−βq then any Young measure minimizer ν = Ω δx ⊗ νx dx for the variational problem (1.1) has finite q-th order moments locally in Ω. Furthermore, there exists a constant c = c(F, p, q), depending on F , p, q only, such that Ω RN ×n
pq
| · | dνx dx q
≤
c p
(1− q )n
|Ω| +
Ω RN ×n
| · |p dνx dx
(1.5)
for all Ω ⊂⊂ Ω, where = dist (Ω , ∂Ω). Remark 1. When δ > 0 the condition (1.4) reads q ≤ β1 log α δ . Possible values of the constants α, β are indicated in Sect. 4. For instance, when p = 2 we may take β = 52 n(n + 2), whereas the corresponding formula for α is more complicated.
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Finally, notice that we do not impose the growth condition (1.2) explicitely in Theorem 1, but that it is a consequence of the hypotheses (1.3) and the assumption that F : Ω × RN ×n → R be locally bounded. Using a standard result on gradient Young measures (cf. [25], Cor. 1.8) in connection with Theorem 1 we may deduce the following corollary. Corollary 1. Assume that the hypotheses of Theorem 1 are satisfied and that {uj } is a minimizing sequence for the variational problem (1.1). Then there exists a new minimizing sequence {vj } such that uj − vj → 0 strongly in W01,p (Ω, RN ) and for each Ω ⊂⊂ Ω the estimate |∇vj |
q
Ω
pq
≤
c
(1− p q )n
|Ω| +
|∇vj |p Ω
holds for all j, where c = c(F, q) and = dist (Ω , ∂Ω). Observe that when we take δ = 0 in (1.3), we obtain (1.5) and the conclusion 1,q of Corollary 1 for any q < ∞. This result is sharp on the scale Wloc . In [11] it is 2×2 shown that there exist integrands F : R → R satisfying (1.3) with p = 2, = 1 and δ = 0 such that Ω F (∇u) admits non-Lipschitz minimizers. The main novelty of the present results compared to the related ones of [5] and [23] is that (1.3) only imposes a condition on F (x, ξ) for large |ξ|. The authors of [5] and [23] impose a condition for all values of ξ. In our setting this would correspond to the hypothesis that |F (x, ξ) − |ξ|p | ≤ δ|ξ|p holds for all x ∈ Ω and ξ ∈ RN ×n . The key to our results is a modification of a criterion for higher integrability first presented in [23]. We refer to Sect. 3 for its statement. As remarked above, our approach, like those of [5, 23], is not restricted to a variational setting, but can equally well be applied to give higher integrability with large exponents of weak solutions to systems of PDEs, see [10] and also [26], Prop. 5.1, for a special case. We briefly recall other related results in the literature. 1,q In [12] it is shown that extremals of variational integrals Ω F (∇u) belong to Wloc for all q < ∞ provided that the integrand is rank-one convex and has the form F (ξ) = |ξ|p + E(ξ), where 1 < p < ∞ and E(ξ)/|ξ|r → 0 as |ξ| → ∞ for some 1 < r < p. The proof is based on a bootstrap argument that uses the results of [7], and it is not possible to treat higher integrability of minimizers in this way under the more general condition that F (ξ)/|ξ|p → 0 as |ξ| → ∞. Other previous works have been mainly concerned with finding W 1,∞ -estimates for minimizers and extremals, either in the scalar case N = 1 by use of different techniques (see e.g. [14], [17] and [34]) or in the general multi-dimensional case when the integrand resembles the p-Dirichlet integrand in a C 2 -sense at infinity. This was done first for p = 2 in [6] and subsequently for p ≥ 2 in [21]. Further refinements along these lines can be found in [33] and [18]. In [31] the author obtains uniform W 1,∞ -estimates for singularly perturbed problems of the type considered in [6].
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The question of higher equi-integrability of minimizing sequences of variational problems without convexity conditions has previously been studied in [29], [4] and [38]. In these papers the authors use Ekeland’s variational principle and Gehring’s Lemma to obtain the higher equi-integrability. As mentioned above, the main novelty of the present approach is that we obtain higher equi-integrability for large exponents and that we can quantify the improved integrability. The organization of the paper is as follows. In Sect. 2 we briefly set the notation and recall the basic facts about Young measures that are used in the remainder of the paper. Sect. 3 contains the criterion for higher integrability together with its proof. Finally, Sect. 4 contains the proof of Theorem 1. 2. Preliminaries In this section we set the notation and we briefly recall some background results about gradient Young measures that are used in the sequel. Throughout the paper a cube Q ⊂ Rn means an open cube with sides that are parallel to the coordinate hyperplanes: Q = Q(x, r) = {y ∈ Rn : max |yi − xi | < r}, 1≤i≤n
where x = (x1 , . . . , xn ) etc. Furthermore, we use the shorthand notation τ Q for the dilated cube Q(x, τ r). On a few occasions we also use euclidean balls: B(x, r) = {y ∈ Rn : |y − x| < r}. (| · | denotes the euclidean norm.) The notation Ω ⊂⊂ Ω means that Ω and Ω are open sets, that the closure Ω is compact and contained in Ω. We use standard notation for maps and function spaces (as in e.g. [16]). In particular, when f : S ⊂ Rn → H is a (Lebesgue-) integrable map defined on a measurable set S and with values in a finite-dimensional inner-product space H, then we denote for each measurable subset T ⊆ S of finite and positive Lebesgue measure |T | = Ln (T ) the average of f over T by one of the following symbols: 1 f (x) dx = − f = fT . |T | T T ∗
In connection with a sequence of maps, the symbols →, and denote strong, weak and weak∗ convergence, respectively. We frequently use the Greek letters ξ, ζ and η to denote matrices, and we consider the matrix space RN ×n with the euclidean norm defined as |ξ|2 = trace(ξ T ξ). The notation and terminology for Young measures follow essentially that of [24], [25] and [32]. Hence a W 1,p gradient Young measure on Ω (henceforth a bounded and open subset of Rn ) is a measure ν on Ω × RN ×n for which there exists a weakly convergent sequence {uj } in W 1,p (Ω, RN ), such that Φ(x, ∇uj (x)) dx → ν, Φ as j → ∞ (2.1) Ω N ×n
→ R which is continuous and vanishes outside of some for each Φ : Ω × R compact set. It is not difficult to show from this definition that the measure ν
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must have the property that ν(O × RN ×n ) = Ln (O) for each open subset O of Ω. A general fact of measure theory then implies that ν can be disintegrated as ν = Ω δx ⊗ νx dx, where for each x in Ω νx is a probability measure on RN ×n . This should be interpreted as follows: For any non-negative Borel function F : Ω × RN ×n → [0, ∞] the function x → νx , F (x, ·) is measurable and F dν = F (x, ·) dνx dx. Ω×RN ×n
Ω RN ×n
General facts about weak convergence of measures imply that p | · | dνx dx ≤ lim inf |∇uj |p . j→∞
Ω RN ×n
(2.2)
Ω
This inequality becomes an equality precisely when the sequence {|∇uj |p } is equiintegrable on Ω. A particular consequence of (2.2) is that the measure νx has a finite p-th moment for a.e. x in Ω. Another result that follows easily from the definition concerns the centres of mass: νx , id = ∇u(x) a.e., where u denotes the weak limit of {uj } in W 1,p (Ω, RN ). The interpretation of the probability measures νx is that they give the limiting distribution of points of the tail {∇uj (y)}j≥k for y near x when k → ∞ (see [2], [32]). The Young measure does not provide any information about possible correlations between oscillations of {∇uj } at different points x and y in Ω. In this sense the Young measure only describes the one-point statistics for the oscillation of the sequence (see [39]). 3. A criterion for higher integrability In this section we generalize a well-known criterion for higher integrability obtained first by Iwaniec (Lemma 2 of [23]). Iwaniec’s approach extended earlier work by Gurov and Reshetnyak [22] to integrals over cubes of different sizes. This extension is crucial for applications to higher integrability of weak solutions to partial differential equations and variational problems. We remark that another related result can be found in [28]. The proof presented below is an adaptation of the proof in [23]. As in [23], a decisive tool for the proof is a local version of the celebrated inequality for the sharp maximal function of Fefferman and Stein. For this we rely on Lemma 4 in [23]. Lemma 1. Let n, d ∈ N, 1 ≤ p < q < ∞, τ ∈ (0, 1), M > 0 and Ω ⊂ Rn be an open and bounded set. Suppose H ∈ Lp (Ω, Rd ) and that for every cube Q ⊂⊂ Ω either (3.1) − |H|p ≤ M Q
or
− |H − Hτ Q |p ≤ ε − |H|p , τQ
where
Q
ε = ε(n, p, q) = (p − 1)1−p 2−p(p+4) 10−5nq .
(3.2)
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Then H ∈ Lqloc (Ω, Rd ) and for each Ω ⊂⊂ Ω we have with = dist (Ω , ∂Ω) the estimate
p q
|H|
q
Ω
q 5(n2 +n) p
where Λ = c(n) · 10
≤
Λ
p
τ n (1− q )n
|H|p + M
(3.3)
Ω
and c(n) is a constant depending on n only.
Remark 2. Lemma 2 of [23] is essentially the above statement with p = 1 and without the option (3.1). (In the formula for ε we interpret (p − 1)1−p as 1 for p = 1.) It is crucial for our approach that (3.1) is included. Remark 3. Lemma 1 could be stated with balls instead of cubes: if for any ball B(x, r) ⊂ Ω either (3.1) or (3.2) holds, then the conclusion (3.3) is still valid (with a slightly different constant c(n)). The proof of Lemma 1 is based on Whitney’s decomposition theorem and the following local version on cubes. Lemma 2. Let 1 ≤ p < q < ∞. Suppose that H ∈ Lp (Ω, Rd ) satisfies the assumptions of Lemma 1. If Q0 ⊂⊂ Ω is a cube, then −
1 2 Q0
pq |H|
q
≤ Cτ −n − (|H|p + M ), Q0
where 2
n
C = C(n, pq ) = 2(n+5) n 2 10
q 5(n2 +n) p
.
(3.4)
We first prove Lemma 1 based on this local estimate, and we present the proof of this result after the proof of Lemma 1. Proof of Lemma 1. Recall that a dyadic cube in Rn is a cube of the form (2k j1 , 2k (1 + j1 )) × · · · × (2k jn , 2k (1 + jn )) ∈ Z are integers. By Whitney’s decomposition theorem ([35] where k, j1 , . . . , jn √ Ch. VI; take c = 3 n in the proof of Th. 1 on p. 167) there exists a disjoint collection Q of dyadic cubes Q ⊂ Ω such that |Ω \ Q| = 0 and 2 diam Q < dist (Q, ∂Ω) ≤ 6 diam Q for all Q ∈ Q. The latter implies in particular that 2Q ⊂ Ω, and, arguing
as in [35], Ch. VI, Prop. 3 on p. 169, we obtain the bounded overlap property Q∈Q 12Q ≤ 6n in Ω. Now let Q = {Q ∈ Q : Q ∩ Ω = ∅}. Clearly, |Ω \ Q | = 0 and by use of Lemma 2 we find pq pq pq
q q q |H| ≤ |H| ≤ |H| ≤ Ω
Q∈Q
Q
Q∈Q
Q
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p p p |Q| q Cτ −n − |H|p + M = C2−n τ −n |Q| q −1 |H| + M .
Q∈Q
2Q
Q∈Q
2Q
Since for Q ∈ Q we have Q ∩ Ω = ∅, it follows that = dist (Ω , ∂Ω) ≤ dist (Q, ∂Ω) + diam Q ≤ 7 diam Q. Hence we may estimate the last sum from above by p
−(1− q )n −n −n √ C2 τ |H|p + M , 7 n 2Q Q∈Q
and by use of the bounded overlap property and the formula (3.4) for C we obtain (3.3). The rest of this section is devoted to the proof of Lemma 2. The proof relies on a few auxiliary results that we recall as we go along. The following result is well-known, and we give the short proof for the convenience of the reader. Lemma 3. Let H ∈ Lp (Ω, Rd ), 1 ≤ p < ∞. Then p1 p1 − |H − HΩ |p ≤ 2 − |H − a|p Ω
Ω
for any constant vector a ∈ R . (The factor 2 can be omitted when p = 2.) d
Proof. Following [23] we deduce from Minkowski’s and H¨older’s inequalities p1 p1 p p − |H − HΩ | = − |(H − a) − (H − a)Ω | Ω
Ω
p1 p1 p p ≤ − |H − a| + |(H − a)Ω | ≤ 2 − |H − a| , Ω
Ω
as asserted.
The rest of this section is devoted to the proof of Lemma 1 under the assumption that p > 1. The case p = 1 is easier and the details are left to the interested reader. Put h = |H|p and let Q ⊂⊂ Ω be a cube. The first goal is to derive the condition (3.5) below for h from the alternative (3.1)-(3.2) for H. If hQ ≤ M , then − |h − hτ Q | ≤ 2τ −n M. τQ
Next assume that hQ > M , so that (3.2) holds. By use of Lemma 3 and the estimate ||ξ|p − |ζ|p | ≤ p|ξ − ζ|(|ξ| + |ζ|)p−1 it follows that p−1 p p − |h − hτ Q | ≤ 2 − ||H| − |Hτ Q | | ≤ 2p − |H − Hτ Q | (|H| + |Hτ Q |) . τQ
τQ
τQ
For δ > 0 we have by Young’s inequality p−1
|H − Hτ Q | (|H| + |Hτ Q |)
≤
1 1 p − 1 p−1 p |H − Hτ Q |p + δ (|H| + |Hτ Q |) , pδ p
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and hence by use of (3.2), (|H| + |Hτ Q |)p ≤ 2p−1 (|H|p + |Hτ Q |p ) and |Hτ Q |p ≤ hτ Q , 1 2ε − |h − hτ Q | ≤ − h + (p − 1)2p+1 δ p−1 − h. δ Q τQ τQ (p−1)/p equates the two coefficients on the right-hand Taking δ = ε/((p − 1)2p ) side and yields − |h − hτ Q | ≤ ε¯(− h + − h) τQ
Q
τQ
with ε¯ = (p − 1)1−1/p 2p ε1/p . It follows that for any cube Q ⊂⊂ Ω − |h − hτ Q | ≤ ε¯ − h + − h + 2τ −n M. τQ
Q
(3.5)
τQ
Fix a cube Q0 ⊂⊂ Ω and let ρ ∈ C 1 (Rn ) be a nonnegative, even function supported in the unit ball and with ρ = 1. For 0 < t < dist(Q0 , ∂Ω) put x−y −n h(y) dy, x ∈ Q0 . t ρ g(x) = gt (x) = t Rn It is clear that 0 ≤ g ∈ L∞ (Q0 ) and that h(x) = limt0 gt (x) for a.e. x ∈ Q0 . Fix 0 < t < dist(Q0 , ∂Ω). Observe that for any cube Q ⊆ Q0 the inequality − |g − gτ Q | ≤ ε¯ − g + − g + 2τ −n M (3.6) τQ
Q
τQ
holds. Indeed, note that (since ρ is even) y − |g − gτ Q | ≤ t−n ρ( ) − |h(x) − hy+τ Q | dxdy. t y+τ Q τQ Rn By (3.5), −
|h − hy+τ Q | ≤ ε¯ −
y+τ Q
h + 2τ −n M,
h+−
y+Q
y+τ Q
and therefore we have (3.6). We use (3.6) to derive a bound on gt Lq¯( 12 Q0 ) which is uniform in t, where q¯ = q/p. The decisive tools for doing this are, as in [23], the local versions of the Hardy-Littlewood and Fefferman-Stein maximal functions. Recall that these functions are defined for a function f ∈ L1 (Q0 ) as f (x) = sup − |f |, x ∈ Q0 , Q0 ⊇Q x Q
and #
f (x) =
sup − |f − fQ |,
Q0 ⊇Q x Q
x ∈ Q0 ,
respectively. (The suprema are taken over all cubes Q ⊆ Q0 that contain x.)
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Theorem 2. Suppose that f # ∈ Lr (Q0 ), 1 ≤ r < ∞. Then f ∈ Lr (Q0 ) and r1 r1 − |f |r ≤ 105nr − |f # |r + 10n+1 − |f |. Q0
Q0
Q0
Proof. We refer to [23], Lemma 4, for the proof. As in [23] we use the auxiliary function d(x)n g(x) for x ∈ Q0 G(x) = 0 else,
where d(x) = dist(x, Rn \ Q0 ). Our next goal is to prove the following assertion: If k ≥ 2n + 1, then n+2 n 2 k n # 2 + 4¯ ε G (x) + 2n (1 + ) g + 22−n τ −n M |Q0 | G (x) ≤ k τ Q0 (3.7) for x ∈ Q0 . Fix x ∈ Q0 and let Q x be a subcube of Q0 . We consider two cases: d(x) ≤ τk diam Q and d(x) > τk diam Q. In the first case we find by elementary estimations supQ dn ≤ (1 + τ −1 k)n nn/2 |Q| and since − |G − GQ | ≤ 2GQ ≤ 2|Q|−1 sup dn g, Q
Q
Q
we get n k − |G − GQ | ≤ 2n 2 (1 + )n g. τ Q Q0
(3.8)
In the other case, d(x) > τk diam Q, so τ −1 Q is a subcube of Q0 and therefore (3.6) holds for τ −1 Q. We get from Lemma 3 and (3.6) (recall that x is a fixed point in Q0 ) that − |G − GQ | ≤ 2 − |dn g − d(x)n gQ | Q
Q
≤ 2 − |dn − d(x)n |g + 2d(x)n ε¯ − Q
τ −1 Q
g + − g + 2τ −n M Q
supQ |dn − d(x)n | + ε¯d(x)n 2¯ εd(x)n − G+ − G + 4d(x)n τ −n M. ≤2 inf Q dn inf τ −1 Q dn τ −1 Q Q By elementary estimations supQ dn ≤ (1 + τ n n k ) d(x) , so
τ n n k ) d(x)
and inf Q dn ≥ (1 −
supQ |dn − d(x)n | + ε¯d(x)n (1 + τk )n − 1 ε¯ ≤ + . inf Q dn (1 − τk )n (1 − τk )n
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Since 0 < τ < 1 and k ≥ 2n√ + 1 it follows that (1 − τ /k)n > (1 − 1/(2n + 1))n = 1 exp(−n log(1 + 2n )) ≥ 1/ e > 1/2 and (1 + τ /k)n − 1 ≤ 2n /k (by use of binomial formula), hence that supQ |dn − d(x)n | + ε¯d(x)n 2n+1 ≤ + 2¯ ε. inf Q dn k Similarly we get that inf τ −1 Q dn ≥ (1 − k1 )n d(x)n , so d(x)n 1 ≤ ≤ 2. inf τ −1 Q dn (1 − k1 )n Finally we have that d(x)n ≤ 2−n |Q0 | and using the Hardy-Littlewood maximal function we can summarize the second case as n+2 2 + 4¯ ε G (x) + 22−n τ −n M |Q0 |. − |G − GQ | ≤ (3.9) k Q Now (3.7) follows easily by addition of (3.8) and (3.9). We use Theorem 2 with f = G, r = q¯ and estimate G# by (3.7) and Minkowski’s inequality to get 1/¯q n+2 1/¯q 2 q¯ 5n¯ q q¯ + 4¯ ε − |G | − |G | ≤ 10 k Q0 Q0 n k + 105n¯q 2n 2 (1 + )n g + 22−n τ −n M |Q0 | + 10n+1 − G. τ Q0 Q0 If we take k = 2n+4 · 105n¯q and keep in mind that ε¯ = (p − 1)1−1/p 2p ε1/p = n+2 2−4 10−5n¯q we achieve that 105n¯q ( 2 k + 4¯ ε) = 1/2. With this choice of k we therefore have 1/¯q q¯ −n n+1 − |G | ≤ Kτ (g + M ) + 2 · 10 − G, Q0
Q0
Q0
n 2 2 = 2(n+3) n 2 105(n +n)¯q . n
where K = K(n, q¯) Recall that G = d g. We eliminate the auxiliary function G by use of the inequalities d(x)n ≤ 2−n |Q0 | for x ∈ Q0 , d(x)n ≥ 4−n |Q0 | for x ∈ 12 Q0 and G ≤ G : 1/¯q 1/¯q n 4n · 2 q¯ q¯ q¯ − G ≤ ≤ − g 1 |Q0 | Q0 2 Q0 22n+n/¯q −n n+1 −n Kτ (g + M ) + 2 · 10 2 g |Q0 | Q0 Q0 and since clearly Kτ −n > 21−n · 10n+1 and (n + 3)2 + nq¯ + 2n + 1 < (n + 5)2 we get 1/¯q q¯ − g ≤ Cτ −n − (g + M ), 1 2 Q0
Q0
Higher integrability of minimizing Young measures n
2
293
2
(n+5) where n 2 105(n +n)¯q . Finally, we recall that g = gt (x) = −nC = C(n, q¯) = 2 t ρ(y/t)h(x − y) dy and pass to the limit t 0 to conclude Rn
−
1/¯q
1 2 Q0
h
q¯
≤ Cτ −n − (h + M ). Q0
Since q¯ = q/p and h = |H|p this finishes the proof.
4. Proof of Theorem 1 The constants α = α(n, p) and β = β(n, p) will be determined in the course of the proof. N ×n Fix a locally bounded Carath´ → R satisfy eodory integrand F : Ω × R ing (1.3), and assume that ν = Ω δx ⊗ νx dx is a minimizing Young measure for the variational problem (1.1). This means that there exists a sequence {uj } ⊂ A such that Ω F (x, ∇uj ) → inf u∈A Ω F (x, ∇u) as j → ∞ and such that {∇uj } generates ν. In the first step of the proof we use Ekeland’s variational principle to find a new sequence {vj } ⊂ A consisting of minimizers of perturbed problems, such that {∇vj } generates ν. In this context, Ekeland’s variational principle was first employed in [29]. Subsequent and related applications have appeared in for example [4, 19, 20, 38]. Let εj = F (x, ∇uj ) − inf F (x, ∇u) u∈A
Ω
Ω
and note that εj → 0 as j → ∞. If we apply Ekeland’s variational principle (see [13], Cor. 6.1, p. 30) to the functional 1,p F (x, ∇u0 + ∇u) when u ∈ W0 (Ω, RN ), Ω I[u] = +∞ else, and the Banach space W01,1 (Ω, RN ) endowed with the norm u = Ω |∇u|, we √ obtain for each j a map vj ∈ A, such that Ω |∇uj − ∇vj | ≤ εj and √ F (x, ∇u) + εj |∇u − ∇vj | F (x, ∇vj ) ≤ Ω
Ω
for all u ∈ u0 + W01,p (Ω, RN ). In particular, the sequence {∇vj } generates the Young measure ν. The second and main step of the proof is to establish the following assertion: There exist constants M > 0, τ ∈ (0, 1) and k ∈ N, such that for any cube Q ⊂ Ω and j ≥ k either (4.1) − |∇vj |p ≤ M Q
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or
− |∇vj − (∇vj )τ Q |p ≤ ε − |∇vj |p , τQ
(4.2)
Q
where ε = ε(n, p, q) is the constant defined in Lemma 1. (We remark that the proof will show that we may take τ = τ (n, p, q) depending on n, p, q only, whereas M and k will depend on the particular integrand F .) Suppose the assertion were false. Then for any choice of τ ∈ (0, 1) there exists a sequence of cubes Qj ⊂ Ω, such that − |∇vj |p → ∞ Qj
−
and
|∇vj − (∇vj )τ Qj |p > ε − |∇vj |p .
τ Qj
Qj
We proceed to show that for a suitable choice of τ ∈ (0, 1) this is impossible. First, note that since F is locally bounded and satisfies (1.3) it also satisfies a growth condition of the form (1.2) for constants ci = ci (F ). Suppose that Qj = Q(xj , rj ) and let Q = (−1/2, 1/2)n . Define 1 p tj = − |∇vj |p , Qj
and wj (x) =
N ×n Fj (x, ξ) = t−p j F (xj + rj x, tj ξ), (x, ξ) ∈ Q × R
1 rj tj
vj (xj + rj x) − (vj )Qj ,
x ∈ Q.
The rescaled integrands Fj are Carath´eodory and if we write Ej (x, ξ) = Fj (x, ξ)− |ξ|p , then by virtue of (1.2) and (1.3) the function Ej satisfies the growth condition |Ej (x, ξ)| ≤ c˜(1 + |ξ|p ) for (x, ξ) ∈ Q × RN ×n and all j, where c˜ depends on c1 , c2 and c3 only. In view of (1.3) (and the above growth condition for |ξ| small) we have for each κ>0 |Ej (x, ξ)| sup ≤ δ. lim sup p j→∞ x∈Q, ξ∈RN ×n κ + |ξ| Fix κ ∈ (0, − δ) and take k = k(κ) ∈ N, such that |Ej (x, ξ)| ≤ (δ + κ)(κ + |ξ|p )
(4.3)
for (x, ξ) ∈ RN ×n and j ≥ k. By definition the maps wj ∈ W 1,p (Q, RN ) satisfy (4.4) (wj )Q = 0, − |∇wj |p = 1, Q (4.5) − |∇wj − (∇wj )τ Q |p > ε τQ
Higher integrability of minimizing Young measures
and
Fj (x, ∇wj ) ≤
Q
Q
295
√ Fj (x, ∇wj + ∇ϕ) + t1−p εj |∇ϕ| j
(4.6)
for all ϕ ∈ W01,p (Q, RN ). We can rewrite the last inequality in terms of the integrand Ej and if we invoke (4.3) we obtain for j ≥ k after a computation that √ εj |∇ϕ| 2(δ+κ)κ+t1−p p p j +δ+κ |∇wj | ≤ −δ−κ |∇wj + ∇ϕ| + −δ−κ spt (ϕ)
spt (ϕ)
spt (ϕ)
(4.7) for ϕ ∈ W01,p (Q, RN ), where spt (ϕ) denotes the support of ϕ. By considering a suitable subsequence we can assume that wj → w in ∗ Lp (Q, RN ), ∇wj ∇w weakly in Lp (Q, RN ) and |∇wj |p Ln µ weakly∗ 0 ∗ 1,p N in C (Q) , where w ∈ W (Q, R ) and µ is a non-negative, finite Borel measure on Q. By weak lower semicontinuity of the norm it follows that |∇w|p ≤ µ(Q) = 1. Q
Take cubes R and S verifying R ⊂⊂ S ⊆ Q and µ(∂R ∪ ∂S) = 0. Fix ϕ ∈ W01,p (R, RN ) and put ϕj = ρ(ϕ + w − wj ), where ρ : Rn → [0, 1] is a C 1 cut-off function which is 1 on R and 0 on Rn \ S. (Extend ϕ by 0 outside R.) Testing (4.7) with ϕj yields for j ≥ k: √ εj |∇ϕj | 2(δ+κ)κ+t1−p j +δ+κ |∇wj |p ≤ −δ−κ |∇wj + ∇ϕj |p + −δ−κ S S S p +δ+κ ≤ −δ−κ |∇w + ∇ϕ| R p−1 p p p−1 p p |wj − w| sup |∇ρ| + 2 (|∇w| + |∇wj | ) + 2 S\R
+ S
√ εj |∇ϕj | 2(δ+κ)κ+t1−p j . −δ−κ
If we pass to the limit j → ∞ we obtain by use of standard results that p +δ+κ |∇w| ≤ µ(S) ≤ −δ−κ |∇w + ∇ϕ|p S R p n n +δ+κ 2p−2 |∇w| L + µ (S \ R) + 2(δ+κ) + −δ−κ 2 −δ−κ κL (S). This inequality is valid for all κ ∈ (0, − δ) and each pair of cubes R ⊂⊂ S ⊆ Q with µ(∂R ∪ ∂S) = 0. Sending κ 0 and R S yields p +δ |∇w| ≤ µ(S) ≤ −δ |∇w + ∇ϕ|p (4.8) S
S
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for all ϕ ∈ W01,p (S, RN ) and all cubes S ⊆ Q with µ(∂S) = 0. (In fact it can be shown that w, in the terminology of [20], is a Q-minimizer for the p-Dirichlet integral with Q = ( + δ)/( − δ).) The next goal is to extract two consequences from (4.8) that will lead to a contradiction with (4.5) and hence prove the assertion. Without loss in generality we can assume that µ(∂Q) = 0. Indeed, otherwise we can consider a concentric subcube rQ with r close to 1 so µ(∂(rQ)) = 0 and replace wj by w ˜j (x) = (wj (rx) − (wj )rQ )/(rsj ) with sj > 0 chosen so Q |∇w ˜j |p = 1. In view of (4.4) and (4.5) there exists s > 0, so s ≤ sj ≤ 1 for all j. The following argument could be applied to a suitable subsequence of w ˜j . Let h ∈ W 1,p (Q, RN ) be the p-harmonic map satisfying h = w on ∂Q. In the sequel we focus on the subquadratic case 1 < p < 2. The details for the remaining cases p ≥ 2 are similar, and rely on the convexity inequality |ξ|p − |ζ|p − p|ζ|p−2 ζ · (ξ − ζ) ≥ 22−p |ξ − ζ|p valid for all ξ, ζ ∈ RN ×n . We leave it to the interested reader to check this. In the subquadratic case the above convexity inequality fails, and instead we shall use the following elementary lemma and Hanner’s inequality. Lemma 4. Let 1 < p < 2, s ∈ [0, 1] and t ∈ [0, 2]. If 2p (1 + (1 + s)p ) ≥ (2 + t)p + (2 − t)p , then t≤
6s p−1 .
Remark 4. It is not hard to show that the largest value for t = t(s) satisfying this inequality obeys t(s) ∼ 4s/(p − 1) as s 0. Proof. Since t → (2 + t)p + (2 − t)p − p(p − 1)2p−2 t2 is convex on [−2, 2] it follows that (2 + t)p + (2 − t)p ≥ 2p+1 + p(p − 1)2p−2 t2 for |t| ≤ 2. The assumption of the lemma therefore implies that 2p (1 + (1 + s)p ) ≥ 2p+1 + p(p − 1)2p−2 t2 , and solving for t2 gives t2 ≤ 4((1 + s)p − 1)/(p(p − 1)). Since (1 + s)p − 1 ≤ (2p − 1)s for s ∈ [0, 1] and (2p − 1)/p ≤ 3/2 when 1 ≤ p ≤ 2 we have t2 ≤ 6s/(p − 1) as stated. The first task is to show that w is W 1,p -close to h. By use of Hanner’s inequality (see [27], Th. 2.5) we obtain p 2p ∇hpLp + ∇wpLp ≥ ∇h + ∇wLp + ∇h − ∇wLp p + ∇h + ∇wLp − ∇h − ∇wLp . Observe that ∇hLp ≤ ∇wLp ≤ (1 + s)∇hLp and 2∇hLp ≤ ∇h + ∇wLp ≤ (2 + s)∇hLp (since h and 2h are p-harmonic and h = w on ∂Q), where 1 +δ p s = −δ − 1.
Higher integrability of minimizing Young measures
297
For later reference we record that 0≤s≤
6α −βq 10 . p
(4.9)
The lower bound is obvious. To verify the upper bound put ϑ = α10−βq and note that δ/ ≤ ϑ, ϑ ∈ [0, 1/2], (1 + t)/(1 − t) is convex and increasing on [0, 1/2], and t1/p is concave on [0, ∞), hence s≤
1+ϑ 1−ϑ
p1
1
− 1 ≤ (1 + 6ϑ) p − 1 ≤
6 ϑ. p
If ∇wLp = 0, then µ(S) = 0 for all cubes S ⊆ Q with µ(∂S) = 0 by (4.8). Applying this to a cube S ⊃ τ Q contradicts (4.5). Consequently, ∇wLp > 0, and therefore ∇hLp > 0. Hence we can normalize in Hanner’s inequality by division with ∇hLp to get p Lp ) 2p (1 + (1 + s)p ) ≥ 2p (1 + ∇w
∇h Lp ≥
∇h+∇w Lp
∇h Lp
+
∇h−∇w Lp
∇h Lp
p
Lp + ∇h+∇w −
∇h Lp
∇h−∇w Lp
∇h Lp
p .
Note that 2 ≤ ∇h + ∇wLp /∇hLp ≤ 2 + s. If 2∇hLp < ∇h − ∇wLp we put r = ∇h − ∇wLp /∇hLp > 2 and consider the consequence 2p (1 + (1 + s)p ) ≥ (2 + r)p > 4p , which is impossible. Hence t = ∇h − ∇wLp /∇hLp ≤ 2 and thus 2p (1 + (1 + s)p ) ≥ (2 + t)p + (2 − t)p . By Lemma 4, ∇h − ∇wLp ≤ ∇hLp
6s p−1
≤
6s p−1
(4.10)
which is the desired first consequence of (4.8). To deduce the second consequence of (4.8), that ∇wj is Lp -close to ∇w for large j, we write up Hanner’s inequality with ∇w, ∇wj : p 2p ∇wj pLp + ∇wpLp ≥ ∇wj + ∇wLp + ∇wj − ∇wLp p + ∇wj + ∇wLp − ∇wj − ∇wLp . As in the above argument we can show that ∇wj − ∇wLp ≤ 2∇wLp holds from a certain step j ≥ j0 . Passing to the limit j → ∞ and noting that 2 ≤ lim sup
∇wj +∇w Lp
∇w Lp
≤2+s
t = lim sup
∇wj −∇w Lp
∇w Lp
∈ [0, 2]
j→∞
we obtain with j→∞
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that 2p (1 + (1 + s)p ) ≥ (2 + t)p + (2 − t)p . Again, by Lemma 4, lim sup ∇wj − ∇wLp ≤ ∇wLp
j→∞
6s p−1
≤
6s p−1 ,
(4.11)
which is the desired second consequence of (4.8). We are now ready for the final step. We need the following regularity result for p-harmonic maps, which is related to Uhlenbeck’s result [37], but in the stated form is due to DiBenedetto and Manfredi [7]. Lemma 5. There exist constants γ ≥ 1 and σ ∈ (0, 1], depending on n and p only, such that if Q ⊂ Rn is a cube and h ∈ W 1,p (Q, RN ) is p-harmonic, i.e. div (|∇h|p−2 ∇hi ) = 0 in Q for i = 1, . . . , N , then p pσ − |∇h − (∇h)rQ | ≤ γr − |∇h|p (4.12) rQ
Q
for all r ∈ (0, 1). (We can take σ = 1 when p = 2.) Proof. The inequality is established on page 1113 √ of [7] with balls instead of cubes. By use of the inclusions Q(x, r/2) ⊂ B(x, r n/2), B(x, 1/2) ⊂ Q(x, 1/2) and Lemma 3 we arrive at (4.12). By Minkowski’s inequality 1 1 p p p p − |∇wj − (∇wj )τ Q | ≤ 2 − |∇wj − ∇w| + τQ
τQ
1 1 p p p p 2 − |∇w − ∇h| + − |∇h − (∇h)τ Q | , τQ
τQ
and hence, by use of (4.10), (4.11) and Lemma 5 1 n p 1 − p 6s ≤ 4τ p p−1 + γ p τ σ. lim sup − |∇wj − (∇wj )τ Q | j→∞
τQ
If we take
τ=
1 − 6s 4γ p p−1
p n+pσ
we obtain 4τ
n −p 6s p−1
pσ n+pσ n 1 6s + γ p τ σ = 2γ (n+pσ)p 4 p−1 ,
and therefore by (4.9), 1 p p ≤ AαB 10−Bβq lim sup − |∇wj − (∇wj )τ Q | j→∞
τQ
Higher integrability of minimizing Young measures
where A = A(n, p) and B = B(n, p) =
pσ 2(n+pσ) .
299
If we take
5n 10n(n + pσ) = pB p2 σ
β= and
1 1 B1 1−p p+5 1 εp = A− B (p − 1) pB 2 B 2 then α = α(n, p) > 0, β = β(n, p) > 0 and 1
α = A− B 10βq
1 p 1 1 p lim sup − |∇wj − (∇wj )τ Q | ≤ εp 2 j→∞ τQ which is in contradiction with (4.5). This proves assertion (4.1)-(4.2). Finally, to finish the proof of Theorem 1 we use Lemma 1. Take τ = τ (n, p, q) ∈ (0, 1), M = M (F, q) > 0 and k = k(F, q) ∈ N so that (4.1)-(4.2) hold for vj when j ≥ k. By Lemma 1 it follows in particular that {|∇vj |p } is locally equi-integrable in Ω, and for Ω ⊂⊂ Ω ⊂⊂ Ω, we get
pq
| · | dνx dx q
Ω RN ×n
Λ τ n
(1− p q )n
≤ lim inf
Ω RN ×n
j→∞
|∇vj |
q
Ω
pq
≤
| · |p + M dνx dx
where = dist (Ω , ∂Ω ). The estimate (1.5) follows easily from this upon letting Ω Ω.
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