Sobolev and Lipschitz regularity for local minimizers of widely ...
arXiv:1604.04189v1 [math.AP] 14 Apr 2016
SOBOLEV AND LIPSCHITZ REGULARITY FOR LOCAL MINIMIZERS OF WIDELY DEGENERATE ANISOTROPIC FUNCTIONALS LORENZO BRASCO, CHIARA LEONE, GIOVANNI PISANTE, AND ANNA VERDE Abstract. We prove higher differentiability of bounded local minimizers to some widely degenerate functionals, verifying superquadratic anisotropic growth conditions. In the two dimensional case, we prove that local minimizers to a model functional are locally Lipschitz continuous functions, without any restriction on the anisotropy.
Contents 1. Introduction 1.1. Overview 1.2. Main results 1.3. Some comments on the proofs 1.4. Plan of the paper 2. Preliminaries 2.1. Notation 2.2. Embedding for anisotropic Sobolev spaces 2.3. Anisotropic Besov-Nikol’ski˘ı spaces 3. A general scheme for improving differentiability 4. Local Sobolev estimate in a particular case 5. Local Lipschitz estimate in dimension two 5.1. Proof of Theorem 1.4 5.2. Uniform Lipschitz estimate Appendix A. Pointwise inequalities References
1 1 2 4 6 6 6 7 9 14 21 25 25 26 30 31
1. Introduction 1.1. Overview. In this paper we continue to investigate differentiability properties of local minimizers of convex functionals, exhibiting wide degeneracies and an orthotropic structure. The model case of functional we want to study is given by ˆ ˆ N X 1 pi 1,p 0 (1.1) F(u; Ω ) = (|uxi | − δi )+ dx + f u dx, u ∈ Wloc (Ω), Ω0 b Ω, pi Ω0 0 Ω i=1
with δi ≥ 0 and pi ≥ 2. We denote p = (p1 , p2 , . . . , pN ) and n o 1,p 1,1 i Wloc (Ω) = u ∈ Wloc (Ω) : uxi ∈ Lploc (Ω), i = 1, . . . , N . The symbol ( · )+ above stands for the positive part. 2010 Mathematics Subject Classification. 35J70, 35B65, 49K20. Key words and phrases. Anisotropic problems, degenerate elliptic equations, Besov-Nikol’ski˘ı spaces. 1
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BRASCO, LEONE, PISANTE, AND VERDE
For p1 = · · · = pN , some results can be found in the recent papers [3] and [4]. We refer to the introduction of [4] for some motivations of this kind of functionals, arising from Optimal Transport problems with congestion effects. Our scope is to generalize these results to the anisotropic case, i.e. to the case where at least one of the exponents pi is different from the others. These functionals pertain to the class of variational problems with non standard growth conditions, first introduced by Marcellini in [17, 18]. Similar functionals have been considered in the past by the Russian school, see for example [12] and [29]. More recently, they have been considered by many authors also in western countries. Among others, we mention (in alphabetical order) Bildhauer, Fuchs and Zhong [1, 2] (where the terminology splitting-type integrals is used), Esposito, Leonetti and Mingione [7], Leonetti [15], Liskevich and Skrypnik [16] and Pagano [22]. However, we point out that the type of degeneracy admitted in (1.1) is heavier than those of the above mentioned references, due to the presence of the coefficients δi ≥ 0 above. We observe that local minimizers of the functional (1.1) are local weak solution of the degenerate elliptic equation N X pi −1 uxi = f. (|uxi | − δi )+ |uxi | xi i=1
The particular case δ1 = · · · = δN = 0 and p1 = · · · = pN = p corresponds to N X
|uxi |p−2 uxi
xi
= f,
i=1
which has been called pseudo p−Laplace equation in the recent literature. Here we prefer to use the terminology orthotropic p−Laplace equation, which seems more adapted and meaningful. 1.2. Main results. Our first result is the Sobolev regularity for some nonlinear functions of the gradient of a bounded local minimizer. For p = (p1 , . . . , pN ), we will use the notation p0 := (p01 , . . . , p0N ), where p0i is the H¨ older conjugate of pi . Theorem 1.1 (Sobolev regularity for bounded minimizers). Let ` ∈ {1, . . . , N − 1} and 2 ≤ p ≤ q. We set p = (p, · · · , p, q, · · · , q ), | {z } | {z } `
and let u ∈
1,p Wloc (Ω)
∩
N −`
L∞ loc (Ω)
be a local minimizer of ˆ N ˆ X 0 F(u; Ω ) = gi (uxi ) dx + i=1
Ω0
f u dx,
Ω0
0
1,p where f ∈ Wloc (Ω) and g1 , . . . , gN : R → R+ are C 2 convex functions such that
1 (|s| − δ)p−2 ≤ gi00 (s) ≤ C (|s|p−2 + 1), + C 1 00 q−2 (|s| − δ)q−2 + 1), + ≤ gi (s) ≤ C (|s| C
s ∈ R, i = 1, . . . , `, s ∈ R, i = ` + 1, . . . , N,
REGULARITY FOR ANISOTROPIC FUNCTIONALS
for some C ≥ 1 and δ ≥ 0. We set Vi = Vi (uxi ),
ˆ
where
Vi (s) = 0
sq
gi00 (τ ) dτ,
3
i = 1, . . . , N.
• If ` = N − 1 and p, q satisfy
p≥N −1
or
p <
(N − 2)2 , N −1
(N − 2) p , (N − 2) − p (N − 2)2 ≤ p < N − 1, N −1
q < (1.2) or
q
N. Finally, for 0 < t < 1 and 1 ≤ p < ∞ we denote by W t,p (RN ) the Sobolev-Slobodecki˘ı space, i. e. n o W t,p (RN ) = u ∈ Lp (RN ) : [u]W t,p (RN ) < +∞ , where
ˆ [u]pW t,p (RN )
ˆ
|u(x) − u(y)|p dx dy. |x − y|N +t p
= RN
Though we will not need this, we recall that class of Besov spaces.
RN t,p W (R)
can be seen a particular instance of the larger
2.2. Embedding for anisotropic Sobolev spaces. We collect here a couple of embedding results that will be needed in the sequel. The first one is well-known1, a proof can be found for example in [27, Theorem 1 & Corollary 1]. Theorem 2.1 (Anisotropic Sobolev embeddings). Let Ω ⊂ RN be an open set, then for every u ∈ W01,p (Ω) we have: (1) if p < N 1 N ˆ X pi p c kukLp∗ (Ω) ≤ , |uxi | i dx i=1
Ω
for a constant c = c(N, p) > 0; (2) if p = N and |Ω| < +∞, for every 1 ≤ χ < ∞ 1 N ˆ pi 1 X c kukLχ (Ω) ≤ |Ω| χ |uxi |pi dx . i=1
Ω
for a constant c = c(N, p, χ) > 0; 1This result is usually attributed to Troisi in the literature of western countries, see [26]. However, Trudinger in
[27] attributes the result for p 6= N to Nikol’ski˘ı, whose paper [21] appeared before [26]. In any case, the methods of proof are different.
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BRASCO, LEONE, PISANTE, AND VERDE
(3) if p > N and |Ω| < +∞ c kukL∞ (Ω) ≤ |Ω|
1 N
− p1
N ˆ X
1
pi
|uxi | dx
pi
,
Ω
i=1
for a constant c = c(N, p) > 0. The next embedding result is stated in [13, Theorem 1]. We provide a proof for the reader’s convenience. Proposition 2.2. Let Ω ⊂ RN be an open set and let p = (p1 , . . . , pN ) be such that 1 < p1 ≤ · · · ≤ pN < p∗ .
(2.2) Then for every E b Ω we have
∗
W 1,p (Ω) ,→ Lp (E),
if p 6= N,
and W 1,p (Ω) ,→ Lχ (E), for every 1 ≤ χ < ∞,
if p = N.
Proof. Let us fix two concentric balls B%0 b BR0 b Ω. For every %0 ≤ % < R ≤ R0 we take a standard cut-off function η ∈ C0∞ (BR ) such that η ≡ 1 on B% , with k∇ηkL∞ (RN ) ≤
C , R−%
for some universal constant C > 0. Let u ∈ W 1,p (Ω), then for every M > 0 we define n o uM = min |u|, M ∈ W 1,p (Ω) ∩ L∞ (Ω), and finally take uM η ∈ W01,p (BR ) ⊂ W01,p (Ω). Let us suppose for simplicity that p < N , by Theorem 2.1 we have ˆ 1∗ 1 N ˆ X pi p pi p∗ ≥c |uM η| dx , |(uM η)xi | dx Ω
Ω
i=1
for some c = c(N, p) > 0. By using the properties of η, with simple manipulations we get ! 1∗ ˆ 1 1 ˆ N ˆ N p X X pi pi C ∗ (2.3) |(uM )xi |pi dx + |uM |pi dx ≥c |uM |p dx , R−% BR Ω B% i=1
i=1
for a possibly different constant c > 0, still depending on N and p only. We now observe that by hypothesis (2.2) we have 1 < pi < p∗ , thus by interpolation in Lebesgue spaces ˆ
pi
|uM | dx
ˆ
1
pi
≤
BR
(1−ϑi ) ˆ |uM | dx
BR
where ϑi =
pi − 1 p∗ ∈ (0, 1). pi p∗ − 1
p∗
|uM | dx BR
ϑ∗i p
,
REGULARITY FOR ANISOTROPIC FUNCTIONALS
9
Thus by Young inequality we get 1 R−%
ˆ
N
pi
|uM | dx BR
pi
1 ˆ (1−ϑi ) # 1−ϑ i τ −ϑi ≤ |uM | dx R−% BR ˆ 1∗ p p∗ +τ |uM | dx ,
"
BR
for every 0 < τ < 1. By choosing τ small enough and using the previous estimate in (2.3), we get ˆ 1 N ˆ N X X pi C pi |uM | dx |(uM )xi | dx + 1 1−ϑi Ω Ω i=1 i=1 (R − %) ! 1∗ ˆ 1∗ ˆ p p c p∗ p∗ + |uM | dx ≥c |uM | dx . 2 BR B% The previous holds for every %0 ≤ % < R ≤ R0 , from [10, Lemma 6.1] we obtain C
N ˆ X i=1
pi
|(uM )xi |
1 dx
Ω
pi
+
ˆ
N X
C
i=1
(R0 − %0 ) 1−ϑi
ˆ |uM | dx ≥
1
Ω
! p∗
|uM | dx
1 p∗
,
B%0
for some constant C = C(N, p) > 0. By arbitrariness of Br0 b BR0 b Ω, for every E b Ω a standard covering argument leads to # ˆ " N ˆ 1∗ 1 ˆ X pi p pi p∗ + |u| dx ≥ |uM | dx , (2.4) C |uxi | dx Ω
Ω
i=1
E
for some constant C = C(N, p, dist(E, ∂Ω)) > 0. In the previous inequality we also used that |uxi | ≥ |(uM )xi |
and
|uM | ≤ |u|,
almost eveywhere on Ω.
If we now take the limit as M goes to +∞ in (2.4), we get the desired result.
Remark 2.3 (Optimality of assumptions). In general we can not take E = Ω or pN ≥ p∗ in the previous result, see [13] for a counter-example. On the contrary, the hypothesis p1 > 1 can be easily removed and we can relax it to p1 ≥ 1. We leave the verification of this fact to the reader. 2.3. Anisotropic Besov-Nikol’ski˘ı spaces. Let ψ ∈ Lp (RN ), for p ≥ 1 and 0 < t ≤ 1 we define the quantities
δhei ψ
(2.5) [ψ]nt,p = sup , i = 1, . . . , N, ∞,i |h|t p N |h|>0
L (R )
and (2.6)
[ψ]bt,p
∞,i
δ2 ψ
he = sup it |h|>0 |h|
,
i = 1, . . . , N.
Lp (RN )
Lemma 2.4. Let 0 < t < 1, then for every ψ ∈ Lp (RN ) we have i 1 C h (2.7) [ψ]bt,p ≤ [ψ]nt,p ≤ [ψ]bt,p + kψkLp (RN ) . ∞,i ∞,i ∞,i 2 1−t
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BRASCO, LEONE, PISANTE, AND VERDE
For t = 1, for every ψ ∈ Lp (RN ) we have 1 [ψ]bt,p ≤ [ψ]nt,p , ∞,i ∞,i 2 p N and there exists ψ0 ∈ L (R ) such that [ψ0 ]b1,p < +∞
[ψ0 ]n1,p = +∞.
and
∞,i
∞,i
Proof. The first inequality in (2.7) is a plain consequence of triangle inequality and invariance by translations of Lp norms. The second one can be proved by using a standard device, see [25, Chapter 2.6]. For t = 1, an instance of function with the properties above can be found in [23, Example page 148]. If t = (t1 , . . . , tN ) ∈ (0, 1]N , by following Nikol’ski˘ı we define the corresponding anisotropic Besov-Nikol’ski˘ı spaces as ) ( N X t,p N∞ (RN ) := ψ ∈ Lp (RN ) : [ψ]nti ,p < +∞ , i=1
and2
( t,p B∞ (RN )
:=
p
N
ψ ∈ L (R ) :
N X
∞,i
) [ψ]bti ,p < +∞ ,
i=1
∞,i
see [20, pages 159–161]. We equip them with the norms kψkN∞ t,p (RN ) := kψkLp (RN ) +
N X i=1
[ψ]nti ,p
and
∞,i
kψkB∞ t,p (RN ) := kψkLp (RN ) +
N X i=1
[ψ]bti ,p . ∞,i
From now on we will always implicitly assume that t1 ≤ t2 ≤ · · · ≤ tN . Before going on, a couple of comments are in order. Remark 2.5 (Comparison of the two spaces). By Lemma 2.4 we get that if 0 < t1 ≤ · · · ≤ tN < 1, then t,p t,p N∞ (RN ) = B∞ (RN ). On the contrary, if ti = 1 for some i ∈ {1, . . . , N }, then t,p t,p N∞ (RN ) ,→ B∞ (RN )
and
t,p t,p N∞ (RN ) 6= B∞ (RN ).
Moreover, we recall that if [ψ]n1,p < +∞,
for some i ∈ {1, . . . , N },
∞,i
then its distributional derivative ψxi belongs to Lp (RN ), see [20, Theorem 4.8]. Remark 2.6. In the isotropic case t1 = · · · = tN = t with 0 < t < 1, we simply denote these t,p t,p spaces by N∞ (RN ) and B∞ (RN ). By Lemma 2.4 the seminorms ψ 7→
N X i=1
[ψ]nt,p
∞,i
and
ψ 7→
N X i=1
[ψ]bt,p , ∞,i
2In [20] this space is denoted by H t and is seen to be a particular instance of a general class of anisotropic Besov p t spaces noted Bpθ , with 1 ≤ θ ≤ ∞.
REGULARITY FOR ANISOTROPIC FUNCTIONALS
11
are equivalent. Moreover, these in turn are equivalent to
2
δh ψ
δh ψ
ψ 7→ sup t or ψ 7→ sup .
t |h|>0 |h| Lp (RN ) |h|>0 |h| Lp (RN ) t,p t,p The next very simple result asserts that N∞ (RN ) and B∞ (RN ) do not change, if in (2.5) and (2.6) the supremum is restricted to 0 < |h| < h0 . The easy proof is left to the reader.
Lemma 2.7. Let 0 < t ≤ 1 and ψ ∈ Lp (RN ), then for every h0 > 0 and every i = 1, . . . , N we have
δhei ψ −t
[ψ]nt,p ≤ sup
|h|t p N + 2 h0 kψkLp (RN ) , ∞,i 00
i = 1, . . . , N,
δ2 ψ
hei = sup t |h|>0 |h|
i = 1, . . . , N.
i
and (2.10)
[ψ]bt,p
∞,i (Ω)
,
Lp (Ω2hei )
Accordingly, we introduce the anisotropic Besov-Nikol’ski˘ı spaces on Ω as ( ) N X t,p N∞ (Ω) := ψ ∈ Lp (Ω) : [ψ]nti ,p (Ω) < +∞ , i=1
∞,i
and ( t,p B∞ (Ω)
:=
p
ψ ∈ L (Ω) :
N X i=1
) [ψ]bti ,p (Ω) < +∞ . ∞,i
REGULARITY FOR ANISOTROPIC FUNCTIONALS
13
Finally, we define n o t,p t,p N∞,loc (Ω) := ψ ∈ Lploc (Ω) : ψ ∈ N∞ (E) for every E b Ω , and n o t,p t,p B∞,loc (Ω) := ψ ∈ Lploc (Ω) : ψ ∈ B∞ (E) for every E b Ω . t,p t,p Remark 2.12. As for the case of RN , the definitions of N∞ (Ω) and B∞ (Ω) do not change if we perform the supremum in (2.9) and (2.10) over 0 < |h| < h0 for some h0 > 0.
Corollary 2.13. Let Ω ⊂ RN be an open set. Under the assumptions of Theorem 2.10 and with the same notations, we have t,p t,p N∞,loc (Ω) ⊂ B∞,loc (Ω) ⊂ Lplocχ (Ω),
for every 1 ≤ χ
0, then there exist x1 , . . . , xk ∈ E such that
E⊂
k [ j=1
B d (xj ). 8
It is sufficient to prove that ψ ∈ Lp χ (Bd/8 (xj )) for every j = 1, . . . , k. We fix one of these balls and omit to indicate the center xj for simplicity. We then take a standard cut-off function t,p η ∈ C0∞ (Bd/4 ) ⊂ C0∞ (Ω) such that η ≡ 1 on Bd/8 . Then we observe that ψ η ∈ B∞ (RN ): indeed, by triangle inequality and (2.1) for every h 6= 0 such that |h| < d/8 we have
2 ψ
δ2 η
δ 2 (ψ η)
δ δ η
hei
he
hei hei hei
η2hei ≤ it ψ +2
|h|ti δhei ψ p N + ti
|h| i p N
p N
|h|ti p N |h| L (R ) L (R ) L (R ) L (R )
1−ti 2
d
δhe ψ ≤4 k∇ηkL∞ kψkLp (B d ) + it , i = 1, . . . , N,
|h| i p 8 2 L (B d ) 2
and the supremum of the latter over 0 < |h| < d/8 is finite, since Bd/2 b Ω by construction. t,p By appealing to Lemma 2.7, we thus get ψ η ∈ B∞ (RN ). We can use Theorem 2.10 and get ψ η ∈ Lp χ (RN ). Since η ≡ 1 on Bd/8 , this gives the desired result.
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BRASCO, LEONE, PISANTE, AND VERDE
3. A general scheme for improving differentiability In this section we consider a slightly more general framework, with respect to that of Theorem 1.1. Namely, we consider a set of C 2 convex functions gi : R → R+ such that 1 (|s| − δ)p+i −2 ≤ gi00 (s) ≤ C (|s|pi −2 + 1), i = 1, . . . , N, (3.1) C for some C ≥ 1, δ ≥ 0 and 2 ≤ p1 ≤ · · · ≤ pN −1 ≤ pN . Remark 3.1. Let us point out the following simple inequality that will be used in what follows: for every a ≤ s ≤ b, we have (3.2) gi00 (s) ≤ Cei gi00 (a) + gi00 (b) + 1 , for some Cei = Cei (pi , δ) ≥ 1. This follows with elementary manipulations, by exploiting (3.1). We leave the details to the reader. 1,p (Ω) a local minimizer of We then consider u ∈ Wloc ˆ N ˆ X 0 F(u; Ω ) = gi (uxi ) dx + i=1
Ω0
f u dx.
Ω0
In particular, u solves (3.3)
N ˆ X
ˆ gi0 (uxi ) ϕxi
dx +
f ϕ dx = 0,
i=1
for every ϕ ∈ W01,p (Ω0 ) and every Ω0 b Ω. For every i = 1, . . . , N , we define ˆ tq Vi = Vi (uxi ), where Vi (t) = gi00 (τ ) dτ. 0
Our aim is to prove that every Vi enjoys some weak differentiability properties. We start with the following result. 0
1,p Proposition 3.2 (Initial gain). Let 2 ≤ p1 ≤ · · · ≤ pN −1 ≤ pN and let f ∈ Wloc (Ω). We suppose that u ∈ L∞ loc (Ω). Then for every i = 1, . . . , N we have p1 pN −1 t,2 Vi ∈ N∞,loc (Ω), ,..., ,1 . where t = pN pN
Proof. We take Br0 b BR0 b Ω a pair of concentric balls centered at x0 and set h0 = (R0 − r0 )/4
and
R=
R0 + r0 . 2
Then we pick ϕ ∈ W01,p (BR ) that we extend it to zero on RN \ BR . For every 0 < |h| < h0 we can insert the test function ϕ−hej (x) in (3.3). With a simple change of variables we get ˆ N ˆ X (3.4) gi0 (uxi )hej ϕxi dx = fhej ϕ dx. i=1
Ω
Ω
REGULARITY FOR ANISOTROPIC FUNCTIONALS
15
By subtracting (3.3) and (3.4) and dividing by |h|, we thus get # " ˆ N ˆ X δhej f gi0 (uxi )hej − gi0 (uxi ) ϕxi dx = ϕ dx. |h| Ω |h| Ω i=1
We now make the following particular choice ϕ = ζ2
δhej u , |h|sj
where sj ∈ (−1, 1] will be chosen below and ζ is the standard cut-off function R − |x − x0 | . ζ(x) = min 1, R − r0 + We obtain
N ˆ X δhej gi0 (uxi ) δhej uxi 2 ζ dx |h| |h|sj i=1 N ˆ X δhej gi0 (uxi ) |ζx | ζ ≤2 i h i=1
ˆ δhej u dx + |h|sj
δhej f h
δhej u 2 |h|sj ζ dx.
Recalling the definition of Vi , using (A.1) in the left-hand side and (A.2) (in combination with (3.2)) in the right-hand side, we obtain 2 N ˆ N ˆ q q δ u X X he δhej Vi δhej Vi 2 j gi00 (uxi )hej + gi00 (uxi ) + 1 |ζxi | ζ sj +1 dx sj +1 sj +1 ζ dx ≤ C |h| 2 |h| 2 |h| 2 i=1 i=1 ˆ δhe f δhe u + j jsj ζ 2 dx. h |h| If we use H¨ older and Young inequalities in the right-hand side, we can absorb the higher-order term. Namely, since we have N ˆ q q δ u X he δhej Vi j gi00 (uxi )hej + gi00 (uxi ) + 1 |ζxi | ζ sj +1 dx sj +1 |h| 2 |h| 2 i=1 2 N ˆ X δhej Vi 2 ≤Cτ sj +1 ζ dx |h| 2 i=1 N ˆ δ u 2 00 C X he j + gi (uxi )hej + gi00 (uxi ) + 1 |ζxi |2 sj +1 dx, τ |h| 2 i=1 where 0 < τ < 1, by choosing τ small enough, we thus get 2 N ˆ N ˆ δ u 2 X X δ V i he he j gi00 (uxi )hej + gi00 (uxi ) + 1 |ζxi |2 sjj+1 dx sj +1 ζ 2 dx ≤ C |h| 2 |h| 2 i=1 i=1 10 p0 pj !1 p ˆ ˆ pj j j δ f δhej u hej +C . sj +1 dx sj +1 dx BR |h| 2 BR |h| 2
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BRASCO, LEONE, PISANTE, AND VERDE
By basic properties of differential quotients, we get for 0 < |h| < h0 ˆ ˆ δ u pj 1−sj pj hej 2 |uxj |pj dx, sj +1 dx ≤ C h0 BR |h| 2 BR0 and similarly ˆ
0 1−sj δ f pj hej sj +1 dx ≤ C h0 2 BR |h| 2
p0j
ˆ
0
|fxj |pj dx. BR0
This yields 2 N ˆ N ˆ δ u 2 X X δ V C hej i 2 he gi00 (uxi )hej + gi00 (uxi ) + 1 sjj+1 dx sj +1 ζ dx ≤ 2 (R − r ) 0 0 |h| 2 |h| 2 i=1 i=1 !1 ! 10 ˆ ˆ pj p pj p0 j 1−sj ux dx fx j dx . + C h0 j j
(3.5)
BR0
BR0
We use again H¨ older inequality in the first term in the right-hand side, so that N ˆ δ u 2 X 00 he gi (uxi )hej + gi00 (uxi ) + 1 sjj+1 dx |h| 2 i=1 !2 pi −2 ˆ N ˆ pi δ u pi pi X pi 00 he j 00 pi −2 . ≤ gi (uxi )hej + gi (uxi ) + 1 dx sj +1 dx BR |h| 2 BR i=1 We now observe that with simple manipulations we have ˆ
00 pi gi (uxi )hej + gi00 (uxi ) + 1 pi −2 dx
ˆ
pi −2
BR
pi
|gi00 (uxi ) + 1|
≤C
pi pi −2
! pi −2 pi
dx
,
BR0
since for every 0 < |h| < h0 we have BR + hej ⊂ BR0 , by construction. Thus from (3.5) we obtain 2 pi !2 ˆ N N ˆ pi X X
δ u δ V C hej hej i 2
gi00 (ux ) + 1 pi dx sj +1 ζ dx ≤ sj +1 i (R0 − r0 )2 L pi −2 (BR0 ) BR |h| 2 |h| 2 i=1 i=1 (3.6) ! 10 ˆ !1 ˆ pj p pj p0 j 1−sj fx j dx ux dx + C h0 . j j B R0
B R0
The first term in the right-hand side is more delicate and we have to distinguish between two cases. Case A: j = N . By hypothesis we have pi ≤ pN for every 1 ≤ i ≤ N . Thus we get ˆ BR
pi δ u N heN sN +1 dx ≤ C R |h| 2
pN −pi pN
ˆ BR
pN ! pi pN δ u heN , sN +1 dx |h| 2
i = 1, . . . , N.
REGULARITY FOR ANISOTROPIC FUNCTIONALS
17
We can then choose sN = 1 so that (sN + 1)/2 = 1 as well. Then from (3.6) we get "N # N ˆ i X 2 N ppN −p X δheN (Vi ) 2 2
C pi 00 N ζ dx ≤
gi (ux ) + 1 pi R0 i h (R0 − r0 )2 L pi −2 (BR0 ) i=1 i=1 ! 10 ! 2 ! 1 ˆ ˆ ˆ pN pN p N pN p0N pN +C . × |fxN | dx |uxN | dx |uxN | B R0
BR0
BR0
Case B: 1 ≤ j ≤ N − 1. This in turn has to be divided in two sub-cases. Case B.1: 1 ≤ i ≤ j. This is similar to Case A, since by hypothesis we have pi ≤ pj . Then for 0 < |h| < h0 we simply have ! pi ˆ ˆ pj p −p δ u pi δ u pj N jp i hej hej j sj +1 dx ≤ C R sj +1 dx BR |h| 2 BR |h| 2 ! pi ˆ pj −pi pj 1−sj N pi pj ≤ C h0 2 R . |uxj |pj dx B R0
Case B.2: j + 1 ≤ i ≤ N . Here we should be more careful. The order of maximal differentiability tj = (sj + 1)/2 is determined here. We set tj = pj /pN as in the statement, we thus get p ˆ ˆ δhej u pi
−pj δhej u j
δhe u pi∞ dx . j |h|tj dx ≤ t p L (BR ) j i BR BR |h| Since pj − tj pi ≥ 0, we further observe that for every 0 < |h| < h0 we have p ˆ ˆ ˆ δhej u pj δhej u j pj −tj pi dx ≤ C hpj −tj pi |uxj |pj dx. dx ≤ h 0 0 h tj pi |h| BR BR BR 0
Moreover kδhej ukL∞ (BR ) ≤ 2 kukL∞ (BR0 ) . By using the previous estimates in (3.6) we thus obtain3 N ˆ X δhej Vi 2 2 |h|tj ζ dx i=1 # " j i 2 (1−tj ) X 2 N ppjj−p
C h0 pi 00
gi (ux ) + 1 pi ≤ R0 kuxj k2Lpj (BR ) i 0 (R0 − r0 )2 L pi −2 (BR0 ) i=1 p pj pj N j X
00
2 ppj 2 p −p 2 1− p C i i N
+ h0 gi (uxi ) + 1 p p−2 kukL∞ (BR ) uxj Lpji(B ) i R0 0 (R0 − r0 )2 L i (BR0 ) i=j+1
2 (1−tj )
fx + C h0 j
p0 L j (B
R0 )
ux pj j L (B
R0 )
,
3It is intended that the second term in the right-hand side is 0 for j = N .
18
BRASCO, LEONE, PISANTE, AND VERDE
for a constant C = C(N, p1 , . . . , pN ) > 0. By taking the supremum over 0 < |h| < h0 , summing over j = 1, . . . , N and recalling that ζ ≡ 1 on Br , we finally conclude that N X j=1
δhej Vi
< +∞, sup tj |h| 0 0, where we used that δ ≥ 1. In order to reconstruct the full gradient p1
∇W14
+ 2s
on the left-hand side, we observe that p1 s + p1 + 2 s W14 2 = p1 x2
p1 s W14 W12 . x2
Then if we fix 1 < q < 2, by H¨ older’s inequality with exponents 2/q and 2/(2 − q), we have !2 ! 2−q 2 ˆ p 2 ˆ p1 s q ˆ q q q 1 s p + 2 s + 1 2−q q 2 4 2 4 ≤ W1 dx . W1 η dx W1 η dx p1 spt(η) x2 x2
REGULARITY FOR ANISOTROPIC FUNCTIONALS
By using the same manipulations as in [3], we thus get ˆ p1 s q ! 2q ˆ p1 2 + 2 ≤ C (1 + s)2 W14 W14 2 η dx η dx x2 x2 (5.5) 2 ˆ p1 q q + s2q q 4 |ηx2 | dx , +C W1
27
ˆ spt(η)
q 2−q
W1
s
! 2−q q
dx
with C = C(p1 , p2 ) > 0. We assume for simplicity that all the balls are centered at the origin. We then fix the radius r0 > 0 as above and define R0 = 2 r0
R1 :=
3 r0 . 2
For r0 < r < R < R1 , we take η ∈ W01,∞ (BR ) to be the standard cut-off function (R − |x|)+ . η(x) = min 1, R−r By multiplying (5.4) and (5.5) ˆ p1 s 2 ˆ W14 + 2 η dx x1
we get
q ! 2q p1 s + W14 2 η dx x2 # " ˆ 2 ˆ pi −2 X 1 s+1 2 s 2 p1 −2 |fε | W1 dx ≤Cδ Wi 2 W1 dx + (s + 1) (R − r)2 BR B R i=1 2−q ˆ ˆ p1 2 q q s 2−q 2 4 × (s + 1) dx W1 W1 dx BR BR x2 # ˆ 2 p1 q q 1 + s2q 4 W1 . + dx 2 (R − r) BR
Then we apply the anisotropic Sobolev inequality of Theorem 2.1 to the compactly supported (p +2 s)/4 function W1 1 η. This yields ˆ p s q∗ 4∗ q 1+ Tq W14 2 η dx ˆ 2 ˆ pi −2 X 1 s+1 2 2 s 2 W W dx + (s + 1) |f | W dx ε 1 1 i (R − r)2 BR i=1 BR ˆ p 2 ˆ 2−q q q 1 s 2−q 2 4 × (s + 1) W1 dx W1 dx ≤ C δ p1 −2
(5.6)
BR
+
1 (R − r)2
ˆ
BR
BR
x2
p1 q + s2q
W1 4
2 q
dx
.
28
BRASCO, LEONE, PISANTE, AND VERDE
The exponents q and q ∗ are given by 4q 4q and q∗ = , 2+q 2−q the constant Tq only depends on q and it degenerates to 0 as q goes to 2. The idea is to use the previous fundamental estimate (5.6) to produce an iterative scheme of reverse H¨older inequalities on shrinking balls. Then we perform a Moser’s iteration in order to conclude. We need to estimate the terms appearing in the right-hand side of (5.6). The crucial difference with respect to [3] is in the first term on the right-hand side of (5.6), i.e. ˆ ˆ 2 ˆ pi −2 p2 −2 p1 X s+1 s 2 2 W2 2 W1 W1s dx. (5.7) W1 dx = W1 W1 dx + Wi q=
i=1
BR
BR
BR
On the contrary, all the other terms are estimated exactly as in [3], thus we omit the details. Let us now focus on the term above, it is useful to introduce the quantity # pi −2 p1 " ˆ 2 pi p1 −2 2 pi 2 pi X R0 2 4 e I(W1 , W2 , fε ; R0 , R1 ) = Wi dx + ∇Wi dx R1 BR1 BR0 i=1 ! 10 ˆ 2 p
+ R0 1
0
|fε |2 p1 dx
p1
.
B R1
e 1 , W2 , fε ; R0 , R1 ) is uniformly bounded, independently of ε. To this aim, as First we claim that I(W for the term containing fε , we observe that by Proposition 2.2 we have the continuous embedding (recall that R1 < R0 ) 0
0
W 1,p (BR0 ) ,→ L2 p1 (BR1 ), thus the term
∗
since p02 < p01 ≤ 2 and 2 p01 < p0 = ˆ
! 2 p01
|fε |
dx
2 p01 p02 , p01 + p02 − p01 p02
1 p01
,
B R1 0
can be uniformly bounded in terms of the W 1,p norm of f on BR0 . The terms containing the p /4 p /4 gradients of W1 1 and W2 2 are more delicate, for them we need Theorem 1.1. Indeed, let us define ˆ tq 00 (s) ds Vi,ε (t) = gi,ε and Vi,ε = Vi,ε ((uε )xi ), i = 1, 2. 0
0 > 0. If we set We observe that Vi,ε : R → R is a locally Lipschitz omeomorphism, with Vi,ε pi fi (t) = δ 2 + (|t| − δ)2+ 4 , t ∈ R, p /4
then we obtain that Wi i
= Φi,ε (Vi,ε ), where −1 Φi,ε (t) = fi (Vi,ε (t)),
t ∈ R.
It is not difficult to see that Φi,ε is a Lipschitz function, with Lipschitz constant independent of ε. Indeed, we have p pi −2 f0i (t) = 0, for |t| < δ and |f0i (t)| ≤ Ci |t| 2 , for |t| ≥ δ.
REGULARITY FOR ANISOTROPIC FUNCTIONALS
29
q p −2 00 (t) ≥ √1 |t| i2 , gi,ε for |t| ≥ δ, Ci for some Ci = Ci (pi , δ) ≥ 1. Thus we get 1 −1 |Φ0i,ε (t)| = f0i (Vi,ε (t)) 0 ≤ Ci , t ∈ R. −1 Vi,ε (Vi,ε (t)) 0 Vi,ε (t) =
By using this observation, we thus obtain ˆ ˆ pi 2 ∇W 4 dx ≤ Li i BR1
|∇Vi,ε |2 dx,
BR1
with Li = Li (pi , δ) > 0. We can now invoke (1.4), in order to bound uniformly the last term. It is only left to observe that the bound in (1.4) also depends on the local L∞ norm of uε . This can be uniformly bounded by appealing to [8, Theorem 3.1], proving the claim. We now come back to estimate the quantities in (5.7). Let us recall that, since we are in dimension N = 2, we have the continuous embedding W 1,2 (BR1 ) ,→ Lϑ (BR1 ) for every 1 ≤ ϑ < +∞. Then by H¨older’s inequality and Sobolev-Poincar´e inequality, exactly as in [3] we get 1 ˆ 2 ˆ p1 p1 p01 s p s 1 2 e 1 , W2 , fε ; R0 , R1 ) R W1 dx . W1 W1 dx ≤ C I(W 0 BR
BR
For the second term we have to be more careful. By using H¨older inequality with exponents p1 p1 p01 , p01 , p1 , p1 − 2 2 we get !1 ! p10 p 2 p01 p 2 p01 p2 −2 p1 ˆ ˆ ˆ p1 p2 −2 1 p1 −2 p2 1 2 + W14 W24 dx W2 2 W1 W1s dx ≤ C dx B R1
BR1
BR
ˆ × BR
W1s p1
dx
1 p1
,
where we further used Young’s inequality and the constant C = C(p1 ) > 0 depends only on p1 . To treat the term into square brakets, we use again Sobolev-Poincar´e inequalities. Namely, we have " # ! 10 p 2 p01 ˆ ˆ 2 p1 p1 p1 2 1 0 p ∇W 4 dx , W14 dx ≤ C R1 1 W12 dx + 1 B R1
B R1
BR1
and ˆ BR1
p 2 p01 p2 −2 p1 −2 2 W24
Thus we obtain ˆ BR
p2 −2 2
W2
p1 p2
! p10
1
dx
W1 W1s dx
p1 −2 p2 p2 −2 p1
2 p1 −2 p2 p01 p2 −2 p1
≤ C R1
"
p2 2
BR1
2 p01
e 1 , W2 , fε ; R0 , R1 ) R ≤ C I(W 0
as well, where we used again that R1 < R0 .
ˆ
W2 dx + ˆ BR
W1s p1
BR1
# p2 2 ∇W 4 dx . 2
dx
1 p1
,
30
BRASCO, LEONE, PISANTE, AND VERDE
By using these estimates in (5.6) and proceeding as in [3] for all the other terms, we obtain (5.8) "ˆ " # 2−q 2q ˆ 2 1 q p1 2−q p1 − p2 R0 +s s p p1 −2 1 2 1 e 1 , W2 , fε ; R0 , R1 ) R W1 ≤Cδ I(W W1 dx dx 0 R−r BR Br 1 # ˆ p1 − p2 s p e 1 , W2 , fε ; R0 , R1 ) R 1 W1 1 dx +(s + 1)2 I(W 0 BR
ˆ
" 2
q 2−q
e 1 , W2 , fε ; R0 , R1 ) × (s + 1) I(W BR
+
R0 R−r
2
2
R0
W1
s
2−q q dx ˆ
2 − p1 −1 q 1
e 1 , W2 , fε ; R0 , R1 ) I(W BR
W1s p1
1# dx
p1
,
for a constant C = C(p1 , q) > 0. The exponent 1 < q < 2 is chosen as 2 p1 q 2 1 q= , so that = p1 and − − 1 = 0. p1 + 1 2−q q p1 By further observing that W1 ≥ 1, from (5.8) we gain 1 ˆ p1 2 s p1 e 1 , W2 , fε ; R0 , R1 )2 dx W1 ≤ C δ p1 −2 I(W Br
" ×
R0 R−r
#2
2 + (s + 1)
2
− p2 R0 1
ˆ BR
W1s p1
dx
2 p1
,
for s ≥ 0. This is an iterative scheme of reverse H¨older inequalities, we can now iterate infinitely many times this estimate, as in [3]. Appendix A. Pointwise inequalities Lemma A.1. Let g : R → R+ be a C 1,1 convex function. Let us set ˆ tp g 00 (τ ) dτ. V (t) = 0
For every a, b ∈ R we have (A.1)
g 0 (a) − g 0 (b) (a − b) ≥ |V (a) − V (b)|2 .
Proof. Without loss of generality, we can assume that a ≥ b. Indeed, g 0 (a) − g 0 (b) and a − b have the same sign, thanks to the monotonicity of g 0 . For a = b there is nothing to prove, so we take a > b. By using Jensen inequality, we have ˆ a 0 0 00 g (a) − g (b) (a − b) = g (t) dt (a − b) b
ˆ ≥
ap
2
g 00 (t) dt
= (V (a) − V (b))2 ,
b
as desired.
REGULARITY FOR ANISOTROPIC FUNCTIONALS
31
Lemma A.2. Let g : R → R+ be a C 1,1 convex increasing function. For every a, b ∈ R we have p 0 g (a) − g 0 (b) ≤ sup (A.2) g 00 (s) |V (a) − V (b)|. s∈[a,b]
Proof. For ε > 0, let us consider the function gε (t) = g(t) + ε t2 . We set ˆ tp Vε (t) = gε00 (τ ) dτ, 0
then we observe that this is a strictly increasing function, thus invertible. Finally, we define Fε (t) = gε0 Vε−1 (t) , which is an increasing function. Indeed, we have q 1 = gε00 (Vε−1 (t)) > 0. Fε0 (t) = gε00 (Vε−1 (t)) 0 −1 Vε (Vε (t)) By basic Calculus, this yields |gε0 (a) − gε0 (b)| = |Fε (Vε (a)) − Fε (Vε (b))| ≤ sup Fε0 (Vε (s)) |Vε (a) − Vε (b)| s∈[a,b]
= sup
p gε00 (s) |Vε (a) − Vε (b)|.
s∈[a,b]
By taking the limit as ε goes to 0, we get the desired conclusion.
|t|p /p,
Remark A.3. When g(t) = the previous inequalities imply the familiar estimates p−2 2 4 p−2 |a|p−2 a − |b|p−2 b (a − b) ≥ (p − 1) 2 |a| 2 a − |b| 2 b . p and p−2 p−2 p−2 p − 1 p−2 p−2 a − |b|p−2 b ≤ 2 |a| 2 + |b| 2 |a| |a| 2 a − |b| 2 b . p References [1] M. Bildhauer, M. Fuchs, X. Zhong, A regularity theory for scalar local minimizers of splitting-type variational integrals, Ann. Sc. Norm. Super. Pisa Cl. Sci., 6 (2007), 385–404. 2 [2] M. Bildhauer, M. Fuchs, X. Zhong, Variational integrals with a wide range of anisotropy, St. Petersburg Math. J., 18 (2007), 717–736. 2 [3] P. Bousquet, L. Brasco, V. Julin, Lipschitz regularity for local minimizers of some widely degenerate problems, to appear on Ann. Sc. Norm. Sup., available at http://cvgmt.sns.it/paper/2515/ 2, 4, 25, 26, 27, 28, 29, 30 [4] L. Brasco, G. Carlier, On certain anisotropic elliptic equations arising in congested optimal transport: local gradient bounds, Adv. Calc. Var., 7 (2014), 379–407. 2 [5] A. Canale, A. D’Ottavio, F, Leonetti, M. Longobardi, Differentiability for bounded minimizers of some anisotropic integrals, J. Math. Anal. Appl., 253 (2001), 640–650. 3, 5 [6] G. Cupini, P. Marcellini, E. Mascolo, Local boundedness of minimizers with limit growth conditions, J. Optim. Theory Appl., 166 (2015), 1-22. 3 [7] L. Esposito, F. Leonetti, G. Mingione, Sharp regularity for functionals with (p, q) growth, J. Diff. Eq., 204 (2004), 5–55. 2 [8] N. Fusco, C. Sbordone, Some remarks on the regularity of minima of anisotropic integrals, Commun. Partial Differ. Equations, 18 (1993), 153–167. 3, 6, 29 [9] M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math., 59 (1987), 245–248. 3 [10] E. Giusti, Metodi diretti nel calcolo delle variazioni. (Italian) [Direct methods in the calculus of variations], Unione Matematica Italiana, Bologna, 1994. 9
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BRASCO, LEONE, PISANTE, AND VERDE
[11] Hong Min-Chun, Some remarks on the minimizers of variational integrals wtih non standard growth conditions, Boll. Un. Mat. Ital. A, 6 (1992), 91–101. 3 [12] ¯I. M. Kolod¯ı˘ı, The boundedness of generalized solutions of elliptic differential equations, Vestnik Moskov. Univ. Ser. I Mat. Meh., 25 (1970), 44–52. 2 [13] S. N. Kruzhkov, ¯I. M. Kolod¯ı˘ı, On the theory of embedding of anisotropic Sobolev spaces, Russian Math. Surveys, 38 (1983), 188-189. 8, 9 [14] A. Kufner, O. John, S. Fuˇc´ık, Function spaces. Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis. Noordhoff International Publishing, Leyden; Academia, Prague, 1977. 6, 11 [15] F. Leonetti, Higher integrability for minimizers of integral functionals with nonstandard growth, J. Differential Equations, 112 (1994), 308–324. 2 [16] V. Liskevich, I. I. Skrypnik, H¨ older continuity of solutions to an anisotropic elliptic equation, Nonlinear Anal., 71 (2009), 1699–1708. 2 [17] P. Marcellini, Regularity and existence of solutions of elliptic equations with p, q−growth conditions, J. Differential Equations, 90 (1991), 1–30. 2 [18] P. Marcellini, Regularity of minimizers of integrals of the Calculus of Variations under non standard growth conditions, Arch. Rational Mech. Anal., 105 (1989), 267–284. 2 [19] P. Marcellini, Un exemple de solution discontinue d’un probl`eme variationnel dans le cas scalaire, preprint n. 11 dell’Ist. Mat. Univ. Firenze (1987). Available at http://web.math.unifi.it/users/marcell/lavori 3 [20] S. M. Nikol’ski˘ı, Approximation of functions of several variables and imbedding theorems. Translated from the Russian by John M. Danskin, Jr. Die Grundlehren der Mathematischen Wissenschaften, Band 205. SpringerVerlag, New York-Heidelberg. 1975. 6, 10, 12 [21] S. M. Nikol’ski˘ı, Imbedding theorems for functions with partial derivatives considered in various metrics, Izd. Akad. Nauk. SSSR, 22 (1958), 321–336. 7 [22] A. Pagano, Higher integrability for minimizers of variational integrals with nonstandard growth, Ann. Univ. Ferrara, 39 (1993), 1–17. 2 [23] E. Stein, Singular integrals and differentiability properties of functions. Princeton Mathematical Series, 30. Princeton University Press, Princeton, N.J. 1970. 10 [24] H. Triebel, Theory of function spaces. III. Monographs in Mathematics, 100. Birkh¨ auser Verlag, Basel, 2006. 6, 12 [25] H. Triebel, Theory of function spaces. II. Monographs in Mathematics, 84. Birkh¨ auser Verlag, Basel, 1992. 10 [26] M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat., 18 (1969), 3–24. 7 [27] N. Trudinger, An imbedding theorem for H 0 (G, Ω) spaces, Studia Math., 50 (1974), 17–30. 7 [28] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), 219–240. 4 [29] N. Uralt’seva, N. Urdaletova, The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations, Vest. Leningr. Univ. Math., 16 (1984), 263–270. 2, 4
(L. Brasco) Dipartimento di Matematica e Informatica ` degli Studi di Ferrara Universita Via Machiavelli 35, 44121 Ferrara, Italy ´matiques de Marseille and Institut de Mathe ´, Marseille, France Aix-Marseille Universite E-mail address: [email protected] (C. Leone & A. Verde) Dipartimento di Matematica “R. Caccioppoli” ` degli Studi di Napoli “Federico II” Universita Via Cinthia, Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy E-mail address: [email protected] E-mail address: [email protected] (G. Pisante) Dipartimento di Matematica e Fisica ` degli Studi di Napoli Seconda Universita Viale Lincoln 5, 81100 Caserta, Italy E-mail address: [email protected]
SOBOLEV AND LIPSCHITZ REGULARITY FOR LOCAL MINIMIZERS OF WIDELY DEGENERATE ANISOTROPIC FUNCTIONALS LORENZO BRASCO, CHIARA LEONE, GIOVANNI PISANTE, AND ANNA VERDE Abstract. We prove higher differentiability of bounded local minimizers to some widely degenerate functionals, verifying superquadratic anisotropic growth conditions. In the two dimensional case, we prove that local minimizers to a model functional are locally Lipschitz continuous functions, without any restriction on the anisotropy.
Contents 1. Introduction 1.1. Overview 1.2. Main results 1.3. Some comments on the proofs 1.4. Plan of the paper 2. Preliminaries 2.1. Notation 2.2. Embedding for anisotropic Sobolev spaces 2.3. Anisotropic Besov-Nikol’ski˘ı spaces 3. A general scheme for improving differentiability 4. Local Sobolev estimate in a particular case 5. Local Lipschitz estimate in dimension two 5.1. Proof of Theorem 1.4 5.2. Uniform Lipschitz estimate Appendix A. Pointwise inequalities References
1 1 2 4 6 6 6 7 9 14 21 25 25 26 30 31
1. Introduction 1.1. Overview. In this paper we continue to investigate differentiability properties of local minimizers of convex functionals, exhibiting wide degeneracies and an orthotropic structure. The model case of functional we want to study is given by ˆ ˆ N X 1 pi 1,p 0 (1.1) F(u; Ω ) = (|uxi | − δi )+ dx + f u dx, u ∈ Wloc (Ω), Ω0 b Ω, pi Ω0 0 Ω i=1
with δi ≥ 0 and pi ≥ 2. We denote p = (p1 , p2 , . . . , pN ) and n o 1,p 1,1 i Wloc (Ω) = u ∈ Wloc (Ω) : uxi ∈ Lploc (Ω), i = 1, . . . , N . The symbol ( · )+ above stands for the positive part. 2010 Mathematics Subject Classification. 35J70, 35B65, 49K20. Key words and phrases. Anisotropic problems, degenerate elliptic equations, Besov-Nikol’ski˘ı spaces. 1
2
BRASCO, LEONE, PISANTE, AND VERDE
For p1 = · · · = pN , some results can be found in the recent papers [3] and [4]. We refer to the introduction of [4] for some motivations of this kind of functionals, arising from Optimal Transport problems with congestion effects. Our scope is to generalize these results to the anisotropic case, i.e. to the case where at least one of the exponents pi is different from the others. These functionals pertain to the class of variational problems with non standard growth conditions, first introduced by Marcellini in [17, 18]. Similar functionals have been considered in the past by the Russian school, see for example [12] and [29]. More recently, they have been considered by many authors also in western countries. Among others, we mention (in alphabetical order) Bildhauer, Fuchs and Zhong [1, 2] (where the terminology splitting-type integrals is used), Esposito, Leonetti and Mingione [7], Leonetti [15], Liskevich and Skrypnik [16] and Pagano [22]. However, we point out that the type of degeneracy admitted in (1.1) is heavier than those of the above mentioned references, due to the presence of the coefficients δi ≥ 0 above. We observe that local minimizers of the functional (1.1) are local weak solution of the degenerate elliptic equation N X pi −1 uxi = f. (|uxi | − δi )+ |uxi | xi i=1
The particular case δ1 = · · · = δN = 0 and p1 = · · · = pN = p corresponds to N X
|uxi |p−2 uxi
xi
= f,
i=1
which has been called pseudo p−Laplace equation in the recent literature. Here we prefer to use the terminology orthotropic p−Laplace equation, which seems more adapted and meaningful. 1.2. Main results. Our first result is the Sobolev regularity for some nonlinear functions of the gradient of a bounded local minimizer. For p = (p1 , . . . , pN ), we will use the notation p0 := (p01 , . . . , p0N ), where p0i is the H¨ older conjugate of pi . Theorem 1.1 (Sobolev regularity for bounded minimizers). Let ` ∈ {1, . . . , N − 1} and 2 ≤ p ≤ q. We set p = (p, · · · , p, q, · · · , q ), | {z } | {z } `
and let u ∈
1,p Wloc (Ω)
∩
N −`
L∞ loc (Ω)
be a local minimizer of ˆ N ˆ X 0 F(u; Ω ) = gi (uxi ) dx + i=1
Ω0
f u dx,
Ω0
0
1,p where f ∈ Wloc (Ω) and g1 , . . . , gN : R → R+ are C 2 convex functions such that
1 (|s| − δ)p−2 ≤ gi00 (s) ≤ C (|s|p−2 + 1), + C 1 00 q−2 (|s| − δ)q−2 + 1), + ≤ gi (s) ≤ C (|s| C
s ∈ R, i = 1, . . . , `, s ∈ R, i = ` + 1, . . . , N,
REGULARITY FOR ANISOTROPIC FUNCTIONALS
for some C ≥ 1 and δ ≥ 0. We set Vi = Vi (uxi ),
ˆ
where
Vi (s) = 0
sq
gi00 (τ ) dτ,
3
i = 1, . . . , N.
• If ` = N − 1 and p, q satisfy
p≥N −1
or
p <
(N − 2)2 , N −1
(N − 2) p , (N − 2) − p (N − 2)2 ≤ p < N − 1, N −1
q < (1.2) or
q
N. Finally, for 0 < t < 1 and 1 ≤ p < ∞ we denote by W t,p (RN ) the Sobolev-Slobodecki˘ı space, i. e. n o W t,p (RN ) = u ∈ Lp (RN ) : [u]W t,p (RN ) < +∞ , where
ˆ [u]pW t,p (RN )
ˆ
|u(x) − u(y)|p dx dy. |x − y|N +t p
= RN
Though we will not need this, we recall that class of Besov spaces.
RN t,p W (R)
can be seen a particular instance of the larger
2.2. Embedding for anisotropic Sobolev spaces. We collect here a couple of embedding results that will be needed in the sequel. The first one is well-known1, a proof can be found for example in [27, Theorem 1 & Corollary 1]. Theorem 2.1 (Anisotropic Sobolev embeddings). Let Ω ⊂ RN be an open set, then for every u ∈ W01,p (Ω) we have: (1) if p < N 1 N ˆ X pi p c kukLp∗ (Ω) ≤ , |uxi | i dx i=1
Ω
for a constant c = c(N, p) > 0; (2) if p = N and |Ω| < +∞, for every 1 ≤ χ < ∞ 1 N ˆ pi 1 X c kukLχ (Ω) ≤ |Ω| χ |uxi |pi dx . i=1
Ω
for a constant c = c(N, p, χ) > 0; 1This result is usually attributed to Troisi in the literature of western countries, see [26]. However, Trudinger in
[27] attributes the result for p 6= N to Nikol’ski˘ı, whose paper [21] appeared before [26]. In any case, the methods of proof are different.
8
BRASCO, LEONE, PISANTE, AND VERDE
(3) if p > N and |Ω| < +∞ c kukL∞ (Ω) ≤ |Ω|
1 N
− p1
N ˆ X
1
pi
|uxi | dx
pi
,
Ω
i=1
for a constant c = c(N, p) > 0. The next embedding result is stated in [13, Theorem 1]. We provide a proof for the reader’s convenience. Proposition 2.2. Let Ω ⊂ RN be an open set and let p = (p1 , . . . , pN ) be such that 1 < p1 ≤ · · · ≤ pN < p∗ .
(2.2) Then for every E b Ω we have
∗
W 1,p (Ω) ,→ Lp (E),
if p 6= N,
and W 1,p (Ω) ,→ Lχ (E), for every 1 ≤ χ < ∞,
if p = N.
Proof. Let us fix two concentric balls B%0 b BR0 b Ω. For every %0 ≤ % < R ≤ R0 we take a standard cut-off function η ∈ C0∞ (BR ) such that η ≡ 1 on B% , with k∇ηkL∞ (RN ) ≤
C , R−%
for some universal constant C > 0. Let u ∈ W 1,p (Ω), then for every M > 0 we define n o uM = min |u|, M ∈ W 1,p (Ω) ∩ L∞ (Ω), and finally take uM η ∈ W01,p (BR ) ⊂ W01,p (Ω). Let us suppose for simplicity that p < N , by Theorem 2.1 we have ˆ 1∗ 1 N ˆ X pi p pi p∗ ≥c |uM η| dx , |(uM η)xi | dx Ω
Ω
i=1
for some c = c(N, p) > 0. By using the properties of η, with simple manipulations we get ! 1∗ ˆ 1 1 ˆ N ˆ N p X X pi pi C ∗ (2.3) |(uM )xi |pi dx + |uM |pi dx ≥c |uM |p dx , R−% BR Ω B% i=1
i=1
for a possibly different constant c > 0, still depending on N and p only. We now observe that by hypothesis (2.2) we have 1 < pi < p∗ , thus by interpolation in Lebesgue spaces ˆ
pi
|uM | dx
ˆ
1
pi
≤
BR
(1−ϑi ) ˆ |uM | dx
BR
where ϑi =
pi − 1 p∗ ∈ (0, 1). pi p∗ − 1
p∗
|uM | dx BR
ϑ∗i p
,
REGULARITY FOR ANISOTROPIC FUNCTIONALS
9
Thus by Young inequality we get 1 R−%
ˆ
N
pi
|uM | dx BR
pi
1 ˆ (1−ϑi ) # 1−ϑ i τ −ϑi ≤ |uM | dx R−% BR ˆ 1∗ p p∗ +τ |uM | dx ,
"
BR
for every 0 < τ < 1. By choosing τ small enough and using the previous estimate in (2.3), we get ˆ 1 N ˆ N X X pi C pi |uM | dx |(uM )xi | dx + 1 1−ϑi Ω Ω i=1 i=1 (R − %) ! 1∗ ˆ 1∗ ˆ p p c p∗ p∗ + |uM | dx ≥c |uM | dx . 2 BR B% The previous holds for every %0 ≤ % < R ≤ R0 , from [10, Lemma 6.1] we obtain C
N ˆ X i=1
pi
|(uM )xi |
1 dx
Ω
pi
+
ˆ
N X
C
i=1
(R0 − %0 ) 1−ϑi
ˆ |uM | dx ≥
1
Ω
! p∗
|uM | dx
1 p∗
,
B%0
for some constant C = C(N, p) > 0. By arbitrariness of Br0 b BR0 b Ω, for every E b Ω a standard covering argument leads to # ˆ " N ˆ 1∗ 1 ˆ X pi p pi p∗ + |u| dx ≥ |uM | dx , (2.4) C |uxi | dx Ω
Ω
i=1
E
for some constant C = C(N, p, dist(E, ∂Ω)) > 0. In the previous inequality we also used that |uxi | ≥ |(uM )xi |
and
|uM | ≤ |u|,
almost eveywhere on Ω.
If we now take the limit as M goes to +∞ in (2.4), we get the desired result.
Remark 2.3 (Optimality of assumptions). In general we can not take E = Ω or pN ≥ p∗ in the previous result, see [13] for a counter-example. On the contrary, the hypothesis p1 > 1 can be easily removed and we can relax it to p1 ≥ 1. We leave the verification of this fact to the reader. 2.3. Anisotropic Besov-Nikol’ski˘ı spaces. Let ψ ∈ Lp (RN ), for p ≥ 1 and 0 < t ≤ 1 we define the quantities
δhei ψ
(2.5) [ψ]nt,p = sup , i = 1, . . . , N, ∞,i |h|t p N |h|>0
L (R )
and (2.6)
[ψ]bt,p
∞,i
δ2 ψ
he = sup it |h|>0 |h|
,
i = 1, . . . , N.
Lp (RN )
Lemma 2.4. Let 0 < t < 1, then for every ψ ∈ Lp (RN ) we have i 1 C h (2.7) [ψ]bt,p ≤ [ψ]nt,p ≤ [ψ]bt,p + kψkLp (RN ) . ∞,i ∞,i ∞,i 2 1−t
10
BRASCO, LEONE, PISANTE, AND VERDE
For t = 1, for every ψ ∈ Lp (RN ) we have 1 [ψ]bt,p ≤ [ψ]nt,p , ∞,i ∞,i 2 p N and there exists ψ0 ∈ L (R ) such that [ψ0 ]b1,p < +∞
[ψ0 ]n1,p = +∞.
and
∞,i
∞,i
Proof. The first inequality in (2.7) is a plain consequence of triangle inequality and invariance by translations of Lp norms. The second one can be proved by using a standard device, see [25, Chapter 2.6]. For t = 1, an instance of function with the properties above can be found in [23, Example page 148]. If t = (t1 , . . . , tN ) ∈ (0, 1]N , by following Nikol’ski˘ı we define the corresponding anisotropic Besov-Nikol’ski˘ı spaces as ) ( N X t,p N∞ (RN ) := ψ ∈ Lp (RN ) : [ψ]nti ,p < +∞ , i=1
and2
( t,p B∞ (RN )
:=
p
N
ψ ∈ L (R ) :
N X
∞,i
) [ψ]bti ,p < +∞ ,
i=1
∞,i
see [20, pages 159–161]. We equip them with the norms kψkN∞ t,p (RN ) := kψkLp (RN ) +
N X i=1
[ψ]nti ,p
and
∞,i
kψkB∞ t,p (RN ) := kψkLp (RN ) +
N X i=1
[ψ]bti ,p . ∞,i
From now on we will always implicitly assume that t1 ≤ t2 ≤ · · · ≤ tN . Before going on, a couple of comments are in order. Remark 2.5 (Comparison of the two spaces). By Lemma 2.4 we get that if 0 < t1 ≤ · · · ≤ tN < 1, then t,p t,p N∞ (RN ) = B∞ (RN ). On the contrary, if ti = 1 for some i ∈ {1, . . . , N }, then t,p t,p N∞ (RN ) ,→ B∞ (RN )
and
t,p t,p N∞ (RN ) 6= B∞ (RN ).
Moreover, we recall that if [ψ]n1,p < +∞,
for some i ∈ {1, . . . , N },
∞,i
then its distributional derivative ψxi belongs to Lp (RN ), see [20, Theorem 4.8]. Remark 2.6. In the isotropic case t1 = · · · = tN = t with 0 < t < 1, we simply denote these t,p t,p spaces by N∞ (RN ) and B∞ (RN ). By Lemma 2.4 the seminorms ψ 7→
N X i=1
[ψ]nt,p
∞,i
and
ψ 7→
N X i=1
[ψ]bt,p , ∞,i
2In [20] this space is denoted by H t and is seen to be a particular instance of a general class of anisotropic Besov p t spaces noted Bpθ , with 1 ≤ θ ≤ ∞.
REGULARITY FOR ANISOTROPIC FUNCTIONALS
11
are equivalent. Moreover, these in turn are equivalent to
2
δh ψ
δh ψ
ψ 7→ sup t or ψ 7→ sup .
t |h|>0 |h| Lp (RN ) |h|>0 |h| Lp (RN ) t,p t,p The next very simple result asserts that N∞ (RN ) and B∞ (RN ) do not change, if in (2.5) and (2.6) the supremum is restricted to 0 < |h| < h0 . The easy proof is left to the reader.
Lemma 2.7. Let 0 < t ≤ 1 and ψ ∈ Lp (RN ), then for every h0 > 0 and every i = 1, . . . , N we have
δhei ψ −t
[ψ]nt,p ≤ sup
|h|t p N + 2 h0 kψkLp (RN ) , ∞,i 00
i = 1, . . . , N,
δ2 ψ
hei = sup t |h|>0 |h|
i = 1, . . . , N.
i
and (2.10)
[ψ]bt,p
∞,i (Ω)
,
Lp (Ω2hei )
Accordingly, we introduce the anisotropic Besov-Nikol’ski˘ı spaces on Ω as ( ) N X t,p N∞ (Ω) := ψ ∈ Lp (Ω) : [ψ]nti ,p (Ω) < +∞ , i=1
∞,i
and ( t,p B∞ (Ω)
:=
p
ψ ∈ L (Ω) :
N X i=1
) [ψ]bti ,p (Ω) < +∞ . ∞,i
REGULARITY FOR ANISOTROPIC FUNCTIONALS
13
Finally, we define n o t,p t,p N∞,loc (Ω) := ψ ∈ Lploc (Ω) : ψ ∈ N∞ (E) for every E b Ω , and n o t,p t,p B∞,loc (Ω) := ψ ∈ Lploc (Ω) : ψ ∈ B∞ (E) for every E b Ω . t,p t,p Remark 2.12. As for the case of RN , the definitions of N∞ (Ω) and B∞ (Ω) do not change if we perform the supremum in (2.9) and (2.10) over 0 < |h| < h0 for some h0 > 0.
Corollary 2.13. Let Ω ⊂ RN be an open set. Under the assumptions of Theorem 2.10 and with the same notations, we have t,p t,p N∞,loc (Ω) ⊂ B∞,loc (Ω) ⊂ Lplocχ (Ω),
for every 1 ≤ χ
0, then there exist x1 , . . . , xk ∈ E such that
E⊂
k [ j=1
B d (xj ). 8
It is sufficient to prove that ψ ∈ Lp χ (Bd/8 (xj )) for every j = 1, . . . , k. We fix one of these balls and omit to indicate the center xj for simplicity. We then take a standard cut-off function t,p η ∈ C0∞ (Bd/4 ) ⊂ C0∞ (Ω) such that η ≡ 1 on Bd/8 . Then we observe that ψ η ∈ B∞ (RN ): indeed, by triangle inequality and (2.1) for every h 6= 0 such that |h| < d/8 we have
2 ψ
δ2 η
δ 2 (ψ η)
δ δ η
hei
he
hei hei hei
η2hei ≤ it ψ +2
|h|ti δhei ψ p N + ti
|h| i p N
p N
|h|ti p N |h| L (R ) L (R ) L (R ) L (R )
1−ti 2
d
δhe ψ ≤4 k∇ηkL∞ kψkLp (B d ) + it , i = 1, . . . , N,
|h| i p 8 2 L (B d ) 2
and the supremum of the latter over 0 < |h| < d/8 is finite, since Bd/2 b Ω by construction. t,p By appealing to Lemma 2.7, we thus get ψ η ∈ B∞ (RN ). We can use Theorem 2.10 and get ψ η ∈ Lp χ (RN ). Since η ≡ 1 on Bd/8 , this gives the desired result.
14
BRASCO, LEONE, PISANTE, AND VERDE
3. A general scheme for improving differentiability In this section we consider a slightly more general framework, with respect to that of Theorem 1.1. Namely, we consider a set of C 2 convex functions gi : R → R+ such that 1 (|s| − δ)p+i −2 ≤ gi00 (s) ≤ C (|s|pi −2 + 1), i = 1, . . . , N, (3.1) C for some C ≥ 1, δ ≥ 0 and 2 ≤ p1 ≤ · · · ≤ pN −1 ≤ pN . Remark 3.1. Let us point out the following simple inequality that will be used in what follows: for every a ≤ s ≤ b, we have (3.2) gi00 (s) ≤ Cei gi00 (a) + gi00 (b) + 1 , for some Cei = Cei (pi , δ) ≥ 1. This follows with elementary manipulations, by exploiting (3.1). We leave the details to the reader. 1,p (Ω) a local minimizer of We then consider u ∈ Wloc ˆ N ˆ X 0 F(u; Ω ) = gi (uxi ) dx + i=1
Ω0
f u dx.
Ω0
In particular, u solves (3.3)
N ˆ X
ˆ gi0 (uxi ) ϕxi
dx +
f ϕ dx = 0,
i=1
for every ϕ ∈ W01,p (Ω0 ) and every Ω0 b Ω. For every i = 1, . . . , N , we define ˆ tq Vi = Vi (uxi ), where Vi (t) = gi00 (τ ) dτ. 0
Our aim is to prove that every Vi enjoys some weak differentiability properties. We start with the following result. 0
1,p Proposition 3.2 (Initial gain). Let 2 ≤ p1 ≤ · · · ≤ pN −1 ≤ pN and let f ∈ Wloc (Ω). We suppose that u ∈ L∞ loc (Ω). Then for every i = 1, . . . , N we have p1 pN −1 t,2 Vi ∈ N∞,loc (Ω), ,..., ,1 . where t = pN pN
Proof. We take Br0 b BR0 b Ω a pair of concentric balls centered at x0 and set h0 = (R0 − r0 )/4
and
R=
R0 + r0 . 2
Then we pick ϕ ∈ W01,p (BR ) that we extend it to zero on RN \ BR . For every 0 < |h| < h0 we can insert the test function ϕ−hej (x) in (3.3). With a simple change of variables we get ˆ N ˆ X (3.4) gi0 (uxi )hej ϕxi dx = fhej ϕ dx. i=1
Ω
Ω
REGULARITY FOR ANISOTROPIC FUNCTIONALS
15
By subtracting (3.3) and (3.4) and dividing by |h|, we thus get # " ˆ N ˆ X δhej f gi0 (uxi )hej − gi0 (uxi ) ϕxi dx = ϕ dx. |h| Ω |h| Ω i=1
We now make the following particular choice ϕ = ζ2
δhej u , |h|sj
where sj ∈ (−1, 1] will be chosen below and ζ is the standard cut-off function R − |x − x0 | . ζ(x) = min 1, R − r0 + We obtain
N ˆ X δhej gi0 (uxi ) δhej uxi 2 ζ dx |h| |h|sj i=1 N ˆ X δhej gi0 (uxi ) |ζx | ζ ≤2 i h i=1
ˆ δhej u dx + |h|sj
δhej f h
δhej u 2 |h|sj ζ dx.
Recalling the definition of Vi , using (A.1) in the left-hand side and (A.2) (in combination with (3.2)) in the right-hand side, we obtain 2 N ˆ N ˆ q q δ u X X he δhej Vi δhej Vi 2 j gi00 (uxi )hej + gi00 (uxi ) + 1 |ζxi | ζ sj +1 dx sj +1 sj +1 ζ dx ≤ C |h| 2 |h| 2 |h| 2 i=1 i=1 ˆ δhe f δhe u + j jsj ζ 2 dx. h |h| If we use H¨ older and Young inequalities in the right-hand side, we can absorb the higher-order term. Namely, since we have N ˆ q q δ u X he δhej Vi j gi00 (uxi )hej + gi00 (uxi ) + 1 |ζxi | ζ sj +1 dx sj +1 |h| 2 |h| 2 i=1 2 N ˆ X δhej Vi 2 ≤Cτ sj +1 ζ dx |h| 2 i=1 N ˆ δ u 2 00 C X he j + gi (uxi )hej + gi00 (uxi ) + 1 |ζxi |2 sj +1 dx, τ |h| 2 i=1 where 0 < τ < 1, by choosing τ small enough, we thus get 2 N ˆ N ˆ δ u 2 X X δ V i he he j gi00 (uxi )hej + gi00 (uxi ) + 1 |ζxi |2 sjj+1 dx sj +1 ζ 2 dx ≤ C |h| 2 |h| 2 i=1 i=1 10 p0 pj !1 p ˆ ˆ pj j j δ f δhej u hej +C . sj +1 dx sj +1 dx BR |h| 2 BR |h| 2
16
BRASCO, LEONE, PISANTE, AND VERDE
By basic properties of differential quotients, we get for 0 < |h| < h0 ˆ ˆ δ u pj 1−sj pj hej 2 |uxj |pj dx, sj +1 dx ≤ C h0 BR |h| 2 BR0 and similarly ˆ
0 1−sj δ f pj hej sj +1 dx ≤ C h0 2 BR |h| 2
p0j
ˆ
0
|fxj |pj dx. BR0
This yields 2 N ˆ N ˆ δ u 2 X X δ V C hej i 2 he gi00 (uxi )hej + gi00 (uxi ) + 1 sjj+1 dx sj +1 ζ dx ≤ 2 (R − r ) 0 0 |h| 2 |h| 2 i=1 i=1 !1 ! 10 ˆ ˆ pj p pj p0 j 1−sj ux dx fx j dx . + C h0 j j
(3.5)
BR0
BR0
We use again H¨ older inequality in the first term in the right-hand side, so that N ˆ δ u 2 X 00 he gi (uxi )hej + gi00 (uxi ) + 1 sjj+1 dx |h| 2 i=1 !2 pi −2 ˆ N ˆ pi δ u pi pi X pi 00 he j 00 pi −2 . ≤ gi (uxi )hej + gi (uxi ) + 1 dx sj +1 dx BR |h| 2 BR i=1 We now observe that with simple manipulations we have ˆ
00 pi gi (uxi )hej + gi00 (uxi ) + 1 pi −2 dx
ˆ
pi −2
BR
pi
|gi00 (uxi ) + 1|
≤C
pi pi −2
! pi −2 pi
dx
,
BR0
since for every 0 < |h| < h0 we have BR + hej ⊂ BR0 , by construction. Thus from (3.5) we obtain 2 pi !2 ˆ N N ˆ pi X X
δ u δ V C hej hej i 2
gi00 (ux ) + 1 pi dx sj +1 ζ dx ≤ sj +1 i (R0 − r0 )2 L pi −2 (BR0 ) BR |h| 2 |h| 2 i=1 i=1 (3.6) ! 10 ˆ !1 ˆ pj p pj p0 j 1−sj fx j dx ux dx + C h0 . j j B R0
B R0
The first term in the right-hand side is more delicate and we have to distinguish between two cases. Case A: j = N . By hypothesis we have pi ≤ pN for every 1 ≤ i ≤ N . Thus we get ˆ BR
pi δ u N heN sN +1 dx ≤ C R |h| 2
pN −pi pN
ˆ BR
pN ! pi pN δ u heN , sN +1 dx |h| 2
i = 1, . . . , N.
REGULARITY FOR ANISOTROPIC FUNCTIONALS
17
We can then choose sN = 1 so that (sN + 1)/2 = 1 as well. Then from (3.6) we get "N # N ˆ i X 2 N ppN −p X δheN (Vi ) 2 2
C pi 00 N ζ dx ≤
gi (ux ) + 1 pi R0 i h (R0 − r0 )2 L pi −2 (BR0 ) i=1 i=1 ! 10 ! 2 ! 1 ˆ ˆ ˆ pN pN p N pN p0N pN +C . × |fxN | dx |uxN | dx |uxN | B R0
BR0
BR0
Case B: 1 ≤ j ≤ N − 1. This in turn has to be divided in two sub-cases. Case B.1: 1 ≤ i ≤ j. This is similar to Case A, since by hypothesis we have pi ≤ pj . Then for 0 < |h| < h0 we simply have ! pi ˆ ˆ pj p −p δ u pi δ u pj N jp i hej hej j sj +1 dx ≤ C R sj +1 dx BR |h| 2 BR |h| 2 ! pi ˆ pj −pi pj 1−sj N pi pj ≤ C h0 2 R . |uxj |pj dx B R0
Case B.2: j + 1 ≤ i ≤ N . Here we should be more careful. The order of maximal differentiability tj = (sj + 1)/2 is determined here. We set tj = pj /pN as in the statement, we thus get p ˆ ˆ δhej u pi
−pj δhej u j
δhe u pi∞ dx . j |h|tj dx ≤ t p L (BR ) j i BR BR |h| Since pj − tj pi ≥ 0, we further observe that for every 0 < |h| < h0 we have p ˆ ˆ ˆ δhej u pj δhej u j pj −tj pi dx ≤ C hpj −tj pi |uxj |pj dx. dx ≤ h 0 0 h tj pi |h| BR BR BR 0
Moreover kδhej ukL∞ (BR ) ≤ 2 kukL∞ (BR0 ) . By using the previous estimates in (3.6) we thus obtain3 N ˆ X δhej Vi 2 2 |h|tj ζ dx i=1 # " j i 2 (1−tj ) X 2 N ppjj−p
C h0 pi 00
gi (ux ) + 1 pi ≤ R0 kuxj k2Lpj (BR ) i 0 (R0 − r0 )2 L pi −2 (BR0 ) i=1 p pj pj N j X
00
2 ppj 2 p −p 2 1− p C i i N
+ h0 gi (uxi ) + 1 p p−2 kukL∞ (BR ) uxj Lpji(B ) i R0 0 (R0 − r0 )2 L i (BR0 ) i=j+1
2 (1−tj )
fx + C h0 j
p0 L j (B
R0 )
ux pj j L (B
R0 )
,
3It is intended that the second term in the right-hand side is 0 for j = N .
18
BRASCO, LEONE, PISANTE, AND VERDE
for a constant C = C(N, p1 , . . . , pN ) > 0. By taking the supremum over 0 < |h| < h0 , summing over j = 1, . . . , N and recalling that ζ ≡ 1 on Br , we finally conclude that N X j=1
δhej Vi
< +∞, sup tj |h| 0 0, where we used that δ ≥ 1. In order to reconstruct the full gradient p1
∇W14
+ 2s
on the left-hand side, we observe that p1 s + p1 + 2 s W14 2 = p1 x2
p1 s W14 W12 . x2
Then if we fix 1 < q < 2, by H¨ older’s inequality with exponents 2/q and 2/(2 − q), we have !2 ! 2−q 2 ˆ p 2 ˆ p1 s q ˆ q q q 1 s p + 2 s + 1 2−q q 2 4 2 4 ≤ W1 dx . W1 η dx W1 η dx p1 spt(η) x2 x2
REGULARITY FOR ANISOTROPIC FUNCTIONALS
By using the same manipulations as in [3], we thus get ˆ p1 s q ! 2q ˆ p1 2 + 2 ≤ C (1 + s)2 W14 W14 2 η dx η dx x2 x2 (5.5) 2 ˆ p1 q q + s2q q 4 |ηx2 | dx , +C W1
27
ˆ spt(η)
q 2−q
W1
s
! 2−q q
dx
with C = C(p1 , p2 ) > 0. We assume for simplicity that all the balls are centered at the origin. We then fix the radius r0 > 0 as above and define R0 = 2 r0
R1 :=
3 r0 . 2
For r0 < r < R < R1 , we take η ∈ W01,∞ (BR ) to be the standard cut-off function (R − |x|)+ . η(x) = min 1, R−r By multiplying (5.4) and (5.5) ˆ p1 s 2 ˆ W14 + 2 η dx x1
we get
q ! 2q p1 s + W14 2 η dx x2 # " ˆ 2 ˆ pi −2 X 1 s+1 2 s 2 p1 −2 |fε | W1 dx ≤Cδ Wi 2 W1 dx + (s + 1) (R − r)2 BR B R i=1 2−q ˆ ˆ p1 2 q q s 2−q 2 4 × (s + 1) dx W1 W1 dx BR BR x2 # ˆ 2 p1 q q 1 + s2q 4 W1 . + dx 2 (R − r) BR
Then we apply the anisotropic Sobolev inequality of Theorem 2.1 to the compactly supported (p +2 s)/4 function W1 1 η. This yields ˆ p s q∗ 4∗ q 1+ Tq W14 2 η dx ˆ 2 ˆ pi −2 X 1 s+1 2 2 s 2 W W dx + (s + 1) |f | W dx ε 1 1 i (R − r)2 BR i=1 BR ˆ p 2 ˆ 2−q q q 1 s 2−q 2 4 × (s + 1) W1 dx W1 dx ≤ C δ p1 −2
(5.6)
BR
+
1 (R − r)2
ˆ
BR
BR
x2
p1 q + s2q
W1 4
2 q
dx
.
28
BRASCO, LEONE, PISANTE, AND VERDE
The exponents q and q ∗ are given by 4q 4q and q∗ = , 2+q 2−q the constant Tq only depends on q and it degenerates to 0 as q goes to 2. The idea is to use the previous fundamental estimate (5.6) to produce an iterative scheme of reverse H¨older inequalities on shrinking balls. Then we perform a Moser’s iteration in order to conclude. We need to estimate the terms appearing in the right-hand side of (5.6). The crucial difference with respect to [3] is in the first term on the right-hand side of (5.6), i.e. ˆ ˆ 2 ˆ pi −2 p2 −2 p1 X s+1 s 2 2 W2 2 W1 W1s dx. (5.7) W1 dx = W1 W1 dx + Wi q=
i=1
BR
BR
BR
On the contrary, all the other terms are estimated exactly as in [3], thus we omit the details. Let us now focus on the term above, it is useful to introduce the quantity # pi −2 p1 " ˆ 2 pi p1 −2 2 pi 2 pi X R0 2 4 e I(W1 , W2 , fε ; R0 , R1 ) = Wi dx + ∇Wi dx R1 BR1 BR0 i=1 ! 10 ˆ 2 p
+ R0 1
0
|fε |2 p1 dx
p1
.
B R1
e 1 , W2 , fε ; R0 , R1 ) is uniformly bounded, independently of ε. To this aim, as First we claim that I(W for the term containing fε , we observe that by Proposition 2.2 we have the continuous embedding (recall that R1 < R0 ) 0
0
W 1,p (BR0 ) ,→ L2 p1 (BR1 ), thus the term
∗
since p02 < p01 ≤ 2 and 2 p01 < p0 = ˆ
! 2 p01
|fε |
dx
2 p01 p02 , p01 + p02 − p01 p02
1 p01
,
B R1 0
can be uniformly bounded in terms of the W 1,p norm of f on BR0 . The terms containing the p /4 p /4 gradients of W1 1 and W2 2 are more delicate, for them we need Theorem 1.1. Indeed, let us define ˆ tq 00 (s) ds Vi,ε (t) = gi,ε and Vi,ε = Vi,ε ((uε )xi ), i = 1, 2. 0
0 > 0. If we set We observe that Vi,ε : R → R is a locally Lipschitz omeomorphism, with Vi,ε pi fi (t) = δ 2 + (|t| − δ)2+ 4 , t ∈ R, p /4
then we obtain that Wi i
= Φi,ε (Vi,ε ), where −1 Φi,ε (t) = fi (Vi,ε (t)),
t ∈ R.
It is not difficult to see that Φi,ε is a Lipschitz function, with Lipschitz constant independent of ε. Indeed, we have p pi −2 f0i (t) = 0, for |t| < δ and |f0i (t)| ≤ Ci |t| 2 , for |t| ≥ δ.
REGULARITY FOR ANISOTROPIC FUNCTIONALS
29
q p −2 00 (t) ≥ √1 |t| i2 , gi,ε for |t| ≥ δ, Ci for some Ci = Ci (pi , δ) ≥ 1. Thus we get 1 −1 |Φ0i,ε (t)| = f0i (Vi,ε (t)) 0 ≤ Ci , t ∈ R. −1 Vi,ε (Vi,ε (t)) 0 Vi,ε (t) =
By using this observation, we thus obtain ˆ ˆ pi 2 ∇W 4 dx ≤ Li i BR1
|∇Vi,ε |2 dx,
BR1
with Li = Li (pi , δ) > 0. We can now invoke (1.4), in order to bound uniformly the last term. It is only left to observe that the bound in (1.4) also depends on the local L∞ norm of uε . This can be uniformly bounded by appealing to [8, Theorem 3.1], proving the claim. We now come back to estimate the quantities in (5.7). Let us recall that, since we are in dimension N = 2, we have the continuous embedding W 1,2 (BR1 ) ,→ Lϑ (BR1 ) for every 1 ≤ ϑ < +∞. Then by H¨older’s inequality and Sobolev-Poincar´e inequality, exactly as in [3] we get 1 ˆ 2 ˆ p1 p1 p01 s p s 1 2 e 1 , W2 , fε ; R0 , R1 ) R W1 dx . W1 W1 dx ≤ C I(W 0 BR
BR
For the second term we have to be more careful. By using H¨older inequality with exponents p1 p1 p01 , p01 , p1 , p1 − 2 2 we get !1 ! p10 p 2 p01 p 2 p01 p2 −2 p1 ˆ ˆ ˆ p1 p2 −2 1 p1 −2 p2 1 2 + W14 W24 dx W2 2 W1 W1s dx ≤ C dx B R1
BR1
BR
ˆ × BR
W1s p1
dx
1 p1
,
where we further used Young’s inequality and the constant C = C(p1 ) > 0 depends only on p1 . To treat the term into square brakets, we use again Sobolev-Poincar´e inequalities. Namely, we have " # ! 10 p 2 p01 ˆ ˆ 2 p1 p1 p1 2 1 0 p ∇W 4 dx , W14 dx ≤ C R1 1 W12 dx + 1 B R1
B R1
BR1
and ˆ BR1
p 2 p01 p2 −2 p1 −2 2 W24
Thus we obtain ˆ BR
p2 −2 2
W2
p1 p2
! p10
1
dx
W1 W1s dx
p1 −2 p2 p2 −2 p1
2 p1 −2 p2 p01 p2 −2 p1
≤ C R1
"
p2 2
BR1
2 p01
e 1 , W2 , fε ; R0 , R1 ) R ≤ C I(W 0
as well, where we used again that R1 < R0 .
ˆ
W2 dx + ˆ BR
W1s p1
BR1
# p2 2 ∇W 4 dx . 2
dx
1 p1
,
30
BRASCO, LEONE, PISANTE, AND VERDE
By using these estimates in (5.6) and proceeding as in [3] for all the other terms, we obtain (5.8) "ˆ " # 2−q 2q ˆ 2 1 q p1 2−q p1 − p2 R0 +s s p p1 −2 1 2 1 e 1 , W2 , fε ; R0 , R1 ) R W1 ≤Cδ I(W W1 dx dx 0 R−r BR Br 1 # ˆ p1 − p2 s p e 1 , W2 , fε ; R0 , R1 ) R 1 W1 1 dx +(s + 1)2 I(W 0 BR
ˆ
" 2
q 2−q
e 1 , W2 , fε ; R0 , R1 ) × (s + 1) I(W BR
+
R0 R−r
2
2
R0
W1
s
2−q q dx ˆ
2 − p1 −1 q 1
e 1 , W2 , fε ; R0 , R1 ) I(W BR
W1s p1
1# dx
p1
,
for a constant C = C(p1 , q) > 0. The exponent 1 < q < 2 is chosen as 2 p1 q 2 1 q= , so that = p1 and − − 1 = 0. p1 + 1 2−q q p1 By further observing that W1 ≥ 1, from (5.8) we gain 1 ˆ p1 2 s p1 e 1 , W2 , fε ; R0 , R1 )2 dx W1 ≤ C δ p1 −2 I(W Br
" ×
R0 R−r
#2
2 + (s + 1)
2
− p2 R0 1
ˆ BR
W1s p1
dx
2 p1
,
for s ≥ 0. This is an iterative scheme of reverse H¨older inequalities, we can now iterate infinitely many times this estimate, as in [3]. Appendix A. Pointwise inequalities Lemma A.1. Let g : R → R+ be a C 1,1 convex function. Let us set ˆ tp g 00 (τ ) dτ. V (t) = 0
For every a, b ∈ R we have (A.1)
g 0 (a) − g 0 (b) (a − b) ≥ |V (a) − V (b)|2 .
Proof. Without loss of generality, we can assume that a ≥ b. Indeed, g 0 (a) − g 0 (b) and a − b have the same sign, thanks to the monotonicity of g 0 . For a = b there is nothing to prove, so we take a > b. By using Jensen inequality, we have ˆ a 0 0 00 g (a) − g (b) (a − b) = g (t) dt (a − b) b
ˆ ≥
ap
2
g 00 (t) dt
= (V (a) − V (b))2 ,
b
as desired.
REGULARITY FOR ANISOTROPIC FUNCTIONALS
31
Lemma A.2. Let g : R → R+ be a C 1,1 convex increasing function. For every a, b ∈ R we have p 0 g (a) − g 0 (b) ≤ sup (A.2) g 00 (s) |V (a) − V (b)|. s∈[a,b]
Proof. For ε > 0, let us consider the function gε (t) = g(t) + ε t2 . We set ˆ tp Vε (t) = gε00 (τ ) dτ, 0
then we observe that this is a strictly increasing function, thus invertible. Finally, we define Fε (t) = gε0 Vε−1 (t) , which is an increasing function. Indeed, we have q 1 = gε00 (Vε−1 (t)) > 0. Fε0 (t) = gε00 (Vε−1 (t)) 0 −1 Vε (Vε (t)) By basic Calculus, this yields |gε0 (a) − gε0 (b)| = |Fε (Vε (a)) − Fε (Vε (b))| ≤ sup Fε0 (Vε (s)) |Vε (a) − Vε (b)| s∈[a,b]
= sup
p gε00 (s) |Vε (a) − Vε (b)|.
s∈[a,b]
By taking the limit as ε goes to 0, we get the desired conclusion.
|t|p /p,
Remark A.3. When g(t) = the previous inequalities imply the familiar estimates p−2 2 4 p−2 |a|p−2 a − |b|p−2 b (a − b) ≥ (p − 1) 2 |a| 2 a − |b| 2 b . p and p−2 p−2 p−2 p − 1 p−2 p−2 a − |b|p−2 b ≤ 2 |a| 2 + |b| 2 |a| |a| 2 a − |b| 2 b . p References [1] M. Bildhauer, M. Fuchs, X. Zhong, A regularity theory for scalar local minimizers of splitting-type variational integrals, Ann. Sc. Norm. Super. Pisa Cl. Sci., 6 (2007), 385–404. 2 [2] M. Bildhauer, M. Fuchs, X. Zhong, Variational integrals with a wide range of anisotropy, St. Petersburg Math. J., 18 (2007), 717–736. 2 [3] P. Bousquet, L. Brasco, V. Julin, Lipschitz regularity for local minimizers of some widely degenerate problems, to appear on Ann. Sc. Norm. Sup., available at http://cvgmt.sns.it/paper/2515/ 2, 4, 25, 26, 27, 28, 29, 30 [4] L. Brasco, G. Carlier, On certain anisotropic elliptic equations arising in congested optimal transport: local gradient bounds, Adv. Calc. Var., 7 (2014), 379–407. 2 [5] A. Canale, A. D’Ottavio, F, Leonetti, M. Longobardi, Differentiability for bounded minimizers of some anisotropic integrals, J. Math. Anal. Appl., 253 (2001), 640–650. 3, 5 [6] G. Cupini, P. Marcellini, E. Mascolo, Local boundedness of minimizers with limit growth conditions, J. Optim. Theory Appl., 166 (2015), 1-22. 3 [7] L. Esposito, F. Leonetti, G. Mingione, Sharp regularity for functionals with (p, q) growth, J. Diff. Eq., 204 (2004), 5–55. 2 [8] N. Fusco, C. Sbordone, Some remarks on the regularity of minima of anisotropic integrals, Commun. Partial Differ. Equations, 18 (1993), 153–167. 3, 6, 29 [9] M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math., 59 (1987), 245–248. 3 [10] E. Giusti, Metodi diretti nel calcolo delle variazioni. (Italian) [Direct methods in the calculus of variations], Unione Matematica Italiana, Bologna, 1994. 9
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(L. Brasco) Dipartimento di Matematica e Informatica ` degli Studi di Ferrara Universita Via Machiavelli 35, 44121 Ferrara, Italy ´matiques de Marseille and Institut de Mathe ´, Marseille, France Aix-Marseille Universite E-mail address: [email protected] (C. Leone & A. Verde) Dipartimento di Matematica “R. Caccioppoli” ` degli Studi di Napoli “Federico II” Universita Via Cinthia, Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy E-mail address: [email protected] E-mail address: [email protected] (G. Pisante) Dipartimento di Matematica e Fisica ` degli Studi di Napoli Seconda Universita Viale Lincoln 5, 81100 Caserta, Italy E-mail address: [email protected]
