Partial regularity for anisotropic functionals of higher order

Vol. 13, N 4, 2007, pp. 692–706 www.esaim-cocv.org

DOI: 10.1051/cocv:2007033

PARTIAL REGULARITY FOR ANISOTROPIC FUNCTIONALS OF HIGHER ORDER

Menita Carozza

and Antonia Passarelli di Napoli

Abstract. ^{We prove aCk,α} partial regularity result for local minimizers of variational integrals of the typeI(u) =

Ωf(D^ku(x))dx, assuming that the integrandf satisfies (p, q) growth conditions.

Mathematics Subject Classification. 35G99, 49N60, 49N99.

Received February 6, 2006. Revised June 8, 2006.

Published online July 20, 2007.

1. Introduction

Higher order variational functionals, emerging in the study of problems from materials science and engineer- ing, have attracted a great deal of attention in last few years [4,5,7]. In particular, the regularity of minimizers of such functionals has been studied very recently. In [15] and [16] the partialC^k,αregularity has been established for quasiconvex integrals with a p-power growth with respect to the gradient and in [3] for convex integrals having subquadratic nonstandard growth condition, only in dimension 2.

The aim of this paper is to establish the partial regularity of minimizers of integral functionals of the type I(u) =

Ωf(D^ku(x))dx (1)

where Ω is a bounded subset ofIRⁿ,u: Ω⊂IRⁿ→IR^N,N≥1,k >1 andf is aC²convex integrand satisfying the non standard growth condition:

C|ξ|^p≤f(ξ)≤L(1 +|ξ|^q) (2)

withp < q, without restriction on the dimension and on the order of derivatives involved, in the superquadratic case.

Nonstandard growth conditions have been introduced by Marcellini, in the scalar case fork= 1. He observed that, even in the scalar case, minimizers of (1) may fail to be regular (see [13, 17, 18]), whenqis too large with respect to p. On the other hand, one can prove regularity of scalar minimizers of (1) if qis not too far away from p(see e.g. [19] and its references). More precisely, in [19] it is shown that if one writes down the Euler equation for the functionalI, under suitable assumptions onpandq, the Moser iteration argument still works, thus leading to asupestimate for the gradientDu of the minimizer.

Keywords and phrases. Partial regularity, non standard growth, higher order derivatives.

1 Dipartimento Pe.Me.Is, Universit`a degli studi del Sannio, Piazza Arechi 2, 82100 Benevento, Italy;carozza@unisannio.it

2 Dipartimento di Matematica e Applicazioni “R. Caccioppoli” Universit`a di Napoli “Federico II”, via Cintia, 80126 Napoli, Italy;antonia.passarelli@unina.it

c EDP Sciences, SMAI 2007 Article published by EDP Sciences and available at http://www.esaim-cocv.org or http://dx.doi.org/10.1051/cocv:2007033

Clearly this approach can not be carried on in the vector valued case, i.e. when N > 1. First regularity results for systems are proved in [1] and [20] under special structure assumptions and in [22] in a more general setting. Moreover, higher integrability results for the gradient of the minimizers of (1) are avalaible in the vectorial case (see the references in [2, 8, 9]).

In this paper we prove that, fork >1, differently from all previous quoted results, iff satisfies (2) and the strong ellipticity assumption

D²f(ξ)η, η ≥γ(1 +|ξ|²)^p−22 |η|² (3)

where

2≤p < q <min

p+ 1, pn n−1

, (4)

a minimizer u ∈ W^k,p(Ω;IR^N) of functional (1) is C^k,α for all α < 1 in an open set Ω₀ ⊂ Ω such that meas(Ω\Ω₀) = 0.

We point out that apart from condition (4), no special structure assumption is needed onf and the condition on the exponents does not depend onk,i.e. the order of derivatives involved.

The proof of our result goes through a more or less standard blow-up argument aimed to establish a decay estimate on the excess function for thek- order derivatives

U(x₀, r) =

Br(x0)|D^ku−(D^ku)_x0,r|²+|D^ku−(D^ku)_x0,r|^pdx.

Here, first order techniques have to be combined with new theoretical arguments needed to face the analytical and geometrical constraints of higher order derivatives. In particular, the essential tool is a lemma due to Fonseca and Mal´y (see [11] and also Lem. 2.4 below) which makes possible to connect in an annulusB_r\B_s twoW^k,p functionsv andwwith a more regular function functionz∈W^k,q(B_r\B_s) withp < q < _n−1^pn .

2. Statements and preliminary lemmas

Let us consider the functional

I(u) =

Ωf(D^ku(x))dx

where Ω is a bounded open subset ofIRⁿ,n≥2. Letf :IR^MN →IR, whereM =^(n+k−1)!_k!(n−1)! andN ≥2, satisfy the following assumptions:

f ∈C² (H1)

C|ξ|^p≤f(ξ)≤L(1 +|ξ|^q) (H2)

D²f(ξ)η, η ≥γ(1 +|ξ|²)^p−22 |η|² (H3) where

2≤p < q <min

p+ 1, pn n−1

. It is well known that

|Df(ξ)| ≤c(1 +|ξ|^q−1). (H4)

We say thatu∈W^k,p(Ω, IR^N) is a minimizer ofI if

I(u)≤I(u+v) for anyv∈u+W₀^k,p(Ω;IR^N).

Remark 2.1. Ifuis a local minimizer of I andφ∈C₀^k(Ω;IR^N) from the minimality condition one has for any ε >0

0≤

Ω[f(D^ku+εD^kφ)−f(D^ku)]dx=ε

Ωdx ₁

0

∂f

∂ξⁱ_α(D^ku+εtD^kφ)D_αφⁱdt

where|α|=k. Dividing this inequality byε, and letting εgo to zero, from (H4) and the assumptionq≤p+ 1 we get

Ω

∂f

∂ξ_αⁱ (D^ku)D_αφⁱdx≥0

and therefore, by the arbitrariness ofφ, the usual Euler-Lagrange system holds:

Ω

∂f

∂ξ_αⁱ (D^ku)D_αφⁱdx= 0 ∀φ∈C₀^k(Ω;IR^N).

The aim of this paper is proving the following

Theorem 2.1. Let f satisfy the assumptions (H1)–(H3)and letu∈W^k,p(Ω;IR^N)be a minimizer of I. Then there exists an open subsetΩ₀ of Ωsuch that

meas(Ω\Ω₀) = 0 and

u∈C^k,α(Ω₀, IR^N) for all α <1.

In what follows, we will denote byua W^k,p(Ω;IR^N) minimizer of the integral functional (1) and assume that its integrandf satisfies (H1)–(H3). We set for everyB_r(x₀)⊂Ω

Br(x0)

g= (g)_x0,r= 1 meas(B_r(x₀))

Br(x0)

g.

Moreover, givenp >1 andu∈W^k,p(Ω;IR^N),k≥1, we will denote byP(y) =P_u(x, R, y) the unique polynomial of degreek−1 such that

Br(x)D^l(u(y)−P(y))dy= 0 l= 1, . . . , k−1.

Its coefficients depend onx, Rand also on the derivatives of u(see [12]). When no confusion will arise, we will omit the dependence ofP onx, R andu.

Next lemma can be found in [11], (Th. 3.3), in a slightly different form.

Lemma 2.2. Let v ∈W^k,p(B₁(0)) and0 < s < r < 1. There exists a linear operator T : W^k,p(B₁(0)) → W^k,p(B₁(0))such that

T v=v on (B₁\B_r)∪B_s and for all µ >0 , for allq < p_n−1ⁿ

Br\Bs

|D^kT v|^{2 1}₂

+µ

Br\Bs

|D^kT v|^{q 1}_q

≤C(r−s)^ρ ⎡

⎣ sup

t∈(s,r)(t−s)⁻¹²

Bt\Bs

|D^kv|^{2 1}₂

+ sup

t∈(s,r)(r−t)⁻¹²

Bt\Bs

|D^kv|^{2 1}₂⎤

⎦

+µ

⎡

⎣ sup

t∈(s,r)(t−s)^−1p

Br\Bt

|D^kv|^{p 1}_p

+ sup

t∈(s,r)(r−t)^−1p

Br\Bt

|D^kv|^p _p¹⎤

⎦

(5)

whereC=C(n, p, q)>0, andρ=ρ(n, p, q)>0.

Let us recall an elementary Lemma proved in [10].

Lemma 2.3. Let ψ be a continuous nondecreasing function on an interval [a, b], a < b. There exist a ∈ [a, a+¹₃(b−a)],b ∈[b−¹₃(b−a), b] such thata≤a < b≤band

ψ(t)−ψ(a)

t−a ≤ 3(ψ(b)−ψ(a)) b−a ψ(b)−ψ(t)

b−t ≤ 3(ψ(b)−ψ(a)) b−a for allt∈(a, b).

Finally, combining the previous two lemmas we obtain a generalization to the case of higher order derivatives of Lemma 2.4 in [10]. We give the proof here for completeness.

Lemma 2.4. Let v, w∈W^k,p(B₁(0))and ¹₄ < s < r <1. Fix p < q < _n−1^np , for all µ >0 andm∈IN there exist a function z∈W^k,p(B₁(0))and ¹₄ < s < s < r< r <1 withr, s depending on v, wandµ, such that

z=v on B_s , z=w on B₁\B_r, (6)

r−s

m ≥r−s≥ r−s 3m and

B_r\B_s

|D^kz|^{2 1}₂

+µ

B_r\B_s

|D^kz|^q _q¹

≤C(r−s)^ρ m^ρ

Br\Bs

1 +

k l=0

|D^lv|²+ k l=0

|D^lw|²+ m² (r−s)²

k−1

l=1

|D^l(v−w)|²

+µ^p

Br\Bs

1 +

k l=0

|D^lv|^p+ k l=1

|D^lw|^p+ m^p (r−s)^p

k−1

l=0

|D^l(v−w)|^{p 1}₂

(7)

whereC=C(n, p, q)>0 andρ=ρ(p, q, n)>0.

Proof. As in Lemma 2.4 in [10], choosem∈IN and set f = 1 +_k

l=0|D^lv|²+_k

l=0|D^lw|²+_(r−s)^m22

_k−1

l=0 |D^l(v−w)|² +µ^p(1 +_k

l=0|D^lv|^p+_k

l=0|D^lw|^p+_(r−s)^mp_pk−1

l=0 |D^l(v−w)|^p).

We may findh∈ {1, ..., m}such that

B_s+h(r−s)

m \B_{s+ (}h−1)(r−s) m

fdx≤ 1 m

Br\Bs

fdx.

Set, fort∈

s+^{(h−1)(r−s)}_m , s+^h(r−s)_m

,

ψ(t) =

Bt\Bs

fdx

which is a continuous increasing function. By Lemma 2.3, there exists [s, r]⊂

s+^{(h−1)(r−s)}_m , s+^h(r−s)_m

such

that r−s

m ≥r−s≥ r−s 3m

and

Bt\Bs

fdx≤3(t−s)m r−s

B_s+h(r−s)

m \B_{s+ (}h−1)(r−s) m

fdx≤3t−s r−s

Br\Bs

fdx, (8)

B_r\Bt

fdx≤3r−t r−s

Br\Bs

fdx (9)

for allt∈(s, r). Set

u=

⎧⎪

⎨

⎪⎩

v(x) if x∈B_s

(r−|x|)v(x)+(|x|−s)w(x)

r−s if x∈B_r\B_s w(x) if x∈B₁\B_r. A direct computation shows that

k l=0

|D^lu|²+µ^q _k

l=0

|D^lu|^p

≤Cf.

If we apply Lemma 2.2 to the function u, we then find z ∈W^k,p(B₁) satisfying (6). Moreover, from (8) and (9), using (5), one readily cheks that

Br\Bs

|D^kz|^{2 1}₂

+µ

Br\Bs

|D^kz|^q _q¹

≤c(r−s)^ρ

|B_r\B_s|¹² (r−s)¹²

Br\Bs

f ¹₂

+|B_r\B_s|^1p (r−s)^1p

Br\Bs

f _p¹

≤c(r−s)^ρ

Br\Bs

f ¹₂

+

Br\Bs

f ¹_p

,

from which (7) follows.

3. The decay estimate

As usual, to get the partial regularity result stated in Theorem 2.1, we need a decay estimate for the excess functionU(x₀, r) defined as follows

U(x₀, r) =

Br(x0 )

[|D^ku−(D^ku)_x0,r|²+|D^ku−(D^ku)_x0,r|^p]dy,

which measures how the k-order derivatives are far from being constant in the ballB_r(x0). The desired decay estimate is established in the next proposition.

Proposition 3.1. Fix M > 0. There exists a constant C_M >0 such that for every 0 < τ < ¹₄, there exists =(τ, M)such that, if

|(D^ku)_x0,r| ≤M and U(x₀, r)≤ then

U(x₀, τ r)≤C_Mτ²U(x₀, r).

Proof. Fix M andτ. We shall determine C_M later. We argue by contradiction assuming that there exists a sequenceB_rh(x_h) satisfying

B_rh(x_h)⊂Ω, |(D^ku)_xh,rh| ≤M, lim

h U(x_h, r_h) = 0, but

U(x_h, τ r_h)> C_Mτ²U(x_h, r_h). (10) Set

A_h= (D^ku)_xh,rh λ²_h=U(x_h, r_h) and letP the polynomial such that

B_rh(xh)D^l(u−P) = 0 l= 0, . . . , k.

Step 1. Blow up. We rescale the functionuin eachB_rh(x_h) to obtain a sequence of functions on B₁(0). Set v_h(y) = 1

λ_hr_h^k[u(x_h+r_hy)−P(x_h+r_hy)], then

D^kv_h(y) = 1

λ_h[D^ku(x_h+r_hy)−A_h].

Clearly we have

(D^lv_h)_0,1= 0 l= 0, . . . , k.

Moreover,

U(x_h, r_h) λ²_h =

B1

[|D^kv_h|²+λ^p−2_h |D^kv_h|^p]dy= 1. (11) Then, passing possibly to a subsequence, we may suppose that

v_h v weakly in W^k,2(B₁;IR^N) (12)

and, since∀h |A_h| ≤M,

A_h→A. (13)

Step 2. v solvesalinear system. Now we show that

B1(0)

∂²f

∂ξ_αⁱ∂ξ_β^j(A)D_βv^jD_αφⁱdy= 0 ∀φ∈C₀^k(B₁;IR^N). (14) Since we assumeq−1≤pwe can write the usual Euler-Lagrange system foru(see Rem. 2.1). Then, rescaling in eachB_rh(x_h), we get for anyφ∈C₀^k(B₁;IR^N) and any 1≤i≤N

B1(0)

∂f

∂ξ_αⁱ (A_h+λ_hD^kv_h)D_αφⁱdy= 0 where|α|=k. Then

1 λ_h

B1(0)[∂f

∂ξ_αⁱ (A_h+λ_hD^kv_h)− ∂f

∂ξ_αⁱ (A_h)]D_αφⁱdy= 0. (15) Let us split

B₁=E_h⁺∪E_h⁻={y∈B₁:λ_h|D^kv_h(y)|>1} ∪ {y∈B₁:λ_h|D^kv_h(y)| ≤1}

then, by (11), we get

|E_h⁺| ≤

E⁺_h λ²_h|D^kv_h|²dy≤λ²_h

B1(0)|D^kv_h|²dy≤cλ²_h. (16) Now, by (H4) and H¨older’s inequality, we observe that

1 λ_h

E⁺_h[Df(A_h+λ_hD^kv_h)−Df(A_h)]Dφdy

≤ c

λ_h|E_h⁺|+cλ^q−2_h

E_h⁺|D^kv_h|^q−1dy

≤cλ_h+c

E_h⁺λ^p−2_h |D^kv_h|^pdy ^q−1_p

λ

2q−p−2

h p |E_h⁺|^p−q+1p ≤cλ_h

where we used again the assumptionq−1≤p.

From this it follows that limh

1 λ_h

E⁺_h[Df(A_h+λ_hD^kv_h)−Df(A_h)]Dφdy= 0. (17) OnE_h⁻ we have

1 λ_h

E_h⁻[Df(A_h+λ_hD^kv_h)−Df(A_h)]Dφdy =

E⁻_h

₁

0 D²f(A_h+sλ_hD^kv_h)D^kv_hDφdsdy

=

E⁻_h

₁

0 [D²f(A_h+sλ_hD^kv_h)−D²f(A_h)]D^kv_hDφdsdy +

E⁻_h D²f(A_h)D^kv_hDφdy.

Note that (16) ensures that χ_E−

h → χ_B1 in L^r(B₁) for all r < ∞and by (11) we have, passing possibly to a subsequence,

λ_hD^kv_h(y)→0 a.e. in B₁. Then, by (12), (13) and the uniform continuity ofD²f on bounded sets, we get

limh

1 λ_h

E⁻_h[Df(A_h+λ_hD^kv_h)−Df(A_h)]Dφdy=

B1

D²f(A)D^kvDφdy.

Collecting (15), (17) and the above equality, we obtain thatv satisfies system (14), which is linear and elliptic with constant coefficients by (H3). By standard regularity results (see [12]), we have for any 0< τ <1

Bτ

|D^kv−(D^kv)_τ|²dy≤cτ²

B1

|D^kv−(D^kv)₁|²dy≤cτ². (18)

Moreover we have

v∈C^∞(B₁;IR^N) (19)

and

λ

p−2p

h (v_h−v)0 weakly in W_loc^k,p(B₁;IR^N).

Step 3. Upper bound. We set

f_h(ξ) = 1

λ²_h[f(A_h+λ_hξ)−f(A_h)−λ_hDf(A_h)ξ]

and, for everyr <1, we consider

I_h,r(w) =

Br

f_h(D^kw)dy.

Note that, by the strong ellipticity assumption (H3), it follows that f_h(ξ)≥0, for any ξ, and remember that v_h is a local minimizer for eachI_h,r. Fix ¹₄ < s <1. Passing to a subsequence we may always assume that

limh [I_h,s(v_h)−I_h,s(v)]

exists.

We shall prove that

limh [I_h,s(v_h)−I_h,s(v)]≤0. (20)

Consider r > s and fix m ∈ IN. Observe that, since v ∈ W^k,p(B₁) and v_h ∈ W^k,p(B₁), Lemma 2.4, with µ=λ

q−2q

h , implies that there existz_h∈W^k,p(B₁) and ¹₄ < s < s_h< r_h< r <1 such that z_h=v on B_sh z_h=v_h on B₁\B_rh

and

B_rh\B_sh|D^kz_h|^{2 1}₂

+λ

q−2q

h

B_rh\B_sh|D^kz_h|^{q 1}_q

≤C(r−s)^ρ m^ρ

Br\Bs

(1 + k l=0

|D^lv|²+ k l=0

|D^lv_h|²+ m² (r−s)²

k−1

l=1

|D^l(v−v_h)|²)

+λ

q−2q p h

Br\Bs

(1 + k l=0

|D^lv|^p+ k l=0

|D^lv_h|^p+ m^p (r−s)^p

k−1

l=1

|D^l(v−v_h)|^p) ¹₂

. (21)

Since by (19),D^kv is locally bounded onB₁ we get

I_h,s(v_h)−I_h,s(v) ≤ I_h,rh(v_h)−I_h,rh(v) +I_h,rh(v)−I_h,s(v)

= I_h,rh(v_h)−I_h,rh(v) +

B_rh\Bs

f_h(D^kv)

≤ I_h,rh(z_h)−I_h,rh(v) +c(r−s)

≤

B_rh\B_sh[f_h(D^kz_h)−f_h(D^kv)] +c(r−s) (22) where we used the minimality ofv_h.

As|fh(ξ)| ≤c(|ξ|²+λ^q−2_h |ξ|^q), we get by (21), using the fact that ^r−s_m <1 and that the quantity on square brackets is greater or equal than 1,

I_h,rh(z_h)−I_h,rh(v) ≤ c

B_rh\B_sh|D^kz_h|²+λ^q−2_h |D^kz_h|^q

≤ c(r−s)^2ρ m^2ρ

Br\Bs

(1 + k l=0

|D^lv|²+ k l=0

|D^lv_h|²+ m² (r−s)²

k−1

l=1

|D^l(v−v_h)|²) ^q₂

+ c(r−s)^2ρ m^2ρ

λ

q−2q p h

Br\Bs

(1 + k

l=0

|D^lv|^p+ k l=0

|D^lv_h|^p+ m^p (r−s)^p

k−1

l=1

|D^l(v−v_h)|^p) ^q₂

= J_h,1+J_h,2.

SinceD^lv_h→D^lv strongly inL²(B₁;IR^N) for everyl < k, we have, using (11) lim sup

h→∞ J_h,1≤cm^−2ρ.

Moreover, since forl= 0, . . . , k λ

p(q−2)

h q

B1

|D^lv_h|^p≤cλ

2(q−p)

h q λ^p−2_h

B1

|D^kv_h|^p≤cλ

2(q−p)

h q

and

λ

p(q−2)

h q

B1

|D^l(v_h−v)|^p≤cλ

p(q−2)

h q

B1

|D^kv_h|^p≤cλ

2(q−p)

h q

we have

limh J_h,2= 0.

Hence we conclude letting firstm→ ∞and then r→sin (22).

Step 4. Lower bound. We shall prove that, fora.e. ¹₄ < r < ¹₂, ift < r then lim sup

h

Bt

|D^kv−D^kv_h|²(1 +λ^p−2_h |D^kv−D^kv_h|^p−2)≤lim

h [I_h,r(v_h)−I_h,r(v)].

For any Borel setA⊂B₁, let us define

µ_h(A) =

A

k l=0

|D^lv_h|²dx.

Passing possibly to a subsequence, sinceµ_h(B₁)≤c, we may suppose

µ_h µ weakly∗ in the sense of measures, whereµis a Borel measure over B₁, with finite total variation. Then fora.e. r <1

µ(∂B_r) = 0

and let us choose such a radiusr. Consider ¹₄ < t < s < r, also such thatµ(∂B_s) = 0, and fixm∈IN. Observe that, as v_h∈W^k,p(B₁) Lemma 2.4 implies that there existz_h∈W^k,p(B₁) and ¹₄ < s < s_h< r_h< r <1 such that

z_h=v_h on B_sh z_h=v_h on B₁\B_rh r_h−s_h≥ r−s

3m and

B_rh\B_sh|D^kz_h|^{2 1}₂

+λ

q−2q

h

B_rh\B_sh|D^kz_h|^{q 1}_q

≤C(r−s)^ρ m^ρ

Br\Bs

1 +

k l=0

|D^lv_h|²

+λ

(q−2)p

h q

Br\Bs

(1 + k

l=0

|D^lv_h|^p) ¹₂

. (23) Passing possibly to a subsequence, we may suppose that

z_h v_r,s weakly in W^k,2(B₁).

and

v_r,s=v in (B₁\B_r)∪B_s.

Moreover, from (23) and the interpolation inequality with ¹_p =^θ₂+^1−θ_q , we deduce that

λ^p−2_h

B1

|D^kz_h|^p ≤ cλ^p−2_h

B1

|D^kz_h|^{2 θp}₂

B1

|D^kz_h|^{q (1−θ)p}₂

≤ cλ^p−2_h

B1

|D^kz_h|^{q (1−θ)p}₂

≤ cλ^p−2_h (cλ^2−q_h )^p−2q−2 ≤c, (24)

sinceθ= ^p−2_q−2^q_p.

Considerζ_h∈C₀^∞(B_rh) such that 0≤ζ_h≤1,ζ_h= 1 onB_sh and|D^lζ_h| ≤ _(rh−s^C_h)l, forl= 0, . . . , k, and set ψ_h =ζ_h(z_h−v_r,s),

wherev_r,s=ρ v_r,s, andρis the usual sequence of mollifiers. Now, settingv=ρ v, we observe that I_h,rh(v_h)−I_h,rh(v) = I_h,rh(v_h)−I_h,rh(z_h) +I_h,rh(z_h)−I_h,rh(v_r,s+ψ_h)

+I_h,rh(ψ_h+v_r,s )−I_h,rh(v_r,s)−I_h,rh(ψ_h) +I_h,rh(v_r,s )−I_h,rh(v) +I_h,rh(ψ_h)

= R_h,1+R_h,2+R_h,3+R_h,4+R_h,5. (25)

To boundR_h,1 we observe that I_h,rh(v_h)−I_h,rh(z_h) =

B_rh\B_shf_h(D^kv_h)−

B_rh\B_shf_h(D^kz_h)≥ −

B_rh\B_shf_h(D^kz_h) on the other hand we have

B_rh\B_shf_h(D^kz_h)≤

B_rh\B_sh|D^kz_h|²+λ^q−2_h |D^kz_h|^q

≤cm^−2ρ

Br\Bs

(1 + k l=0

|D^lv_h|²) +λ

q−2q p h

Br\Bs

(1 + k

l=0

|D^lv_h|^p) ^q₂

and then arguing as we did in Step 3 to bound J_h,1 we get lim sup

h

B_rh\B_shf_h(D^kz_h)≤Cm^−2ρ hence, lettingh→ ∞we get

lim inf

h R_h,1≥ −Cm^−2ρ. (26)

We obtain that R_h,2 =

B_rh\B_shf_h(D^kz_h)−f_h(D^kψ_h+D^kv_r,s )

≥ −c

B_rh\B_sh|D^kψ_h+D^kv_r,s |²+λ^q−2_h |D^kψ_h+D^kv_r,s|^q

≥ −c

B_rh\B_sh|D^kz_h|²+λ^q−2_h |D^kz_h|^q+|D^kv_r,s |²+λ^q−2_h |D^kv_r,s |^q

− c

B_rh\B_sh(

k−1

l=0

m^2(k−l)

(r−s)^2(k−l)|D^l(z_h−v_r,s )|²+λ^q−2_h

k−1

l=0

m^q(k−l)

(r−s)^q(k−l)|D^l(z_h−v_r,s )|^q)

= −S_h,1−S_h,2

where we used the boundr_h−s_h≥^r−s_3m. Denoting byP_lthe polynomial of degreek−1 such that

B1

(D^l(P_l−z_h)) = 0,