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From (P.3), (P.4) and (9.3.29) we get Z 0 −a dt Z B(σ) ∂u ∂t 2 dx_≤ ≤ c(ν, K, U, λ, σ, a, b, m, n)   1+ X |α|<m Z 0 −bkf α k0,B(3σ)dt 2 + Z 0 −b|u| 2 m,B(3σ)dt   . The last inequality and (9.3.25) allows us to conclude the proof.

applying Theorem 7.2.1, it follows u_{∈ L}2_{(−a, 0, H}m+1(B(σ), RN₎₎ and Z 0 −a|u| 2 m+1,B(σ)dt≤ ≤ c(ν, K, U, λ, σ, a, b, m, n)   1 + X |α|<m Z 0 −bkf α k0,B(3σ)dt 1+ϑ 2 + Z 0 −b|u| 2 m,B(3σ)dt   . (9.3.25) Finally we have to prove that u ∈ H1_{(−a, 0, L}2_{(B(σ), R}N_{)) and inequality (9.2.5).}

From inequality (9.2.3) we have Z 0 −a dt Z B(σ)kD ′′_uk4_dx ≤ ≤ c(ν, K, U, λ, σ, a, b, m, n)   1 + X |α|<m Z 0 −bkf α k0,B(3σ)dt 1+ϑ 2 + Z 0 −b|u| 2 m,B(3 σ)dt    (9.3.26) then we have D′′u_{∈ L}4_{(B(σ) × (−a, 0), R}′′). (9.3.27) Moreover, bearing in mind that, for |α| < m, aα

(X, p) satisfies (P.3), and for |α| = m, aα(X, p) satisfies (P.4), we have

Dαaα_{(X, p) ∈ L}2 B_{(σ) × (−a, 0), R}N
_{∀α : |α| ≤ m} (9.3.28) Recalling the definition of weak solution, for every ϕ ∈ C∞

0 (Q, RN), proceeding as in [24], we have Z 0 −a dt Z B(σ)  u     ∂ ϕ ∂ t  dx= X |α|≤m Z 0 −a dt Z B(σ) (Dα_aα (X, Du)| ϕ ) dx, (9.3.29)

and, bearing in mind (9.3.28), we obtain that ∃∂u

∂t ∈ L

2 _B

every |h| < h0, it follows Z B(2σ)kτ i,hD′uk2kD′′uk2dx≤ Z B(2σ)kτ i,hD′uk4dx 1 2Z B(2σ)kD ′′_uk4 dx 1 2 ≤ ≤ |h|2 k D′′u_k2_0,4,B(5 2σ)k D ′′_uk2 0,4,B(2σ) ≤ |h|2 | u | 4 m,4,B(5 2σ) .

Integrating in (−b∗_{,0), from (9.3.23) it follows}

Z 0 −a dt Z B(σ) k τi,hD′′uk2 dx≤ ≤ c(ν, K, U, λ, σ, a, b, m, n)|h|2   1+ X |α|<m Z 0 −bkf α k0,B(3σ)dt 1+ϑ 2 + Z 0 −b|u| 2 m,B(3σ)dt   . (9.3.24) If h0 ≤ |h| < σ₂,for every i = 1, 2, . . . , n we easily obtain

Z 0 −a dt Z Bσ)kτ i,hD′′uk2dx≤4 Z 0 −a dt Z B(3σ)kD ′′_uk2 dx_{≤ 4}h 2 h2₀ Z 0 −a dt Z B(3σ)kD ′′_uk2 dx_≤ ≤ c(ν, K, U, λ, σ, a, b, m, n)h2 Z 0 −b |u| 2 m,B(3σ)dt ≤ ≤ c(ν, K, U, λ, σ, a, b, m, n)|h|2   1 + X |α|<m Z 0 −bkf α k0,B(3σ)dt 1+ϑ 2 + Z 0 −b|u| 2 m,B(3σ)dt   . It is then proved, for every |h| < σ

2 and every i ∈ {1, 2, . . . , n} , that

Z 0 −a dt Z B(σ)kτ i,hD′′uk2 dx≤ ≤ c(ν, K, U, λ, σ, a, b, m, n) |h|2   1 + X |α|<m Z 0 −bkf α k0,B(3σ)dt 1+ϑ 2 + Z 0 −b|u| 2 m,B(3σ)dt   ,

Multiplying each term for ρ2

µ and integrating respect to (−b∗,−1µ) and applying

(9.3.5), we achieve Z −µ1 −b∗ ρ2_µdt Z B(5 2σ)  |fα | + kD′′u_k2 τ_i,−hDα ψ2mτi,hu
dx ≤ ≤ _{4 c(K, m, n)}ν Z −1µ −b∗ ρ2_µ dt Z B(2σ) ψ2m_{k τ}i,hD′′uk2 dx+ +c(ν, K, U, λ, σ, a, b, m, n)h2   1+ X |α|<m Z 0 −bkf α k0,B(3σ)dt 1+ϑ 2 + Z −1 µ −b∗|u| 2 m,B(3σ)dt   . Taking into consideration the last inequality and the properties of the function ψ, from (9.3.21) we deduce Z −2µ −a dt Z B(σ)kτ i,hD′′uk2 dx≤ ≤ c(ν, K, U, λ, σ, a, b, m, n) h2   1 + X |α|<m Z 0 −bkf α k0,B(3σ)dt 1+ϑ 2 + Z −µ1 −b∗ |u| 2 m,B(3σ)dt   + + c(ν, K, σ, m, n) Z −1 µ −b∗ dt Z B(2σ) ψ2mρ2_{µ k τ}i,hD′uk2 k D′′uk2 dx.

From which, passing the limit µ → ∞, we get Z 0 −a dt Z B(σ)kτ i,hD′′uk2 dx≤ ≤ c(ν, K, λ, σ, m, n)h2   1 + X |α|<m Z 0 −bkf α k0,B(3σ)dt 1+ϑ 2 + Z 0 −b∗|u| 2 m,B(3σ)dt   + + c(ν, K, σ, m, n) Z 0 −b∗ dt Z B(2σ) ψ2mρ2_µkτi,hD′uk2kD′′uk2 dx. (9.3.23)

Let us now estimate the last term in (9.3.23). Using H¨older inequality, applying Theorem 7.2.2 (for p = 4, B(5

2σ) instead of B(σ) and t = 4

Let us focus our attention on the last term, taking into account that from (9.3.4), for a. e. t ∈ (−b∗_{,0), we have} u_{(·, t) ∈ H}m,4  B ₅ 2σ  , RN 

then using H¨older and Young inequalities, for every α such that |α| < m, for every ε >0, it follows Z B(5 2σ) (|fα | + kD′′_uk2 ) kτi,−hDα ψ2mτi,hu
k dx ≤ ≤ Z B(3σ)|h| −2 _τ i,−hDα ψ2mτi,hu
2 dx 1 2 !Z B(5₂σ) h2_(|fα_{| + kD}′′u_k2)2dx "1 2 ≤ ≤ ε₂|h|−2 Z B(3σ) τi,−hDα ψ2mτi,hu
2 dx + c(ε) h2 Z B(5₂σ)(|f α |2+ kD′′u_k4) dx. Furthermore, for every α such that |α| < m, from Theorem 7.2.2 for every h ∈ R with |h| < h0 and for every ε > 0, we have

ε 2|h| −2Z B(3σ) τ_i,−hDα ψ2mτi,hu
2 dx_≤ ε 2 Z B(7 2σ) D′′ _ψ2m_τ i,hu
2 dx _≤ ≤ ε Z B(2σ) ψ2m _{k τ}i,hD′′uk2 dx + c(σ, ε) Z B(2σ)k τ i,hD′uk2 dx≤ ≤ ε Z B(2σ) ψ2m _{k τ}i,hD′′uk2 dx + c(σ, ε) h2 Z B(3σ)kD ′′_uk2 dx

the last inequality follows, as before, applying Theorem 7.2.2 for p = 2. Let us now choose ε = ν 4 c(K,m,n), it ensures Z B(5 2σ)  |fα | + kD′′u_k2 τ_i,−hDα ψ2mτi,hu
dx ≤ ≤ _{4 c(K, m, n)}ν Z B(2σ) ψ2m_kτi,hD′′uk2dx +c(ν, K, σ, m, n)h2 Z B(3σ)|f α |2dx_+|u|2_m,B(3σ)+|u|4_m,4,B(5 2σ)  . (9.3.22)

Then, from (9.3.9) estimating the terms A, B, C, D and E, for every ε > 0, we have ν Z −µ1 −b∗ dt Z B(2σ) ψ2mρ2_{µ kτ}i,hD′′uk2 dx≤ ≤n3ε + c(K, U, m, n)_{|h| + h}2_{+ |h|}λ_{+ |h|}2λoZ −1µ −b∗ dt Z B(2σ) ψ2mρ2_µkτi,hD′′uk2dx+ +c(K, σ, a, b, m, n, ε) h2 Z −µ1 −b∗ dt Z B(3 σ)  1 + kD′′u_k2 dx+c(σ, a, b, n, )Kh2+ + c(K, σ, m, n, ε) Z −µ1 −b∗ dt Z B(2σ) ψ2mρ2_µkτi,hD′uk2kD′′uk2 dx+ + c(K, m, n)X |α|<m Z −µ1 −b∗ ρ2_µdt Z B(5₂σ)(|f α |+ kD′′u_k2_)kτi,−hDα ψ2mτi,hu
k dx. (9.3.20)

We observe that the function

h_{−→ c(K, U, σ, m, n)}_{|h| + h}2_{+ |h|}λ_{+ |h|}2λ

is continuous in the origin, then ∃ h0(ν, K, U, λ, σ, m, n), 0 < h0 < min{1,σ₂}, such

that for every |h| < h0, we have

c(K, U, σ, m, n)_{|h| + h}2_{+ |h|}λ_{+ |h|}2λ< ν 4. For each integer i = 1, . . . , n , for ε = ν

12 and every h such that |h| < h0(< 1), it

follows ν 2 Z −1 µ −b∗ dt Z B(2σ) ψ2mρ2_µkτi,hD′′uk 2 dx_≤ ≤ c(ν, K, σ, a, b, m, n) |h|2 Z −1 µ −b∗ dt Z B(3 σ)  1 + kD′′u_k2 dx+ + c(ν, K, σ, m, n) Z −1 µ −b∗ dt Z B(2σ) ψ2mρ2_µkτi,hD′uk2kD′′uk2 dx + + c(K, m, n) X |α|<m Z −1 µ −b∗ ρ2_µdt Z B(5 2σ) (|fα | + kD′′u_k2_{) kτ}i,−hDα ψ2mτi,hu
kdx. (9.3.21)

The term B can be estimated, for every ε > 0, as follows |B| ≤nε+ c(K, U, m, n)_|h|λ_{+ |h|}2λo Z −µ1 −b dt Z B(2σ) ψ2mρ2_µkτi,hD′′uk2 dx+ + c(K, σ, m, n, ε)h2 Z −µ1 −b dt Z B(3σ)kD ′′_uk2 dx+ + c(K, m, n, ε) Z −µ1 −b dt Z B(2σ) ψ2mρ2_µkτi,hD′uk2kD′′uk2 dx. (9.3.16)

Let us consider the term C, for every ε > 0, we have |C| ≤ε+ c(K, m, n) h2_{+ |h|}
Z −1µ −b dt Z B(2σ) ψ2mρ2_µkτi,hD′′uk2 dx+ + c(K, σ, m, n, ε) h2 Z −1µ −b dt Z B(3σ)  1 + kD′′u_k2 dx. To estimate the term D, we firstly observe that

(ρ′_µρµ)(t)                = 0 _{if t ≤ −b, −a ≤ t ≤ −µ}2, t_{≥ −}1_µ ≤ b−a1 if − b ≤ t ≤ −a ≤ 0 if − 2µ ≤ t ≤ − 1 µ (9.3.17)

then, using Theorem 7.2.2, we obtain

D = Z Q ψ2mρ′_µρµkτi,huk2dX = = Z −a −b dt Z B(2σ) ψ2mρ′_µρµkτi,huk2dx+ Z −1µ −µ2 dt Z B(2σ) ψ2mρ′_µρµkτi,huk2dx≤ ≤ _b 1 − a Z −a −b dt Z B(2σ) kτi,huk2dx ≤ h2 b_{− a} Z −a −b dt Z B(3σ) kDiuk2dx. (9.3.18) Finally, using (P.3) condition, the term E can be expressed as follows

|E| ≤ c(K, m, n) X |α|<m Z −1 µ −b ρ2_µdt Z B(5₂σ)(|f α | + kD′′u_k2_)kτi,−hDα ψ2mτi,hu
k dx. (9.3.19)

we have ν Z −µ1 −b dt Z B(2σ) ψ2mρ2_{µ kτ}i,hD′′uk2 dx= ν Z −µ1 −b dt Z B(2σ) ψ2mρ2_µ X |α|=m kτi,hDαuk2dx≤ ≤ Z Q ψ2mρµ X |α|=|β|=m N X k=1  τi,hDβuk(X)
g∂aα ∂pβ_k     ρµτi,hDαu  dX _≤ ≤ A + B + C + D + E, (9.3.9) where A_{= −} X |α|=|β|=m X γ<α N X k=1 Z Q cαγ(ψ)ψmρ2µ  τi,hDβuk(X)
g∂aα ∂pβ_k    (τi,hDγu)(X)  dX, (9.3.10) B _{= −} X |α|=m X |β|<m N X k=1 Z Q  τi,hDβuk(X)
g∂aα ∂pβ_k     Dα ψ2mρ2µτi,hu(X)
 dX, (9.3.11) C _{= −h} X |α|=m Z Q  g∂aα ∂xi     Dα ψ2mρ2µτi,hu)(X)
 dX, (9.3.12) D = Z Q ψ2m ρ′_µρµk τi,huk2dX, (9.3.13) E _{= −}X |α|<m Z Q aα_{(X, Du) |τ}i,−hDα ψ2mρ2µτi,hu

dX. (9.3.14)

We observe that, for every ε > 0, we have |A| ≤ ε Z −1µ −b dt Z B(2σ) ψ2mρ2_µkτi,hD′′uk2dx+c(K, σ, m, n, ε) h2 Z −1µ −b dt Z B(3σ)  1 + kD′′u_k2dx. (9.3.15)

Then, equality (9.3.8) becomes Z Q X |α|=m  h∂ag α ∂xi + X |β|≤m N X k=1 τi,hDβuk(X)
g∂aα ∂pβ_k    Dα ψ2mρµ[(ρµτi,hu) ∗ gs]
 dX = = Z Q ψ2mρ′_µ(τi,hu|(ρµτi,hu)∗gs) dX + Z Q τi,hu| ψ2mρµ[(ρµτi,hu) ∗ gs]′
dX₋ X |α|<m Z Q

aα_{(X, Du) |τi,−h}Dαψ2mρµ[(ρµτi,hu)∗gs]

dX.

Taking into account, for α : |α| = m, that Dα ψ2mρµ[(ρµτi,hu) ∗ gs]
= ψ2mρµ[(ρµτi,hDαu) ∗ gs]+ψmρµ X γ<α cαγ(ψ) [(ρµτi,hDγu) ∗ gs] where |cαγ(ψ)| ≤ c(m, n) σm−|γ| , we obtain Z Q ψ2mρµ X |α|=|β|=m N X k=1  τi,hDβuk(X)
g∂aα ∂pβ_k     (ρµτi,hDαu) ∗ gs  dX = = − X |α|=|β|=m X γ<α N X k=1 Z Q  τi,hDβuk(X)
g∂aα ∂pβ_k     ψmρµcαγ(ψ) [(ρµτi,hDγu) ∗ gs]  dX₋ − X |α|=m X |β|<m N X k=1 Z Q  τi,hDβuk(X)
g∂aα ∂pβ_k     Dα ψ2mρµ [ (ρµτi,hu) ∗ gs]
 dX₋ − h X |α|=m Z Q  g∂aα ∂xi    Dα ψ2mρµ[(ρµτi,hu) ∗ gs]
 dX+ Z Q ψ2mρ′_µ(τi,hu|(ρµτi,hu)∗gs) dX+ + Z Q τi,hu| ψ2mρµ[(ρµτi,hu) ∗ g′s]
dX₋X |α|<m Z Q

aα_{(X, Du) |τi,−h}Dαψ2mρµ[(ρµτi,hu) ∗ gs]

dX.

For s → +∞, using ellipticity condition (3.6), symmetry hypothesis, convolution property of gs and that

lim s→+∞ Z Q τi,hu| ψ2mρµ[(ρµτi,hu) ∗ gs′]
dX = 0,

Let i be a positive integer, i ≤ n, and h a real number such that |h| < σ

2. For every

µ >2_{a and for every s > max{µ,T}1_{−b} let us define the following “test function”} ϕ(X) = τ_i,−hψ2mρµ[(ρµτi,hu) ∗ gs]

, _{∀X = (x, t) ∈ Q.} (9.3.7) Substituting in (9.2.1) the above defined function ϕ, we have

Z

Q

X

|α|=m

τi,haα(X, Du) |Dα ψ2mρµ[(ρµτi,hu) ∗ gs]

dX = = Z Q τi,hu| ψ2m{ρµ[(ρµτi,hu) ∗ gs]}′
dX₋ −X |α|<m Z Q aα_{(X, Du) |τ}i,−hDα  ψ2mρµ[(ρµτi,hu) ∗ gs]
dX. (9.3.8) For every α : |α| = m and a. e. X = (x, t) ∈ Q, we have

τi,haα(X, Du(X)) = aα x+ hei, t, Du(x + hei, t)

− aα_{(X, Du(X)) =} = Z 1 0 d dη a α _{x+ ηhe}i_{, t, Du(X) + η τ} i,hDu(X)
dη = = h Z 1 0 ∂ ∂xi aα x+ ηhei_{, t, Du(X) + η τ} i,hDu(X)
dη+ + X |β|≤m N X k=1 τi,hDβuk(X)
Z 1 0 ∂ ∂pβ_ka α _x+ηhei_{, t, Du(X)+ητ} i,hDu(X)
dη= = h∂ag α ∂xi + X |β|≤m N X k=1 τi,hDβuk(X)
g∂aα ∂pβ_k,

where, if b = b(X, p), for simplicity of notation, we set ˜b(X) =Z 1 0 b x+ ηhei_{, t, Du(X) + η τ} i,hDu(X)
dη.

then, using (9.3.1), written with θ = 1 − λ 2, and (9.3.2)–(9.3.4), we have Z 0 −b∗kuk 4 m,4,B(5 2σ) dt_{≤ c(σ)} Z 0 −b∗kuk 4 m,4+ 4λ n−λ,B(52σ) dt_≤ ≤ c(θ, λ, σ, m, n) Z 0 −b∗kD ′′_uk2 1−λ 2,B(52σ) k u_k2 Cm−1,λ_(B(5 2σ),RN) dt_≤ ≤ c(ν, K, U, λ, σ, m, n)   1+ X |α|<m Z 0 −bkf α k0,B(3σ)dt 1+ϑ 2 + Z 0 −b|u| 2 m,B(3σ) dt   , (9.3.5) then it follows the requested inequality (9.2.3).

Proof. Let us fix B(3σ) = B(x0_{,3σ) ⊂⊂ Ω, a, b ∈ (0, T ) with a < b and h ∈ R such}

that |h| < σ2, set b∗ = a+b

2 and let ψ(x) ∈ C0∞(Rn) a real function satisfying the

following properties 0 ≤ ψ ≤ 1 in Rn_{, ψ = 1 in B(σ), ψ = 0 in R}n

\ B(2σ), kDψk ≤ c σ

in Rn_.

Let us also define the function ρµ(t), for µ > 2_a, µ integer, the following real function

ρµ(t) =                        1 _{if − a ≤ t ≤ −µ}2 0 _{if t ≤ −b and t ≥ −}1_µ t+b b−a if − b < t < −a −(µt + 1) if − µ2 < t <− 1 µ. (9.3.6)

Moreover set {gs(t)} the sequence of symmetric regularizing functions such that

gs(t) ∈ C₀∞(R), gs(t) ≥ 0, gs(t) = gs(−t), supp gs ⊂  −1_s,1 s  , Z R gs(t) dt = 1.

and Z 0 −b∗|D ′′_u|2 ϑ,B(5 2σ) dt≤ ≤ c(ν, K, U, ϑ, λ, σ, a, b, m, n)   1 + X |α|<m Z 0 −b kfα k0,B(3σ)dt 1+ϑ 2 + Z 0 −b∗|u| 2 m,B(3σ)dt   . (9.3.1) Hence, we remark that u ∈ Cm−1,λ_{(Ω, R}N

), then, it results, for a. e. t ∈ (−b∗_,0),

u_{(x, t) ∈ H}m+ϑ  B  5 2σ  , RN  ∩Cm−1,λ ! B  5 2σ  , RN " , _{∀ 0 < ϑ < 1, ∀ B(3σ) ⊂⊂ Ω.} Then, from Theorem 7.2.4 with Ω = B 5

2σ

, _{1 − λ < θ < 1, for δ =} 1₂, and for a.e. t _{∈ (−b}∗,0) : u_{(x, t) ∈ H}m,p  B  5 2σ  , RN  , and there exists a constant c = c(θ, λ, σ, m, n) such that

kukm,p,B(5₂σ) ≤ ckuk 1 2 m+θ,B(5 2σ) k u_k12 Cm−1,λ_(B(5 2σ),RN) , where p = 4 + _{n−2(θ+λ−1)}8(θ+λ−1) >4. The choice θ = 1 − λ

2(> 1 − λ) ensures that for a. e. t ∈ (−b∗,0) we have

u_{(x, t) ∈ H}m,p  B ₅ 2σ  , RN  , with p = 4 + 4λ n_{− λ}, ∀B(3σ) ⊂⊂ Ω. (9.3.2) and kukm,p,B(5₂σ) ≤ c(θ, λ, σ, m, n)kuk 1 2 m+1−λ2,B(52σ) k u_k12 Cm−1,λ_(B(5 2σ),RN) , (9.3.3) where p = 4 + 4λ n−λ >4.

Then we have, for a. e. t ∈ (−b∗_{,0), the following inclusion between Sobolev}

spaces u_{(x, t) ∈ H}m,p  B  5 2σ  , RN  ⊂⊂ Hm,4  B  5 2σ  , RN  (9.3.4)

and the following estimate holds Z 0 −akuk 4 m,4,B(σ)dt≤c(ν, K, U, λ, σ, a, b, m, n)   1+ X |α|<m Z 0 −bkf α k0,B(3 σ)dt 1+ϑ 2 + Z 0 −b|u| 2 m,B(3 σ)dt    (9.2.3)

where K = sup_QkD′_{uk and U = kuk}

Cm−1,λ_(Q,˜RN₎.

Theorem 9.2.2. (main result). _{If u ∈ L}2_{(−T, 0, H}m_{(Ω, R}N_{)) ∩ C}m−1,λ_{(Q, R}N_),

0 < λ < 1, is a weak solution of the system (6) and if the assumptions (P.1) – (3.6)

hold, then ∀B(3σ) = B(x0_{,3σ) ⊂⊂ Ω, ∀a, b ∈ (0, T ), a < b it results}

u_{∈ L}2_{(−a, 0, H}m+1(B(σ), RN_{)) ∩ H}1_{(−a, 0, L}2(B(σ), RN)) (9.2.4)

and the following estimate holds

Z 0 −a ( |u|2m+1,B(σ)+     ∂u ∂t     2 0,B(σ) ) dt_≤ ≤ c(ν, K, U, λ, σ, a, b, m, n)   1 + X |α|<m Z 0 −bkf α k0,B(3 σ)dt 2 + Z 0 −b|u| 2 m,B(3 σ)dt    (9.2.5)

where K = sup_QkD′_{uk and U = kuk}

Cm−1,λ_(Q,RN₎.

9.3

Proofs of the main results

Proof. Let us observe that, using Theorem 2.III in [11], for every 0 < ϑ < 1 and

b∗₌ a+b 2 ,we have u_{∈ L}2  −b∗,0, Hm+ϑ  B  5 2σ  , RN  ,

we have kaαk + n X r=1 ∂ aα ∂ xr + N X k=1 X |β|<m ∂ aα ∂ pβ_k ≤ M(K) (1 + kp ′′_{k) ,} N X k=1 X |β|=m ∂ aα ∂ pβ_k ≤ M(K); (3.6) ∃ ν = ν(K) > 0 such that: N X h,k=1 X |α|=|β|=m ∂ aα h(X, p) ∂ pβ_k ξ α h ξ β k ≥ ν(K) X |β|=m ξβ 2 N = νkξk 2_, for every ξ = ( ξα

) ∈ R′′ _{and for every (X, p) ∈ Q × R, with kp}′_{k ≤ K .}

If the coefficients aα_{satisfy condition (3.6) we say that the system (6) is strictly}

elliptic in Ω.

9.2

Main results

We say a function u ∈ L2_{(−T, 0, H}m_{(Ω, R}N

) ∩ Cm−1,λ_{(Q, R}N_{), N positive integer and}

0 < λ < 1, weak solution in Q to the nonlinear parabolic system of order 2m X |α|≤m (−1)|α|Dαaα(X, Du) + ∂ u ∂ t = 0 if Z Q  X |α|≤m (aα (X, Du)| Dα_ϕ ) −  u_|∂ ϕ ∂ t  dX = 0, _{∀ϕ ∈ C0}∞(Q, RN_{). (9.2.1)} Theorem 9.2.1._{. If u ∈ L}2_{(−T, 0, H}m(Ω, RN )) ∩ Cm−1,λ_{(Q, R}N_{), 0 < λ < 1, is}

a weak solution of the system (6) and if the assumptions (P.1) – (3.6) hold, then

∀B(3σ) = B(x0_{,3σ) ⊂⊂ Ω, ∀a, b ∈ (0, T ), a < b, it results}

and p = {pα_}

|α|≤m, pα ∈ RN, the generic point of R. If p ∈ R, we set p = (p′, p′′)

where p′ _{= {p}α }|α|<m ∈ R′ = Q|α|<mRNα, p′′ = {pα}|α|=m ∈ R′′ = Q|α|=mRNα,and kpk2 ₌ X |α|≤m kpα k2 N, kp′k2 = X |α|<m kpα k2 N, kp′′k2 = X |α|=m kpα k2 N. We consider, as usual, Di = ∂ ∂xi , i = 1, . . . , n; Dα = Dα1 1 D α2 2 . . . Dnαn; Du _{= {D}αu_}|α|≤m, D′u _{= {D}αu_}|α|<m, D′′u _{= {D}αu_}|α|=m.

Let us consider the following differential nonlinear variational parabolic system of order 2m :

X

|α|≤m

(−1)|α|Dαaα(X, Du) + ∂ u

∂ t = 0 (9.1.1)

where aα_{(X, p) = a}α_{(X, p}′_{, p}′′_{) are functions of Λ = Q × R in R}N_{, satisfying the}

following conditions:

(P.1) for every α : |α| < m and every p ∈ R, the function X −→ aα_{(X, p), defined in}

Q having values in RN_{, is measurable in X;}

(P.2) for every α : |α| < m and every X ∈ Q, the function p −→ aα_{(X, p), defined in}

R having values in RN_{, is continuous in p;}

(P.3) for every α : |α| < m and every (X, p) ∈ Λ, such that kp′_{k ≤ K, we have}

kaα

(X, p)k ≤ M(K)|fα

(X)| + kp′′k2,

where fα

∈ L2_(Q);

(P.4) for every α : |α| = m, the function aα_{(X, p}′_{, p}′′_{), defined in Q × R having values}

Nonlinear parabolic systems

In this chapter, we investigate differentiability of the solutions of nonlinear parabolic systems of order 2 m in divergence form of the following type

X

|α|≤m

(−1)|α|Dαaα(X, Du) + ∂ u ∂ t = 0.

The results are achieved inspired by the papers [23] and [25]. This chapter can be viewed as a continuation of the study of regularity properties for solutions of elliptic systems started in [15] and continued in [16] and [18], and also as a generalization of the paper [7] where regularity properties of the solutions of nonlinear elliptic systems of order 2m with quadratic growth are reached.

9.1

Problem formulation

Let us set m, N positive integers, α = (α1, . . . , αn) a multi-index and |α| = α1+

. . .+ αn the order of α. We denote by R the Cartesian product

R = Y

|α|≤m

RN_α

where K = sup

Ω

kD′_uk.

Therefore, because we are exactly in the same situation studied in n. 3 Chapt. IV of [4], we get the conclusion.

and, ∀α : |α| = m, Gαs_{(x, Du) = −}∂a α_{(x, Du)} ∂xs − X |β|<m N X k=1 (DsDβuk) ∂aα_{(x, Du)} ∂pβ_k . (8.4.4) Let us also assume in (8.4.2) θ = Dsϕ with ϕ ∈ C₀∞(Ω0, RN), summing from 1 to

n _{respect to s, we gain that the function u ∈ H}m+1(Ω0, RN) ∩ Cm−1,λ(Ω0, RN) is

solution of the following quasilinear system of order 2 (m + 1) Z Ω0 X |α|=|β|=m n X r,s=1 Bαr,βs(x, Du)DsDβu|DrDαϕ
dx= = Z Ω0 X |α|=m n X s=1 (Gαs_{(x, Du) + δ} αs X |β|<m aβ_{(x, Du)|D}βDsϕ) dx , ∀ϕ ∈ C₀∞(Ω0, RN) (8.4.5) where Bαrβs= δrsAαβ. (8.4.6)

We point out that system (8.1.1) is strictly monotone but, because of aα _{∈ C}1_(Ω×

R, RN

), for |α| = m, this condition is equivalent to that of strict ellipticity. Let us prove that the same is also true of system (8.4.5) with the same ellipticity constant ν. _{Indeed, thanks to (8.4.6) and (8.4.3), for every system {η}αs_}

α,s=1,2,...,n of vectors of RN_{, we have} X |α|=|β|=m n X r,s=1 Bαr,βsηβs| ηαr
= n X s=1 X |α|=|β|=m Aαβηβs| ηαs
= = n X s=1 X |α|=|β|=m N X h,k=1 Ahk_αβη_hβsη_kαs= n X s=1 X |α|=|β|=m N X h,k=1 ∂aα h ∂pβ_kη βs h η αs k ≥ ν n X s=1 X |α|=m kηαs k2N .

Moreover from the hypotheses (E.3) and (E.4) it follows kGαs+ δαs X |β|<m aβ_{(x, Du)k ≤ c(K){1 +} X |α|<m |fα| + kD′′u_k2_},

m.

Theorem 8.4.1. _{Let u ∈ H}m_{(Ω, R}N_{) ∩ C}m−1,λ_{(Ω, R}N_{), 0 < λ < 1, a weak solution}

of the system (8.1.1), are true the hypotheses (E.1), (E.2), (E.4), (E.5), (E.3) for

fα_{∈ L}n−2λ2n (Ω), |α| < m, and aα(x, Du) ∈ C1(Ω × R, RN) for |α| = m, . Then, there

exists a closed set Ω0⊂ Ω, such that

Hn−q(Ω0) = 0 for a number q > 2, u ∈ Cm,γ(Ω \ Ω0, RN) for a suitable γ ∈ (0, 1) ,

where H_n−q(Ω0) is the (n − q)-dimensional Hausdorff measure of Ω0.

Proof. of Theorem 4.1. Let us fix a positive number s, s ≤ n, and assume in the

definition of weak solution (9.2.1) ϕ = Dsθ, for θ ∈ C0∞(Ω0, RN), Ω0 ⊂⊂ Ω, we have

Z

Ω0

X

|α|≤m

(Dsaα(x, Du)|Dαθ) dx = 0, ∀ θ ∈ C₀∞(Ω0, RN) . (8.4.1)

we can write the derivatives: Dsaα(x, Du) = ∂aα ∂xs + X |β|<m N X k=1 (DsDβuk) ∂aα ∂pβ_k + X |β|=m N X k=1 (DsDβuk) ∂aα ∂pβ_k. Applying the previous theorem we have that u ∈ Hlocm+1(Ω, RN), thus we are able

to write (8.4.1) as follows Z Ω0 X |α|=|β|=m Aαβ(x, Du)DsDβu|Dαθ
dx= = Z Ω0 {X |α|=m (Gα,s (x, Du)|Dα_θ )− X |α|<m (aα (x, Du)|Dα_D sθ)}dx, ∀θ ∈ C0∞(Ω0, RN) (8.4.2) where ∀α, β : |α| = |β| = m, Aαβ = {Ahkαβ} , Ahkαβ = ∂aα h(x, Du) ∂pβ_k , h, k = 1, . . . , N (8.4.3)

Let us now estimate the last term using the H¨older inequality Z Q(2σ)kτ i,hD′uk 2 kD′′u_k2 dx_≤ Z Q(2σ)kτ i,hD′uk 4 dx 1 2 Z Q(2σ)kD ′′_uk4 dx 1 2 , Then, applying Theorem 7.2.2 (for p = 4, Q(5₂σ) instead of Q(σ) and t = 4₅), for every |h| < h0, it follows Z Q(2σ)kτ i,hD′uk2kD′′uk2 dx≤ h2kD′′uk2_0,4,Q(5 2σ)kD ′′_uk2 0,4,Q(2σ) ≤ h 2 |u|4m,4,Q(3σ) . (8.3.52) From (8.3.51) and (8.3.52), for every i (1 ≤ i ≤ n) and every |h| < h0, we gain the

following estimate Z Q(σ) kτi,hD′′uk2 dx≤ c(ν, K, σ, m, n) h2{1+( X |α|<m kfα k0,Q(3σ))2+|u| 2 m,Q(3σ)+|u| 4 m,4,Q(3σ)}. If h0 ≤ |h| < σ₂,as in (8.3.19), we have that Z Q(σ)kτ i,hD′′uk2dx≤4 Z Q(3σ)kD ′′_uk2 dx_≤4h 2 h2 0 Z Q(3σ)kD ′′_uk2 dx_{≤ c(ν, K, U, λ, σ, m, n)h}2_|u|2_m,Q(3σ)≤ ≤ c(ν, K, U, λ, σ, n) h2{ 1+(X |α|<m kfα k0,Q(3σ))2+|u| 2 m,Q(3σ)+|u| 4 m,4,Q(3σ)}, ∀i = 1, 2, . . . , n .

It is then proved, for every |h| < σ

2 and every i ∈ {1, 2, . . . , n} , that

Z Q(σ)kτ i,hD′′uk2 dx≤ c(ν, K, U, λ, σ, m, n) h2{1+( X |α|<m kfαk0,Q(3σ))2+|u| 2 m,Q(3σ)+|u| 4 m,4,Q(3σ)},

applying Theorem 7.2.1, it follows (8.2.9) and (8.2.10).

8.4

. Partial H¨

older continuity of higher order

deriva-tives

As application of the previous differentiability properties for solutions of system (8.1.1) we have the following result of partial H¨older continuity of derivatives of order

Exploiting Theorem 8.2.3 we can achieve that u ∈ Hlocm,4(Ω, RN), then we can estimate

the last term as follows X |α|<m Z Q(3σ)(|f α | + kD′′u_k2) τi,−hDα ψ2mτi,hu
dx ≤ ≤ X |α|<m ( Z Q(3σ)|h| −2 _τ i,−hDα ψ2mτi,hu
2 dx)12( Z Q(3σ) h2_(|fα_{| + kD}′′u_k2)2dx)12 ≤ ≤ ₂ε|h|−2 X |α|<m Z Q(3σ) τ_i,−hDα ψ2mτi,hu
2 dx + c(ε) h2 X |α|<m Z Q(3σ)(|f α |2+ kD′′u_k4) dx .

Furthermore, from Theorem 7.2.2 (for p = 2, Q(7₂σ) instead of Q(σ) and t = 6₇), for every h ∈ R con |h| < h0 and every ε > 0, we have

ε 2|h| −2Z Q(3σ) τi,−hDα ψ2mτi,hu
2 dx_≤ ε 2 Z Q(7₂σ) D′′ _ψ2m_τ i,hu
2 dx _≤ ≤ ε Z Q(2σ) ψ2m _{k τ}i,hD′′uk 2 dx + c(σ, ε) Z Q(2σ)k τ i,hD′uk 2 dx_≤ ≤ ε Z Q(2σ) ψ2m _{k τ}i,hD′′uk2 dx + c(σ, ε) h2 Z Q(3σ)kD ′′_uk2 dx. the last inequality follows, as before, applying Theorem 7.2.2 (for p = 2, Q(3σ) instead of Q(σ) and t = 2₃). Let us now choose ε = ν

4 c(K), it ensure X |α|<m Z Q(3σ)  |fα | + kD′′_uk2 _τ i,−hDα ψ2mτi,hu
dx ≤ ≤_4c(K)ν Z Q(2σ) ψ2m_kτi,hD′′uk2dx+ c(ν, K, σ, m)h2   ( X |α|<m kfαk0,Q(3σ))2+ |u| 2 m,Q(3σ)+ |u| 4 m,4,Q(3σ)   . Taking into consideration the last inequality and the properties of the function ψ,

from (8.3.50) we deduce Z Q(σ) kτi,hD′′uk2dx≤c(ν, K, σ, m, n)h2{1+( X |α|<m kfα k0,Q(3σ))2+|u| 2 m,Q(3σ)+|u| 4 m,4,Q(3σ)}+ + c(ν, K, σ, n) Z Q(2σ)kτ i,hD′uk2kD′′uk2 dx . (8.3.51)

we assure that ϑ∗ _{∈ (0, 1) exists and is such that} 2(1+ϑ∗_)n n−2ϑ∗_λ > 4. Let us set p∗ in  4,2(1+ϑ_n−2ϑ∗∗)n_λ  ,from (9.3.2), we have u_{∈ H}m,p∗(Q(σ), RN ) , ∀Q(σ) ⊂⊂ Ω from which, because of p∗_{>4, it follows}

u_{∈ H}m,4(Q(σ), RN) . (8.3.49)

We end the conclusion remarking that (8.3.49) is true for every Q(σ) ⊂⊂ Ω.

8.3.2

Proof of local differentiability result in

H

space

Proof. of Theorem 3.4. _{Let us consider ψ(x) ∈ C}∞

0 (Rn) the cut-off function

above defined in (8.3.1), Q(4σ) ⊂⊂ Ω a generic cube, i ≤ n a positive integer and h a real number such that |h| < σ

2. Carrying on as in the proof of Theorem 8.2.1, we

obtain ν 2 Z Q(2σ) ψ2m_kτi,hD′′uk2dx≤c(ν, K, σ, m, n) h2 Z Q(3σ)  1 + kD′′u_k2dx+ + c(ν, K, σ, m, n) Z Q(2σ) kτi,hD′uk2kD′′uk2 dx+ +c(K)X |α|<m Z Q(3σ)(|f α |+kD′′_uk2_{) τ}

i,−hDα ψ2mτi,hu
dx.

and we reach the inequality |D′′u_|2_ϑ i,Q(4−iρ)≤ c(ν, K, U, ϑ, λ, ρ, m, n)(1 + ( X |α|<m kfα k0, 2n n−2λ,Q(4ρ)) 1+ϑi−1_{+ |u|}2 m,Q(4ρ)). (8.3.47) Let us fix arbitrarily x0 ∈ Ω, Q(σ) = Q(x0, σ) ⊂⊂ Q(σ0) = Q(x0, σ0) ⊂⊂ Ω and

assume ρ = σ0−σ

8 .The set of cubes

F = Q(y0,4−i−1ρ), y0 _{∈ Q(σ)} is an open cover of Q(σ), let us then extract the finite cover

Q(y(1)_,4−i−1_{ρ) , Q(y}(2)_,4−i−1_{ρ) , . . . , Q(y}(t)_,4−i−1_ρ).

After that, set Ωk = Q(y(k),4−i−1ρ) ∩ Q(σ), k = 1, 2, . . . , t, t

[

k=1

Ωk = Q(σ), Q(y(k),4ρ) ⊂⊂ Q(σ0) ⊂⊂ Ω , ∀k = 1, 2, . . . , t ,

from (8.3.47) and Theorem 7.1.6 (if Ω = Q(σ), ϑ = ϑi and σ = ₄3ρi+1), we have

|D′′u_|2_ϑ i,Q(σ)≤ c(ν, K, U, ϑ, λ, σ, σ0, m, n){1+( X |α|<m kfα k0,_n−2λ2n ,Q(σ0)) 1+ϑi−1 _{+ |u|}2 m,Q(σ0)}

we gain (8.2.6) and (9.3.1) bearing in mind that ϑi−1< ϑ≤ ϑi.

To prove (8.2.8) we remark that u ∈ Cm−1,λ_{(Ω, R}N_{), then}

u_{∈ H}m+ϑ(Q(σ), RN

) ∩ Cm−1,λ(Q(σ), RN_),

∀ 0 < ϑ < 1, ∀ Q(σ) ⊂⊂ Ω. In addition, Theorem 7.1.7 ensures that

u_{∈ H}m,p(Q(σ), RN), _{∀ 1 ≤ p <} 2(1 + ϑ)n

n_{− 2ϑλ} , ∀ 0 < ϑ < 1, ∀Q(σ) ⊂⊂ Ω. (8.3.48) and observing that

lim ϑ→1− 2(1 + ϑ)n n_{− 2ϑλ} = 4n n_{− 2λ} > 4 ,

i) ϑs = 1 − (1 − ϑ0)s+1;

ii) 0 < ϑs< ϑs+1<1 ;

iii) ϑs+1− ϑs= ϑ0(1 − ϑ0)s+1;

iv) ϑs+1< ϑs+ λ₂(1 − ϑs) ;

v) qs= 2(1+ϑ_n−2ϑs_s)n_λ < _n−2λ4n .

It ensure that fα _{∈ L}qs_{2 (Ω), for every s = 0, 1, 2, . . . and every α such that |α| < m.}

Due to lim

s→+∞ϑs = 1, fixing arbitrarily ϑ ∈ (ϑ0,1) exists a positive integer i = i(ϑ, λ)

such that ϑi−1 < ϑ≤ ϑi <1.

Additionally, from Theorem 8.2.1 we deduce

u_{∈ H}m+ϑ0_{(Q(4ρ), R}N_{) ∩ C}m−1,λ_{(Q(4ρ), R}N_{), ∀Q(4ρ) ⊂⊂ Ω} and |D′′u_|2_ϑ 0,Q(ρ)≤ c(ν, K, U, λ, ρ, m, n)(1 + X |α|<m kfα k0,_n−2λ2n ,Q(4ρ)+ |u| 2 m,Q(4ρ)). (8.3.45)

Exploiting Theorem 8.2.2 for ϑ = ϑ0 q= q0, ϑ′ = ϑ1 and Ω = Q(4ρ), as well as iv)

and v) for s = 0, we have

u_{∈ H}m+ϑ1_{(Q(ρ), R}N_{) ∩ C}m−1,λ_{(Q(ρ), R}N₎ and |D′′u_|2_ϑ1,Q(4−1_ρ)≤c(ν, K, U, λ, ρ, m, n)(1 +( X |α|<m kfα k0,q0₂,Q(ρ))1+ϑ0+|u| 2 m,Q(ρ)+|D′′u| 2 ϑ0,Q(ρ)) ≤ ≤c(ν, K, U, λ, ρ, m, n) ( 1 + (X |α|<m kfα k0,_n−2λ2n ,Q(4ρ)) 1+ϑ0_{+ |u|}2 m,Q(4ρ)).

making use i times of Theorem 8.2.2 we establish, ∀Q(4ρ) ⊂⊂ Ω , so that

then, for every h : 0 < |h| < 2σ is integrable in [−2σ, 2σ] the second member of n X i=1 1 |h|1+2ϑ′ Z Q(σ)kτ i,hD′′uk2 dx ≤ c(ν, K, U, ϑ, λ, m, n, σ)· · {1+(X |α|<m kfαk0,q₂,Q(3σ))1+ϑ+|u| 2 m,Q(3σ)+|D′′u| 2 ϑ,Q(3σ)} 1 |h|1+2ϑ′−2ϑ−λ(1−ϑ) (8.3.43) and thus also the first one is integrable.

It is then proved that

n X i=1 Z 2σ −2σ dh |h|1+2ϑ′ Z Q(σ)kτ i,hD′′uk2 dx ≤ c(ν, K, U, ϑ, ϑ′, λ, m, n, σ)· · {1+(X |α|<m kfα k0,q₂,Q(3σ)) 1+ϑ + |u|2m,Q(3σ)+|D′′u| 2 ϑ,Q(3σ)}, ∀ 0 < ϑ′ < ϑ+ λ 2(1 − ϑ). (8.3.44) Because of u ∈ Hm_{(Ω, R}N_{) from Theorem 7.1.3, we have}

D′′u_{∈ H}ϑ′_{(Q(σ), R”)} and |D′′u_|2_ϑ′_,Q(σ)≤c(n) n X i=1 Z 2σ −2σ dh |h|1+2ϑ′ Z Q(σ)kτ i,hD′′uk2 dx ≤ ≤ c(ν, K, U, ϑ, ϑ′, λ, m, n,σ_){1+(X |α|<m kfα k0,q₂,Q(3σ)) 1+ϑ +|u|2m,Q(3σ)+|D′′u| 2 ϑ,Q(3σ)},

we achieve our goal.

Proof. of Theorem 3.3. Let us fix ϑ0 = λ₄ and make a point of the geometric

series

1 + (1 − ϑ0) + (1 − ϑ0)2+ · · · + (1 − ϑ0)r+ · · · .

For s = 0, 1 , . . . , let us set ϑs = ϑ0 s

P

r=0(1 − ϑ0

)r_{. We achieve, for every s = 0, 1, . . . ,}

Therefore for every ε > 0 and every 2 < p < min(4, q) we reach ν 2 Z Q(2σ) ψ2m_kτi,hD′′uk2 dx ≤ c(ν, K, σ, m, n)h2 Z Q(5 2σ)  1 + kD′′u_k2 dx+ + c(ν, K, U, ϑ, λ, m, n, σ) | h |2ϑ + 2λ(1−ϑ) n| D′′u_|2_ϑ,Q(5 2σ) + 1 o + + 2ε Z Q(2σ) ψ2m_{k τ}i,hD′′uk2 dx + c(σ, ε) h2 Z Q(5 2σ) k D′′u_k2 dx+ + c(K, U, p, ε) |h|p−2+λ(2−p2) Z Q(5₂σ) ((X |α|<m |fα|)p2 + k D′′ukp) dx.

Let us now set in the last inequality ε = ν

8 and p = 2(1 + ϑ) ∈ 2, min(4, q)

.We have, for every h : |h| < h0(< 1), that

ν 4 Z Q(2σ) ψ2m_{k τ}i,hD′′uk2 dx ≤ c(ν, K, σ, m, n)h2 Z Q(5₂σ)  1 + kD′′u_k2 dx+ + c(ν, K, U, ϑ, λ, m, n, σ) | h |2ϑ+2λ(1−ϑ)n| D′′u _|2_ϑ,Q(5 2σ) + 1 o + + c(ν, K, U, ϑ) |h|2ϑ+λ(1−ϑ) Z Q(5₂σ) ((X |α|<m |fα|)1+ϑ+kD′′u_k2(1+ϑ)_{) dx ≤} ≤ c(ν, K, U, ϑ, λ, m, n, σ) | h |2ϑ+λ(1−ϑ)· ·{1+|u|2m,Q(5 2σ)+|D ′′_u|2 ϑ,Q(5 2σ)+|u| 2(1+ϑ) m,2(1+ϑ),Q(5 2σ) +(X |α|<m kfαk0,q₂,Q(5 2σ)) 1+ϑ }. From (8.3.34), for |h| < h0,we gain

n X i=1 Z Q(σ)kτ i,hD′′uk2 dx ≤ ≤ c(ν, K, U, ϑ, λ, m, n, σ) |h|2ϑ+λ(1−ϑ){1+(X |α|<m kfα k0,q 2,Q(3σ)) 1+ϑ +|u|2m,Q(3σ)+|D′′u| 2 ϑ,Q(3σ)}. (8.3.42) The procedure if h0 ≤ |h| < 2σ is similar to the one used in the proof of Theorem

8.2.1 .

Combining both results we obtain that (8.3.42) is true for |h| < 2σ. Let us now choose 0 < ϑ′ _{< ϑ+}λ

inequality, for 2 < p < min(4, q), we carry out |D|≤ X |α|<m Z Q(5₂σ)[M (K)(|f α |+kD′′u_k2) τi,−hDα ψ2mτi,hu
4 p−1_][ τ i,−hDα ψ2mτi,hu
2− 4 p ]dx≤ ≤ c(K, p) X |α|<m !Z Q(5₂σ)  |fα |p2 + kD′′ukp τ_i,−hDα(ψ2mτ_i,hu) 4−p p p 2 _dx "2 p · · !Z Q(5 2σ) τi,−hDα ψ2mτi,hu
2 dx "p−2 p = = c(K, p) X |α|<m !Z Q(5₂σ)|h| p−2 |fα |p2 + kD′′ukp τ_i,−hDα ψ2mτ_i,hu
4−p 2 _dx "2 p · · !Z Q(5₂σ)|h| −2 _τ i,−hDα ψ2mτi,hu
2 dx "p−2 p . (8.3.40)

The use of the suitable consequence of Young inequality ab ≤ εa1+s_+ε−1

sb1+1s,denoting with s= 2 p_{− 2} , a= !Z Q(5₂σ)|h| −2 _τ i,−hDα ψ2mτi,hu
2 dx "p−2 p , b= c(K, p) !Z Q(5 2σ) |h|p−2|fα |p2 + kD′′ukp τ_i,−hDα ψ2mτ_i,hu
4−p 2 _dx "2 p

and the hypothesis u ∈ Cm−1,λ_{(Ω, R}N_{) allows us to have}

|D| ≤ ε |h|−2 X |α|<m Z Q(5₂σ) τi,−hDα ψ2mτi,hu
2 dx _{+ c(K, U, p, ε) |h|}p−2+λ(2−p2)_· · X |α|<m Z Q(5₂σ)  |fα |p2 + kD′′ukp  dx , _{∀ε > 0, ∀ 2 < p < min(4, q). (8.3.41)} Thus we also need Theorem 7.2.2 to obtain

ε_|h|−2 X |α|<m Z Q(5₂σ) τi,−hDα ψ2mτi,hu
2 dx _{≤ ε} Z Q(3σ) D′′ _ψ2m_τ i,hu
2 dx _≤ ≤ 2ε Z Q(2σ) ψ4m_kτi,hD′′uk2 dx + c(σ, ε) Z Q(2σ) ψ2m_kτi,hD′uk2 dx ≤ ≤ 2ε Z Q(2σ) ψ2m _kτi,hD′′uk2 dx + c(σ, ε) h2 Z Q(5₂σ) kD ′′_uk2 dx .

the last inequality is obtained considering that kτi,hD′uk 2ϑp p−2 _{∈ L}p−22ϑ (Q(2σ)) , kτ_i,hD′uk 2(1−ϑ)p p−2 _∈ Lp−2(1+ϑ)p−2 (Q(2σ)) , 2ϑ p−2+ p−2(1+ϑ)

p−2 = 1. From Theorem 7.2.2 for t = 45 and Q(52σ) in place of Q(σ),

∀h ∈ R, |h| < σ

2,we attain the inequality

Z Q(2σ)kτ i,hD′ukp dx 2ϑ p ≤ |h|2ϑ !Z Q(5 2σ) kD′′u_kp dx "2ϑ p , _{∀ p, q : 2(1+ϑ) < p < q .} (8.3.36) Using the hypothesis u ∈ Cm−1,λ_{(Ω, R}N_{) we deduce, for every 2(1 + ϑ) < p < q, that}

Z Q(2σ)kτ i,hD′uk 2(1−ϑ)p p−2(1+ϑ) _dx p−2(1+ϑ) p ≤ U2(1−ϑ)|h|2λ(1−ϑ)[mis Q(2σ)]p−2(1+ϑ)p _. (8.3.37) From (8.3.35)–(8.3.37) we reach Z Q(2σ)

ψ2m_kτi,hD′uk2kD′′uk2dx≤ c(U, ϑ, n, p, σ) |h|2ϑ+2λ(1−ϑ)|u|2(1+ϑ)_m,p,Q(5 2σ)

, _{∀ 2(1+ϑ)< p< q,} that, for p = 1 + ϑ + q₂ and combined with (8.3.34) for ρ = 5

2σ and for p = 1 + ϑ + q 2, gives Z Q(2σ) ψ2m_kτi,hD′uk 2 kD′′u_k2 dx_≤ ≤ c(K, U, ϑ, λ, n, σ) |h|2ϑ+2λ(1−ϑ)n|D′′u_|2_ϑ,Q(5 2σ)+ 1 o , (8.3.38) ν 2 Z Q(2σ) ψ2m_kτi,hD′′uk2 dx≤ c(ν, K, σ, m, n)h2 Z Q(5₂σ)  1 + kD′′u_k2 dx+ + c(ν, K, U, ϑ, λ, n, m, σ) |h|2ϑ+2λ(1−ϑ)n|D′′u_|2_ϑ,Q(5 2σ)+ 1 o + | D | . (8.3.39) Let us focus our attention on the term D. Combining (E.3), (8.3.31) and H¨older

and Z Q(ρ) D′′u_{− (D}′′u)_Q(ρ) p dx_{≤ c(ϑ, λ, m, n, p) (mis Q(ρ))}1−pq [D′u] pϑ 1+ϑ λ,Ω |D ′′_u|1+ϑp ϑ,Q(ρ) . (8.3.32) Thus we deduce that

| u |2(1+ϑ)m,p,Q(ρ) = Z Q(ρ)k D ′′_ukp dx 2(1+ϑ) p ≤ ≤ 2(p−1)2(1+ϑ)p Z Q(ρ) D′′u_−(D′′u)_Q(ρ) pdx+ Z Q(ρ) (D′′u)_Q(ρ) pdx 2(1+ϑ) p ≤ ≤ c(U, ϑ, λ, m, n, p)  |D′′_u|2 ϑ,Q(ρ)+ (D′′u)_Q(ρ) 2(1+ϑ)  ≤ ≤ c(U, ϑ, λ, m, n, p) n| D′′u_|2_{ϑ,Q(ρ) + | u |}2(1+ϑ)_m,Q(ρ) o. (8.3.33) Using interpolation inequality contained in Theorem 7.1.4 we derive

|u|2(1+ϑ)m,Q(ρ) ≤ c(ϑ, n)

n

|D′′u_|2_{ϑ,Q(ρ)kukm−1,Q(ρ)}2ϑ + ρ−2(1+ϑ)_kuk2(1+ϑ)_m−1,Q(ρ)o_≤ ≤ c(K, ϑ, m, n, ρ) n|D′′u_|2_ϑ,Q(ρ) + 1o

then, ∀ Q(ρ) ⊂⊂ Ω and ∀2 ≤ p < q = 2(1+ϑ)nn−2ϑλ , we have

|u|2(1+ϑ)m,p,Q(ρ)≤ c(K, U, ϑ, λ, m, n, p, ρ) n |D′′_u|2 ϑ,Q(ρ)+ 1 o . (8.3.34) By (8.3.31), the hypothesis u ∈ Cm−1,λ_{(Ω, R}N_{) and H¨older inequality, for every}

2(1 + ϑ) < p < q, it follows Z Q(2σ) ψ2m_kτi,hD′uk2kD′′uk2dx≤ Z Q(2σ)kD ′′_ukp dx 2 p Z Q(2σ)kτ i,hD′uk 2p p−2 _dx p−2 p = (8.3.35) = Z Q(2σ)k D ′′_{u k}p dx 2 pZ Q(2σ)k τ i,hD′uk 2ϑp p−2 _{k τ} i,hD′uk 2(1−ϑ)p p−2 _dx p−2 p ≤ ≤ Z Q(2σ)kD ′′_ukp dx 2 pZ Q(2σ)kτ i,hD′uk p dx 2ϑ pZ Q(2σ)kτ i,hD′uk 2(1−ϑ)p p−2(1+ϑ)_dx p−2(1+ϑ) p

Recalling that u ∈ Cm−1,λ_{(Ω, R}N_{), we have}

kτi,hD′u(x)k ≤ U |h|λ, ∀x ∈ Q(2σ). (8.3.27)

Moreover applying Theorem 7.2.2, for p = 2, t = 4

5 and Q(52σ) in replacement of

Q_{(σ), for every h ∈ R such that |h| <} σ₂, we achieve Z Q(2σ)kτ i,hD′uk2 dx≤ h2 Z Q(5₂σ)kD ′′_uk2 dx, i= 1, 2, . . . , n. (8.3.28) From (8.3.24) – (8.3.28) it follows ν Z Q(2σ) ψ2m_kτi,hD′′uk2 dx≤ {3ε + c(K, U, σ, m, n)(|h| + h2+ |h|λ+ |h|2λ)}· · Z Q(2σ) ψ2m_kτi,hD′′uk2 dx + c(K, σ, m, n, ε) h2 Z Q(5₂σ)  1 + kD′′u_k2 dx+ + c(K, σ, m, n, ε) Z Q(2σ) ψ2m_{k τ}i,hD′uk 2 k D′′u_k2 dx_{+ | D | , ∀ε > 0.} (8.3.29) As in the proof of Theorem 8.2.1 there exists h0(ν, K, U, λ, σ, m, n), 0 < h0 <

min{1, σ 2}, such that c(K, U, σ, m, n)_{|h| + h}2_{+ |h|}λ_{+ |h|}2λ< ν 4, |h| < h0 then, for ε = ν 12 we have ν 2 Z Q(2σ) ψ2m_kτi,hD′′uk2 dx≤ c(ν, K, σ, m, n)h2 Z Q(5₂σ)  1 + kD′′_uk2 dx+ + c(ν, K, σ, m, n) Z Q(2σ) ψ2m_kτi,hD′uk2kD′′uk2dx+ |D| . (8.3.30)

Let us now estimate the last two terms in (8.3.30).

Applying Theorem 7.1.7 we have, ∀ Q(ρ) = Q(x0_{, ρ) ⊂⊂ Ω and 2 ≤ p < q =} 2(1+ϑ)n n−2ϑλ ,

that

and |D′′u_|2_{ϑ,Q(σ) ≤ c(n)} n X i=1 Z 2σ −2σ dh |h|1+2ϑ Z Q(σ)kτ i,hD′′uk2 dx, ∀ 0 < ϑ < λ 2. (8.3.23) From (8.3.22), (8.3.23) and (8.3.21) we reach the conclusion.

Proof. of Theorem 3.2. Let us fix x0 ∈ Ω and the cube Q(4σ) = Q(x0,4σ) ⊂⊂ Ω.

Let us also consider a positive integer i ≤ n, and a real number h such that |h| < σ 2.

As in the proof of Theorem 8.2.1, for every ε > 0, we have

ν Z Q(2σ) ψ2m_kτi,hD′′uk2 dx≤ ε Z Q(2σ) ψ2m_kτi,hD′′uk2 dx+ + c(K, σ, m, n, ε) Z Q(2σ) kτi,hD′uk2 dx + B + C + D , (8.3.24)

where ψ is the above defined cut-off function (see (8.3.1)) and the terms B, C and D are considered in (8.3.7) – (8.3.9).

The terms |B| and |C| can be estimated , ∀ε > 0, as follows |B| ≤ε+ c(K, σ, m, n) _{|h| + h}2
Z Q(2σ) ψ2m_kτi,hD′′uk2 dx+ + c(K, σ, m, n, ε)h2 Z Q(2σ) ψ2m_{1 + kD}′′u_k2 dx+ Z Q(2σ)kτ i,hD′uk2 dx, (8.3.25) |C| ≤ Z Q(2σ) n ε+ c(K, σ, m, n)_kτi,hD′uk + kτi,hD′uk2 o ψ2m_kτi,hD′′uk2 dx+ + c(K, σ, m, n, ε) Z Q(2σ)kτ i,hD′uk2 dx+c(K, σ, m, n, ε) Z Q(2σ) ψ2m_kτi,hD′uk2kD′′uk2 dx, (8.3.26) similarly to the proof of Theorem 8.2.1.

Otherwise if h0 ≤ |h| < 2σ, we get for i = 1, 2, . . . , n, Z Q(σ)kτ i,hD′′uk2dx≤ 2 Z Q(σ) D′′_{u x+ he}i
2_{dx+ 2} Z Q(σ)kD ′′_u(x)k2 dx_≤ ≤ 2 Z Q(3σ) k D ′′_{u(x) k}2 dx+ 2 Z Q(σ) k D ′′_{u(x) k}2 dx_≤ ≤ 4 Z Q(3σ) k D ′′_uk2 dx_{≤ 4}|h| λ hλ 0 Z Q(3σ)k D ′′_uk2 dx_≤ ≤c(ν, K, U, λ, σ, m, n)|h|λ{1+ X |α|<m kfα k0,1,Q(3σ)+|u| 2 m,Q(3σ)} (8.3.19)

From (8.3.18) and (8.3.19), for every 0 < |h| < 2σ, it follows that

n X i=1 1 |h|1+2ϑ Z Q(σ)kτ i,hD′′uk2 dx≤ ≤ c(ν, K, U, λ, σ, m, n)   1 + X |α|<m kfαk0,1,Q(3σ)+ |u| 2 m,Q(3σ)    1 |h|1+2ϑ−λ. (8.3.20) The hypothesis 0 < ϑ < λ

2 assures that 1 + 2ϑ − λ < 1, then the function of the

variable h that appears in the second member of (8.3.20) is integrable in [−2σ, 2σ], it implies the integrability in [−2σ, 2σ] of the left term of inequality (8.3.20) and it follows n X i=1 Z 2σ −2σ dh |h|1+2ϑ Z Q(σ)kτ i,hD′′uk2 dx≤ ≤ c(ν, K, U, ϑ, λ, σ, m, n)   1 + X |α|<m kfα k0,1,Q(3σ)+ |u| 2 m,Q(3σ)    . (8.3.21)

Finally, recalling that u ∈ Hm_{(Ω, R}N_{), from (8.3.21) it follows that D}′′_{u satisfy the}

hypotheses of Theorem 7.1.3, we can conclude that

|D| ≤ X |α|<m Z Q(3σ)ka α_{(x, D}′_{u, D}′′_{u)k τ} i,−hDα ψ2mτi,hu
dx ≤ ≤ c(K)X |α|<m Z Q(3σ)  |fα

| + kD′′u_k2 τ_i,−hDα ψ2mτi,hu
dx.

(8.3.14) On the other hand, using the hypothesis that u ∈ Cm−1,λ_{(Ω, R}N_{), we easily obtain}

τi,−hDα ψ2mτi,hu

(x) ≤ 2 U |h|λ, _{∀x ∈ Q(3σ).} (8.3.15) From (8.3.14) and (8.3.15) we have

|D| ≤ c(K, U, m) |h|λ  X |α|<m Z Q(3σ)|f α | + kD′′u_k2 dx   . (8.3.16) From (8.3.5), (9.3.15), (9.3.16), (9.3.19), choose ε = ν 12 we deduce that ν Z Q(2σ) ψ2m_kτi,hD′′uk2 dx ≤ nν 4 + c(ν, K, U, m, n)  |h| + h2+ |h|λ+ |h|2λo· · Z Q(2σ) ψ2m_kτi,hD′′uk 2 dx+c(ν, K, U, σ, m, n)(h2_+|h|λ_+|h|2λ) Z Q(3σ) (1+X |α|<m |fα |+kD′′u_k2)dx. (8.3.17) Because of the continuity of the function h −→ c(K, U, σ, m, n)|h| + h2_{+ |h|}λ

+ |h|2λ in the origin, ∃ h0(ν, K, U, λ, σ, n), 0 < h0 < min{1, σ}, such that for every |h| < h0,

we have

c(K, U, σ, m, n)_{|h| + h}2_{+ |h|}λ_{+ |h|}2λ< ν 4. Let us consider, at first, that |h| < h0 <1.

Recalling that 0 < λ < 1 we have h2 _{+ |h|}λ

+ |h|2λ ≤ 3 |h|λ and taking into consideration that ψ(x) = 1 in Q(σ), from (8.3.17), it follows, for i = 1, 2, . . . , n,

ν 2 Z Q(σ)kτ i,hD′′uk2 dx≤ c(ν, K, U, σ, m, n) |h|λ{1 + X |α|<m kfαk0,1,Q(3σ)+ |u| 2 m,Q(3σ)}. (8.3.18)

Similarly, using (E.4), for the term C we have |C| ≤ Z Q(2σ) X |α|=m kaα_{(x, D}′_{u(x) + τ}

i,hD′u(x), D′′u(x) + τi,hD′′u(x)) −

− aα_{(x, D}′_{u(x), D}′′_{u(x) + τ} i,hD′′u(x))k Dα _ψ2m_τ i,hu
dx ≤ ≤c(K, n) Z Q(2σ)kτ

i,hD′uk (1+kD′′uk+kτi,hD′′uk) c(σ, m)ψ2m−1kτi,hD′uk+ψ2mkτi,hD′′uk

dx= = Z Q(2σ) [ψm_kτi,hD′′uk] [c(K, n)ψmkτi,hD′uk (1 + kD′′uk)] dx + + c( K, n ) Z Q(2σ) ψ2m _{k τ}i,hD′uk k τi,h D′′uk2 dx+ + Z Q(2σ)[c(σ, m)kτ i,hD′uk]  c(K, n)ψ2m−1_kτi,hD′uk(1+kD′′uk+kτi,hD′′uk)  dx. Because of u ∈ Cm−1,λ_{(Ω, R}N_{), for every ε > 0, it follows}

|C| ≤ε Z Q(2σ) ψ2m_kτi,hD′′uk2dx+c(K, n, ε) Z Q(2σ) ψ2m_kτi,hD′uk2  1+kD′′_uk2_dx+ + c(K, U, n) |h|λ Z Q(2σ) ψ2m_kτi,hD′′uk2dx+ c(σ, m) Z Q(2σ)kτ i,hD′uk2dx+ + c(K, n) Z Q(2σ) ψ2m_{k τ}i,hD′uk2  1 + k D′′u_k2_{+ k τ}i,hD′′uk2  dx _≤ ≤nε+ c (K, U, n)_{| h |}λ_{+ | h |}2λo Z Q(2σ) ψ2m_{k τ}i,h D′′uk2 dx + + c(K, σ, m, n, ε) Z Q(2σ)kτ i,hD′uk 2 dx_{+ c(K, U, n, ε)|h|}2λ Z Q(2σ)kD ′′_uk2 dx. Using again (8.3.11), we get

|C| ≤nε+ c(K, U, n)_|h|λ_{+ |h|}2λo Z Q(2σ) ψ2m_kτi,hD′′uk2 dx+ + c(K, U, σ, m, n, ε)h2_{+ |h|}2λ Z Q(3σ)kD ′′_uk2 dx , _{∀ε > 0.} (8.3.13) Finally, let us estimate the terms D. For the hypothesis (E.3), we have

From (8.3.10) and (8.3.11), we have |A| ≤ ε Z Q(2σ) ψ2m_kτi,hD′′uk2 dx+ c(K, σ, m, n, ε) h2 Z Q(3σ)kD ′′_uk2 dx , _{∀ε > 0 .} (8.3.12) From (E.4) we can majorize the term B as follows

|B| ≤ Z Q(2σ) X |α|=m aα_{(x + he}i_{, D}′_{u(x) + τ}

i,hD′u(x), D′′u(x) + τi,hD′′u(x)) −

− aα_{(x, D}′_{u(x) + τ}

i,hD′u(x), D′′u(x) + τi,hD′′u(x))k

Dα _ψ2m_τ i,hu
dx ≤ ≤c(K, n) |h| Z Q(2σ)(1+kD ′′_uk+kτ i,hD′′uk)  2mk σ ψ 2m−1_kτ i,hD′uk+ψ2mkτi,hD′′uk  dx_≤ ≤c(K, m, n) |h| Z Q(2σ) ψ2m_kτi,hD′′uk2 dx+ Z Q(2σ)  c_{(K, m, n) |h| ψ}m_{(1 + kD}′′u_k)  ψm_kτi,hD′′uk  dx+ Z Q(2σ)[c(σ, m) kτ i,hD′uk] [c(K, n) |h| ψm(1+kD′′uk+kτi,hD′′uk)] dx.

Then, for every ε > 0, we have |B| ≤ c(K, m, n) |h| Z Q(2σ) ψ2m_kτi,hD′′uk2 dx+ ε Z Q(2σ) ψ2m_kτi,hD′′uk2 dx+ + c(K, m, n, ε)h2 Z Q(2σ) ψ2m_1+kD′′u_k2dx+c(σ, m) Z Q(2σ)kτ i,hD′uk2dx+ + c(K, m, n) h2 Z Q(2σ) ψ2m_{1 + kD}′′u_k2_{+ kτ}i,hD′′uk 2 dx = = ε+ c(K, m, n) h2 _{+ |h|}
Z Q(2σ) ψ2m _kτi,hD′′uk2 dx + +c(K, m, n, ε)h2 Z Q(2σ) ψ2m_(1+kD′′u_k2)dx+c(σ, m) Z Q(2σ)kτ i,hD′uk2dx.

Using (8.3.11), we can estimate |B| as follows |B| ≤ε+ c(K, m, n) h2_{+ |h|}
Z Q(2σ) ψ2m_kτi,hD′′uk2 dx+ + c(K, σ, m, n, ε) h2 Z Q(3σ)  1 + kD′′_uk2 dx, _{∀ε > 0.}

B _{= −} Z Ω X |α|=m aα(x + hei_{, D}′_{u(x) + τ}

i,hD′u(x), D′′u(x) + τi,hD′′u(x)) −

− aα_{(x, D}′_{u(x) + τ}

i,hD′u(x), D′′u(x) + τi,hD′′u(x)) | Dα ψ2mτi,hu

dx, (8.3.7) C _{= −} Z Ω X |α|=m (aα_{(x, D}′_{u(x) + τ}

i,hD′u(x), D′′u(x) + τi,hD′′u(x)) −

− aα_{(x, D}′_{u(x), D}′′_{u(x) + τ} i,hD′′u(x)) | Dα ψ2mτi,hu

dx, (8.3.8) D_{= −}X |α|<m Z Ω aα(x, D′_{u(x), D}′′_{u) | τ} i,−hDα ψ2mτi,h u

dx. (8.3.9)

Let us estimate the terms A, B, C and D.

Applying hypothesis (E.5) and the properties of the function ψ, we have |A| ≤ 2m Z Q(2σ) ψ2m−1 X |α|=m kaα_{(x, D}′_{u(x), D}′′_{u(x) + τ} i,hD′′u(x)) − − aα_{(x, D}′_{u(x), D}′′_{u(x)) k ( D}α_{ψ) τ} i,hu dx ≤ ≤ c( K, m ) Z Q(2σ) ψ2m−1X |α|=m |Dα_ψ | kτi,hD′′uk kτi,hD′uk dx ≤ ≤ c (K, σ, m, n) Z Q(2σ) ψ2m−1_kτi,hD′′ukkτi,hD′uk dx ≤ ≤ c (K, σ, m, n) Z Q(2σ) ψm _kτi,hD′′uk kτi,hD′uk dx.

Then, for every ε > 0, we have |A| ≤ ε Z Q(2σ) ψ2m_kτi,hD′′uk2 dx+ c(K, σ, m, n, ε) Z Q(2σ)kτ i,hD′uk2 dx , (8.3.10)

On the other hand, using Theorem 7.2.2 for p = 2, t = 2₃ and Q(3σ) instead of Q_{(σ), for every h ∈ R such that |h| < 1 −}2₃
3σ = σ, we have

Z Q(2σ)kτ i,huk2_N dx≤ h2 Z Q(3σ)kD iuk2_N dx , i= 1, 2, . . . , n . (8.3.11)

formula (9.3.8) becomes: Z Ω ψ2m X |α|=m (aα_{(x, D}′_{u(x), D}′′_{u(x) + τ}

i,hD′′u(x)) − aα(x, D′u(x), D′′u(x))|τi,hDαu) dx =

(8.3.4) = −2m Z Ω ψ2m−1X |α|=m

(aα(x, D′u(x), D′′u(x) + τi,hD′′u(x))−aα(x, D′u(x), D′′u(x))| (Dαψ) τi,hu)dx−

− Z

Ω

X

|α|=m

(aα(x, D′u(x)+τi,hD′u(x), D′′u(x)+τi,hD′′u(x))−aα(x, D′u(x), D′′u(x)+τi,hD′′u(x))|

| Dα ψ2mτi,hu

dx₋ Z Ω X |α|=m

aα(x + hei, D′u(x) + τi,h D′u(x), D′′u(x) + τi,h D′′u(x)) −

− aα_{( x, D}′_{u(x) + τ}

i,hD′u(x), D′′u(x) + τi,hD′′u(x) ) | Dα ψ2m τi,hu

dx ₋ − X |α|<m Z Ω aα( x, D′u, D′′u_{) | τ}i,−hDα ψ2mτi,hu

dx.

Using hypotheses (E.5) we can minimize the first member of (8.3.4), as follows Z

Ω

ψ2m X

|α|=m

(aα(x, D′u(x), D′′u(x) + τi,hD′′u(x)) − aα(x, D′u(x), D′′u(x))|τi,hDαu) dx =

= Z

Q(2σ)

ψ2m (a(x, D′u(x), D′′u(x) + τi,hD′′u(x)) − a(x, D′u(x), D′′u(x)) | τi,hD′′u) dx ≥

≥ ν Z Q(2σ) ψ2m _kτi,hD′′uk2 dx, then we obtain ν Z Q(2σ) ψ2m_kτi,hD′′uk2 dx ≤ A + B + C + D , (8.3.5) where A_{= −2m} Z Ω ψ2m−1 X |α|=m

(aα(x, D′u(x), D′′u(x) + τi,hD′′u(x)) −

Let us also consider i ≤ n a positive integer, h a real number, |h| < σ, and let us also set ϕ= τi,−h ψ2mτi,hu
, (8.3.2) it follows that ϕ ∈ Hm

0 (Ω, RN) ∩ Hm−1,∞(Ω, RN). From (9.2.1), written for this “test

function” ϕ, it follows

Z

Ω

X

|α|=m

τi,haα(x, Du) |Dα ψ2mτi,hu

dx = −X |α|<m Z Ω aα_{(x, Du) |τ}i,−hDα ψ2mτi,h u

dx. (8.3.3)

On the other hand, for every α such that |α| = m and for a. e. x ∈ Q(2σ), it follows:

τi,haα(x, Du(x)) = τi,haα(x, D′u(x), D′′u(x))

= aα_{(x + he}i_{, D}′_{u(x + he}i_{), D}′′_{u(x + he}i

)) − aα_{(x, D}′_{u(x), D}′′_u(x))

= aα_{(x + he}i_{, D}′_{u(x) + τ}

i,hD′u(x), D′′u(x) + τi,hD′′u(x)) − aα(x, D′u(x), D′′u(x))

=aα(x + hei_{, D}′_{u(x) + τ}

i,hD′u(x), D′′u(x) + τi,hD′′u(x))

− aα(x, D′u(x) + τi,hD′u(x), D′′u(x) + τi,hD′′u(x))]

+ [aα_{(x, D}′_{u(x) + τ}

i,hD′u(x), D′′u(x) + τi,hD′′u(x))

− aα_{(x, D}′_{u(x), D}′′_{u(x) + τ}

i,hD′′u(x))]

+[ aα_{(x, D}′_{u(x), D}′′_{u(x) + τ}

i,hD′′u(x)) − aα(x, D′u(x), D′′u(x))] .

Regarding in mind that

Dα(ψ2m_τ

8.2.2

Local differentiability result in

H

space

Let us now apply the previous local differentiability properties in Hm+ϑ_{(Ω, R}N_{), 0 <}

ϑ < 1 to reach the main objective of this chapter (see also [18]).

Theorem 8.2.4. _{(Main result) If u ∈ H}m_{(Ω, R}N_{) ∩ C}m−1,λ_{(Ω, R}N_{), 0 < λ < 1, is}

a weak solution of the system (8.1.1) satisfying the hypotheses (E.1), (E.2), (E.4),

(E.5) and, for fα

∈ Ln−2λ2n (Ω) assumption (E.3), then

u_{∈ Hloc}m+1(Ω, RN_{) (8.2.9)}

and, for every cube Q(4σ) ⊂⊂ Ω, the following inequality is true

|u|2m+1,Q(σ) ≤ c(ν, K, U, λ, σ, m, n)  1 + (X |α|<m kfαk0,Q(4σ))2+ |u| 2 m,Q(4σ)+ |u| 4 m,4,Q(4σ)  , (8.2.10) where K = sup Ω

kuk and U = kukCm−1,λ_(Ω,RN₎.

8.3

Proofs of main goals

In this section we give the proof of the main results of this chapter.

8.3.1

Proofs of local differentiability results in

H

spaces

We start with the proof of local differentiability results in Hm+ϑ_{(Ω, R}N_{), 0 < ϑ <} λ 2.

Proof. of Theorem 8.2.1 Let us choose x0 ∈ Ω and a generic cube Q(4σ) =

Q(x0_{,4σ) ⊂⊂ Ω, let ψ(x) ∈ C}∞

0 (Rn) a cut-off function having the following

proper-ties:

0 ≤ ψ ≤ 1 in Rn_{, ψ= 1 in Q(σ), ψ = 0 in R}n

\ Q(2σ), kDψk ≤ k σ in R

where |D′′_u|2 ϑ,Q(σ)= P |α| = m |Dα_u|2 ϑ,Q(σ) , K = sup Ω kD′_{uk and U = kuk} Cm−1,λ_(Ω,RN₎. Theorem 8.2.2. _{If u ∈ H}m+ϑ(Ω, RN_{) ∩ C}m−1,λ_{(Ω, R}N_{), 0 < ϑ, λ < 1, is a weak}

solution of the system (8.1.1), the assumptions (E.1), (E.2), (E.4) , (E.5) and the condition (E.3) with fα_{∈ L}q_{2(Ω), q =} 2(1+ϑ)n

n−2ϑλ , are true, we have

u_{∈ Hloc}m+ϑ′(Ω, RN_), ∀ϑ′ ∈  0, ϑ + λ 2(1 − ϑ)  , (8.2.4)

and, for every cube Q(4σ) ⊂⊂ Ω, we have the following inequality

|D′′u_|2_ϑ′_,Q(σ)≤ c(ν, K, U, ϑ, ϑ′, λ, σ, m, n) ·     1 +   X |α|<m kfα k0,₂q,Q(4σ)   1+ϑ + |u|2m,Q(4σ)+ |D′′u| 2 ϑ,Q(4σ)     , (8.2.5) where |D′′_u|2 ϑ,Q(σ)= P |α| = m|D α_u |2ϑ,Q(σ) , K = sup Ω kD′_{uk and U = kuk} Cm−1,λ_(Ω,RN₎.

Applying an iterative method, we have the following result. Theorem 8.2.3. _{If u ∈ H}m(Ω, RN

) ∩ Cm−1,λ_{(Ω, R}N_{), 0 < λ < 1, is a weak solution}

of the system (8.1.1), the hypotheses (E.1), (E.2), (E.4), (E.5) and the condition

(E.3) with fα _{∈ Ln−2λ}2n _{(Ω) are verified, then}

u_{∈ Hloc}m+ϑ(Ω, RN) , _{∀ ϑ : 0 < ϑ < 1.} (8.2.6)

Moreover, for every cube Q(σ) ⊂⊂ Q(σ0) ⊂⊂ Ω, we have

|D′′u_|2_ϑ,Q(σ) ≤ c(ν, K, U, ϑ, λ, σ, σ0, m, n)     1 +  X |α|<m kfαk0, 2n n−2λ,Q(σ0)   1+ϑ + |u|2m,Q(σ0)     , (8.2.7) where K = sup Ω kD′_{uk and U = kuk} Cm−1,λ_(Ω,RN₎. Moreover u_{∈ Hloc}m,4(Ω, RN) . (8.2.8)

(E.5) for every (x, p′_{) ∈ Ω×R}′_{, the functions p}′′_{−→ a}α_{(x, p}′_{, p}′′_{), |α| = m, are strictly}

monotone with non-linearity q = 2, so that there exist two positive constants

M_{(K) and ν(K) such that ∀(x, p}′_{) ∈ Ω × R}′,_{with kp}′_{k ≤ K, and ∀p}′′, q′′ _{∈ R}′′, we obtain:

ka(x, p′, p′′_{) − a(x, p}′, q′′_{)k ≤ M(K) kp}′′_{− q}′′_{k ,} (a(x, p′, p′′_{) − a(x, p}′, q′′_)|p′′_{− q}′′_{) ≥ ν(K) kp}′′_{− q}′′_k2 .

Remark 8.1.1. We point out that the assumptions (E.1) — (E.5) are more general

than the one used by Campanato and Cannarsa in [7].

8.2

Main results

Let Ω be an open bounded set in Rn

, n ≥ 2.

We say weak solution of the system (9.2) a function u ∈ Hm_{(Ω, R}N_{) ∩ L}∞_{(Ω, R}N₎

such that Z Ω X |α|≤m (aα (x, Du)| Dα_ϕ ) dx = 0, ∀ ϕ ∈ Hm 0 (Ω, RN) ∩ Hm−1,∞(Ω, RN). (8.2.1)

8.2.1

Local fractional differentiability results

Let us now state the local fractional differentiability results (see also [18]).

Theorem 8.2.1. _{If u ∈ H}m_{(Ω, R}N_{) ∩ C}m−1,λ_{(Ω, R}N_{), 0 < λ < 1, is a weak solution}

of the system (8.1.1) and the assumptions (E.1) — (E.5) are satisfied, then

u_{∈ Hloc}m+ϑ(Ω, RN_), ∀ϑ ∈  0,λ 2  , (8.2.2)

moreover, for every cube Q(4σ) ⊂⊂ Ω, we have the following inequality

|D′′u_|2_{ϑ,Q(σ) ≤ c(ν, K, U, ϑ, λ, σ, m, n)}  1 + X |α|<m kfα k0,1,Q(4σ)+ |u| 2 m,Q(4σ)   , (8.2.3)

and p = {pα_}

|α|≤m, pα ∈ RN, the generic point of R. If p ∈ R, we set p = (p′, p′′)

where p′ _{= {p}α }|α|<m ∈ R′ = Q|α|<mRNα, p′′ = {pα}|α|=m ∈ R′′ = Q|α|=mRNα,and kpk2 ₌ X |α|≤m kpα k2 N, kp′k2 = X |α|<m kpα k2 N, kp′′k2 = X |α|=m kpα k2 N. We consider, as usual, Di = ∂ ∂xi , i= 1, . . . , n; Dα = Dα1 1 D α2 2 . . . Dnαn.

Let us consider the following differential nonlinear variational system of order 2m : X

|α|≤m

(−1)|α|_Dα_aα_{(x, Du) = 0 (8.1.1)}

where aα_{(x, p) = a}α_{(x, p}′_{, p}′′_{) are functions of Λ = Ω × R in R}N_{, satisfying the}

following conditions:

(E.1) for every α and for every p ∈ R, the function x −→ aα_{(x, p), defined in Ω}

having values in RN_{, is measurable in x;}

(E.2) for every α and for every x ∈ Ω, the function p −→ aα

(x, p), defined in R

having values in RN_{, is continuous in p;}

(E.3) for every α, such that |α| < m, for every (x, p′_{, p}′′_{) ∈ Ω × R, with kp}′_k

N ≤ K, we have: kaα_{(x, p}′_{, p}′′_{)k ≤ M(K)}  |fα (x)| + X |α|=m kpα k2N   = M(K)_|fα (x)| + kp′′k2, where fα ∈ L1_(Ω);

(E.4) for every x ∈ Ω, ∀y ∈ Qx,√1 ndx



∀p′_{, q}′ _{∈ R}′_{, where kp}′_{k , kq}′_{k ≤ K and for}

every p′′_{∈ R}′′_{, we have:}

ka(x, p′, p′′_{)k ≤ M(K) (1 + kp}′′_k)

ka(x, p′_{, p}′′_{) − a(y, q}′_{, p}′′_{)k ≤ M(K) (kx − yk + kp}′_{− q}′_{k) (1 + kp}′′_{k) ;}

where a (x, p) ≡ (aα_{(x, p))}

Nonlinear elliptic systems

We continue the study of regularity properties for solutions of elliptic systems started in [15] and continued in [18] (see also [16]), proving, in a bounded open set Ω of Rn_{, local differentiability and partial H¨older continuity of the weak solutions u of}

nonlinear elliptic systems of order 2m in divergence form X

|α|≤m

(−1)|α|Dαaα(x, Du) = 0.

Specifically, are generalized the results obtained by Campanato and Cannarsa, con-tained in [7], under the hypothesis that the coefficient aα_{(x, Du) , are strictly}

mono-tone with nonlinearity q = 2.

8.1

Problem formulation

Let us set m, N positive integers, α = (α1, . . . , αn) a multi-index and |α| = α1+

. . .+ αn the order of α. We denote by R the Cartesian product

R = Y

|α|≤m

RN

α