Model problems from nonlinear elasticity : partial regularity results

ESAIM: Control, Optimisation and Calculus of Variations

Vol. 13, N^o 1, 2007, pp. 120–134 www.edpsciences.org/cocv

DOI: 10.1051/cocv:2007007

MODEL PROBLEMS FROM NONLINEAR ELASTICITY:

PARTIAL REGULARITY RESULTS

Menita Carozza

and Antonia Passarelli di Napoli

Abstract. In this paper we prove that every weak and strong local minimizer u∈ W^1,2(Ω,IR³) of the functional

I(u) =

Ω|Du|²+f(AdjDu) +g(detDu),

whereu: Ω⊂IR³→IR³,f grows like|AdjDu|^p,ggrows like|detDu|^q and 1< q < p <2, isC^1,α on an open subset Ω₀ of Ω such thatmeas(Ω\Ω₀) = 0. Such functionals naturally arise from nonlinear elasticity problems. The key point in order to obtain the partial regularity result is to establish an energy estimate of Caccioppoli type, which is based on an appropriate choice of the test functions. The limit casep=q≤2 is also treated for weak local minimizers.

Mathematics Subject Classification. 35J50, 35J60, 73C50.

Received May 11, 2005.

1. Introduction

Let us consider integral functionals of the Calculus of Variations of the type I(u) =

Ω

F(Du)dx ,

where Ω⊂IRⁿ andu: Ω→IR^N.

An interesting class of integral functionals which naturally arise from problems of nonlinear elasticity [3] is the one of polyconvex functionals,i.e. functionals in which the integrand is a convex function of the minors of the matrix [Du]. It is well known that polyconvex functionals are also quasiconvex, but they often satisfy anisotropic growth conditions which are not recovered by the results concerning the quasiconvex case [1, 5, 6, 8, 9, 13, 14].

For this reason [10, 11] have considered polyconvex integrals with anisotropic growth conditions, which are close to the typical examples arising from nonlinear elasticity theory. A model case included in the results of [10] is, forn=N = 3

I(u) =

Ω|Du|²+|Du|^p+|AdjDu|^p+|detDu|^p

Keywords and phrases. Nonlinear elasticity, partial regularity, polyconvexity.

1 Dipartimento Pe.Me.Is., Piazza Arechi II, 82100 Benevento, Italy;carozza@unisannio.it

2 Dipartimento di Matematica e Appl. “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.edpsciences.org/cocvor http://dx.doi.org/10.1051/cocv:2007007

where p > 2. They proved that absolute minimizers ofI(u) are C^1,α except for a closed subset of Ω of zero Lebesgue measure. Recall that a function u is an absolute minimizer of I(u) if I(u) ≤ I(u+ϕ) for every ϕ∈C_o^∞(Ω).

Motivated by a recent paper of Ball [4], in [7, 15] partial regularity has been proved for a new class of minimizers of I(u), when F is a quasiconvex integrand. In particular, they consider W^1,p¯ local minimizers, defined as follows

Definition 1.1. Let 1≤p≤p¯≤+∞. A mapu∈W^1,p(Ω; IR^N) is aW^1,p¯(Ω; IR^N) local minimizer ofI(v), if there exists δ >0 such thatI(u)≤I(v) wheneverv∈u+W_o^1,p(Ω; IR^N) and||Dv−Du||p¯≤δ.

We will refer to a W^1,p¯(Ω; IR^N) local minimizer ofI(v) with 1≤p <¯ ∞ as astrong local minimizerand to a W^1,∞(Ω; IR^N) local minimizer as aweak local minimizer. It has been noted in [15] that, if p= ¯p, the study of partial regularity of strong local minimizers can be reduced to the study of absolute minimizers. For this reason we confine ourselves to the case ¯p > p. Since in [4] the study of this class of minimizers is proposed for polyconvex integral functionals, as a natural continuation of the results in [15], in this paper we proveC^1,α partial regularity for weak and strong local minimizers of polyconvex functionals of the type

Ω|Du|²+f(AdjDu) +g(detDu),

where u: Ω⊂IR³→IR³, f grows like|AdjDu|^p, g grows like|detDu|^q and 1< q < p <2. We mention that, under the same assumptions onf andgof Theorem A below, the regularity result for absolute minimizers has been obtained in [16].

Theorem A. Let us consider the functional

I(v) =

Ω|Dv|²+f(AdjDv) +g(detDv), (1.1)

and suppose thatf : IR^3×3→IR,g: IR→IRare C² convex functions satisfying the following assumptions (H1) c₁(µ²+|z|²)^p−22 |ξ|²≤f_zizj(z)ξ_iξ_j≤c₂(µ²+|z|²)^p−22 |ξ|²,

(H2) c₃(µ²+t²)^q−22 ≤g(t)≤c₄(µ²+t²)^q−22 ,

where 1< q < p <2 and µ≥0. Let 2≤p¯≤ ∞and assume that u∈W^1,2(Ω,IR³)∩W_loc^1,p¯(Ω,IR³)be aW^1,p¯ local minimizer of I(v).

If p¯=∞, we assume in addition that

limsup_R→0+||Du−(Du)_x,R||_L^∞(B(x,R))< δ (1.2) holds locally uniformly inx∈Ω, withδ as in Definition 1. Then, there existsα∈(0,1)such thatu∈C^1,α(Ω0) for some open subset Ω0 ofΩwith meas(Ω\Ω0) = 0.

As far as we know, no results are available, even for absolute minimizers, if p =q ≤ 2. However, in the special case of weak local minimizers, we are able to deal also with this assumption. Namely we have

Theorem B. Let f, g satisfy the same assumptions as in Theorem A with 1 < p = q ≤ 2. Let u ∈ W_loc^1,∞(Ω,IR³) be a weak local minimizer for the functional (1.1) such that

limsup_R→0+||Du−(Du)_x,R||_L^∞(BR(x))< δ (1.3)

holds locally uniformly inx∈Ω, withδ as in Definition 1. Then, there existsα∈(0,1)such thatu∈C^1,α(Ω0) for some open subset Ω0 ofΩwith meas(Ω\Ω0) = 0.

We have restricted our study to the case n=N = 3, which is the most significant from the point of view of the applications, thus avoiding the heavy technicalities needed for the general case n≥3,N ≥2, which in any case can carried on without any new idea. A fundamental tool needed to prove partial regularity is a new Caccioppoli type estimate (see Lem. 2.3).

The difficulties here are twofold. The first one is due to the anisotropic growth of the functional which requires the use of suitable test functions obtained (as in [10]) by interpolating the values ofuon the boundary.

The second difficulty comes from the definition of local minimizers which imposes a bound on theW^1,p¯norm of the test functions. Kristensen and Taheri [15] discovered that this restriction is responsible of the fact that in the proof of the Caccioppoli estimate it is impossible the iteration on small radii and thus they are lead to an inequality which does not involve any two concentric balls. Here, in a different way, we also obtain a pre-iterated form of the Caccioppoli inequality involving only two balls of radiiRand ^R₂.

This weaker form of the usual Caccioppoli estimate is however enough to establish the decay estimate by the use of an extra new iteration argument.

2. The energy estimate

Let Ω be a bounded open subset of IR³. IfA∈Hom(IR³,IR³) we set

∧0A= 1, ∧1A=A, ∧2A= adjA, ∧3A= detA.

Following [11], for everyA, B∈Hom(IR³,IR³) and fork= 2,3, we shall write,

∧k(A+B) = k i=0

∧k−iA ∧iB, (2.1)

where ∧_k−iA ∧_iB denotes a suitable linear combination of products of a component of∧_k−iAtimes a com- ponent of∧_iB. The explicit expression of∧_k−iA ∧_iB will not be needed in the sequel.

Definition 2.1. Let Ω be a smooth bounded domain of IR³ and p > 1. The Sobolev class ∧³W^1,p(Ω; IR³) consists of all functions u∈W^1,p(Ω; IR³) such that∧iDu∈L^p(Ω) for every 1≤i≤3.

Remark that the Sobolev class∧³W^1,p is not a linear space. Indeed, if u, v∈ ∧³W^1,p it may happen that u+v∈ ∧³W^1,p.

In what followsB_r(x) will be the ball centered in xof radiusr. If no confusion arises,B_r will stands for a ball centered in 0 of radius r. Whenr = 1, we may useB instead of B1. The letter C will denote a generic constant whose value may change from line to line.

Lemma 2.2. Let αbe a constant and let f_h, g_h be two sequences of functions in L¹(B)such that f_h→α for a.e. x∈B,g_h→g weakly inL^1+η(B), for someη >0. Assume that

B|f_hg_h|^1+δdx <∞, whereδ≥η. Then f_hg_h→αg weakly inL^1+η(B).

Proof. See [11], Lemma 5.5.

If Ω ⊂ IR³ is a bounded open set and λ ∈ (0,1) is a real number, we consider, for v ∈ W^1,2(Ω,IR³), the functional

J_λ(v) =

Ω[|Dv|²+λ^α|AdjDv|^p+λ^β|detDv|^q]dx (2.2) where 0< ^q_p <^β_α and 1< q < p <2.

If Q≥1 is a real number we say that u∈W^1,2(Ω,IR³) is aW^1,p¯(Ω,IR³) Q-local minimizer ofJ_λ if there exists aδ >0 such thatJ_λ(u)≤QJ_λ(ϕ) for anyϕ∈u+W₀^1,2(Ω,IR³) with

||Dϕ−Du||L^p¯≤δ. (2.3)

We prove the following Caccioppoli type estimate

Lemma 2.3. Let 2 ≤p¯≤ ∞ and let u∈W^1,2(Ω,IR³)∩W_loc^1,p¯(Ω,IR³)be a W^1,p¯ Q- local minimizer of J_λ. If p¯=∞, assume also that

limsup_R→0+||Du||_L^∞(B(x,R)) < δ (2.4) where δ is the number appearing in (2.3). Then there exist a constant c depending only on Q and a radius R¯ depending only onδ and aθ∈(0,1)such that for anyR <R,¯ B_R⊂⊂Ω

J_λ(u;BR

2)≤θJ_λ(u;B_R) + c R²

BR\BR 2

|u−u_R|²dx

+cλ^αR^5−3p

BR

|Du|²dx_p

+cλ^{βp−αqp−q} R^{3−2(p−q)3pq}

BR

|Du|²dx_2(p−q)^pq .

Proof. FixB_R⊂Ω and define E_R=

ρ∈R

2, R :

∂Bρ

|Du|^p¯dH²≤ 8 R

BR\BR 2

|Du|^p¯dx

and

∂Bρ

|Du|²dH²≤ 8 R

BR\BR 2

|Du|²dx

.

We have that R

2, R

\E_R=C1∪ C2 where

C1=

ρ∈R 2, R

:

∂Bρ

|Du|^p¯dH²≥ 8 R

BR\BR 2

|Du|^p¯

C2=

ρ∈R 2, R

:

∂Bρ

|Du|²dH²≥ 8 R

BR\BR 2

|Du|²

.

One easily gets that

meas(C_i)<R 8 and then

meas(E_R)≥R

4· (2.5)

Now letω=_|x|^x and fora.e. ρ∈(^R₂, R) consider the functionω→u(ρω). For every 1< m <2 we get

∂Bρ

u−u_∂Bρ^m∗dH²

m1∗

≤c

∂Bρ

|Du|^mdH²

m1

(2.6) where as usualm^∗= ₂²_−m^m denote the Sobolev exponent ofm.

For eachρ∈E_R, following [11], we define the function

ϕ(rω) =

⎧⎪

⎨

⎪⎩

u_∂Bρ r≤^R₄

ρ−ρ−r^R₄u_∂Bρ +^r−_ρ−^R_R⁴

4u(ρω) ^R₄ ≤r≤ρ

u(rω) ρ≤r≤R

(2.7)

and observe that, for ^R₄ < r < ρ

|Dϕ(rω)| ≤c

|u(ρω)−u_∂Bρ|

ρ−^R₄ +|Du(ρω)| (2.8)

|AdjDϕ(rω)| ≤c|u(ρω)−u_∂Bρ|

ρ−^R₄ |Du(ρω)|+c|AdjDu(ρω)| (2.9)

|detDϕ(rω)| ≤c|u(ρω)−u_∂Bρ|

ρ−^R₄ |AdjDu(ρω)|. (2.10)

If ¯p <∞, by (2.8) and the assumptionu∈W_loc^1,p¯, one easily gets that there exists a ¯R= ¯R(δ) such that

BR

|Du−Dϕ|^p¯dx=

Bρ

|Du−Dϕ|^p¯dx≤ c

Bρ\BR 4

|u(ρω)−u_∂Bρ|^p¯ (ρ−^R₄)^p¯ dx+c

BR

|Du|^p¯dx

≤ c

(ρ−^R₄)^{p−¯ 1}

∂Bρ

|u(ρω)−u_∂Bρ|^p¯dH²+c

BR

|Du|^p¯dx

≤ cρ^p¯ (ρ−^R₄)^{p−¯ 1}

∂Bρ

|Du|^p¯dH²+c

BR

|Du|^p¯dx

≤ cR^{p−¯ 1} (ρ−^R₄)^{p−¯ 1}

BR\BR 2

|Du|^p¯dx+c

BR

|Du|^p¯dx≤c

BR

|Du|^p¯

where we used also Poincar´e inequality and the fact that ^R₂ < ρ < R. Then, previous inequality ensures thatϕ is an admissible test function, by the absolute continuity of the integral for ¯p <∞and by (2.4) if ¯p=∞.

Using (2.8), (2.9), (2.10) and the fact thatuis a W^1,p¯Q- local minimizer we get

Bρ

[|Du|²+λ^α|AdjDu|^p+λ^β|detDu|^q]dx≤Q

Bρ

[|Dϕ|²+λ^α|AdjDϕ|^p+λ^β|detDϕ|^q]dx

≤ c R

∂Bρ

|u−u_∂Bρ|²dH²+cR

∂Bρ

|Du|²dH² + cλ^α

R^p−1

∂Bρ

|u−u_∂Bρ|^p|Du|^pdH²+cλ^αR

∂Bρ

|AdjDu|^pdH² + cλ^β

R^q−1

∂Bρ

|u−u_∂Bρ|^q|AdjDu|^qdH² (2.11)

where we used again that ^R₂ < ρ < R. Now, we observe that

∂Bρ

|u−u_∂Bρ|^p|Du|^pdH²≤

∂Bρ

|Du|²dH²

p2

∂Bρ

|u−u_∂Bρ|^2−p2p dH²

2−p2

≤c

∂Bρ

|Du|²dH²

p2

∂Bρ

|Du|^pdH² ≤cR^2−p

∂Bρ

|Du|²dH²

p

, (2.12)

where we used H¨older inequality and (2.6). Moreover using Young’s inequality and (2.6) again we get cλ^β

R^q−1

∂Bρ

|u−u_∂Bρ|^q|AdjDu|^qdH²

≤ cλ^{βp−αqp−q} R^p−qpq−1 ∂Bρ

|u−u_∂Bρ|^p−qpq dH² +cλ^αR

∂Bρ

|AdjDu|^pdH²

≤ cλ^{βp−αqp−q}

R^p−qpq−1 _∂Bρ|Du|^{2p−2q+pq2pq} dH²

2p−2q+pq 2(p−q)

+cλ^αR

∂Bρ

|AdjDu|^pdH²

≤cλ^{βp−αqp−q} R^3−p−qpq

∂Bρ

|Du|²dH²

2(p−q)pq

+cλ^αR

∂Bρ

|AdjDu|^pdH² . (2.13)

Inserting the inequalities (2.12) and (2.13) in (2.11) we obtain

Bρ

[|Du|²+λ^α|AdjDu|^p+λ^β|detDu|^q]dx

≤ c R

∂Bρ

|u−u_∂Bρ|²dH²+cR

∂Bρ

|Du|²dH²+cλ^αR

∂Bρ

|AdjDu|^pdH²

+cλ^αR^3−2p

∂Bρ

|Du|²dH²

p

+cλ^{βp−αqp−q} R^3−p−qpq

∂Bρ

|Du|²dH²

2(p−q)pq

. (2.14)

Recalling that ρis inE_R, we get

Bρ

[|Du|²+λ^α|AdjDu|^p+λ^β|detDu|^q]dx

≤c R

∂Bρ

|u−u_∂Bρ|²dH²+cR

∂Bρ

|Du|²dH²+cλ^αR

∂Bρ

|AdjDu|^pdH²

+cλ^αR^3−3p

BR\BR 2

|Du|²dx

p

+cλ^{βp−αqp−q} R^{3−2(p−q)3pq}

BR\BR 2

|Du|²dx

2(p−q)pq

. (2.15)

Integrating (2.15) with respect toρinE_R, using (2.5) we obtain R

BR 2

[|Du|²+λ^α|AdjDu|^p+λ^β|detDu|^q]dx

≤ c R

BR\BR 2

|u−u_R|²dx+cR

BR\BR 2

|Du|²dx+cλ^αR

BR\BR 2

|AdjDu|^pdx

+cλ^αR^4−3p

BR

|Du|²dx _p

+cλ^{βp−αqp−q} R^{4−2(p−q)3pq}

BR

|Du|²dx _2(p−q)^pq

. (2.16)

Dividing inequality (2.16) byR and using the standard trick of “hole-filling” we get J_λ(u;BR

2)≤θJ_λ(u;B_R) + c R²

BR\BR 2

|u−u_R|²dx

+cλ^αR^3−3p

BR

|Du|²dx_p

+cλ^{βp−αqp−q} R^{3−2(p−q)3pq}

BR

|Du|²dx_2(p−q)^pq

whereθ= _c+1^c ∈(0,1)i.e. the conclusion follows.

3. Proof of Theorem A

Let us consider the excess function U(x, r) =

Br(x)|Du−(Du)_r|²+|Adj(Du−(Du)_r)|^p+|det(Du−(Du)_r)|^q (3.1) we want to establish, as usual, a decay estimate forU(x, r). More precisely we have the following lemma.

Lemma 3.1. Letu∈W^1,2(Ω,IR³)∩W_loc^1,p¯(Ω,IR³)be aW^1,p¯local minimizer of I (satisfying (1.3) ifp¯= +∞).

For any M >0 andτ∈(0,¹₂)there exist two constantsc(M)andε(τ, M)such that, if

|(Du)r| ≤M and U(x, r)< ε, (3.2)

then

U(x, τr)≤c(M)τ^µU(x, r), (3.3)

for someµ independent ofM andτ.

Proof. Step 1(Blow up). FixM >0 andτ∈(0,¹₂). Arguing by contradiction, we assume that there exists a sequenceB_rh(x_h)⊂⊂Ω such that

|(Du)_xh,rh| ≤M and λ²_h=U(x_h, r_h)→0, (3.4)

but U(x_h, τr_h)

λ²_h > c(M)τ^µ, (3.5)

for somec(M) to be determined later. Setting A_h= (Du)_xh,rh and v_h=u(x_h+r_hy)−(u)_xh,rh−r_hA_hy

λ_hr_h (3.6)

for ally∈B1(0), we have

B1(0)|Dv_h|²+λ²_h^p−2|Adj(Dv_h)|^p+λ³_h^q−2|det(Dv_h)|^q = 1 (3.7) and (v_h)0,1 = 0. Passing to a subsequence and using the divergence structure of minors, we may assume, without loss of generality, that

Dv_h→Dv w−L²(B1) v_h→v s−L²(B1) λ_h^2p−2p AdjDv_h→0 w−L^p(B1)

λ_h^3q−2q detDv_h→0 w−L^q(B1)

A_h→A a_h→a. (3.8)

We introduce the rescaled functionals f_h(ξ) = 1

λ²_h[f(Adj(A_h+λ_hξ))−f(Adj(A_h))−Df(Adj(A_h))(Adj(A_h+λ_hξ)−Adj(A_h))]

and

g_h(s) = 1

λ²_h[g(det(A_h+λ_hs))−g(det(A_h))−g(det(A_h))(det(A_h+λ_hs)−det(A_h))]

and for anyh,we set

I_h(w) =

B1

|Dw|²+f_h(Dw) +g_h(Dw). (3.9)

It is easy to check thatI_h(v_h)≤I_h(v_h+ϕ) providedϕ∈W₀^1,p¯(B₁, R^N) and

||Dϕ||p¯≤δ_h= _δ

λhr_h^np¯ p <¯ ∞

λδh p¯=∞.

Step 2(v solves a linear system). By formula (2.1) one easily deduces that, for everyφ∈C₀^∞(B1), d

dt ∧i(A_h+λ_h(Dv_h+tDφ))|t=0 =∧i−1(A_h+λ_hDv_h) λ_hDφ.

Then the minimality of v_himplies that they solve the Euler Lagrange systems:

B1(0)

Dv_hDφdx+

B1(0) 1 0

D²f(Adj(A_h) +t(Adj(A_h+λ_hDv_h)−Adj(A_h)))dt

×(λ_hAdj(Dv_h) +A_h Dv_h)·[(A_h+λ_hDv_h) Dφ]dx +

B1(0) 1 0

g(det(A_h) +t(det(A_h+λ_hDv_h)−det(A_h)))dt

×(Adj(A_h) Dv_h+λ_hDv_h adj(Dv_h))·[Adj(A_h+λ_hDv_h) Dφ]dx= 0 (3.10) for allφ∈C₀^∞(B₁). Lettingh→ ∞, using (3.8) and Lemma 2.2 we get

0 =

B1(0)

DvDφ+D²f(AdjA)(A Dv)(A Dφ) +D²g(detA)(AdjA Dv)(AdjA Dφ)dx

thenvsolves a linear elliptic system, with constant coefficients. By standard regularity result (see [12]) we have that for anyσ∈(0,¹₂]

Bσ

|Dv−(Dv)_σ|²≤cσ²

B1

|Dv−(Dv)_σ|²≤cσ² (3.11)

and

|(Dv)2σ−(Dv)_σ|²≤cσ² (3.12)

where the constantc depends only onM. Setting

w_h(y) =v_h(y)−(Dv_h)_σy−(v_h)2σ (3.13) and using the fact thatv_h minimizes the functional (3.9), one easily see thatw_h minimizes the functional

w→

B1

|Dw|²+f_h(Dw+ (Dv_h)_σ) +g_h(Dw+ (Dv_h)_σ). (3.14) Now, we claim that

|f_h(ξ)| ≤c(M)(|ξ|²+λ²_h^p−2|Adjξ|^p) (3.15)

|g_h(ξ)| ≤c(M)(|ξ|²+λ³_h^q−2|detξ|^q).

Namely, by definition off_h, we have that f_h(ξ) = 1

λ²_h[f(Adj(A_h+λ_hξ))−f(Adj(A_h))−Df(Adj(A_h))(Adj(A_h+λ_hξ)−Adj(A_h))]

= 1

λ²_h[f(AdjA_h+λ²_hAdjξ+A_h λ_hξ)−f(Adj(A_h))−Df(Adj(A_h))(λ²_hAdjξ+A_h λ_hξ)]

and setting

ζ=λ²_hAdjξ+A_h λ_hξ we can write

λ²_h|f_h(ξ)|= 1

0 (1−s)D²f(AdjA_h+sζ)ζζds

≤ |ζ|² 1

0

µ²+|AdjAh+sζ|^2p−2₂

≤c|ζ|²(µ²+|AdjA_h|²+|ζ+ AdjA_h|²)^p−22 , (3.16) where we used Lemma 2.1 in [2] and assumption (H2). On the other hand we observe that if

|λ²_hAdj(ξ)| ≤ |A_h λ_hξ|, then by (3.16) we have

λ²_h|f_h(ξ)| ≤cλ²_h|A_h ξ|²≤c(M)λ²_h|ξ|². If

|λ²_hAdj(ξ)|>|A_h λ_hξ|, then by (3.16)

λ²_h|f_h(ξ)| ≤cλ²_h^p|Adj(ξ)|^p.

The second inequality in (3.15) is analogue. Using Lemma 2.1 in [2] again and assumptions (H1) and (H2) we obtain

|f_h(ξ)| ≥c(M)(|ξ|²+λ²_h^p−2|Adjξ|^p),

|g_h(ξ)| ≥c(M)(|ξ|²+λ³_h^q−2|detξ|^q).

Hence the functional defined at (3.14) is equivalent to the following w→

B1

|Dw|²+λ²_h^p−2|Adj(Dw)|^p+λ³_h^q−2|det(Dw)|^q

and thenw_h is aW^1,p¯Q-local minimizer ofJ_λh(B1) for someQ=Q(M), withα= 2p−2 andβ = 3q−2.

Step 3(Conclusion). Rescaling the excess function defined by (3.1), we get U(x_h, σr_h) =

B_σrh(xh)|Du−(Du)_σrh|²+|Adj(Du−(Du)_σrh)|^p+|det(Du−(Du)_σrh)|^q

=

Bσ(0)

λ²_h|Dv_h−(Dv_h)_σ|²+λ²_h^p|Adj(Dv_h−(Dv_h)_σ)|^p+λ³_h^q|det(Dv_h−(Dv_h)_σ)|^p

=

Bσ(0)

λ²_h|Dw_h|²+λ²_h^p|Adj(Dw_h)|^p+λ³_h^q|det(Dw_h)|^p.

Since w_h satisfies the assumptions of Lemma 2.3 (in particular if ¯p= +∞ assumption (1.3) implies that w_h satisfies (2.4) withδ_h= _λ^δ

h), forhsufficiently large we obtain U(x_h, σr_h)

λ²_h =

Bσ(0)|Dw_h|²+λ²_h^p−2|Adj(Dw_h)|^p+λ³_h^q−2|det(Dw_h)|^q

≤θU(x_h,2σr_h) λ²_h + c

σ²

B2σ

|w_h−(w_h)2σ|²

+cσ^5−3pλ²_h^p−2

B2σ

|Dw_h|²

p

+cσ^{3−2(p−q)3pq} λ_h^{p−qpq −2}

B2σ

|Dw_h|² _2(p−q)^pq

. (3.17)

Passing to the limit ash→ ∞in (3.17) and using (3.8), (3.11), (3.12) and the assumption 1< p < q <2 we get lim sup

h→∞

U(x_h, σr_h)

λ²_h ≤θlim sup

h→∞

U(x_h,2σr_h) λ²_h + c

σ²

B2σ

|v−(v)2σ−(Dv)_σy|²dy

≤θlim sup

h→∞

U(x_h,2σr_h) λ²_h + c

σ²

B2σ

[|v−(v)_2σ−(Dv)_2σy|²+|(Dv)2σy−(Dv)_σy|²]dy

≤θlim sup

h→∞

U(x_h,2σr_h) λ²_h +c

B2σ

|Dv−(Dv)2σ|²dy+cσ²

≤θlim sup

h→∞

U(x_h,2σr_h)

λ²_h +c1(M)σ², (3.18)

where we used Poincar´e inequality. Setting forσ∈(0,¹₂] ϕ(σ) = lim sup

h→∞

U(x_h, σr_h) λ²_h inequality (3.18) can be rewritten as

ϕ(σ)≤θϕ(2σ) +c1(M)σ² and one can easily check that the function

ψ(σ) =ϕ(σ) +c1(M)σ² is such that, for allσ∈(0,¹₂],

ψ(σ)≤max θ,1

2

ψ(2σ) =γψ(2σ) 0< γ <1. (3.19)

An iteration procedure yields that, for anyk= 0,1,2, . . . andσ∈(0,¹₂] ψσ

2^k

≤γ^kψ(σ). (3.20)

Now, takeτ in the interval ₁

2^k+1;₂¹_k

and observe, by the definition ofψ, that ψ(τ)≤Cψ

1 2^k

. (3.21)

Putting together estimates (3.20) and (3.21), we obtain ϕ(τ)≤ψ(τ)≤γ^k−1ψ

1 2

= 2^{(k−1) log2γ}ψ 1

2

= 2^k+1

4

log₂γ

ψ 1

2

≤ 1

4

log₂γ 1 τ

log₂γ

ψ 1

2

= 1

γ²τ^−log2γψ 1

2

≤C₂(M)τ^µ. (3.22)

Inequality (3.22) contradicts (3.5) if we choosec(M) larger thenC2(M).

We now are in position to give the proof of Theorem A, that relies on a standard iteration argument involving the excess function.

Lemma 3.2. Let 0< α <1 and M >0. Then there exist τ ∈(0,¹₂)and ε >0 both depending on αand M such that if

B(x, r)⊂Ω, |(Du)r| ≤M, and U(x, r)< ε then

U(x, τ^lr)≤(τ^l)^αU(x, r) for every l∈N.

Proof. See Lemma 6.1 [10].

Lemma 3.3. Let u∈ ∧_kW^1,p(Ω). Then

r→lim0

Br(x)| ∧_i(Du−(Du)_r|^p= 0 for almost everyx∈Ωand1≤i≤k.

Proof. See Lemma 6.2 [10].

The proof of Theorem A is now consequence of a iteration procedure based on Lemma 4.1. The singular set turns out to be contained in the complement of

Ω_o=

x∈Ω : lim

r→0(Du)_r= 0, lim

r→0

Br(x)|Adj(Du−(Du)_r|^p= 0,

r→lim0

Br(x)|det(Du−(Du)_r|^q= 0

. (3.23)

4. The limit case

In this section we treat the limit case in which the growth exponentsp=q≤2. Also in this case the partial regularity of weak local minimizers is based on a decay estimate for the excess function

U(x, r) =

Br(x)|Du−(Du)_r|²+|Adj(Du−(Du)_r)|^p+|det(Du−(Du)_r)|^p. (4.1) The proof of Theorem B differs from the one of Theorem A only in the Caccioppoli type estimate.

Lemma 4.1. Let p≤2, andu∈W^1,2(Ω,IR³)∩W_loc^1,∞(Ω,IR³)be aW^1,∞ Q- local minimizer of J_λ(v) =

Ω|Dv|²+λ^2p−2|AdjDv|^p+λ^3p−2|detDv|^p such that

limsup_R→0+||Du||_L^∞(BR(x)) < δ

λ, (4.2)

where δ is the number appearing in (1.3). Then there exist a constant c depending only on Q, a radius R¯ depending only onδ and a numberθ∈(0,1) such that for anyR <R, we have¯

J_λ(u;BR

2)≤θJ_λ(u;B_R) + c R²

BR\BR 2

|u−u_R|²+cλ^2p−2R^5−3p

⎛

⎝

BR\BR 2

|Du|²

⎞

⎠

p

,

if p <2,

J_λ(u;BR

2)≤θJ_λ(u;B_R) + c R²

BR\BR 2

|u−u_R|², if p= 2.