Calc. Var. (2009) 35:215–238 DOI 10.1007/s00526-008-0203-4
Calculus of Variations
Partial regularity for polyconvex functionals depending on the Hessian determinant Menita Carozza · Chiara Leone · Antonia Passarelli di Napoli · Anna Verde
Received: 15 January 2007 / Accepted: 9 September 2008 / Published online: 1 October 2008 © Springer-Verlag 2008
Abstract We prove a C 2,α partial regularity result for local minimizers of polyconvex variational integrals of the type I (u) = |D 2 u|2 + g(det(D 2 u))d x, where is a bounded 2,2 () and g ∈ C 2 (R) is a convex function, with subquadratic open subset of R2 , u ∈ Wloc growth. Mathematics Subject Classification (2000)
35G99 · 49N60 · 49N99
1 Introduction The study of the partial regularity for minimizers of integral functionals of the type f (D k u),
where is a bounded open subset of Rn , n ≥ 2, u : ⊂ Rn → R N and D k u = (D α u i )|α|=k stands for the tensor of all k th order weak partial derivatives of the function u, has attracted a great interest in the last few years [5,9,24,28], since they naturally arise in various mathematical models in engineering and material sciences. Among them,
M. Carozza (B) Dipartimento Pe.Me.Is, Università degli studi del Sannio, Piazza Arechi 2, 82100 Benevento, Italy e-mail:
[email protected] C. Leone · A. Passarelli di Napoli · A. Verde Dipartimento di Matematica e Applicazioni “R.Caccioppoli”, Università di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy e-mail:
[email protected] A. Passarelli di Napoli e-mail:
[email protected] A. Verde e-mail:
[email protected]
123
216
M. Carozza et al.
we recall the gradient theories of phases transitions within elastic regimes [12,29,31], the study of equilibria of micromagnetic materials where it is useful the knowledge of second order energies [11,13,31], the theories of second order structured deformations [32] and the Blake–Zisserman model for image segmentation theory [10]. A natural assumption for the existence of minimizers of such integral functionals is the k-quasiconvexity of the integrand [1,26,16,30], introduced by Meyers [30], which means that the following inequality is satisfied for any ϕ ∈ C0∞ (; R N ): f z 0 + D k ϕ − f (z 0 ) d x ≥ 0
for any constant tensor z 0 . Therefore it is interesting to investigate the regularity properties of minimizers of variational integrals, under this assumption. In case of quasiconvex integrals of the first order, satisfying standard growth conditions, the theory of the regularity has been initiated in the fundamental paper by Evans [17] and then developed for example in [2,6–8,14,19,23]. Partial C k,α regularity results for minimizers of higher order variational problems in case the integrand is a k-quasiconvex function such that c|ξ | p ≤ f (ξ ) ≤ C(1 + |ξ | p ) p > 1, were recently obtained in [24] and [28]. However all known natural examples of quasiconvex functions (as in the theory of nonlinear elasticity introduced by Ball [4]) are, in fact, polyconvex, i.e. convex function of the minors of the matrix D k u. It is well known that polyconvex functionals are also quasiconvex, but they often satisfy nonstandard growth conditions, which are not recovered by the above mentioned results concerning the quasiconvex case. For this reason, here we consider the following polyconvex integral 2 2 D u + g det D 2 u I (u) = dx (1)
which is close to the typical examples of nonlinear elasticity. Here and in what follows is an open bounded subset of R2 , u : ⊂ R2 → R, det(D 2 u) is the Hessian determinant and g is a C 2 convex function satisfying the following growth conditions: p−2 p−2 c1 µ2 + t 2 2 ≤ g (t) ≤ c2 µ2 + t 2 2 with 1 < p < 2, for some µ > 0 and some positive constants c1 , c2 . Partial C 1,α regularity results for minimizers of polyconvex functionals depending on the gradient of the minimizer have been extensively investigated. The study has been initiated by Fusco and Hutchinson in [20] and [21], where the authors established partial C 1,α regularity results for minimizers of polyconvex functionals whose model was I (u) = |Du|2 + |det Du|2 d x (2)
and u : ⊂ → R2 . Then, other contributions to the study of the where ⊂ regularity of minimizers of polyconvex functionals have been given in [15,25,27,33]. The main difficulty in dealing with polyconvex functionals is that the set of the functions for which the integrand is finite is not a linear space, even for (2). This corresponds to the fact that if A and B are 2 × 2 matrices , we cannot improve the powers in the estimate det(A + B)2 ≤ c (det A)2 + (det B)2 + |A|4 + |B|4 . R2
123
R2
Partial regularity for polyconvex functionals depending on the Hessian determinant
217
Thus, if z is a function obtained by smoothly interpolating between u and v, one cannot estimate I (z) in terms of I (u) and I (v). In [20], the authors introduced a new comparison function construction that allows estimates of this form by suitably interpolating between boundary values of the functions u and v. To apply this technique, in the case of second order derivatives, one should instead interpolate between the boundary values of the gradients Du and Dv, but unfortunately the function thus obtained is not, in general, the gradient of a W 2,2 () function. Here we overcome this difficulty using a result due to Fonseca and Malý (see [18]) which enables us to connect in the annulus Br \Bs two W 2,2 functions with a W 2,2 p function, with 1 < p < 2. The proof of our result involves a blow up argument, that is used to prove a decay estimate on the excess function 2 2 2 U (x0 , r ) = D u − D 2 u x ,r + V det D 2 u − D 2 u x ,r dy, 0
0
Br (x0 ) p−2
where the function V is defined as V (t) = (µ2 + t 2 ) 4 t. It is worth pointing out that the use of such function is due to the subquadratic growth of the function g, an assumption needed in order to use the result of [18]. The blow-up procedure involves, as usual, a sequence of bad balls B(x h , rh ) and a corresponding sequence of rescaled functions vh defined on B(0, 1). We first show that the rescaled functions vh converge in some weak sense to a limit function v, then we establish that v satisfies a linear equation (see Sect. 3, Proposition 3.1, Steps 2 and 3). The key point is to show that vh converge to v also in a strong sense (see Sect. 3, Proposition 3.1, Steps 4 and 5), so that the functions vh share with v a good decay estimate. We remark that the proof avoid any use of Caccioppoli inequalities and strongly uses the fact that u is a minimizer.
2 Statements and preliminary Lemmas Let us consider the functional I (u, ) =
2 2 D u + g det D 2 u d x
(3)
where is an open bounded subset of R2 , u : ⊂ R2 → R, and det D 2 u is the Hessian determinant. Let 1 < p < 2, and g : R → R satisfy for a suitable µ > 0 the following assumptions: g ∈ C 2 (R)
(H1)
p−2 p−2 c1 µ2 + t 2 2 ≤ g (t) ≤ c2 µ2 + t 2 2 ,
(H2)
for every t ∈ R and some positive constants c1 , c2 > 0. 2,2 () is a local minimizer of I if Definition 2.1 A function u ∈ Wloc
I (u, O) ≤ I (v, O) for any open O and any v ∈ u + W02,2 (O). The aim of this paper is proving the following
123
218
M. Carozza et al.
2,2 Theorem 2.1 Let g satisfy the assumptions (H1), (H2) and let u ∈ Wloc () be a local minimizer of I . Then there exists an open subset 0 of such that
meas(\0 ) = 0 and u ∈ C 2,α (0 ), for all 0 < α < 1. 2,2 In what follows, we will denote by u ∈ Wloc () a local minimizer of the integral functional (3) and assume that its integrand satisfies (H1), (H2). We set, for every Br (x0 ) ⊂
u = (u)x0 ,r = Br (x0 )
1 meas(Br (x0 ))
u. Br (x0 )
We will often omit the centre of the ball, thus writing only u r and Br . Finally, we will denote by P(y) = Pu (x, R, y) = A + (B, y) + (C y, y) the polynomial of degree 2 such that Dl (u(y) − P(y))dy = 0, B R (x)
for l = 0, 1, 2, so B = (Du)x,R , C =
1 2 D u x,R , 2
and
A = (u)x,R − ((C y, y))x,R ,
(4)
(see [22], page 79). When no confusion will arise, we will omit the dependence of P on x, R and u. We begin by giving the following basic inequality (see Lemma 2.1 in [3]). Lemma 2.2 For every continuous function f : [0, 1] → [0, 1], for every γ ∈ − 21 , 0 , and µ ≥ 0 we have 1 f (t) dt ≤
1
f (t)(µ2 + |η + t (ξ − η)|2 )γ dt
0
(µ2
0
+ |η|2
+ |ξ |2 )γ
≤
8 , 2γ + 1
(5)
for all ξ, η not both zero if µ = 0. For t ∈ R, let us define the following function p−2 V (t) = µ2 + t 2 4 t.
(6)
Next statement (see Lemma 2.1 in [6]) contains some useful properties of the function V .
123
Partial regularity for polyconvex functionals depending on the Hessian determinant
219
Lemma 2.3 Let p ∈ (1, 2) and V be the function defined in (6), then for every t ∈ R, λ > 0 (i)
1 + µ2
p−2 4
p−2
p p min |t|, |t| 2 ≤ |V (t)| ≤ min µ 2 |t|, |t| 2
p (ii) |V (λt)| ≤ max λ, λ 2 |V (t)| (iii) |V (t + s)| ≤ c( p, µ)[|V (t)| + |V (s)|] (iv)
p |t − s| ≤ 2
|V (t) − V (s)| p−2 ≤ c( p)|t − s| µ2 + |t|2 + |s|2 4
(7)
(v) |V (t) − V (s)| ≤ c( p)|V (t − s)|
µ2 + 4M 2 (vi) |V (t − s)| ≤ c( p) µ2
2−4 p
|V (t) − V (s)| i f |s| ≤ M.
In [6] the lemma is proved in case µ = 1, but it is easy to adapt that proof to the case µ > 0. The following lemma (the proof can be found in [9], Lemma 2.4) is a slight generalization to the case of higher order derivatives of Lemma 2.4 in [18]. Lemma 2.4 Let v, w ∈ W 2,2 (B1 (0)) and 41 < s < r < 1. Let 2 < q < 4; then, for all λ > 0 and m ∈ N, there exists a function z ∈ W 2,2 (B1 (0)) and 41 < s < s < r < r < 1 with r , s depending on v, w and λ, such that z = v on Bs , z = w on B1 \Br , r −s r −s ≥ r − s ≥ m 3m
(8)
and ⎛ ⎜ ⎝
Br \Bs
⎞1 2
⎛
⎜ |D z| ⎠ + λ ⎝ 2
⎞1
2⎟
q
q⎟
|D z| ⎠ 2
Br \Bs
⎡
(r − s)ρ ⎢ ≤C (1 + λ2 ) ⎣ mρ
1+
Br \Bs
2
|Dl v|2 +
l=0
m2 + (r − s)2
1
2
|Dl w|2
l=0
|D (v − w)| l
2
⎤1 2
⎥ ⎦ ,
(9)
l=0
where C = C(q) > 0 and ρ = ρ(q) > 0 .
123
220
M. Carozza et al.
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 function U (x0 , r ) defined as follows 2 2 2 U (x0 , r ) = D u − D 2 u x ,r + V det D 2 u − D 2 u x ,r dy, 0
0
Br (x0 )
which measures how the second order derivatives are far from being constant in the ball Br (x0 ). 2,2 Proposition 3.1 Let u ∈ Wloc () be a local minimizer of I . Fix M > 0. There is a constant C M,µ = C(M, µ) > 0 such that for every 0 < τ < 41 , there exists = (τ, M, µ) such that, for every Br (x0 ) ⊂ , if 2 D u x ,r ≤ M and U (x0 , r ) ≤
0
then U (x0 , τr ) ≤ C M,µ τ 2 U (x0 , r ). Proof Fix M and τ . We shall determine C M,µ later. We argue by contradiction assuming that there exists a sequence of balls Brh (x h ) ⊂ satisfying 2 D u x ,r ≤ M, lim U (x h , rh ) = 0, h
h
h→∞
but U (x h , τrh ) > C M,µ τ 2 U (x h , rh ). Set
Ah = D 2 u x
h ,r h
(10)
, λ2h = U (x h , rh ),
and let P the polynomial relative to u and Brh (x h ), whose coefficients are defined in (4). Step 1: Blow up. We rescale the function u in each Brh (x h ) to obtain a sequence of functions defined in B1 (0). Set vh (y) =
1 [u(x h + rh y) − P(x h + rh y)], λh rh2
so that 1 2 D u(x h + rh y) − Ah . λh
D 2 vh (y) = We have also
Moreover, U (x h , rh ) = λ2h
123
D l vh
0,1
= 0 for l = 0, 1, 2.
2 2 dy = 1 |D 2 vh |2 + λ−2 h |V det λh D vh | B1
(11)
Partial regularity for polyconvex functionals depending on the Hessian determinant
221
Then, passing possibly to a subsequence, we may suppose that ⎧ D 2 vh D 2 v ⎪ ⎪ ⎨ Dvh → Dv ⎪ ⎪ ⎩ vh → v
Ah → A,
(12)
weakly in L
2
2×2
B1 , R , 2 B1 , R ,
strongly in
L2
strongly in
L 2 (B1 ),
λδh D 2 vh → 0 a.e. in B1 for any δ > 0, V det λh D 2 vh 2 dy ≤ λ2 . h
(13)
(14) (15)
B1
Finally, by (15), using (7)i of Lemma 2.3, we deduce det λh D 2 vh p ≤ C,
(16)
B1
where C depends only on p and µ. Step 2: v solves a linear equation. Define the normalized function 1 gh (ξ ) = 2 g(det(Ah + λh ξ )) − g(detAh ) − g (detAh )(det(Ah + λh ξ ) − detAh ) . λh (17) The corresponding normalized functional is defined as 2 2 D w + gh D 2 w dy. Ih (w) = B1
We have that vh is a local minimizer of Ih amongst functions in W 2,2 (B1 ), with det D 2 w ∈ p L (B1 ) which agree with vh outside some compact subset of B1 . The proof of the minimality of vh can be easily obtained following the lines of the proof of Lemma 5.4 in [20]. We report it here for the sake of completeness . Let us consider w ∈ W 2,2 (B1 ), with H (w) ∈ L p (B1 ) and w = vh outside some compact h set K B1 . Then, for z ∈ B(x h , rh ), the function w ∗ (z) = λh rh2 w( z−x rh ) + P(z) belongs to 2,2 ∗ p ∗ W (B(x h , rh )) and H (w ) ∈ L (B(x h , rh )). Moreover w agrees with u outside a compact set well contained in B(x h , rh ). Since u is a minimizer of I , it follows that 1 1 |D 2 vh |2 + 2 g det Ah + λh D 2 vh ≤ |D 2 w|2 + 2 g det Ah + λh D 2 w . λh λh B1
B1
In fact, the resulting boundary integrals involving the functions w, vh and their gradients vanish, since w and vh coincide outside the compact set K B1 , as well as their gradients. But we also have that det Ah + λh D 2 vh = det Ah + λh D 2 w , B1
B1
123
222
M. Carozza et al.
or, equivalently
H (u) =
B(x h ,rh )
H (w ∗ ).
B(x h ,rh )
Now, since the Hessian can be written in divergence form, to prove this fact, it is enough to show that if X ∈ L 1 (B1 ) is a vector valued function, X = 0 outside some compact contained in B(x h , rh ), and divX ∈ L 1 (where divX is defined in the distributional sense), then B(xh ,rh ) divX = 0. Actually, this can be done mollifying X in the usual way, using the Divergence Theorem for the mollified functions and then passing to the limit. At this point we can consider the Euler-Lagrange equation for vh . To do this we need to calculate the following derivative d det Ah + λh D 2 vh + t D 2 ϕ = λh Ah + λh D 2 vh D 2 ϕ, dt t=0 for ϕ ∈ Cc∞ (B1 ), where is the bilinear form defined by A B = det(A + B) − det A − det B. Hence vh solves the Euler–Lagrange equation: 0 = 2D 2 vh D 2 ϕ B1
1 + λh
⎞ ⎛ 1 ⎝ g det Ah + t det Ah + λh D 2 vh − det Ah dt ⎠ 0
B1
· det Ah + λh D 2 vh − det Ah Ah + λh D 2 vh D 2 ϕ
D 2 vh D 2 ϕ
=2 B1
1 + λh
⎛ 1 ⎞ ⎝ g det Ah + t det Ah + λh D 2 vh − det Ah dt ⎠
B1
0
B1
0
· det Ah + λh D 2 vh − det Ah Ah D 2 ϕ ⎞ ⎛ 1 1 ⎝ g det Ah + t det Ah + λh D 2 vh − det Ah dt ⎠ + λh · det Ah + λh D 2 vh − det Ah λh D 2 vh D 2 ϕ
= Ih + I Ih + I I Ih , for all ϕ ∈ Cc∞ (B1 ). Letting h → ∞, we want to show that v solves the following linear elliptic equation: 2D 2 v D 2 ϕ + g (det A) A D 2 v A D 2 ϕ = 0, (18) B1
for all ϕ ∈
123
Cc∞ (B1 ).
Partial regularity for polyconvex functionals depending on the Hessian determinant
223
To do this fact we will prove that, for all ϕ ∈ Cc∞ (B1 ), ⎧ ⎪ ⎪ lim Ih = 2 D 2 v D 2 ϕ, ⎪ ⎪ h→∞ ⎪ ⎪ ⎪ ⎨ B1 lim I Ih = g (det A) A D 2 v A D 2 ϕ , ⎪ ⎪ h→∞ ⎪ ⎪ B1 ⎪ ⎪ ⎪ ⎩ lim I I I = 0. h→∞
(19)
h
Thanks to (13), the first limit in (19) is obvious. Let us deal with I Ih . Since det(Ah + λh D 2 vh ) − det Ah = det(λh D 2 vh ) + Ah λh D 2 vh ,
(20)
we can write it as the sum of two parts ⎞ ⎛ 1 I Ih = ⎝ g det Ah + t det Ah + λh D 2 vh − det Ah dt ⎠ 0
B1
·λh det D 2 vh Ah D 2 ϕ ⎛ ⎞ 1 2 + ⎝ g det Ah + t det Ah + λh D vh − det Ah dt ⎠ 0
B1
· A h D 2 vh
Ah D 2 ϕ
= I Ih1 + I Ih2 . Hence, by (H2) we get I Ih1 ≤ µ
p−2 2
λh M D 2 ϕ L ∞ (B1 )
|D 2 vh |2 ,
B1
which, thanks to (13), tends to zero, sinceλh → 0. 1 As far as I Ih2 is concerned, we have that 0 g det Ah + t det Ah + λh D 2 vh − det Ah p−2
2 dt tends to g (det A) a.e. in B1 from (12) and (14). Moreover, itis bounded by µ . Since by (12) and (13), Ah D 2 vh Ah D 2 ϕ tends to A D 2 v A D 2 ϕ weakly in L 2 (B1 ), it is easy to see that lim I Ih2 = g (det A) A D 2 v A D 2 ϕ .
h→∞
B1
Finally, we consider I I Ih ; using (H2), Lemma 2.2, and (7)iv of Lemma 2.3 we can estimate it in the following way: 2 p−2 I I Ih ≤ µ 4 D 2 ϕ L ∞ (B1 ) V (det Ah + λh D 2 vh − V (det Ah ) ||D 2 vh p B1 ≤ c( p, µ) D 2 ϕ L ∞ (B1 ) V det Ah + λh D 2 vh − det Ah ||D 2 vh , B1
where in the last inequality we used (7)v . Recalling (20), applying (7)iii , Hölder inequality, (7)i , and (15) we obtain
123
224
M. Carozza et al.
I I Ih ≤ c( p, µ) D ϕ
2
V det λh D 2 vh + V Ah λh D 2 vh D 2 vh
L ∞ (B1 ) B1
⎛
⎜ ≤ c( p, µ) D 2 ϕ L ∞ (B1 ) ⎝
⎞1 ⎛ ⎞1 2 2 2 2 ⎟ ⎜ ⎟ 2 2 V det λh D vh ⎠ ⎝ D vh ⎠
B1
B1
+ c( p, µ) D 2 ϕ L ∞ (B1 )
|Ah λh D 2 vh ||D 2 vh |
B1
≤ c D 2 ϕ L ∞ (B1 ) λh , which tends to zero as h tends to infinity. Then (19) is proved, and by classical regularity results (see for instance [22], Theorem 2.1 and Remark 2.3 pp. 78–79) it now follows that v ∈ C ∞ (B1 ), and
(21)
2 D v − D 2 v 2 dy ≤ C(M, µ)τ 2 , τ
(22)
Bτ
for any τ ∈ (0, 1). Step 3: An estimate of gh (ξ ). We claim that 1 |V (det(Ah + λh ξ ) − det Ah )| λ2h 2 p−2 |ξ |2 p . ≤ gh (ξ ) ≤ C( p, M) µ p−2 |ξ |2 + λh
c(µ, M, p)
(23)
Let us prove first the upper bound. From the definition of gh , from (H2), and using Lemma 2.2, we have 1 λ2h gh (ξ ) =
(1 − t)g (det(Ah + λh ξ ) + (1 − t)(det Ah − det(Ah + λh ξ ))dt
0
· (det(Ah + λh ξ ) − det Ah )2 ≤ c( p)(µ2 + (det(Ah + λh ξ ))2 + (det Ah )2 ) = c( p)(µ2 + (z + det Ah )2 + (det Ah )2 )
p−2 2
p−2 2
(det(Ah + λh ξ ) − det Ah )2
z2,
with z = det(Ah + λh ξ ) − det Ah = λ2h detξ + λh Ah ξ . Now we observe that if λ2h |detξ | ≤ λh |Ah ξ | then λ2h gh (ξ ) ≤ c( p)µ p−2 |z|2 ≤ 2 c( p)M 2 µ p−2 λ2h |ξ |2 . On the other hand, if λ2h |detξ | > λh |Ah ξ |
123
(24)
Partial regularity for polyconvex functionals depending on the Hessian determinant
225
then
λ2h gh (ξ )
z2 ≤ c( p) µ + 2 2
p−2 2
2p
z 2 ≤ c( p)|z| p ≤ c( p)λh |ξ |2 p ,
and the right inequality of (23) is proved. To prove the lower bound, let us come back to the first equality in (24). Using again (H2) and Lemma 2.2, we obtain p−2 2
λ2h gh (ξ ) ≥ c( p)(µ2 + (det(Ah + λh ξ ))2 + (det Ah )2 )
(det(Ah + λh ξ ) − det Ah )2
≥ c( p)|V (det(Ah + λh ξ )) − V (det Ah )|2 ≥ c(µ, M, p)|V (det(Ah + λh ξ ) − det Ah )|2 , where we used (7)iv and (7)vi (remember that det Ah ≤ M 2 ) for the last two inequalities. Step 4: Upper bound. We consider, for every r < 1, 2 D w(y)2 + gh D 2 w(y) dy. Ih,r (w) = Br
Note that, since g (t)
≥ 0, the previous integrand is non negative; moreover vh is a minimizer of Ih,r , amongst functions w ∈ W 2,2 (Br ) which agree with vh outside some compact subset of Br . Fix 14 < s < 1. Passing to a subsequence we may always assume that lim [Ih,s (vh ) − Ih,s (v)]
h→∞
exists. We shall prove that lim [Ih,s (vh ) − Ih,s (v)] ≤ 0.
(25)
h→∞
Consider r > s and fix m ∈ N. Observe that, since v ∈ W 2,2 (B1 ) and vh ∈ W 2,2 (B1 ), p−1
Lemma 2.4, with λ = λh p , implies that there exist z h ∈ W 2,2 (B1 ) and r < 1 such that z h = v on Bsh
1 4
< s < sh < r h <
z h = vh on B1 \ Brh
and ⎛ ⎜ ⎝
⎞1
p−1 p
⎟ |D 2 z h |2 ⎠ + λh
Brh\Bsh
≤C
⎛
2
⎡
(r − s)ρ ⎢ ⎣ mρ
⎜ ⎝
⎞
1 2p
⎟ |D 2 z h |2 p ⎠
Brh\Bsh
1+
2
|Dl v|2 +
l=0
Br\Bs
+
m2 (r − s)2
2
|Dl vh |2
l=0
|D(v − vh )| + 2
m2 (r − s)2
|v − vh |
2
⎤1 2
⎥ ⎦ ,
(26)
123
226
M. Carozza et al.
since we may always suppose that λh ≤ 1. Note that, by (21), D 2 v is locally bounded on B1 ; so, using the minimality of vh and (23), we obtain Ih,s (vh ) − Ih,s (v) ≤ Ih,rh (vh ) − Ih,rh (v) + Ih,rh (v) − Ih,s (v) = Ih,rh (vh ) − Ih,rh (v) + |D 2 v|2 + gh D 2 v Brh\Bs
≤ Ih,rh (z h ) − Ih,rh (v) + c( p, M, µ) (r − s) |D 2 z h |2 + gh D 2 z h + c( p, M, µ) (r − s). ≤
(27)
Brh\Bsh
r −s Thanks to (23), we get by (26), using the fact that < 1 and that the quantity in the m square brackets is greater than or equal to 1, 2 p−2 |D 2 z h |2 + λh |D 2 z h |2 p Ih,rh (z h ) − Ih,rh (v) ≤ c(µ, p, M) Brh\Bsh
≤ c(µ, p, M)
+
2
(r
m 2ρ
|Dl vh |2 +
l=0
⎡
− s)2ρ
⎢ ⎣
1+
Br \Bs
m2 (r
− s)2
2
|Dl v|2
l=0
⎤p
⎥ |D(v − vh )|2 + |v − vh |2 ⎦ (28)
Thanks to (13), we get lim sup[Ih,rh (z h ) − Ih,rh (v)] ≤ c(µ, p, M, r, s)m −2ρ , h→∞
and, passing to the limit as h tends to infinity in (27), (25) is proved letting first m → ∞ and then r → s. Step 5: Lower bound. We shall prove that, for a.e. 41 < r < 21 , if τ < r , then 2 D v − D 2 v h 2 c(µ, p, M) lim sup h→∞
+λ−2 h |V
Bτ
det Ah + λh D 2 v − D 2 vh − det Ah )|2
≤ lim [Ih,r (vh ) − Ih,r (v)]. h→∞
For any Borel set A ⊂ B1 , let us define νh (A) =
2
|Dl vh |2 d x .
A l=0
Passing possibly to a subsequence, since νh (B1 ) ≤ c, we may suppose that νh ν weakly∗ in the sense of measures,
123
Partial regularity for polyconvex functionals depending on the Hessian determinant
227
where ν is a Borel measure over B1 , with finite total variation. Then for a.e. r < 1 ν(∂ Br ) = 0; let us choose such a radius r . Consider 41 < s < r , also such that ν(∂ Bs ) = 0, and fix m ∈ IN . Observe that, as vh ∈ W 2,2 (B1 ), Lemma 2.4 implies that there exist z h ∈ W 2,2 (B1 ) and 41 < s < sh < rh < r < 21 such that z h = vh on Bsh
z h = vh on B1 \Brh r −s r h − sh ≥ 3m
and ⎛ ⎜ ⎝
Brh\Bsh
⎞1
⎛
2
p−1 p
⎟ |D 2 z h |2 ⎠ + λh ⎡
(r − s)ρ ⎢ ≤C ⎣ mρ
⎜ ⎝
⎞
1 2p
⎟ |D 2 z h |2 p ⎠
Brh\Bsh
1+
Br \Bs
2
⎤1 2 ⎥ |Dl vh |2 ⎦ .
(29)
l=0
Passing possibly to a subsequence, we may suppose that z h vr,s weakly in W 2,2 (B1 ), sh → s , rh → r < 21 , and vr,s = v in (B1 \ Br ) ∪ Bs Consider ζh ∈ Cc∞ (Brh ) such that 0 ≤ ζh ≤ 1, ζh = 1 on Bsh and |Dl ζh | ≤ l = 0, 1, 2, and set
C , (rh −sh )l
for
), ψh = ζh (z h − vr,s
= ρ v where vr,s
r,s is the convolution between the usual sequence of mollifiers ρ and vr,s . Now, setting v = ρ v, we observe that
Ih,rh (vh ) − Ih,rh (v ) = Ih,rh (vh ) − Ih,rh (z h )
+ Ih,rh (z h ) − Ih,rh vr,s + ψh
+ Ih,rh (ψh + vr,s ) − Ih,rh (vr,s ) − Ih,rh (ψh )
+ Ih,rh (vr,s ) − Ih,rh (v )
+ Ih,rh (ψh )
= Rh,1 + Rh,2 + Rh,3 + Rh,4 + Rh,5 .
(30)
In order to simplify the presentation, we treat the terms Rh, j j = 1, . . . , 5 in separate steps.
123
228
M. Carozza et al.
Estimate of Rh,1 To bound Rh,1 , first we observe that
|D 2 vh |2 + gh D 2 vh −
Ih,rh (vh ) − Ih,rh (z h ) = Brh\Bsh
|D 2 z h |2 + gh D 2 z h
Brh\Bsh
|D 2 z h |2 + gh D 2 z h .
≥− Brh\Bsh
On the other hand we have |D 2 z h |2 + gh D 2 z h ≤ c(µ, p, M) Brh\Bsh
2 p−2
|D 2 z h |2 + λh
Brh\Bsh
⎡
⎢ ≤ c(µ, p, M, r, s)m −2ρ ⎣
|D 2 z h |2 p ⎤p
(1 +
Br\Bs
2
⎥ |Dl vh |2 )⎦
l=0
so that
|D 2 z h |2 + gh D 2 z h ≤ c(µ, p, M, r, s)m −2ρ .
lim sup h→∞
Brh\Bsh
Hence we have lim inf Rh,1 ≥ −c(µ, p, M, r, s)m −2ρ
(31)
h→∞
Estimate of Rh,2 As far as the term Rh,2 is concerned, it yields Rh,2 =
2 2 ε 2
| − gh D 2 ψh + D 2 vr,s |D z h | + gh D 2 z h − |D 2 ψh + D 2 vr,s
Brh\Bsh
≥ −c(µ, p, M)
2 |D 2 ψh + D 2 vr,s | + λh
2 p−2
2p |D 2 ψh + D 2 vr,s |
Brh\Bsh
≥ −c(µ, p, M)
2 p−2
|D 2 z h |2 + λh
Brh\Bsh
−c(µ, p, M) Brh\Bsh
1 l=0
1 l=0
123
2 p−2
2p |D 2 vr,s |
m 2(2−l)
|Dl (z h − vr,s )|2 (r − s)2(2−l)
2 p−2 +λh
= −Sh,1 − Sh,2
2 |D 2 z h |2 p + |D 2 vr,s | + λh
m 2(2−l)
|Dl (z h − vr,s )|2 p (r − s)2(2−l)
(32)
Partial regularity for polyconvex functionals depending on the Hessian determinant
229
−s where we used the bound rh − sh ≥ r3m . Thanks to the Sobolev embedding Theorem we l l have that, for l = 0, 1, D z h → D vr,s strongly in L 2 p , so that
lim sup Sh,2 ≤ c(µ, p, M) h→∞
1 l=0
2 l
D vr,s − vr,s .
m 2(2−l) (r − s)2(2−l)
B1 2
To bound Sh,1 , observe that, for every h,
2 |D 2 vr,s | ≤c
Brh\Bsh
2 |D 2 vr,s − D 2 vr,s |
|D 2 vr,s |2 + c Br\Bs
B1 2
≤ c lim inf j→∞
2 |D 2 vr,s − D 2 vr,s |
Br\Bs
2
B1 2
≤ c lim inf j→∞
|D z j | + c 2
|D v j |2 + 2
(Br\Bs )\(Br j \Bs j )
|D z j | + c
+ c lim sup j→∞
2
2
Br j \Bs j
2 |D 2 vr,s − D 2 vr,s | .
(33)
B1 2
As usual, we estimate the second integral using (29), while the first one is less than or equal to c ν(Br \ Bs ). Moreover the term
2 p−2
|D 2 z h |2 + λh
|D 2 z h |2 p
Brh\Bsh
can be treated as in Step 4 dealing with (28). Hence lim inf Rh,2 ≥ −c(µ, p, M, r, s)m −2ρ − c ν(Br \ Bs ) h→∞
2 | + − c |D 2 vr,s − D 2 vr,s B1 2
− c(µ, p, M)
1 l=0
m 2(2−l) (r − s)2(2−l)
|Dl (vr,s − vr,s )|2 .
(34)
B1 2
Estimate of Rh,3 Our aim is to prove that
lim sup Rh,3 ≥ −c(µ, M) D 2 vr,s h→∞
L2
B1 2
D 2 ψ
L2 B 1
(35)
2
123
230
M. Carozza et al.
where ψ ε is the weak limit in W 2,2 (B1 ) of ψhε , which we can always suppose to exist. We first observe that 1 ε Rh,3 = 2 g det Ah + λh D 2 ψhε + D 2 vr,s − g det Ah + λh D 2 ψhε λh Brh
ε −g det Ah + λh D 2 vr,s + g(det Ah )] 1 ε + det Ah − 2 g (det Ah ) det Ah + λh D 2 ψhε + D 2 vr,s λh Brh
ε − det Ah + λh D 2 ψhε −det Ah + λh D 2 vr,s
ε D 2 ψhε D 2 vr,s
+2 Brh
1 = 2 λh
1
d ε g t det Ah + λh D 2 ψhε + D 2 vr,s +(1 − t)det Ah + λh D 2 ψhε dt
Brh 0
ε + (1 − t)det Ah dt −g t det Ah + λh D 2 vr,s ε ε g (det Ah )D 2 ψhε D 2 vr,s +2 D 2 ψhε D 2 vr,s − Brh
= Jh −
Brh
g (det Ah )D
2
ψhε
ε D 2 vr,s
Brh
+2
ε D 2 ψhε D 2 vr,s
Brh
≥ Jh − c(M) D
2
ε ψhε L 2 (B 1 ) D 2 vr,s
L 2 (B 1 ) 2
(36)
2
Let us deal with Jh . Computing the derivative and developing the determinants we get 1 Jh =
ε g t det Ah + λh D 2 ψhε + D 2 vr,s + (1 − t)det Ah + λh D 2 ψhε
0
ε ε ε + (1 − t)det Ah det(λh D 2 vr,s dt ) + Ah λh D 2 vr,s − g t det Ah + λh D 2 vr,s 1 +
2 2 ε ε +(1 − t)det Ah +λh D 2 ψhε · λh D ψh g t det Ah +λh D 2 ψhε + D 2 vr,s
0 ε D 2 vr,s
= Jh1 + Jh2 .
ε Denoting by ηh (t) = t det Ah + λh D 2 ψhε + D 2 vr,s + (1 − t)det Ah + λh 2 ε βh (t) = t det Ah + λh D vr,s + (1 − t)det Ah , again Jh1 can be written as 1 g (sηh (t) + (1 − s)βh (t)) det λh D 2 ψhε + Ah λh D 2 ψhε Jh = [0,1]2
123
(37) D 2 ψhε and
ε ε ε ds dt +tλ2h D 2 vr,s D 2 ψhε · det(λh D 2 vr,s ) + λh Ah D 2 vr,s
Partial regularity for polyconvex functionals depending on the Hessian determinant
231
ε g (sηh (t) + (1 − s)βh (t)) det λh D 2 ψhε + tλ2h D 2 vr,s D 2 ψhε
= [0,1]2
ε ε · det λh D 2 vr,s + λh Ah D 2 vr,s ds dt ε ds dt g (sηh (t) + (1 − s)βh (t)) Ah λh D 2 ψhε det λh D 2 vr,s + [0,1]2
+
ε ds dt. g (sηh (t) + (1 − s)βh (t))λ2h Ah D 2 ψhε Ah D 2 vr,s
[0,1]2
(38) Since, by (H2), g (sηh (t) + (1 − s)βh (t)) is bounded by a constant c = c(µ, p) and, by (12) and (14), converges to g (det A) a.e. in B1 , it is easily seen that the last term in (38), ε ). integrated in Brh and divided by λ2h , converges to B g (det A)(A D 2 ψ ε )(A D 2 vr,s r
On the other hand, the first two terms, integrated in Brh and divided by λ2h , tend to zero as h ε is in L ∞ (B , R2×2 ), tends to infinity, since g is bounded by a constant c = c(µ, p), D 2 vr,s 1 and D 2 ψhε is uniformly bounded in L 2 (B 1 , R2×2 ). 2 Hence we proved that 1 ε lim sup 2 Jh1 ≥ −c(µ, M) D 2 ψ ε L 2 (B 1 ) D 2 vr,s
L 2 (B 1 ) . 2 2 h→∞ λh
2
(39)
Brh
The last term to consider is
1 λ2h
Brh
Jh2 .
1 ε + (1 − t)det (Ah + λh We will denote by G h = 0 g t det Ah + λh D 2 ψhε + D 2 vr,s ε , so that G converges to g (det A) a.e. in B , D 2 ψhε dt and by Fh = D 2 ψhε D 2 vr,s h 1 ε weakly in L 2 (B , R2×2 ). Moreover while Fh tends to D 2 ψ ε D 2 vr,s 1 1 2 J = G h Fh . h λ2h
Brh
Brh
We will prove that G h Fh is uniformly bounded in L 1+σ (B1 , R2×2 ) for some σ > 0, so that 1 ε lim 2 Jh2 = g (det A)D 2 ψ ε D 2 vr,s . (40) h→∞ λ h Brh
Br
Using the growth condition on g , it is easy to prove that |g (t)| ≤ c( p) 1 + |t| p−1 . Thus, 1 ε 1 + |t det Ah + λh D 2 ψhε + D 2 vr,s |G h Fh | ≤ c( p) 0
ε | +(1 − t)det Ah + λh D 2 ψhε | p−1 · |D 2 ψhε ||D 2 vr,s
123
232
M. Carozza et al.
ε p−1 ≤ c( p) 1 + |det Ah + λh D 2 ψhε | p−1 + det λh D 2 vr,s ε p−1 2 ε 2 ε D ψh D vr,s + Ah + λh D 2 ψhε λh D 2 vr,s % p−1 2 ε % 2 ε ε D ψ %D v ≤ c( p, M) D 2 ψhε % D 2 vr,s + det λh D 2 ψhε r,s h p−1 2 ε 2 ε ε ε +|λh D 2 ψhε | p−1 |D 2 ψhε ||D 2 vr,s | + |det λh D 2 vr,s |D ψh ||D vr,s
|
ε p−1 2 ε ε ε p−1 2 ε ε D ψh ||D 2 vr,s + λ2h D 2 ψhε ||D 2 vr,s D ψh ||D 2 vr,s . + λh D 2 vr,s (41)
At this point, using (16) it is not difficult to prove that G h Fh is uniformly bounded in 2p 2p > 1. L 3 p−2 (B1 , R2×2 ), where 3p − 2 Putting together (39) and (40) we get
lim sup Jh ≥ −c(µ, M)||D 2 vr,s || h→∞
||D 2 ψ || . L2 B 1 L2 B 1 2
(42)
2
Hence inserting (42) in (36) , we conclude with (35). Estimate of Rh,4 To bound Rh,4 we observe that 2 2 D v + g h D 2 v − D 2 v 2 − g h D 2 v
Rh,4 = r,s r,s Brh\Bs
2 2 D v + gh D 2 v ≥ −c (M, p, µ) |Br \ Bs− |.
≥− Brh\Bs−
Then lim inf Rh,4 ≥ −c(M, p, µ)|Br \ Bs− |. h→∞
(43)
Estimate of Rh,5 Estimate (23), together with the definitions of ψh and z h , implies Rh,5 = Ih,rh ψh
2 2 D ψ + gh D 2 ψ
= h h Brh
≥
2
D v − D 2 v h 2
Bτ
+c (µ, M, p)
V (det Ah + λh D 2 v − D 2 vh − det Ah )2 , λ−2 h
Bτ
for small enough and for every radius τ <
123
1 4
< rh .
(44)
Partial regularity for polyconvex functionals depending on the Hessian determinant
233
Conclusion of the lower bound Passing to a subsequence we may suppose that lim sup Rh,5 = lim Rh,5 . h→∞
h→∞
Therefore returning to (30), from (31), (34), (35), (43) and (44) we get ⎛ ⎜ lim inf [Ih,r (vh ) − Ih,r (v )] ≥ lim sup ⎝ D 2 v − D 2 vh |2 h→∞
h→∞
+c(µ, M, p)
Bτ
⎞
V det Ah + λh D 2 v − D 2 vh − det Ah 2 ⎟ λ−2 ⎠ h
Bτ
−c(M, p, µ) |Br \Bs− | − c ν (Br \Bs ) − c (µ, M) D 2 vr,s
L 2 (B 1 ) ||D 2 ψ || 2
−c(µ, p, M, r, s)m −2ρ −c
2 |D 2 vr,s − D 2 vr,s | −
B1
1
m 2(2−l)
l=0
(r −s)2(2−l)
2
L2 B 1 2
2 l
D vr,s −vr,s . B1 2
(45) We would like to pass to the limit as ε tends to zero in (45). Before doing that, let us note that ε 2 ε 2 2 2 lim sup |Ih,r (v ) − Ih,r (v)| ≤ |D v | − |D v| h→∞ Br 2 ε +c(µ, M) (46) |D v | + |D 2 v| |D 2 v ε − D 2 v|, Br
2 ε 2 2 2 2 2 |D v − D vh | − |D v − D vh | lim h→∞ Bτ 2 ε 2 ε 2 2 2 2 2 = |D v | − |D v| − 2D v D v − D v , Bτ
(47)
and
lim sup λ−2 h h→∞
V det Ah + λh (D 2 v ε − D 2 vh − det Ah 2
Bτ
2 − V det Ah + λh D 2 v − D 2 vh − det Ah ≤ c(µ, M)ω(ε), (48)
where ω(ε) goes to zero as ε tends to zero. The proof of (47) is obvious, while to get (46) it is enough to use the definition of gh , the growth condition on g and (12). We postpone the proof of (48) to the Appendix.
123
234
M. Carozza et al.
On the other hand, arguing as in (32), (33), and (34) to bound Rh,2 , and considering that ε a.e. in B and ψ ε = 0 a.e. in B \ B , we have ψ ε = vr,s − vr,s 1 s r 1 2 2 2 m 2(2−l) l
D ψ | ≤ c(µ, p, M) |D vr,s − vr,s (r − s)2(2−l) l=0
B1
+c
B1 2
|D vr,s − 2
2 D 2 vr,s |
+ c(µ, p, M, r, s)m −2ρ + c ν(Br \Bs ).
(49)
B1 2
At this point, thanks to (46), (47), and (48) we can pass to the limit, as → 0, in (45), and using (49) we get lim inf [Ih,r (vh ) − Ih,r (v)] h→∞ 2 2 2 ≥ lim sup D 2 v− D 2 vh |2 +c(µ, M, p)λ−2 h |V det A h +λh D v− D vh −det A h h→∞
Bτ
− c(M, p, µ) |Br \Bs | − c ν(Br \Bs ) − c(µ, p, M, r, s)m −2ρ , Finally, letting m tend to infinity and s tend to r we obtain 2 2 2 lim sup D 2 v − D 2 vh |2 + c(µ, M, p)λ−2 h |V det A h + λh D v − D vh − det A h h→∞
Bτ
≤ lim[Ih,r (vh ) − Ih,r (v)]. h
Step 6: Conclusion of the proof of Proposition 3.1. From the previous two steps we conclude that, for any Bτ , with 0 < τ < 41 , 2 D v − D 2 vh 2 = 0, lim
(50)
h→∞
1 h→∞ λ2 h
lim
Bτ
V det Ah + λh D 2 v − D 2 vh − det Ah )2 = 0.
(51)
Bτ
In order to obtain a contradiction to (10), we first use (50) to compute U (x h , τrh ) = lim h→∞ h→∞ λ2h lim
2 2 1 2 D vh − D 2 vh + 2 V det λh D 2 vh − D 2 vh τ τ λh
Bτ
2 2 2 = . D v − D 2 v + lim sup λ−2 V det λh D 2 vh − D 2 vh h τ
Bτ
We will prove that lim sup λ−2 h h→∞
h→∞
τ
Bτ
2 2 V det λh D vh − D 2 vh = 0, τ
Bτ
Let us prove (52). First observe that (51) is equivalent to 1 V (det(λh (D 2 v − D 2 vh )) + Ah λh (D 2 v − D 2 vh )2 = 0. lim 2 h→∞ λ h Bτ
123
(52)
(53)
Partial regularity for polyconvex functionals depending on the Hessian determinant
235
We can now estimate Vh : 1 V (det λh D 2 vh − D 2 v + det λh D 2 v − (D 2 vh )τ Vh = 2 λh Bτ
2 +λ2h D 2 vh − D 2 v D 2 v − D 2 vh τ 2 1 V (det λh D 2 vh − D 2 v 2 + V (det λh D 2 v − D 2 vh ≤ c( p) 2 τ λh Bτ
2 +V λ2h D 2 vh − D 2 v D 2 v − D 2 vh τ = Vh,1 + Vh,2 + Vh,3 , having used (7)iii . Since we may suppose λh ≤ 1, (7)ii yields 2 2 2 p−2 = 0, V det D v − D 2 vh lim sup Vh,2 ≤ lim sup c( p)λh τ h→∞
h→∞
Bτ
since 2 p − 2 > 0 and the constant vector (D 2 vh )τ converges. Moreover, by (7)i , we have 2 2 p−2 D v − D 2 vh p |D 2 v − D 2 vh | p = 0, lim sup Vh,3 ≤ lim sup c( p)λh τ h→∞
(54)
h→∞
(55)
Bτ
thanks to (50). As far as Vh,1 is concerned, we add and subtract into the argument of V the quantity Ah λh (D 2 v − D 2 vh ) and we use (7)iii , (7)i , and (53) to get lim sup Vh,1 h→∞
1 ≤ lim sup c( p) 2 λ h→∞ h
2 V det λh D v − D 2 vh + Ah λh D 2 v − D 2 vh 2
Bτ
2 2 1 2 + lim sup c( p) 2 V − Ah λh D v − D vh λh h→∞ Bτ 2 ≤ lim sup c(µ, p) |Ah |2 D 2 v − D 2 vh = 0, h→∞
(56)
Bτ
the last limit being zero thanks to (50). Combining (54), (55), and (56) we finally get (52). Then, thanks to (22), U (x h , τrh ) lim = h→∞ λ2h
2 D v − D 2 v 2 ≤ C(M, µ)τ 2 , τ
Bτ
which contradicts (10), if we choose C M,µ = 2C(M, µ).
123
236
M. Carozza et al.
4 Proof of Theorem 2.1 In this section we finally prove our main Theorem stated in Sect. 2. The proof rests on a standard iteration argument involving U (x0 , r ), essentially based on Proposition 3.1. Lemma 4.1 Let 0 < α < 1 and M > 0, then there exist 0 < τ < depending on α and M such that if B(x, r ) ⊂ , | D 2 u x,r | ≤ M
1 4
and ε > 0, both
and U (x, r ) ≤ ε then 2α U (x, r ) U x, τ l r ≤ τ l for each l ∈ N. Proof The proof relies on the same arguments of Lemma 6.1 in [20], iterating Proposition 3.1 and adapting the proof to the different structure of U (x, r ). Before ending we still need a result, which can be easily obtained arguing as in Lemma 6.2 of [20]. 2,2 Lemma 4.2 Let u ∈ Wloc () be a function such that V (det (D 2 u)) ∈ L 2loc (), then 2 V det D u − (D 2 u)x,r 2 d x = 0 lim r →0 B(x,r )
for almost every x ∈ . The proof of the main Theorem is now a consequence of a standard iteration procedure based on Lemma 4.1 (for more details see [22]). The singular set turns out to be contained in the complement of 0 =
⎧ ⎪ ⎨ ⎪ ⎩
x ∈ : lim(D 2 u)x,r = D 2 u(x) and lim r
r →0 B(x,r )
⎫ ⎪ ⎬
2 dx = 0 V det D 2 u − D 2 u ⎪ x,r ⎭
so that by Lemma 4.2 we have | \ 0 | = 0
5 Appendix In this section we want to prove that −2 V det Ah + λh D 2 v ε − D 2 vh − det Ah 2 lim sup λh h→∞ Bτ 2 − V det Ah + λh D 2 v − D 2 vh − det Ah ≤ c(µ, M)ω(ε),
123
(57)
Partial regularity for polyconvex functionals depending on the Hessian determinant
237
To this aim we observe that 2 1 V det Ah + λh D 2 v ε − D 2 vh − det Ah 2 λ h
Bτ
2 2 2 − V det Ah + λh D v − D vh − det Ah 1 V det λh D 2 v ε − D 2 vh + Ah λh D 2 v ε − D 2 vh ≤ c( p, µ) 2 λh Bτ 2 −det λh D v − D 2 vh − Ah λh D 2 v − D 2 vh × V det λh D 2 v ε − D 2 vh + V Ah λh D 2 v ε − D 2 vh +V det λh D 2 v − D 2 vh + V Ah λh D 2 v − D 2 vh 1 V det λh D 2 v ε − λ2 D 2 v ε − D 2 v D 2 vh = c( p, µ) 2 h λh Bτ +Ah λh D 2 v ε − D 2 v − det λh D 2 v · βhε 1 V det λh D 2 v ε + V λ2 D 2 v ε − D 2 v D 2 vh ≤ c( p, µ) 2 h λh Bτ +V Ah λh D 2 v ε − D 2 v + V det λh D 2 v · βhε 1 = c( p, µ) 2 αhε · βhε (58) λh Bτ
having used (7)v and (7)iii . By (7)i , (7)ii , (7)iii , and (15) we deduce that there is a constant C independent of ε such that 1 lim sup 2 (βhε )2 ≤ C, (59) h→∞ λh Bτ
while, using again (7)i , it yields 2 1 lim sup 2 (αhε )2 ≤ c(µ, M) D 2 v ε − D 2 v , h→∞ λh Bt
Bt
so that lim lim sup
ε→0 h→∞
1 λ2h
αhε
2
= 0.
(60)
Bt
Inserting (59) and (60) in (58), we conclude with (57).
References 1. Acerbi, E., Fusco, N.: Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86, 125–145 (1984) 2. Acerbi, E., Fusco, N.: A regularity theorem for minimizers of quasiconvex integrals. Arch. Ration. Mech. Anal. 99, 261–281 (1987) 3. Acerbi, E., Fusco, N.: Regularity for minimizers of non-quadratic functionals: the case 1 < p < 2. J. Math. Anal. Appl. 140, 115–135 (1989)
123
238
M. Carozza et al.
4. Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 69, 337–503 (1977) 5. Bildhauer, M., Fuchs, M.: Higher order variational problems with non-standard growth condition in dimension two: plates with obstacles. Annal. Acad. Scie. Fennicae Mathematica 26, 509–518 (2001) 6. Carozza, M., Fusco, N., Mingione, G.: Partial regularity of minimizers of quasiconvex integrals with subquadratic growth. Ann. Mat. Pura Appl. (4) 175, 141–164 (1998) 7. Carozza, M., Passarelli di Napoli, A.: A regularity theorem for quasiconvex integrals: the case 1 < p < 2. Proc. Royal Soc. Edinburgh 126 A, 1181–1199 (1996) 8. Carozza, M., Passarelli di Napoli, A.: Partial regularity of local minimizers of quasiconvex integrals with subquadratic growth. Proc. Royal Soc. Edinburgh 133 A, 1249–1262 (2003) 9. Carozza, M., Passarellidi Napoli, A.: Partial regularity for anisotropic functionals of higher order. ESAIM: COCV Control Optim. and Calc. Var. 13(4), 692–706 (2007) 10. Carriero, M., Leaci, A., Tomarelli, F.: Strong minimizers of Blake–Zisserman functional. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 15, 257–285 (1997) 11. Choksi, R., Khon, R.V., Otto, F.: Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy. Comm. Math. Phys. 201, 61–79 (1999) 12. Conti, S., Fonseca, I., Leoni, G.: A convergence result for the two gradient theory of phase. Comm. Pure Appl. Math. 55, 857–936 (2002) 13. DeSimone, A.: Energy minimizers for large ferromagnetic bodies. Arch. Ration. Mech. Anal. 125, 99–143 (1993) 14. Duzaar, F., Grotowski, J.F., Kronz, M.: Regularity of almost minimizers of quasi-convex integrals with subquadratic growth. Annali di Mat. 184, 421–448 (2005) 15. Esposito, L., Mingione, G.: Partial regularity for minimizers of degenerate polyconvex energies. J. Convex Anal. 8, 1–38 (2001) 16. Esposito, L., Mingione, G.: Relaxation for higher order integrals below the natural growth exponent. Diff. Int. Equations 15, 671–696 (2002) 17. Evans, L.: Quasiconvexity and partial regularity in the calculus of variations. Arch. Ration. Mech. Anal. 95, 227–252 (1984) 18. Fonseca, I., Malý, J.: From Jacobian to Hessian: distributional form and relaxation. Riv. Mat. Univ. Parma 7, 4*, 45–74 (2005) 19. Fusco, N., Hutchinson, J.: C 1,α partial regularity of functions minimizing quasiconvex integrals. Manu. Math. 54, 121–143 (1985) 20. Fusco, N., Hutchinson, J.: Partial regularity in problems motivated by nonlinear elasticity. SIAM J. Math. Anal. 22, 1516–1551 (1991) 21. Fusco, N., Hutchinson, J.: Partial regularity and everywhere continuity for a model problem from non linear elasticity. Bull. Austr. Math. Soc. 22, 1516–1551 22. Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Anal. Math. Stud., vol. 105, Princeton University Press, NJ (1983) 23. Giaquinta, M., Modica, G.: Partial regularity of minimizers of quasiconvex integrals. Ann. Inst. H. Poincaré 3, 1516–1551 (1986) 24. Guidorzi, M.: A remark on partial regularity of minimizers of quasiconvex integrals of higher order. Rend. Ist. Mat. Trieste XXXII, 1–24 (2000) 25. Guidorzi, M.: Partial regularity in nonlinear elasticity. Manu. Math. 107, 25–41 (2002) 26. Guidorzi, M., Poggiolini, L.: Lower semicontinuity for quasiconvex integrals of higher order. NoDEA Nonlinear Diff. Eqs. Appl. 6, 227–246 (1999) 27. Hamburger, C.: Partial regularity of minimizers of polyconvex variational integrals. Calc. Var. 18, 221–241 (2003) 28. Kronz, M.: Partial regularity results for minimizers of quasiconvex functionals of higher order. Ann. Inst. H. Poincaré - Anal. Non linéaire 19(1), 81–112 (2002) 29. Kohn, R.V., Müller, S.: Surface energy and microstructure in coherent phase transiction. Comm. Pure Appl. Math. 47, 405–435 (1994) 30. Meyers, N.: Quasi-convexity and lower semicontinuity of multiple integrals of any order. Trans. Am. Math. Soc. 119, 125–149 (1965) 31. Müller, S.: Variational models for microstructures and phase transiction. Lectures Notes MPI Leipzig, (1998) 32. Owen, D.R., Paroni, R.: Second order structured deformations. Arch. Ration. Mech. Anal. 155, 215–235 (2000) 33. Passarelli di Napoli, A.: A regularity result for a class of polyconvex functionals. Ricerche Mat. 379–393 (1999)
123
