On a $Γ$-limit of Willmore functionals with additional curvature penalization term
Heiner Olbermann

 




  


  
 
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H. Olbermann] Heiner
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On a Γ-limit of Willmore functionals with additional curvature
  penalization term
    [
    December 30, 2023
==================================================================================





We consider the Willmore functional on graphs, with an
additional penalization of the area where the curvature is
non-zero. Interpreting the penalization parameter as a Lagrange multiplier, this
corresponds to the Willmore functional with a constraint on the area where
the graph is flat. Sending the
penalization parameter to ∞ and rescaling suitably, we derive the limit
functional in the sense of Γ-convergence. 




§ INTRODUCTION





 §.§ Motivation: A constrained Willmore problem


The motivation for the present paper comes from  a constrained Willmore
problem. More precisely, let us  consider smooth hypersurfaces M⊂^3  of a fixed topology,  with 
constraints on the
amount of surface area where the surface is flat and non-flat
respectively. Here, by flat we mean that  the second fundamental form vanishes. Within this class, we are
interested in the variational problem

    inf_M∫_M (H^2-2 K) ̣̋^2=inf_M∫_M(κ_1^2+κ_2^2)̣̋^2 ,

where κ_1,
κ_2 denote  the principal curvatures, H=κ_1+κ_2  the mean curvature, K=κ_1κ_2 the Gauss curvature,  and ^̋2 is the
two-dimensional Hausdorff measure.



Here we are going to simplify this problem in two ways: First, we are not going
to consider arbitrary surfaces, but only  graphs.  Secondly, we are going to
replace the constraint of having a fixed amount of non-flat surface area by a
penalization of the non-flat part. This is the usual attempt of capturing
constraints via the introduction of Lagrange multipliers. We will however not be   able to prove a rigorous equivalence between the constrained variational
  problem and the problem involving Lagrange multipliers.

The latter consists in, for  λ>0 and a set of graphs M   with fixed
surface area, in  the variational problems

    inf_M∫_M (H^2-2 K) ̣̋^2+λ^̋2({x∈ M:S_M≠ 0}) ,

where S_M denotes the second fundamental form of M. Additionally, boundary conditions or other constraints on M may be imposed.



Obviously, the shape of  minimizers  for such
a problem depend on the  penalization parameter λ. One expects that the
concentration of curvature increases with  λ,  i.e.,  the area where the
surface is flat
 becomes larger as λ increases (for configurations of low energy). The main purpose of the
present paper is a rigorous investigation of the limit λ→∞ for
the variational problem (<ref>).






 §.§ Statement of main result

For any Borel set U⊂^n, 
let
(U) denote the set of signed Radon measures on U.   We denote
by (U;^p) the ^p valued  Radon measures on U. Let ^n× n_sym denote the symmetric n× n
matrices, and let
(U;^n× n_) denote the space of measures with values in the symmetric matrices, i.e., {μ∈(U;^n×
  n):μ_ij=μ_ji for i≠ j}. 
For μ∈(U;^p), let |μ| denote the total variation
measure. 



For μ∈(U;^p), we have by the Radon-Nikodym differentiation Theorem (see
Theorem <ref> below) that for |μ|-almost every x∈ U, the derivative
μ̣/|̣μ| exists.
For any one-homogeneous function h:^p→ and any μ∈(U;^p), we may hence define 

    h(μ)=h(μ̣/|̣μ|)|̣μ| .

This is a well defined Borel measure.





For ξ∈, let τ_1(ξ), τ_2(ξ) denote the eigenvalues
of ξ. 
We set

    ρ^0(ξ):=∑_i=1^2 |τ_i(ξ)| .

We will repeatedly use the following estimates:

    |ξ|≤ρ^0(ξ)≤ 2|ξ| .

Note that ρ^0:→ is sublinear and positively one-homogeneous.



Let Ω⊂^2 be an open  bounded  set
with smooth boundary,
 and let u∈ BH(Ω); that is the space of u∈
W^1,1(Ω) such that ∇ u∈ BV(Ω;^2), where ∇ u
denotes the approximate gradient of u. 
We will use the usual notation for the BV function ∇ u:
J_∇ u denotes the jump set of ∇ u. On J_∇ u, there
exists a measurable function ν_∇ u with values in S^2 such that ∇ u has
well defined limits ∇ u^± on both sides of the hyperplane defined by
ν_∇ u. The set S_∇ u is the singular set of D∇ u, i.e., the set where
D∇ u is not absolutely continuous w.r.t. Ł^2. (The operator “D”
denotes the distributional derivative.)  Furthermore, C_∇
  u:=S_∇ u∖ J_∇ u. We have the decomposition

    D∇ u=∇^2 u Ł^2+(∇ u^+-∇ u^-)⊗ν_∇ u J_∇ u+
    D^s∇ u  C_∇
      u .

Let C_0(Ω) denote the completion of C_c^0(Ω) with respect to the sup-norm. We say that a sequence μ_j∈ℳ(Ω) converges weakly * in the sense of measures to μ∈ℳ(Ω) if for any f∈ C_0(Ω), we have

    ∫_Ω fμ̣_j→∫_Ωfμ̣ .

The convergence of vector-valued measures is defined analogously.
For a sequence u_j∈ BV(Ω), we say that u_j→ u weakly * in BV if
u_j→ u in W^1,1 and D u_j→ Du weakly * in the sense of measures.







For v∈^2 and
ξ∈^2× 2, we define

    (v)   =1/√(1+|v|^2)([  v; -1 ]) ,
    (v)   =𝕀_2× 2+v⊗ v ,
    (v,ξ)   =1/√(1+|v|^2)ξ
    
      S(v,ξ)   =(v)^-1/2(v,ξ)(v)^-1/2 .

By this definition, S(∇ u(x),∇^2 u(x))∈ is the second fundamental form (or
shape operator) of the graph of u at u(x) in matrix form (supposing u is
sufficiently smooth at x); its eigenvalues are the
principal curvatures of the graph.
Let F_λ:^2× 2→ be defined by

    F_λ(ξ)= 0     if ξ=0
    
    |ξ|^2+λ    else.

We define _λ: W^2,2(Ω)→[0,+∞) by

    _λ(u)=λ^-1/2∫_ΩF_λ(S(∇ u,∇^2
      u))√(1+|∇ u|^2)x̣ .

Note that up to a normalizing factor, for smooth functions u, the right hand side is precisely the functional introduced in the
previous subsection,

    λ^1/2_λ(u)=∫_(u)(H^2-2 K)̣̋^2+λ^̋2({x∈(u):S_(u)≠ 0})  ,

where (u) denotes the graph of u.
For u∈ W^2,2(Ω), the right hand side in (<ref>) is 
 finite, since the Willmore integrand
√(1+|∇ u|^2)|S|^2 is bounded from above by |∇^2 u|^2 (see
Lemma <ref> below).
Let arccos:[-1,1]→[0,π] be the inverse function of
cos:[0,π]→[-1,1], and for v=(v_1,v_2)^T∈^2, let v^=(-v_2,v_1)^T.  We define : BH(Ω)→[0,+∞) by

    (u)   =2∫_Ωρ^0(S(∇ u,∇^2 u))√(1+|∇ u|^2)x̣
         +2∫_C_∇ uρ^0(S(∇
          u,Ḍ∇ u/|̣D∇ u|))√(1+|∇ u|^2)|̣D∇ u| C_∇ u
         +2∫_J_∇ uarccos (∇ u^+)· (∇
        u^-)√(1+|ν_∇ u^·∇ u|^2)̣̋^1 .

Again, the right hand side always exists and is finite, since ρ^0(S(v,ξ))√(1+|v|^2)≤
2|ξ|, and hence the integrands can be estimated by the Lebesgue regular, jump
and Cantor part of the measure ρ(D∇ u) respectively.
Finally, let us write =BH(Ω)∩ W^1,∞(Ω).



Our main result is the following theorem, which establishes the
Γ-convergence _λ→ in the weak-* topology of BH(Ω).




  
  
  (i) Let u_λ be a sequence in W^2,2(Ω) with
    lim sup_λ→∞_λ(u_λ)<∞, ∫_Ω
    u_λx̣=0 and ∇ u_λ_L^∞<C. Then there exists a
    subsequence (no relabeling) and u∈ such that

    
    u_λ→ u   in W^1,1(Ω),   ∇ u_λ→∇
    u weakly * in BV(Ω;^2) .


  (ii)
Let u_λ, u be as in (<ref>). Then we have

    lim inf_λ→∞_λ(u_λ)≥(u) .


  (iii)
Let u∈. Then there exists a sequence u_λ such that (<ref>)
is fulfilled and 

    lim sup_λ→∞_λ(u_λ)≤(u) .

  



  
  
  (i) For u∈ C^2(Ω), the limit functional  can be written as

    (u)=∫_(u)2ρ^0(S_(u))̣̋^2 ,

where  (u) denotes the graph of u. 
The formula for  from the statement of the theorem is a generalization
for surfaces whose second fundamental form is a measure. We note that graphs of
functions in BH(Ω) do not belong to the class of curvature varifolds as
defined in  <cit.>. The latter do
not allow for a Cantor part in the curvature measure. 

  (ii) For the “geometrically linearized” functionals

    𝒢_λ(u)=λ^-1/2∫_Ω F_λ(∇^2 u)x̣

we have shown in <cit.> that the limit functional (again in
the sense of Γ-convergence) is given by 𝒢(u)=2∫_Ω(ρ^0(D∇ u)). Here we merely replace the
second derivative ∇^2 u by the second fundamental form S(∇
u,∇^2 u). However, the presence of lower order terms makes the analysis
more difficult for several reasons. There exist a few different techniques for the
proof of lower semicontinuity of integral functionals that depend on lower order
terms starting from the results without those terms, see
<cit.>. These
techniques do not work here since we consider  the convergence ∇
u_λ→∇ u weakly * in BV (and not in W^1,p with p>1 as in
the quoted references). The lower semicontinuity in BV for integral functionals that
depend on lower order terms has been treated in <cit.>. Their
technique cannot be applied in a straightforward way here either, the reason
being that for fixed λ the integrands of our functionals have 2-growth
at infinity. Our technique will be a modification of the one from
<cit.>, choosing a cutoff that despite the 2-growth does not increase
the energy by too much. 

Carrying on with the comparison of our result with the one in <cit.>,
we would like to point out that we are able to determine the form of the Γ-limit
on the jump part explicitly, which is not possible in the general situation
treated in <cit.>. This requires the solution of a certain variational
problem that we obtain through some geometric considerations (see Section
<ref>).

Concerning the upper bound, this is more difficult here than in
<cit.> again because of the presence of  lower order
terms. In that reference, the upper bound follows directly from well known
properties of approximations of BV functions by mollification. Here, we need
to keep track of the behavior of the lower order terms in this approximation
process, for which we need to use some results on the fine properties of BV functions.


  (iii)
The requirement ∇ u_λ_L^∞<C in the
  compactness part of the theorem (statement (i)) may seem unnatural. Without such an
  assumption however, we are not able to obtain control of the BH-norm from
  the energy alone. This can be seen by considering graphs of functions with
  almost vertical parts. The energy of these almost vertical parts can be made
  arbitrarily small.  In this way, we might obtain functions of arbitrarily
  large L^1 norm with bounded energy. This can be considered as an artefact of
  the restriction to graphs, and shows that a geometric description would be
  more appropriate. This will be the topic of future work.

The requirement ∫_Ω u_λx̣=0 is included in the statement (i) to enforce the
convergence u_λ→ u in L^1. Without such an assumption, we would
still have the convergence ∇ u_λ→∇ u weakly * in BV (for
a subsequence).







 §.§ Scientific context

Vesicles of polyhedral shape play an important role in biology. Examples are
virus capsids <cit.>, carboxysomes <cit.>, cationic-ionic vesicles, and assembled
supramolecular structures <cit.>. 
In <cit.>, a model
for the formation of polyhedral structures 
based on minimization of the free elastic  energy of topologically spherical
shells has been suggested. In the model, the free energy
is a function of the deformation of the shell, and the material distribution of
the two elastic components that the shell is made of. 










Elastic inhomogeneities are known to exist in many virus
capsids and for carboxysomes; in both of these cases, the vesicle shell is made
up of different protein types. In <cit.>,
 it has been suggested that the inhomogeneities can  act as the
driving force for faceting.
In this reference, it is assumed that the vesicle wall consists of two
components, with different elastic properties (“soft” and “hard”), and the amount of soft and hard material available for the formation of the vesicle is fixed.
The variational problems (<ref>) and (<ref>),
interpreted as minimization problems for the free elastic energy, are models for
such two-phase vesicles. Following this interpretation, we investigate here the
limit in which the contrast between soft and hard phase is large (the hard
phase does not bend at all), and there is
a very small amount of soft material.



 §.§ Comparison to the analogous one-dimensional problem


Consider the following variational problem, which is a lower dimensional analogue for problem
(<ref>), with the topology fixed to be that of a
sphere instead of a graph:

    inf{∫_Mκ^2 ṣ+λ^̋1({x∈ M:κ≠ 0}):M homeomorphic to S^1, ^̋1(M)=2π}

Pulling the penalization term into the integral, we obtain

    inf{∫_Mf̃_λ(κ) ṣ :M homeomorphic to S^1, ^̋1(M)=2π} ,

with

    f̃_λ(κ)=λ+κ^2    if κ≠ 0
    
    0    else.

It is well known that such a problem requires relaxation to guarantee the
existence of minimizers. The relaxed problem is obtained by  replacing the integrand
with its convex lower semicontinuous envelope, 

    f̃_λ^**(κ)=2√(λ)|κ|    if |κ|≤√(λ)
    λ+κ^2    else.

We see immediately  that the integrands λ^-1/2f̃_λ^**
are monotone decreasing, and converge to the function κ↦
2|κ|. From this convergence, one  deduces without difficulty the
Γ-convergence of the respective integral functionals, with respect to weak *
convergence of the curvatures. The limit
functional ^(1):M↦ 2∫_M|κ|ṣ is also defined for  one-spheres whose
curvature is only a measure. Note that there is a large set of minimizers for
^(1): Any one-sphere with non-negative curvature will be a
minimizer. 



The situation in dimension two is completely different: From Theorem
<ref>, it is natural to conjecture that one may define a limit
functional in the sense of Γ convergence that for smooth surfaces is
given by (<ref>).
For surfaces
of convex bodies, this
functional  is the same as the
total mean curvature. For sufficiently smooth surfaces, it is
known that the only minimizer of this functional within the class of topological
two-spheres is the round sphere, see
<cit.>. 



 §.§ Some notation, plan of the paper

The
symbol “C” is used as follows: A statement such as “f≤ Cg” is shorthand for
“there exists a constant C>0  such that f≤
Cg”. The value of C may change within the same line. 
For f≤ Cg, we also write f≲ g.

 For ξ∈, 
let τ_i(ξ), i=1,2 denote the eigenvalues of  ξ. We
denote the operator norm of ξ by 

    |ξ|_∞=max(|τ_1(ξ)|,|τ_2(ξ)|) .

The two-dimensional Lebesgue measure is denoted by Ł^2, the d-dimensional
Hausdorff measure by ^̋d. For x=(x_1,x_2)^T∈^2, we write
x^=(-x_2,x_1)^T.

The following objects will be defined in ^n, but most often we are going to consider  the case n=2. It will be obvious from the context when this special choice is made  and we will not mention it explicitly. (The symbol Ω will always denote a two-dimensional domain.) Let Q=[-1/2,1/2]^n and for ν∈
S^n-1={x∈^n:|x|=1}, let Q_ν be a closed cube of sidelength one in ^n, centered in the origin,  with one of its
sides parallel to ν. (This defines Q_ν uniquely for n=2.) For a  set K∈^n, x_0∈^n and ρ>0,  we write K(x_0,ρ)= x_0+ρ K.
By O(t), we denote terms f(t) that satisfy lim sup_t→∞ t^-1|f(t)|<∞. We fix a radially symmetric function η∈ C^∞_c(^n) such that η⊂ B(0,1) and ∫_^nηx̣=1, and define η_:=^-nη(·/).



The plan of the paper is as follows: In Section <ref>, we will collect a number of theorems from the literature that we will apply later on. In Section <ref>, we prove  some auxiliary lemmas, that will be used in Section <ref>, which contains the proof of the main theorem. In an attempt to increase readability, we have separated the part of the proof concerning the upper bound for points in the jump part from the rest  into a section on its own,  Section <ref>. 


§ PRELIMINARIES




 §.§ Measures and BV functions

Let U⊂^n be open.














  Let λ, μ be Radon measures in U with μ≥ 0. Then there
  exists a Borel set E⊂ U with μ(E)=0 such that for any x_0∈μ∖ E we have

    lim_ρ↓ 0λ(x_0+ρ K)/μ(x_0+ρ
      K)=λ̣/μ̣(x_0)

for any bounded convex set K containing the origin. Here, the set E is
independent of K.




  
Let u∈ BV(U;^m) and for a bounded convex open set K containing the
origin, and let ξ be the density of Du with respect to |Du|,
ξ=(̣Du)/(̣|Du|).
For x_0∈ (|Du|), assume that ξ(x_0)=η⊗ν with η∈^m,
ν∈^n, |η|=|ν|=1, and for
ρ>0 let 

    v^(ρ)(y)=ρ^n-1/|Du|(x_0+ρ K)(u(x_0+ρ
    y)-_x_0+ρ Ku(x')x̣') .

Then for every σ∈ (0,1) there exists a sequence ρ_j converging to
0 such that v^(ρ_j) converges in L^1(K;^m) to a function v∈
BV(K;^m) which satisfies |Dv|(σK)≥σ^n and can be
represented as 

    v(y)=ψ(y·ν)η

for a suitable non-decreasing function ψ:(a,b)→, where a=inf{y·ν:y∈ K} and b=sup{y·ν:y∈ K}.







When considering the blow-up of measures, the following special case of Theorem
0.1 in <cit.> will be useful:


  
  Let {μ_j}_j,μ∈ℳ(U;^p), such that
  
    μ_j→μ   and   
        |μ_j|→|μ|   weakly * in the sense of measures.

  Furthermore, let h:^p→ be positively one-homogeneous. Then
  
    h(μ_j)→ h(μ)     weakly * in the sense of measures.



We recall that BH(U) denotes the set of functions u∈ W^1,1(U) such that
∇ u∈ BV(U;^n). 


The set 
BH(U)
can be made into a normed space by setting

    u_BH(U)=u_W^1,1(U)+|D∇ u|(U) .

We say that a sequence u_j∈ BH(U) converges weakly * to u∈
BH(U) if u_j→ u in W^1,1(U) and D∇ u_j→ D∇  u weakly * in
(U;^n× n).





Let u_j be a bounded
  sequence in BH(U). Then there exists a subsequence (no relabeling) and
  u∈ BH(U) such that

    u_j→ u   weakly * in BH(U) .



Now let us assume that U has smooth boundary. The trace operator

    γ_0:u↦ u|_∂ U

is linear surjective as a map W^1,1( U )→ L^1(∂ U ) and also as a map BV( U )→
L^1(∂  U ). For the spaces
W^2,1( U ) and BH( U ), we may also  consider the operator

    γ_1:u↦∇ u|_∂ U · n ,

where n denotes the unit outer normal of ∂ U.
The following theorem combines statements from    Chapter 2 and  the appendix of <cit.>.

  
  
  
  (i) The operator 
(γ_0,γ_1) is linear surjective  as a map 

    BH( U )→γ_0(W^2,1( U ))× L^1(∂ U ) .


  (ii) There exists  a continuous right inverse 

    γ_0(W^2,1( U ))× L^1(∂ U )→ W^2,1( U ) .

  



For more on the space BH, see e.g. <cit.>.




 §.§ Relaxation of integral functionals that depend on higher
  derivatives

A
function f:^m× n^k→ is called k-quasiconvex if


    f(ξ)=inf{∫_[-1/2,1/2]^nf(ξ+∇^k φ)x̣:φ∈
      W^k,∞_0([-1/2,1/2]^n;^m)} ,

see <cit.>.  



The so-called  k-quasiconvexification of f:^m× n^k→ is given by
the right hand side above,

    Q_kf(ξ)=inf{∫_[-1/2,1/2]^n
    f(ξ+∇^kφ)x̣: φ∈ W^k,∞_0([-1/2,1/2]^n;^m)} .

In the case k=1, one obtains the relaxation of integral
functionals u↦∫ f(∇ u)x̣ by replacing f by its
quasiconvex envelope Q_1f. 





































 §.§ Blow-up method


The main tool in our proof will be the so-called blow-up method. In the context
of lower semicontinuity of integral functionals in BV, this has been developed
by Fonseca and Müller.


 Let f:^m×^m× n→ be 
   quasiconvex and positively one-homogeneous[When comparing our
     statement of the theorem with the one in <cit.>, note that the assumption that f is positively one-homogeneous implies that the
  recession function for f is identical to f.] in the second argument. Assume that
 v_j→ v weakly * in BV(U) and  f(v_j,∇
  v_j)Ł^n→μ weakly * in the sense of measures, and that 
  ζ_2,ζ_3 are defined as the Radon-Nikodym derivatives

    ζ_2=μ̣/(̣|D^sv|
    C_v),  ζ_3=μ̣/(̣^̋1 J_v) .

Then
  
    ζ_2(x_0)   ≥ f(v(x_0),Ḍv/|̣Dv|(x_0))
           for
        |D^sv| C_v a.e. x_0∈Ω
    ζ_3(x_0)   ≥ K_f(v^+(x_0),v^-(x_0),ν_v(x_0))    for |D^sv|
        J_v a.e. x_0∈Ω ,

  where
  
    K_f(a,b,ν)=inf{   ∫_Q_νf(w,∇ w)x̣: w∈ W^1,1(Q_ν), 
        w(x)=a  for  x·ν=+1/2,   w(x)=b  for  x·ν=-1/2} .










§ SOME AUXILIARY LEMMAS




 §.§ Relaxation and quasiconvexification


 We consider the following integrands, defined for v∈^2, ξ∈: 

    f_λ(v,ξ)=λ^-1/2 F_λ(S(v,ξ))√(1+|v|^2) .

This choice implies _λ(u)=∫_Ω f_λ(∇ u,∇^2 u)x̣.





In order to find the lower semicontinuous envelope of _λ,  we will need to
determine  the 2-quasiconvexification of f_λ. In
principle this is contained in <cit.>, and the
appendix of <cit.> contains a detailed proof of the case
v=0. Hence we only point out the modifications that are
necessary with respect to the latter; these changes can be found in the appendix to the
present paper. 




Let v∈^2.  The 2-quasiconvexification of f_λ(v,·)=λ^-1/2F_λ( S(v,·))√(1+|v|^2) is given by


    Q_2f_λ(v,ξ)=√(1+|v|^2)2 ρ^0(S(v,ξ))-2| S(v,ξ)|/√(λ)    if ρ^0(S(v,ξ))≤√(λ)
    |S(v,ξ)|^2/√(λ)+√(λ)    else.






In the sequel we use the notation 

  
    g_λ(ξ)   = 2(ρ^0(ξ)-|ξ|/√(λ))    if
        ρ^0(ξ)≤√(λ)
    |ξ|^2/√(λ)+√(λ)    else.
    
      h_λ(v,ξ)   =Q_2 f_λ(v,ξ)
    =g_λ(S(v,ξ))√(1+|v|^2) .








 §.§ Properties of h_λ


The following straightforward estimate will be used repeatedly:


  Let v∈^2,ξ∈^2× 2. Then

    |S(v,ξ)|^2≤ (1+|v|^2)^-1|ξ|^2 .



  This follows easily from the observation that g(v)^-1 is a symmetric matrix
  with eigenvalues 1 and (1+|v|^2)^-1.






In the following lemma, we collect some properties of g_λ. 







  (i) Let M>1. There exists a constant C=C(M) such that whenever
  A,B∈ with |A|≤ M|B|, we have
  
    g_λ(A)≤ C  g_λ(B) .


  (ii) For A,B∈, we have

    |g_λ(A)-g_λ(B)|≤ C|A-B|(1+|A|+|B|/√(λ)) .


  (iii)  For every λ>0, we have
  
    g_λ(ξ)≥ 2 |ξ|_∞ .




We prove (i) by case distinction:
If √(λ)≥ρ^0(A), then we have

    g_λ (A)≤
       2ρ^0(A)≤ 4|A|≤ 4M|B|≤ 4M g_λ(B) .

If ρ^0(B)≤√(λ)≤ρ^0(A), then we have

    g_λ(A)= √(λ) +|A|^2/√(λ)≤
    2|A|+|A|^2/M^-1|A|≤ 3M^2 |B|≤ 3M^2 g_λ(B) .

If √(λ)≤min(ρ^0(A),ρ^0(B)), then 

    g_λ(A)=√(λ) +|A|^2/√(λ)≤ M^2 g_λ(B).

This completes the proof of (i).



To prove (ii) it suffices to observe that g_λ is piecewise
differentiable. A direct computation yields 

    |∇ g_λ(A)|≤ C(1+|A|/√(λ))

almost everywhere, which immediately implies (ii).



Finally we prove (iii). For ξ=0, the inequality is trivial. So let ξ≠
0, and denote the eigenvalues of ξ by τ_1,τ_2.  For ρ^0(ξ)≤√(λ), we have

    g_λ(ξ)   =2(ρ^0(ξ)-|ξ|/√(λ))
       ≥ 2(ρ^0(ξ)-|ξ|/|ξ|_∞)
       ≥ 2 (ρ^0(ξ)-min(|τ_1|,|τ_2|))
       = 2 |ξ|_∞ .

For ρ^0(ξ)≥√(λ), we have by the Cauchy-Schwarz inequality,

    g_λ(ξ)=√(λ)+|ξ|^2/√(λ)≥ 2|ξ|√(λ)/√(λ)≥ 2|ξ|_∞ .

This proves the lemma.

 



















In the following lemma, we introduce the following notation: The pointwise limit of h_λ for λ→∞ is

    G(v,ξ)=2 ρ^0(S(v,ξ))√(1+|v|^2) .





We have that 
  
    |h_λ(v,ξ)-h_λ(ṽ,ξ)|   ≤ C|v-ṽ|
        max(h_λ(v,ξ),h_λ(ṽ,ξ)) 
    |f_λ(v,ξ)-f_λ(ṽ,ξ)|   ≤ C|v-ṽ|
        max(f_λ(v,ξ),f_λ(ṽ,ξ))
    |G(v,ξ)-G(ṽ,ξ)|   ≤ C|v-ṽ|
        max(G(v,ξ),G(ṽ,ξ))

for all v,ṽ∈^2, ξ∈, where the constants C do not
depend on λ.


We recall that S(v,ξ) is given explicitly by 

    S(v,ξ)=(1+|v|^2)^-1/2(𝕀+v⊗ v)^-1/2ξ(𝕀+v⊗
      v)^-1/2 .

We claim that

    |∇_v S(v,ξ)|≲S(v,ξ)/√(1+|v|^2) .

Indeed, 
noting that

    (𝕀+v⊗ v)^-1/2= 1/√(1+|v|^2)v⊗
      v/|v|^2+v^⊗ v^/|v|^2 ,

this follows from a direct calculation, which we omit here.
Now we may estimate the partial derivative of h_λ(v,ξ) using the chain
rule and Lemma <ref> (ii),

    |∇_v h_λ(v,ξ)|   = |g_λ(S(v,ξ))∇_v
        √(1+|v|^2)
        + √(1+|v|^2)∇ g_λ(S(v,ξ))∇_v S(v,ξ)|
       ≲
      g_λ(S(v,ξ))+√(1+|v|^2)(1+S(v,ξ)/√(λ))S(v,ξ)/√(1+|v|^2)
       ≲ g_λ(S(v,ξ))
       ≤ h_λ(S(v,ξ)) .

The analogous claim for f_λ is trivial for ξ=0, and follows from
(<ref>) and the chain rule for ξ≠ 0. The inequality for G is obtained from the one for h_λ by taking the limit λ→∞.



The following lemma will provide the proof of the lower bound once the
additional complication of the lower
order terms has been treated. 


  
Let Ω⊂^2 be open and bounded, v_0∈^2, ξ_0∈^2× 2,
  w_λ→ 0 in L^1(Ω) as λ→∞, and ∇ w_λ_L^1<C. Then

    lim inf_λ→∞∫_Ω h_λ(v_0,ξ_0+∇ w_λ)x̣≥
    2Ł^2(Ω)ρ^0(S(v_0,ξ_0))√(1+|v_0|^2) .




Up to details, the proof is identical to the proof of Lemma 6.2 (i)
in  <cit.>. There it is proved that

    lim inf_λ→∞∫_Ω g_λ(ξ_0+∇ w_λ)x̣≥
    2Ł^2(Ω)ρ^0(ξ_0) .

In that proof, one only needs to replace g_λ with
g_λ(S(v_0,·)). Apart from the additional dependence of some of the
constants “C” on v_0 that appear in the proof, all arguments go through unchanged.




 §.§ Blow-up of higher order gradients


Theorem <ref> describes the behavior of integrands depending on gradients under
the blow-up procedure. This  will not be quite enough for our purposes: For the jump part, our
proof will take advantage of the fact that we consider the second fundamental form of
the graph, which in turn means that we need to consider integrands that depend on first and
second derivatives.



  
  Let Ω⊂^2 be open and bounded.
Assume that f:^2×^2× 2→ fulfills the following
properties:


  (i) f is quasiconvex and positively one-homogeneous in the second
  argument with f(v,ξ)≤ C|ξ|

  (ii) The functional u↦∫_Ω f(∇ u,∇^2 u)x̣ is continuous in W^2,2(Ω)

Furthermore assume that u_λ is a sequence in W^2,2(Ω), 
 u_λ→ u weakly * in BH(Ω),   f(∇ u_j,∇^2
  u_j)Ł^n→μ weakly * in the sense of measures, and that 
  ζ_3 is defined as the Radon-Nikodym derivative

    ζ_3=μ̣/(̣^̋1 J_∇ u) .

Then
  
    ζ_3(x_0)   ≥K̃_f(∇ u^+(x_0),∇ u^-(x_0),ν_∇ u(x_0))
           for |D^s∇ u|
        J_∇ u a.e. x_0∈Ω ,

  where
  
    K̃_f(a,b,ν)=inf{∫_Q_νf(∇ w,∇^2 w)x̣: w∈𝒜_a,b,ν}

and

    𝒜_a,b,ν   ={ w∈
      C^∞(Q_ν):
             w(x)=a· x  in some neighborhood of
      {x∈∂ Q_ν:x·ν=1/2} ,
             w(x)=b· x  in some neighborhood of
      {x∈∂ Q_ν:x·ν=-1/2} ,
           ∇^k w(x+ν^)=∇^k w(x)  for   x·ν^=-1/2 and  k=1,2,…} .








We write ν≡ν_∇ u(x_0).
With 

    u_λ^(ρ)(x)=ρ^-1(u_λ(x_0+ρ
      x)-u_λ(x_0)) ,      U(x)= ∇ u^+(x_0)· x    
        if x·ν≥ 0
    ∇ u^-(x_0)· x    if  x·ν<
      0 ,

we have that for |D∇ u| J_∇ u almost every x_0,
lim_ρ→ 0lim_λ→∞u_λ^(ρ)= U in W^1,1(Q_ν), see Theorem 3.77
in
<cit.>. Additionally,  

    ζ_3(x_0)=lim_ρ→ 0lim_λ→∞ρ^-1∫_Q_ν(x_0,ρ)f(∇ u_λ,∇^2
    u_λ)x̣ .

Choose ρ_j→ 0,λ_j→∞ such that u_λ_j^(ρ_j)→
U in W^1,1(Q_ν) and 

    ζ_3(x_0)   =lim_j→∞ρ_j^-1∫_Q_ν(x_0,ρ_j)f(∇
      u_λ_j,∇^2
      u_λ_j)x̣ .

We write
u_j:=u_λ_j^(ρ_j). 



We set U_j:=η_ρ_j*U. U_j is affine on the slices orthogonal to ν.
With this notation, we have

    ζ_3(x_0)=lim_j→∞∫_Q_νf(∇
      u_j,∇^2
      u_j)x̣ .

Hence it remains  to show

    K̃_f(∇ u^+(x_0),∇ u^-(x_0),ν)≤lim_j→∞∫_Q_νf(∇
      u_j,∇^2
      u_j)x̣ .

By the continuity assumption (ii), we may assume that u_j∈ C^∞(Q_ν)
in the proof of (<ref>).




For l∈, let K_l∈ be the smallest integer that satisfies

    K_l>lsup_j{u_j_W^2,1+U_j_W^2,1} ,

and

    α_l:=max(1/l,sup{u_j-U_j_W^1,1:j>l}) ,    
    s_l:=α_l/K_l .

Note that α_l→ 0 as l→∞.
For i=0,…,K_l, let

    Q_i,l=(1-α_l+i  s_l)Q_ν .

Consider a family of cut-off functions {φ_i,l:i=1,…,K_l} with 

    φ_i,l∈ C_c^∞(Q_i,l) ,   0≤φ_i,l≤ 1 ,  φ_i,l=1
     on 
    Q_i-1,l ,  ∇^k φ_i,l_L^∞=O(s_l^-k) for  k=1,2 .

For j>l, we define 

    ũ_j^i,l:=φ_i,l u_j+(1-φ_i,l) U_j .

We have that ũ_j^i,l∈𝒜_∇ u^+(x_0),∇
  u^-(x_0),ν (for j large enough).
On Q_i,l∖ Q_i-1,l, we have 

    ∇^2 ũ_j^i,l   = (u_j-U_j)∇^2φ_i,l+∇
      (u_j-U_j)⊗∇φ_i,l
          + ∇φ_i,l⊗∇
      (u_j-U_j)+φ_i,l∇^2(u_j-U_j)+∇^2 U_j .


Now we may estimate, for every i=1,…, K_l,

    ∫_Q_ν f(∇ũ_j^i,l,∇^2 ũ_j^i,l)x̣   ≤∫_Q_i-1,l f(∇ u_j,∇^2 u_j)x̣
     + C∫_Q_i,l∖ Q_i-1,l|∇^2 ũ_j^i,l|x̣
          + C∫_Q_ν∖ Q_i,l|∇^2 U_j|x̣
       ≤∫_Q_i-1,l f(∇ u_j,∇^2 u_j)x̣
          + C∫_Q_i,l∖ Q_i-1,ls_l^-2|u_j-U_j|+s_l^-1|∇ u_j-∇
     U_j|+|∇^2 u_j|+|∇^2 U_j|x̣
          + C∫_Q_ν∖ Q_i,l|∇^2 U_j|x̣

We write T_i,l=Q_i,l∖ Q_i-1,l, and  choose an increasing sequence j(l) with j(l)>l such that for every i=1,…,K_l,

    _T_i,l|u_j-U_j|x̣   <s_l^2
    _T_i,l|∇ u_j-∇ U_j|x̣   
      <s_l .

This is possible by u_j-U_j_W^1,1→ 0.









With the help of these estimates,
the second error term in (<ref>) for j=j(l) can be estimated as follows,

    ∫_T_i,l   s_l^-2|u_j-U_j|+s_l^-1|∇ u_j-∇
      U_j|+|∇^2 u_j|+|∇^2 U_j|x̣
       ≤ C (∇^2 u_j_L^1(T_i,l)+ ∇^2 U_j_L^1(T_i,l)+s_l) .

Summing over all i and averaging, we obtain

    1/K_l∑_i=1^K_l∫_Q_ν f(∇ũ_j^i,l,∇^2
      ũ_j^i,l)x̣   ≤∫_Q_ν f(∇ u_j,∇^2 u_j)x̣
      + C/K_l∫_Q_ν|∇^2 U_j|x̣
          +C/K_l∫_Q_ν(|∇^2 u_j|+|∇^2 U_j|+1)x̣+C s_l

Since the error terms vanish for l→∞, we can choose
i=i(l)∈{1,…,K_l} such that

    lim inf_l→∞∫_Q_ν f(∇ũ_j^i,l,∇^2
      ũ_j^i,l)x̣≤lim_j→∞∫_Q_ν f(∇  u_j,∇^2
       u_j)x̣ .

Since ũ_j^i,l∈𝒜_∇ u^+(x_0),∇ u^-(x_0),ν, the last equation proves
(<ref>).





 §.§ Geometric considerations




We will need to apply Lemma <ref> to the following particular
choice of integrand:

    G_∞(v,ξ)=2| S(v,ξ)|_∞√(1+|v|^2) .


By some geometric considerations, we are able to determine  K̃_G_∞ in Lemma <ref> below. We start with a preparatory
lemma. The assumptions are chosen such that we may apply the lemma to graphs of functions
in _a,b,ν as defined in Lemma <ref> with ν=e_2, see Figure <ref>.



  
Let M be an oriented C^2 submanifold of ^3 with the following
properties:


  (i) M is diffeomorphic to a square

  (ii) There exists l>0 and for each x_1∈[0,l] there exists a C^2
  curve γ_x_1 contained in {x_1}×[0,1]× with its two endpoints
  in {x_1}×{0}× and {x_1}×{1}× respectively, such that

    M=⋃_x_1∈[0,l]γ_x_1 .


  (iii) There exist _0,_1∈ S^2 such that the for each x_1∈
  [0,l], the surface normals in the endpoints of γ_x_1 are given by
  _0,_1 respectively.

Then 

    ∫_M|S_M|_∞̣̋^2≥ l arccos_0·_1 ,

and equality holds if  any two  curves γ_x_1, γ_x_1' are
parallel translations of each other
in x_1 direction, and
their curvature does not change sign.


Looking at the slices for  x_1=constant,  we have that

    ∫_M|S_M|_∞̣̋^2 ≥∫_0^l
      ∫_γ_x_1|S_M|_∞̣̋^1 .

Denoting by N_x_1 a differentiable choice of a normal to M along γ_x_1, we have that the derivative of the normal
DN_x_1 fulfills

    |DN_x_1|_∞≤
    |S_M|_∞ .
 
Hence, by the fundamental theorem of calculus, and letting
_S^2(·,·) denote the geodesic distance on S^2,

    ∫_γ_x_1|S_M|_∞̣̋^1≥∫_γ_x_1|D N_x_1|̣̋^1≥_S^2(_0,_1)=
     arccos_0·_1 .

The claimed inequality follows. If the curves γ_x_1 are parallel translations of
each other in x_1-direction  and
their curvature does not change sign, then the inequalities become sharp.









  Let a,b∈^2, ν∈ S^1 with a·ν^=b·ν^, and
  G_∞(v,ξ)=2|S(v,ξ)|_∞√(1+|v|^2). Then with K̃
  defined as in the statement of Lemma <ref>, we have that

    K̃_G_∞(a,b,ν)=2√(1+|a·ν^|^2)arccos(a)·(b)  .



Let w∈_a,b,ν. After a rotation of the coordinate system, we may
assume that ν= e_2 and a_1=b_1. Let M_1 denote the graph of w. By
applying a suitable 
Euclidean motion (namely, a rotation
 with axis parallel to e_2 and a translation), we may map w|_[0,1]×{0} to
[0,√(1+a_1^2)]×{(0,0)} and   w|_[0,1]×{1} to
[0,√(1+a_1^2)]×{(1,0)} respectively, see Figure <ref>. Let us denote the
resulting submanifold of ^3 by M_2. By the periodicity 
of ∇^k w for k∈{1,2} in x_1-direction, we may translate M_2∩ [0,l]×[0,1]× in x_1-direction by
l=√(1+a_1^2), and the resulting set will still be a C^2 submanifold,  with ∫_M_3|S_M_3|_∞̣̋^2=∫_M_1|S_M_1|_∞̣̋^2. To
M_3, we may apply Lemma <ref> to obtain the claimed lower bound. If ∇ w
is constant in x_1 direction and e_2·∇ w is monotone in x_2 direction, the second
part of that lemma yields that the bound is also attained.








§ PROOF OF THE MAIN THEOREM




 §.§ Compactness








Using ∇ u_λ_L^∞<C, we have that 

    |∇^2 u_λ|≤ C |S(∇
    u_λ,∇^2 u_λ)|

By Lemma <ref> (iii), we have that

    |ξ|≤ g_λ(ξ)   for all ξ∈ .

From (<ref>) and (<ref>) it follows that

    |∇^2 u_λ|≤ h_λ(∇ u_λ,∇^2 u_λ) ,

and hence

    lim sup∇^2 u_λ_L^1(Ω)≤ C .

By Theorem <ref>, we obtain the weak * convergence in BH for
a subsequence.







 §.§ Lower bound




  The main tool of the proof is the blowup technique by Fonseca and Müller. We
  have that
  
    D∇ u= ∇^2 u Ł^2 + D^s∇ u C_∇ u+(∇ u^+-∇
      u^-)⊗ν_∇ u^̋1
      J_∇ u .

In the sequel, we write ν≡ν_∇ u.
  
  
  
  
  
  
  
  
  After choosing a subsequence, we may assume that lim_λ→∞_λ(u_λ)=lim inf_λ→∞_λ(u_λ),
  without increasing the lim inf.  Recalling the definition (<ref>) of
  h_λ, we have that h_λ=Q_2f_λ≤
  f_λ, and hence there exists a Radon measure μ such that (after passing to a
  further subsequence)
  
    h_λ(∇ u_λ,∇^2 u_λ)Ł^2→μ   weakly *
        in the sense of measures.

  Let ζ_1,ζ_2,ζ_3 denote the Radon-Nikodym derivative of μ
  with respect to Ł^2, |D^s∇ u| C_∇ u and ^̋1
  J_∇ u respectively. By the non-negativity of μ, we have
  
    μ≥ζ_1 Ł^2 +ζ_2 |D^s∇ u| C_∇ u+ ζ_3
      ^̋1 J_∇ u .

  We will show that


    ζ_1(x)≥   2ρ^0(S(∇ u(x),∇^2
          u(x)))√(1+|∇ u|^2)   for Ł^2-a.e.  x∈Ω
    ζ_2(x)≥   2ρ^0(S(∇ u(x),(̣D∇ u)/|̣D∇ u|(x)))√(1+|∇ u|^2)   for
      |D^s∇ u|-a.e.  x∈ C_∇ u
    ζ_3(x)≥   2arccos((∇ u^+)·(∇
        u^-))√(1+|ν^·∇ u|^2)   for
      ^̋1-a.e.  x∈ J_∇ u .

This will prove the lower bound.

We will first prove (<ref>).  

We write v_λ=∇ u_λ.  
 For Ł^2-almost every x_0, we may
choose a sequence (_j)_j∈ converging to zero, such that μ(∂
Q(x_0,_j))=0 for every j∈. When we write → 0 in the sequel, we
actually mean the limit j→∞ for such a sequence. Also, we will drop the index j in our notation. For every , we have

    lim_λ→∞∫_Q(x_0,) h_λ(v_λ,∇
    v_λ)x̣ = μ(Q(x_0,)) .

Moreover,

    lim_→ 0lim_λ→∞ Dv_λ(Q(x_0,))/|D
      v|(Q(x_0,))=Ḍ v/|̣D v|(x_0) .

Note that by Theorem <ref> we have

    ζ_1(x_0)=   lim_→ 0μ(Q(x_0,))/Ł^2(Q(x_0,))
    
        =   lim_→ 0lim_λ→∞_Q(x_0,)h_λ(v_λ,∇ v_λ)x̣ .

We write v_0:=v(x_0). For  small enough, define w_λ,:Q→^2 by

    w_λ,(x)=^-1(v_λ(x_0+ x)-v_0) .

Furthermore let w_0(x)=∇ v_0· x. Using a change of variables and the
Cauchy-Schwarz inequality we have

    lim_→ 0lim_λ→∞w_λ,-w_0_L^1(Q)   =
        lim_→ 01/∫_Q|v(x_0+ x)-v_0-∇ v_0· x|x̣
       = lim_→ 01/^3∫_Q(x_0,)|v(x)-v_0-∇
        v_0·
        (x-x_0)|x̣
       ≤lim_→ 01/^2(∫_Q(x_0,)|v(x)-v_0-∇
          v_0·
          (x-x_0)|^2x̣)^1/2
       =0 .

The last equality above holds for Ł^2-almost every
x_0 by the remark below Theorem 3.83 in
<cit.> (which is a slightly stronger variant of approximate differentiability). 
Also note that we have

    _Q(x_0,)h_λ(v_λ,∇ v_λ)x̣=∫_Qh_λ(v_0+ w_λ,(x),∇ w_λ,(x))x̣ .


By (<ref>) and (<ref>), it is possible to choose ≡(λ) depending on λ such that
with w_λ:=w_λ,(λ) we have that

    lim_λ→∞w_λ-w_0   =0
    lim_λ→∞∫_Qh_λ(v_0+ w_λ(x),∇ w_λ)x̣   =ζ_1(x_0) .


We need to modify w_λ such that we get a suitable L^∞-bound for fixing
the lower order terms. Namely, we are going to construct w̃_λ such that
w̃_λ_L^∞≤^-1/2, and


    lim inf_λ→∞∫_Q h_λ(v_0+w̃_λ,∇w̃_λ)x̣≤lim inf_λ→∞∫_Q h_λ(v_0+ w_λ,∇  w_λ)x̣ .


Let K_λ be the largest integer smaller than log_2 ^-1/2. For
k=1,…,K_λ, we define

    E_k^λ:={x∈ Q: 2^k-1<|w_λ-w_0|≤ 2^k} .

Next we choose k_λ∈{1,…, K_λ} such that with E^λ:=E_k_λ^λ, we have

    ∫_E^λ(1+h_λ(v_0+ w_λ,∇  w_λ))x̣≤ K_λ^-1∫_Q (1+h_λ(v_0+ w_λ,∇  w_λ))x̣

and we define φ_λ:[0,∞)→ such that

    φ_λ(x)=   1 for  x≤ 2^k_λ-1
    φ_λ(x)=   0 for  x≥ 2^k_λ
    
      |φ_λ'|≤   2^2-k_λ .

Now we set

    w̃_λ=w_0+φ_λ(|w_λ-w_0|)(w_λ-w_0) .

Note that w̃_λ-w_0_L^∞≤^-1/2 by
construction, 










and w̃_λ=w_0 on
{x∈Ω:|w_λ-w_0|≥ 2^k_λ}. We have that

    lim inf_λ→∞∫_Q h_λ(v_0+w̃_λ,∇w̃_λ)x̣   ≤lim inf_λ→∞∫_{|w_λ-w_0|≤ 2^k_λ-1}
        h_λ(v_0+ w_λ,∇ w_λ)x̣
          +  ∫_E^λ h_λ(v_0+w̃_λ,∇w̃_λ)x̣
          + ∫_{|w_λ-w_0|≥ 2^k_λ} h_λ(v_0+ w_0,∇
        w_0)x̣ .


We claim that


    ∫_E^λ h_λ(v_0+w̃_λ,∇w̃_λ)x̣≤
      C(v_0,∇ w_0) ∫_E^λ(1+h_λ(v_0+ w_λ,∇ w_λ))x̣ .

Indeed, we have that on E^λ, | w_λ-w̃_λ|≲^1/2, and hence

    |(v_0+ w_λ)-(v_0+w̃_λ)|   ≤ C(v_0) ^1/2 ,
    |√(1+|v_0+ w_λ|^2)-√(1+|v_0+w̃_λ|^2)|   ≤
        C ^1/2 .

Also,

    |∇w̃_λ|   = |∇ w_0+φ_λ' (w_λ-w_0)⊗∇
        |w_λ-w_0|+φ_λ∇ (w_λ-w_0)|
       ≤ C( |∇ w_0|+ (2^-k_λ|w_λ-w_0|+1)|∇ (w_λ-
        w_0)|)
       ≤ C(∇ w_0) (|∇ w_λ|+1) .

Hence, we have that

    |S(v_0+w̃_λ, ∇w̃_λ)|≤ C(v_0,∇ w_0)(|S(v_0+ w_λ,
      ∇  w_λ)|+1)

and it follows from Lemma <ref> (i) that


    g_λ(S(v_0+w̃_λ, ∇w̃_λ))
      ≤ 
      C (v_0,∇ w_0)(g_λ(S(v_0+w̃_λ,
        ∇  w_λ))+1) .


Our claimed inequality (<ref>) now follows from (<ref>) and
(<ref>).



Using (<ref>), (<ref>) and the fact K_λ→∞ as λ→∞,
as well as w_λ→ w_0 in L^1, the right hand side of (<ref>) can be
estimated from above by

    lim inf_λ→∞(∫_Q h_λ(v_0+ w_λ,∇ w_λ)x̣
        + C(v_0,∇ w_0)Ł^2({|w_λ-w_0|≥ 2^k_λ})
        |∇ w_0|)
    
      = lim inf_λ→∞∫_Q h_λ(v_0+ w_λ,∇ w_λ)x̣ .

This proves (<ref>).


Now we have by Lemma <ref> and Lemma <ref>,

    lim inf_λ→∞∫_Q h_λ(v_0+w̃_λ,∇w̃_λ)x̣   ≥lim inf_λ→∞1-C(v_0,∇ w_0)^1/2/1+C(v_0,∇ w_0)^1/2∫_Q
      h_λ_λ(v_0,∇w̃_λ)x̣
       ≥ 2 √(1+|v_0|^2)ρ^0(S(v_0,∇ w_0)) .

This proves equation (<ref>).



Recall G(v,p)=2ρ^0(S(v,p))√(1+|v|^2), and
G_∞(v,p)=2|S(v,p)|_∞√(1+|v|^2). 







Let v_λ→ v weakly * in BV.  We have by Lemma <ref> (iii) that

    h_λ(v_λ,∇ v_λ)≥ G_∞(v_λ,∇ v_λ)

for all λ.  By Theorem <ref>, we have that for
|D^sv| C_v almost every point x_0∈Ω,

    ζ_2(x_0)≥ G_∞(v(x_0),Ḍv/|̣Dv|(x_0))

which proves (<ref>), since Ḍv/|̣Dv|(x_0) is rank one, and
hence

    2ρ^0(S(v(x_0),Ḍv/|̣Dv|(x_0)))=2|S(v(x_0),Ḍv/|̣Dv|(x_0))|_∞ .

By Lemma <ref>, we have in a similar fashion for |D^sv| J_v
almost every x_0∈Ω,

    ζ_3(x_0)≥K̃_G_∞(v^+(x_0),v^-(x_0),ν(x_0)) .

By Lemma <ref>, it follows

    ζ_3(x_0)≥ 2√(1+|ν^· v(x_0)|^2)arccos(v^+(x_0))·(v^-(x_0)) .

This proves (<ref>) and completes the proof of the lower bound.




 §.§ Upper bound

For the proof of the upper bound, we will need a modification of the well known
result in the calculus of variations that states that the relaxation of integral
functionals with suitable integrands that depend on x,u,∇ u is obtained
by the quasiconvexification of the integrand with respect to the gradient
variable. Here, we will need the analogous result for integrands that depend on
∇ u, ∇^2 u. 



  Let 1≤ p<∞, and let f:^m× 2×^m× 2× 2→
  such that 
  
    .
    0≤   f(v,ξ)≤ C (1+|ξ|^p)
    
    |f(v,ξ)-f(ṽ,ξ)|≤   C|v-ṽ|max (f(v,ξ),f(ṽ,ξ)) 
    
    |Q_2f(v,ξ)-Q_2f(ṽ,ξ)|≤   C|v-ṽ| max (Q_2f(v,ξ),Q_2f(ṽ,ξ))
    } ∀
    v,ṽ∈^2× n, ξ∈^m× 2× 2.

Furthermore, let Ω⊂^2 be open and bounded with smooth boundary, u∈ W^2,p(Ω)  and δ>0. Then there
exists w∈ W^2,p(Ω;^m) with

    u-w_W^1,p(Ω;^m)   <δ
    ∫_Ω f(∇ w,∇^2 w)x̣   < ∫_Ω Q_2f(∇ u,∇^2 u)x̣+δ .




For the proof of the proposition, we are going to use 


Let Ω⊂^2 be open and bounded with smooth boundary, and let p∈
[1,∞).  Furthermore let u∈ C^3(Ω) and δ>0. Then there exists w∈ 
  W^2,∞(Ω) and Ω_w⊂Ω such that Ω_w is
  the union of mutually disjoint closed cubes, w is
  piecewise a polynomial of degree 2 on Ω_w, and furthermore

    u-w_W^2,p(Ω)   <δ ,
    w_W^2,∞   ≲u_W^2,∞
    ∫_Ω∖Ω_w(1+|∇^2u|^p+|∇^2w|^p)x̣   <δ .




We remark that the inequality w_W^2,∞≲u_W^2,∞ does not appear in the statement of Lemma A.3 in <cit.>, but is clear from the proof there. Also, the assumptions on the domain Ω that we make here are sufficient; the assumption of simple connectedness made in <cit.> does not play a role in the proof of that lemma. Finally, we note in passing that neither there nor here the assumption that the dimension of the domain is n=2 is of any relevance; the respective statements hold true for general n just as well.



First we recall the well known fact that rank-one convex functions are locally
Lipschitz continuous (see e.g. <cit.>). 
This holds true in particular
for Q_2f(v,·) for any v, and hence the assumption (<ref>)
implies that Q_2f is locally Lipschitz continuous in both arguments.
More precisely, with the assumed
growth properties for f, we  have Q_2f(v,ξ)≤ C(1+|ξ|^p) and
hence 

    |Q_2(v,ξ)-Q_2(v,ξ̃)|≤ C|ξ-ξ̃|(1+|ξ|^p-1+|ξ̃|^p-1)   ∀ξ,ξ̃∈^m× n^2

where C is some constant that is independent of v,ξ,ξ̃ (see
Proposition 2.32 in 
<cit.>).



We set u_:=η_* u and claim that

    lim_→ 0∫_Ω Q_2f(∇ u_,∇^2 u_)x̣=∫_Ω
    Q_2f(∇ u,∇^2 u)x̣ .

Indeed,
we have that  u_→ u in W^2,p, and hence by (<ref>) and the
assumption (<ref>), we have

    ∫_Ω   |Q_2f(∇ u_,∇^2 u_)- Q_2f(∇ u,∇^2 u)|x̣
       ≤∫_Ω |Q_2f(∇ u_,∇^2 u_)-
      Q_2f(∇ u_,∇^2 u)|
       + |Q_2f(∇ u_,∇^2 u)-
      Q_2f(∇ u,∇^2 u)|x̣
       ≤ C∫_Ω |∇^2 u_-∇^2 u| (1+|∇^2
      u|^p-1+|∇^2 u_|^p-1)+max(|∇ u_-∇ u|,1) (1+|∇^2 u|^p)
      x̣

For → 0, the integral on the right hand side  converges to 0, proving the claim (<ref>).





























Let Δ>0. We choose u_ such that u-u__W^2,p<Δ.
By Lemma <ref>, there exists w_∈ W^2,∞ and a union of
disjoint closed cubes Ω_w⊂Ω such that each component of  w_ is a polynomial
of degree 2 on each of the cubes, and 

    w_-u__W^2,p   <Δ
    
      |Ω∖Ω_w|(1+w__W^2,∞^p)   <Δ .

By the same kind of estimate as in (<ref>), we obtain that additionally, we may choose w_, Ω_w
such that

    ∫_Ω Q_2f(∇ w_,∇^2 w_)x̣<∫_Ω Q_2f(∇
    u_,∇^2 u_)x̣+Δ .

Moreover, we may choose the cubes to be so small that on each cube Q̃ with center x_0 in the collection,

    sup_x∈Q̃|∇ w_(x)-∇ w_(x_0)|<Δ .

















































Let Q̃ be a cube where the components of w_ are quadratic polynomials, with midpoint
x_0 and sidelength r. Choose ξ∈ W_0^2,∞(Q̃) such that

    ∫_Q̃ f(∇
    w_(x_0),∇^2 w_(x_0)+∇^2ξ)x̣≤vol(Q̃) Q_2f(∇ w_(x_0),∇^2 w_(x_0)) +Δ/N ,

where N is the total number of cubes. 
Let us write ξ̃(x)=ξ(x_0+rx) for x∈ [-1/2,1/2]^2, and define ξ̃ on ^2 by
1-periodic extension.
For M∈, let

    ξ_M(x)=M^-2ξ̃(M(x-x_0)) .

We have that  ξ_W^1,∞→ 0 for M→∞, ∇^2ξ_L^∞=∇^2ξ_M_L^∞,  and 

    ∫_Q̃ f(∇
    w_(x_0),∇^2 w_(x_0)+∇^2ξ)x̣=∫_Q̃ f(∇
    w_(x_0),∇^2 w_(x_0)+∇^2ξ_M)x̣ .

We choose  M
so large that ∇ξ_M_L^∞<Δ.
This implies 

    ∇ w_+∇ξ_M-∇ w_(x_0)_L^∞(Q̃)<2Δ .

Using the local Lipschitz continuity in the first argument of f as assumed in (<ref>),

    ∫_Q̃ f(∇
    w_+∇ξ_M,∇^2 w_(x_0)+∇^2ξ_M)x̣<
    1+CΔ/1-CΔ∫_Q̃ f(∇
    w_(x_0),∇^2 w_(x_0)+∇^2ξ_M)x̣ .

We repeat the same for all cubes Q̃ in Ω_w, obtaining a corrector
function ξ_Q̃∈ W^2,∞_0(Q̃) in each of them.
We set w̅= w_+∑_Q̃ξ_Q̃. Denoting by x_Q̃
the center of the cube Q̃, we have

    ∫_Ω f(∇w̅ ,∇^2 w̅)x̣   ≤∑_Q̃∫_Q̃f(∇w̅,∇^2w̅)x̣
    +∫_Ω∖Ω_wf(∇w̅,∇^2w̅)x̣
       ≤∑_Q̃1+CΔ/1-CΔ∫_Q̃f(∇ 
    w_(x_Q̃),∇^2w̅)x̣+C ∫_Ω∖Ω_w(1+|∇^2w̅|^p)x̣
       ≤∑_Q̃1+CΔ/1-CΔ(∫_Q̃Q_2f(∇ 
    w_
    (x_Q̃),∇^2 w_)x̣+Δ/N)+CΔ
       ≤(1+CΔ/1-CΔ)^2 ∫_Ω Q_2f(∇
    w_,∇^2 w_)x̣+CΔ .

Here we used again (<ref>) in combination with the assumption (<ref>) to obtain the
last inequality.
By u-w__W^2,p< 2Δ and (<ref>), this last estimate proves the claim
by choosing Δ small enough.












Just as for the lower  bound, we will use the blow-up method for the proof of
the upper bound, in combination with a suitable mollification.
We assume that we are given a sequence λ_j→∞, and we will prove
that for any subsequence, there exists a further subsequence fulfilling the
upper bound. We omit the index j from our notation and write λ→∞ for j→∞.



Step 1:
Let Ω̂ be
some neighborhood of Ω in ^2.
By Theorem <ref>,  there exists a function v∈
W^2,1(Ω̂∖Ω)  such that the traces γ_0(v),γ_1(v)
on ∂Ω
with respect to Ω̂∖Ω
are identical with γ_0(u), γ_1(u) (up to the appropriate sign
change for γ_1). By Theorem 3.84 in <cit.>, the function
w:=uχ_Ω+vχ_Ω̂∖Ω is an element of BH(Ω̂), with
|D∇ w|(∂Ω)=0.




For >0 small enough, we have {x∈^2:(x,Ω)<}⊂Ω̂,
and we may set u_(x)=(η_ * w)(x) for x∈Ω. Here
=(λ,u) is chosen such that 

    S(∇ u_,∇^2 u_)_L^∞≤∇^2 u__L^∞< √(λ)/2 ,

and (λ,u)→ 0 as λ→∞.
Such a choice of  is possible by the standard estimate

    ∇^2 u__L^∞(Ω)≤ C^-2|D∇ w|(Ω̂)

where C=sup |η|; the first inequality in (<ref>) being
valid by Lemma <ref>.
Writing u_λ=u_(λ,u), v_λ=∇ u_λ, v=∇ u and recalling the definition
(<ref>) of
h_λ,  we have that

    h_λ(v_λ,∇ v_λ)   ≤ 2√(1+|v_λ|^2)ρ^0(S(v_λ,∇ v_λ))
       ≤ 2 |∇ v_λ| .

Hence we have that after
passing to a subsequence, there exists a measure μ such that

    .h_λ(v_λ,∇ v_λ)Ł^2Ω   →μ
    ∇ v_λŁ^2   → Dv
    
    |∇ v_λ|Ł^2   → |Dv|} weakly * in the sense of measures

Additionally, it follows from  |D∇ w|(∂Ω)=0 and
(<ref>) that
μ(∂Ω)=0, and hence 
    lim_λ→∞∫_Ω
      h_λ(v_λ,∇ v_λ)x̣= μ(Ω) .




Step 2. 
For every continuous non-negative function φ∈ C_0(Ω), we have

    ∫_Ωφ μ̣   =
      lim_λ→∞∫_Ωφ h_λ(
      v_λ,∇
      v_λ)x̣
       ≤  2lim_λ→∞∫_Ωφ |∇ v_λ| x̣
       = 2∫_Ωφ |̣Dv| .

Hence, μ is absolutely continuous with respect to |Dv|, and
the measure μ can be decomposed into mutually singular measures,

    μ=ζ_1Ł^2+ζ_2 |D^sv| C_v+ζ_3 ^̋1 
    J_v  .

We will prove

    ζ_1(x)≤   2ρ^0(S(v(x),∇ v(x)))√(1+|v|^2)   for Ł^2-a.e.  x∈Ω
    ζ_2(x)≤   2ρ^0(S(v(x),Ḍv/|̣Dv|(x)))√(1+|v|^2)   for
      |D^sv|-a.e.  x∈ C_v
    ζ_3(x)≤   2arccos((v^+)·(
        v^-))√(1+|ν^· v|^2)   for ^̋1-a.e.  x∈ J_v .

Once we have proved these inequalities, we have proved

    lim sup_λ→∞∫_Ω h_λ(v_λ,∇ v_λ)x̣≤(u) .

The upper bound then follows by Q_2 f_λ=h_λ and Proposition
<ref>. Indeed, the assumptions of the proposition (with p=2) are
fulfilled for f_λ by
Lemma <ref>.



Step 3. To prove (<ref>),
we use that at Ł^2-almost every x_0,  v=∇ u is approximately differentiable, i.e., 

    lim_r→ 01/r_Q(x_0,r)|v(x)-v(x_0)-∇ v(x_0)·(x-x_0)|x̣=0 .

In particular, we may assume that x_0 is a Lebesgue point of v,

    lim_r→ 0 |v(x)-v(x_0)|x̣=0 .

We define

    v^(ρ)(x)   =ρ^-1(v(x_0+ρ x)-v(x_0))
    
        V(x)   =∇ v(x_0)·(x-x_0)

and have by (<ref>) that

    v^(ρ)→ V  in L^1(Q(0,2))

 and by Theorem <ref> that

    .
          Dv^(ρ)   →∇ v(x_0)Ł^2
    
          |Dv^(ρ)|   → |∇ v(x_0)|Ł^2} weakly * in the sense of measures.

Now we choose ρ(λ) with (λ,u)<ρ(λ)/2.
In the sequel, we write ρ≡ρ(λ). We set

    v^(ρ)_λ(x)=ρ^-1(v_λ(x_0+ρ x)-v(x_0))

and have

    v^(ρ)_λ→ V  in L^1(Q(0,2))

 and 
  
    .∇ v^(ρ)_λŁ^2   →∇ v(x_0)Ł^2
    
          |∇ v^(ρ)_λ|Ł^2   → |∇ v(x_0)|Ł^2} weakly * in the sense of measures.

Moreover,

    sup_x∈ Q(x_0,ρ) |v_λ(x)-v(x_0)|
           ≤sup_ Q(x_0,ρ) |η_(λ,u)*(v-v(x_0))|
       ≤v_λ-v(x_0)_L^1(Q(x_0,2ρ))
    
             → 0 for λ→∞ .





By the Radon-Nikodym Theorem,

    ζ_1(x_0)=lim_λ→∞_Q(x_0,ρ)
      h_λ(v_λ,∇ v_λ)x̣ .


We recall that G:^2×→ is defined by

    G(ξ,p)=2ρ^0(S(ξ,p))√(1+|ξ|^2) .

By Lemma <ref> we have that
  
    |G(ξ,p)-G(ξ̃,p)|≲ |ξ-ξ̃| max
      (G(ξ,p),G(ξ̃,p)) .


By combining  (<ref>) with the definition (<ref>) of h_λ,
 we have that h_λ( v_λ,∇ v_λ)≤ G(
 v_λ,∇ v_λ), and hence


    ζ_1(x_0)≤lim sup_λ→∞_Q(x_0,ρ)
      G(v_λ,∇ v_λ)x̣ .


By (<ref>) and (<ref>), we obtain


    ζ_1(x_0)   ≤lim sup_λ→∞_Q(x_0,ρ)
        G(v(x_0),∇ v_λ)x̣
       =lim sup_λ→∞∫_Q(0,1) G(v(x_0),∇
        v_λ^(ρ))x̣ .

  By Theorem <ref> and (<ref>), we finally get
  
    ζ_1(x_0)   ≤∫_Q(0,1) G(v(x_0),∇ v(x_0))x̣
       =G(v(x_0),∇ v(x_0)) ,

which proves (<ref>).
  

  
  











Step 4. 
For |D^sv| C_v-almost every x_0, we have that by Alberti's rank one Theorem <cit.> and Theorem <ref>
the following holds true:  There exists a sequence
ρ_l↓ 0 and 
a  monotone function ψ∈ BV(-1/2,1/2) 
such that the rescaled functions 

    v^(ρ_l)(x):=ρ_l/|Dv|(Q_ξ(x_0,ρ_l))(
      v(x_0+ρ_l x)-_Q_ξ(x_0,ρ_l)v (x')x̣')

converge for l→∞ in L^1(Q_ξ;^2) to
the function 

    Ψ:x↦ξψ(x·ξ) ,

where ξ∈ S^1 fulfills (̣Dv)/|̣Dv|(x_0)=ξ⊗ξ.
Also, from Theorem <ref>, we have that
  
    .  Dv^(ρ_l)   → DΨ
     |Dv^(ρ_l)|   → |DΨ|} weakly * in the sense of measures.


From now on, in order to alleviate the notation, we are going to omit the index
l from ρ_l, and we write lim_ρ→ 0 for lim_l→∞.
As a consequence of the convergence of v^(ρ)→Ψ in L^1,  we have in particular that x_0 is a Lebesgue point of v,

    lim_ρ→ 0_Q(x_0,ρ)|v-v(x_0)|x̣=0 .


  
















By |Dv^(ρ)|(Q_ξ)=1 and the lower semicontinuity of total variation, we have
that |DΨ|(Q_ξ)=|Dψ|(-1/2,1/2)≤ 1.



































Now we choose  ρ(λ)→ 0 such that μ(∂
Q(x_0,ρ(λ)))=0, /ρ→ 0, and 

    ζ_2(x_0)=
    lim_λ→ 01/|D^sv
      |(Q_ξ(x_0,ρ(λ)))∫_Q_ξ(x_0,ρ(λ))h_λ(v_λ,∇ v_λ)x̣ .



Writing ρ≡ρ(λ), we note that

    (v^(ρ))_/ρ(x)   =ρ/|Dv|(Q_ξ(x_0,ρ))v_λ(x_0+ρ x)
    (∇ (v^(ρ))_/ρ)(x)
         =ρ^2/|Dv|(Q_ξ(x_0,ρ))(∇ v_λ)(x_0+ρ x) .



  As in the previous step, using the fact that x_0 is a Lebesgue point for v, we obtain
    
    sup_x∈ Q(x_0,ρ)|v_λ(x)-v(x_0)|→ 0  as λ→ 0 .





















Using h_λ≤ G (see again step 3),
we have that


    lim sup_λ→∞   1/|D^sv
          |(Q_ξ(x_0,ρ))∫_Q_ξ(x_0,ρ) h_λ(v_λ
          ,∇ v_λ)x̣
       ≤lim sup_λ→∞1/|D^s
          v|(Q_ξ(x_0,ρ))∫_Q_ξ(x_0,ρ) G(v_λ
          ,∇ v_λ)x̣ .

By a change of variables, we obtain

    ∫_Q_ξ(x_0,ρ) G(v_λ,∇ v_λ)x̣=ρ^2∫_Q_ξG(v_λ
          (x_0+ρ y),∇ v_λ
          (x_0+ρ y))ỵ .

Combining this with (<ref>) and (<ref>), we obtain

    lim sup_λ→∞   1/|D^sv
          |(Q_ξ(x_0,ρ))∫_Q_ξ(x_0,ρ) h_λ(v_λ
          ,∇ v_λ)x̣
       ≤lim sup_λ→∞∫_Q_ξ G(v_λ(x_0+ρ y)
          ,∇(v^(ρ))_/ρ)ỵ .

  













By (<ref>), this yields

    lim sup_λ→∞   1/|D^sv
          |(Q_ξ(x_0,ρ))∫_Q_ξ(x_0,ρ) h_λ(v_λ
          ,∇ v_λ)x̣
       ≤lim sup_λ→∞∫_Q_ξ G(v(x_0)
          ,∇(v^(ρ))_/ρ)x̣ .

By (<ref>) and /ρ→ 0, we have that


    .∇(v^(ρ))_/ρŁ^2   → DΨ
    |∇(v^(ρ))_/ρ|Ł^2   → |DΨ|} weakly * in the sense of measures.
  
By Theorem <ref>, it follows that

    lim sup_λ→∞∫_Q_ξ G(v(x_0)
          ,∇(v^(ρ))_/ρ)x̣   = ∫_Q_ξG(v(x_0)
          ,ξ⊗ξ)|̣DΨ| 
       ≤ G(v(x_0)
          ,ξ⊗ξ) ,



















































which (together with (<ref>)) proves
 (<ref>).




Step 5. For ^̋1 J_∇ u-almost every x_0, we have that by Theorem <ref>
the following holds true:
The rescaled functions 

    v^(ρ)(x)=v
      (x_0+ρ x)-_Q_ν(x_0,ρ)v(x')x̣'

converge in L^1(2Q_ν;^2) to
the function 

    Ψ:x↦ v^+(x_0)    if  x·ν >0
    
    v^-(x_0)     if  x·ν≤ 0 .


Additionally we have, for every β>0, 

    lim_ρ→ 0∫_(1+β)Q_ν|∇ v^(ρ)|x̣=|DΨ|((1+β)Q_ν) .















Now we choose ρ(λ)  such that ρ→ 0,
and /ρ→ 0  as λ→∞. 

Then we   may write, again using h_λ≤ G,

    lim_λ→ 01/ρ∫_Q_ν(x_0,ρ) h_λ(v_λ
      ,∇ v_λ)x̣   ≤lim inf_λ→ 01/ρ∫_Q_ν(x_0,ρ) G(v_λ
      ,∇ v_λ)x̣
       ≤lim inf_λ→ 0∫_Q_νρ G(v_λ
       (x_0+ρ x),∇ v_λ(x_0+ρ x))x̣
       =lim inf_λ→ 0∫_Q_ν G(η_/ρ*v^(ρ),
          ∇η_/ρ*v^(ρ))x̣ .


Since /ρ→ 0, we have that

    η_/ρ*v^(ρ)   →Ψ   in L^1(Q_ν)
    lim_λ→∞∫_Q_ν|∇η_/ρ*v^(ρ)|x̣   =|DΨ|(Q_ν) .

By Proposition <ref>, which will be proved in Section <ref> below, this implies 

    lim sup_λ→∞∫_Q_ν    G(η_/ρ*v^(ρ), ∇η_/ρ*v^(ρ))x̣
       ≤ 2√(1+|ν^· v(x_0)|^2)arccos((v^+(x_0))·(v^-(x_0))) ,

and  combining the latter with (<ref>) proves (<ref>), since the left hand side of (<ref>) is just ζ_3(x_0).
















































§ PROOF OF THE UPPER BOUND FOR THE BLOW-UP OF THE JUMP PART



We recall that G(v,ξ)=2ρ^0(S(v,ξ))√(1+|v|^2). In the present section, we write I=[-1/2,1/2].



  Let v_j∈ C^1( Q) such that v_j→ 0 in W^1,1(Q). Furthermore, let P:^2→ be the projection P(x_1,x_2)=x_1, Δ>0 and for j∈,
  
    A_j:={x∈ Q:|v_j(x)|>Δ} .

  Then Ł^1(P(A_j))→ 0 as j→∞.


  By the continuity of the trace operator W^1,1(Q)→ L^1(∂ Q), we have that
  
    v_j(·,-1/2)→ 0,   v_j(·,1/2)→ 0    in L^1(I) .

  Setting
  
    J^1_j   :={t∈ I: |v_j(t,-1/2)|+|v_j(t,1/2)|>Δ/2} ,
    
        J^2_j   :={t∈ I:∫_I|∂_x_2v_j(t,x_2)|x̣_2>Δ/2} ,

  we have that P(A_j)⊂ J^1_j∪ J^2_j and Ł^1(J_j^1)+Ł^1(J_j^2)→ 0. This proves the claim.  



  Let a_1∈ and  w∈ C^1(I). Then
  
    ∫_-1/2^1/2G   ( ([  a_1; w(t) ]),([     0     0;     0 w'(t) ]))ṭ
       ≤ 2 √(1+a_1^2)(arccos([     a_1; w(-1/2) ])·([     a_1; w(+1/2) ])
          +2∫_-1/2^1/2|min(0,w'(t))|ṭ) .






First we recall the geometric meaning of the integral: Let
  
    W(x_1,x_2)=∫_0^x_2w(t)ṭ+a_1 x_1 .

  Then we have that
  
    ∫_-1/2^1/2 G( ([  a_1; w(t) ]),([     0     0;     0 w'(t) ]))ṭ       =∫_Q G(∇ W,∇^2 W)x̣
       = ∫_ W|_Q 2ρ^0(S_ W|_Q)̣̋^2 .

As in the proof of Lemma <ref>, we may rotate, cut and glue the graph W_Q to obtain the graph of the function

    W̃:[-√(1+a_1^2)/2,√(1+a_1^2)/2]×[-1/2,1/2]→ ,

defined by

    W̃(x_1,x_2)=1/√(1+a_1^2)∫_0^x_2 w(t)ṭ ,

without changing the “energy”, i.e., W̃ satisfies

    ∫_ W|_Q 2ρ^0(S_ W|_Q)̣̋^2=∫_W̃|_Q̃ 2ρ^0(S_W̃|_Q)̣̋^2 ,

  where we have written Q̃=[-√(1+a_1^2)/2,√(1+a_1^2)/2]×[-1/2,1/2]. Now W̃ is constant in x_1-direction, which implies that the normal vector is contained in the x_2-x_3 plane,
  
    (∇W̃)∈{(0,x_2,x_3)^T:x_2^2+x_3^2=1} ,

  which implies
  
    ρ^0(S_W̃)=(1+|∂_x_2W̃|^2)^-1/2|∂_x_2(∇W̃)| .


  


  This leaves us, after the passage back to a one-dimensional setting, with the following  calculation:
  
    ∫_W̃|_Q̃ 2ρ^0(S_W̃|_Q)̣̋^2 
         =
          2√(1+a_1^2)∫_-1/2^1/2|∂_t ([               0; w(t)/√(1+a_1^2) ])|ṭ .


We observe ((0,w(t)/√(1+a_1^2))^T)=((0,w(t))^T).  By giving {(0,x_2,x_3)^T:x_2^2+x_3^2=1}≃ S^1 an orientation (the one corresponding to increasing w(t)), the total variation of the curve
((0,w(·))^T) can be estimated from above by the distance of its endpoints on S^1, plus twice the integral of |∂_t((0,w(t))^T)| over the region where ∂_t((0,w(t))^T) is anti-parallel to the orientation:


    ∫_-1/2^1/2|∂_t ([    0; w(t) ])|ṭ   ≤arccos([       0; w(-1/2) ])·([       0; w(+1/2) ])
         +2∫_{t:w'(t)≤ 0}|∂_t ([    0; w(t) ])|ṭ
       ≤arccos([     a_1; w(-1/2) ])·([     a_1; w(+1/2) ])
          +2       ∫_-1/2^1/2|min(0,w'(t))|ṭ .


In the last inequality, we have used the fact that the angle between the normals at the endpoints does not change when applying a rotation, and |∂_t(w̃(t))|≤|∂_t w̃(t)|. This completes the proof of  the lemma.




  
  Let a∈^2, ν∈ S^1, t∈, b=a+tν, and Ψ:Q_ν→^2 defined by
 
  
    Ψ(x)=a    if  x·ν≤ 0
    b    if x·ν>0 .

  Assume that v_j∈ C^1(Q_ν;^2) is a sequence such that
  
    v_j   →Ψ   in  L^1(Q_ν;^2)
    ∫_Q_ν|∇ v_j|x̣   → |DΨ|(Q_ν)=|t| .

    Then
    
    lim sup_j→∞∫_Q_ν G(v_j,∇ v_j)x̣≤ 2√( 1+|a·ν^|^2)arccos(b)·(a) .




  We may assume without loss of generality that t≥ 0, ν=e_2,  which implies b=(a_1,a_2+t)^T. Let P:^2→ be defined by  P(x_1,x_2)=x_1.

Fix Δ>0. For j∈, we will split Q≡ Q_ν into a “good” and a “bad” set, and estimate the energy on these sets separately. We write v_j=((v_j)_1,(v_j)_2)^T and define the bad set by setting

    Ã_j   :={x∈ Q:  |(v_j)_1( x)-a_1|>Δ} ,
    
      A_j   :=P^-1(P(Ã_j))

The assumptions of the present proposition imply in particular that

    lim inf_j→∞∫_Q|∇ (v_j)_2|x̣≥ |DΨ_2|(Q_ν)=t ,

and hence
  
    (v_j)_1-a_1   → 0    in W^1,1(Q)
    ∂_x_1(v_j)_2   → 0    in L^1(Q) .











  Next we claim that there exists a sequence β_j↓ 0 such that for any (measurable) J⊂ [-1/2,1/2] and any j∈, we have that 
    
    | ∫_J |v_j(x_1,1/2)-v_j(x_1,-1/2)|x̣_1- tŁ^1(J)|≤β_j .

  To prove this claim, we note that we thanks to  the continuity of the trace operator W^1,1(Q)→ L^1(∂ Q), we have
  
    .
        v_j(·,-1/2)   → a 
    
            v_j(·,1/2)   → b 
          }   in L^1(I) .

  Hence, with
  
    β_j:= ∫_-1/2^+1/2 |v_j(x_1,1/2)-v_j(x_1,-1/2)-(b-a)|x̣_1 ,

  we obtain the claim (<ref>) by the triangle inequality.

  Additionally, by possibly increasing β_j, but still retaining the property β_j↓ 0, we may achieve
  
    |∫_Q|∇ v_j|x̣-t|≤β_j .

    
  We set I_j=P(A_j). By (<ref>) and Lemma <ref>, we have that Ł^1(I_j)→ 0.
  We are now in position to estimate the contribution of the bad sets A_j:
    
    lim sup_j→∞∫_A_j|∇ v_j|x̣    =lim sup_j→∞|∫_A_j|∇ v_j|x̣-tŁ^1(I_j)|
       ≤lim sup_j→∞|∫_Q|∇ v_j|x̣-t|
        +|∫_Q∖ A_j|∇ v_j|x̣-tŁ^1(I∖ I_j)|
       =0

where we have used in the last inequality that by (<ref>) and (<ref>),

    tŁ^1(I∖ I_j)-β_j≤∫_Q∖ A_j|∇ v_j|x̣≤ t+β_j ,

    which yields t=lim_j→∞∫_Q∖ A_j|∇ v_j|x̣. The estimate (<ref>) implies that
    
    lim sup_j→∞∫_A_j G(v_j,∇ v_j)x̣=0 .

    We now turn our attention to the contribution of the good sets
    
    E_j:=Q∖ A_j .

    By Lemma <ref>, we have that on E_j,
      
    |G(v_j,∇ v_j)   -G((a_1,(v_j)_2)^T, ∇ v_j)|
       ≤ CΔmax(G(v_j,∇ v_j),G((a_1,(v_j)_2)^T, ∇ v_j)) .

    By the subadditivity of ξ↦ G(v,ξ), G(v,ξ)≤ C|ξ|, ∇ (v_j)_1→ 0 in L^1 and ∂_x_1 (v_j)_2→ 0 in L^1 (see (<ref>)), we have that
    
    lim sup_j→∞   ∫_E_j 
          G((a_1,(v_j)_2)^T, ∇ v_j)x̣
       ≤lim sup_j→∞∫_E_j G((a_1,(v_j)_2)^T, ([             0             0;             0 ∂_x_2 (v_j)_2 ]))x̣ .

    Combining (<ref>) and (<ref>), we obtain
    
    lim sup_j→∞   ∫_E_j 
    G(v_j, ∇ v_j)x̣
       ≤1+CΔ/1-CΔlim sup_j→∞∫_E_j G((a_1,(v_j)_2)^T, ([             0             0;             0 ∂_x_2 (v_j)_2 ]))x̣ .

By Lemma <ref>, we obtain

    1-CΔ/1+CΔlim sup_j→∞   ∫_E_j 
      G(v_j, ∇ v_j)x̣
       ≤
    2√(1+a_1^2)∫_-1/2^1/2arccos(v_j(x_1,-1/2))·(v_j(x_1,1/2))x̣_1
         +4√(1+a_1^2)∫_Q|min(0,∂_x_2(v_j)_2)|x̣ .

Since v_j(·,-1/2)→ a and v_j(·,1/2)→ b in L^1(I), and 

    ∫_Q|min(0,∂_x_2(v_j)_2)|x̣→ 0 ,

  the proof of the proposition is completed by sending Δ→ 0.







§ PROOF OF PROPOSITION <REF>



  First we prove √(1+|v|^2)g_λ(S(v,·))≤ Q_2 f_λ(v,·). Indeed, we prove
  the slightly stronger claim √(1+|v|^2)g_λ(S(v,·))≤ Q_1 f(v,·),
  following the proof of Theorem 6.28 in <cit.>, where this is proved
  for λ=1 and v=0. The modifications that are necessary with respect
  to that proof are minor, so we will be brief. 

First one shows that g_1(S(v,·)) is polyconvex
  by defining

    θ(t)=2t     if t≤ 1
    1+t^2     else,

and convex functions H_±:^2× 2×→ by

    H_±(ξ,A)=θ((|S(v,ξ)|^2± 2
        S(v,ξ))^1/2)∓2A/(1+|v|^2)^2 .

Defining the convex function H(ξ,A)=max (H_+(ξ,A),H_-(ξ,A)),  one can then show
that g_1(S(v,ξ))=H(ξ,ξ) and hence g_1(S(v,·)) is polyconvex. Furthermore,
we have that g_λ(S(v,·))=√(λ) g_1(S(v,·/√(λ))) and
hence g_λ(S(v,·)) is polyconvex for every λ>0. The
inequality √(1+|v|^2)g_λ(S(v,·))≤ f_λ(v,·) can be verified from the definitions. This
proves √(1+|v|^2) g_λ(S(v,·))≤ Q_2 f_λ(v,·).
It
remains to show the opposite inequality. To do so, we make the following
definition:

  Let f:→. We say that f is symmetric rank one convex if

    f(tξ_1+(1-t)ξ_2)≤ t f(ξ_1)+(1-t) f(ξ_2)

for all t∈[0,1], and for all ξ_1,ξ_2∈ such that ξ_1-ξ_2=αη⊗η for
some α∈, η∈^n.

Furthermore, for f:→, we set

    f(ξ):=sup{g(ξ):g≤ f  and  g  is symmetric rank one
      convex} .


By Lemma B.5 of <cit.>, we have Q_2f_λ(v,·)≤ f_λ(v,·). In Lemma <ref> below, the inequality f_λ(v,·)≤ h_λ(v,·) is proved. This completes the proof
of the proposition.





  We have

    f_λ(v,·)≤ h_λ(v,·) .



We write f̅_λ:=(1+|v|^2)^-1/2f_λ(v,·), and need to show
f̅_λ≤ g_λ(S(v,·)).

Let ξ∈. Then there exists an orthonormal basis ẽ_1,ẽ_2 and x,y∈ such that

    S(v,ξ)=xẽ_1⊗ẽ_1+y ẽ_2⊗ẽ_2 .

We may assume |x|+|y|< √(λ), since otherwise we know
f̅_λ(ξ)=g_λ(S(v,ξ))=√(λ)+x^2+y^2/√(λ). Similarly, we may assume
0<|x|+|y|, since otherwise f̅_λ(ξ)=g_λ(S(v,ξ))=0.
Let
α,β∈ (0,1) to be chosen later, and set

    S_1=([ 0 0; 0 0 ]) ,  
    S_2=([ x/α   0;   0   0 ]) ,  
    S_3=([   x   0;   0 y/β ]) ,

where all matrices are in the ẽ_1,ẽ_2 basis. Observing that  the map
σ↦ S(v,σ)  is linear and invertible, we may
choose ξ_i such that S(v,ξ_i)=S_i for i=1,2,3. 

Note that β S_3+(1-β)(α S_2+(1-α) S_1))=S(v,ξ), and
S_3-(α S_2+(1-α)S_1),S_2-S_1 are both symmetric-rank-one. By
linearity, we also have that βξ_3+(1-β)(αξ_2+(1-α)
ξ_1))=ξ and that ξ_3-(αξ_2+(1-α)ξ_1),ξ_2-ξ_1 are
symmetric-rank-one. (These relations provide a “laminate” of order two.)
By Lemma B.7 in <cit.> (which states that f̅_λ may be estimated from above by laminates of order k for any k∈), 
we have

    f̅_λ(ξ)≤   βf̅_λ(ξ_3)+(1-β)(αf̅_λ(ξ_2)+(1-α) f̅_λ(ξ_1))
    
      =   β(√(λ)+x^2+y^2/β^2/√(λ))+(1-β)α(√(λ)+x^2/α^2/√(λ)) .

Now we assume |x|>0. The right hand side in the last estimate is convex in α; it attains its minimum at α=
|x|/√(λ). Hence, 

    f̅_λ(ξ)≤   β(√(λ)+x^2+y^2/β^2/√(λ))+(1-β)2|x|
    
      =    2|x|+β/√(λ)(√(λ)-|x|/√(λ))^2+y^2/β/√(λ)

Choosing β=|y|/(√(λ)-|x|), we obtain

    f̅_λ(ξ)≤ 2(|x|+|y|-|xy|/√(λ))=g_λ(S(v,ξ)) .

It remains to prove the claim for the case |x|=0. Then we have 

    f̅_λ(ξ)≤   βf̅_λ(ξ_3)+(1-β) f̅_λ(ξ_1)
    
      =   β(√(λ)+x^2+y^2/β^2/√(λ)) .

Again setting β=|y|/(√(λ)-|x|), we obtain the same conclusion as before.
This proves the lemma.






alpha








