The regularisation of the <span class="mathjax-formula">$N$</span>-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions

DOI:10.1051/cocv:2008039 www.esaim-cocv.org

THE REGULARISATION OF THE N-WELL PROBLEM BY FINITE ELEMENTS AND BY SINGULAR PERTURBATION ARE SCALING EQUIVALENT

IN TWO DIMENSIONS

Andrew Lorent

Abstract. LetK:=SO(2)A1∪SO(2)A2. . . SO(2)AN whereA1, A2, . . . , AN are matrices of non- zero determinant. We establish a sharp relation between the following two minimisation problems in two dimensions. Firstly theN-well problem with surface energy. Letp∈[1,2], Ω⊂R² be a convex polytopal region. Define

I^p(u) =

Ωd^p(Du(z), K) +D²u(z)²dL²z

and let AF denote the subspace of functions in W^2,2(Ω) that satisfy the affine boundary condition Du=F on∂Ω (in the sense of trace), whereF∈K. We consider the scaling (with respect to) of

m^p := inf

u∈AF

I^p(u).

Secondly the finite element approximation to the N-well problem without surface energy. We will show there exists a space of functionsD^h_F where each functionv∈ D^h_F is piecewise affine on a regular (non-degenerate)h-triangulation and satisfies the affine boundary condition v=lF on∂Ω (wherelF

is affine withDlF =F) such that for

αp(h) := inf

v∈D_F^h

Ωd^p(Dv(z), K) dL²z

there exists positive constantsC1<1<C2(depending onA1, . . . , AN,p) for which the following holds true

C1αp√

≤m^p ≤ C2αp√

for all >0. Mathematics Subject Classification.74N15.

Received June 7, 2007. Revised January 15, 2008.

Published online June 24, 2008.

1. Introduction

The main goal of this paper is to show the equivalence in two dimensions (in the sense of scaling) of two different regularisations of a non-convex variational problem that forms a model of crystalline microstructure,

Keywords and phrases. Two wells, surface energy.

1 MIS MPG, Inselstrasse 22, 04103 Leipzig, Germany. andrew.lorent@sns.it

Article published by EDP Sciences c EDP Sciences, SMAI 2008

specifically regularisation by second order gradients (otherwise known as singular perturbation) and regularisa- tion by discretationvia finite elements.

We focus on the simplest problem with non-trivial symmetries, the N-well problem in two dimensions. To set the scene let us take the Ball-James [3,4], Chipot-Kinderlehrer [6] approach to crystal microstructure. We have an energy function I on the space of deformationsu: Ω⊂R³→R³ which has the form

I(u) =

ΩW(Du(x)) dL²x, (1.1)

where W is the stored energy density function that describes the various properties of the material. The functionW has its minimum on a set of matrices known as thewells

K=SO(3)A1∪SO(3)A2. . . SO(3)A^N. (1.2) Roughly speaking theA1, A2, . . . , A^N are symmetry related and represent the lattice states of the material.

Sincew must be invariant with respect to rotation of the ambient space the wellsK must have form (1.2).

FunctionalI is minimised over the space of functions that have affine boundary conditionF∈K.

A key point is that functionalI is not weakly lower semi-continuous. Minimising sequences form finer and finer oscillations, as is to be expected in any model designed to capture properties of microstructure.

Surprisingly for certain choices of K of the form (1.2) in two or three dimensions, the quasiconvex hull (see [27] for precise definitions and more information) ofK (which we denote K^qc) is sufficiently rich to allow for the existence ofF ∈K^qc\K for which there exists an exact minimiser ofI over a space of function with boundary conditions F. Specifically if K = SO(2)∪SO(2)H where H =

λ 0 0 μ

and μλ ≥ 1 [35], or K =SO(2)A1∪SO(2)A2. . . SO(2)Ak where A1, A2, . . . , Ak satisfy a certain condition [14], or if K are the so call cubic to tetragonal wellsK=SO(3)U1∪SO(3)U2∪SO(3)U3where forλ >1

U1=

⎛

⎝λ² 0 0 0 _λ¹ 0 0 0 _λ¹

⎞

⎠, U2=

⎛

⎝

1

λ 0 0

0 λ² 0 0 0 ¹_λ

⎞

⎠ andU3=

⎛

⎝

1

λ 0 0

0 ¹_λ 0 0 0 λ²

⎞

⎠

[15], then in these cases there is an exact minimiser toI for someF ∈K^qc\K. This follows from work of M¨uller- Sver´ˇ ak [29,30], Sychev [33,34], Kirchheim [19,20] and Conti-Dolzmann-Kirchheim [11], see also Dacorogna- Marcellini [12] for a different approach to some related problems. The approach of M¨uller-ˇSver´ak uses the theory of “convex integration” (denoted by CI from this point) developed by Gromov, it is one of the simplest results of the theory.

FunctionalI does not constrain oscillations of the gradient, it does not give a length scale or any restriction on the fine geometry of the microstructure. For many materials, the observed length scale of the microstructure is many orders larger than the atomic scale and for these materials functional I is only a first approximation.

To overcome this the following adaption of the functional I is commonly made, see [27], Section 6, I(u) =

Ω

W(Du(z)) +D²u(z)²dL²z.

Roughly speaking this is a regularisation of I that constrains the minimiser u of I to have less than M interfaces when typically M will be a negative power of that depends on K and W. For example if we take K = SO(2)∪SO(2)H (with det (H) = 1) and W(·) ∼ d(·, K) then using the characterisation of Sver´ˇ ak [35] (as will be explained later) we have the upper bound of infI≤c⁶¹. Letv(z) :=u(√

z)⁻¹², then

⁻¹2ΩD²v² ≤m⁻¹ ≤c⁻⁵⁶. Now v satisfies the elliptic Euler Lagrange equation divDW(Dv) + Δ²v = 0 which by standard elliptic regularity meansDvis Holder with Holder exponentO(1), thus each interface running through⁻¹²Ω contributesO

⁻¹²

to

⁻¹2ΩD²v²so we have at mostM ≤c¹²⁻⁵⁶ =c⁻¹³ such interfaces.

There have been a number of studies of simplified versions of functional I [8,22,26]. However these works focus on the case where I acts on scalar functions and the wells of I are given by two matrices. In this case (scaling) sharp upper and lower bounds have been proved. For functional with wells that have rotational invariance,i.e. of the form (1.2), nothing is known about the energy of minimisers.

Another way to constrain oscillation in the gradient is to minimiseI directly over the space of functions that are piecewise affine on ahsized triangular grid. This is known as the finite element approximation ofI. There have been many studies of finite element approximations to functionals of the form I, again for the simplified case where the wells are given by sets of two or three matrices [5,7,23,24].

Our main achievement in this paper is to show that for the specific stored energy functionW(·)∼d^p(·, K) (for somep∈[1,2]) these two regularisations are scaling equivalent.

For the case where the wells ofIare given by sets of two or three matrices (andW(·)∼d(·, K)) it is possible to calculate the scaling of the energy ofIand the scaling of the energy of the finite element approximation toI [7,23]. To be more specific given matricesA,B with rank (A−B) = 1 using methods from [7] it can be shown that for wellsK1={A, B}the functionalI¹minimised over the space of functions that are piecewise affine on a h-sized triangular grid¹and have affine boundary conditionF0=μ0A+ (1−μ0)B (for someμ0∈(0,1)) scales like √

h. Strictly speaking the functional studied in [7] acts on scalar functions but the method works for the case stated above with minor modifications. In [23] three rank-1 connected matrices were considered, expanding on the methods of [7] it was shown in [23] that if functionalI²has wellsK2=

−1 0

0 −1

,

−1 0

0 1

,

1 0 0 0

(andW(·)∼d(·, K2)) then over the space of piecewise affine functions with boundary conditionF1=

0 0 0 0 the energy of functionalI²scales likeh¹³. Using very similar methods to [7] and [23] it is possible to show that functionals I^a :=

Ωd (Du, K^a) +D²u² fora = 1,2 are such that their energy scales like infI¹ ∼⁴¹ and infI²∼¹⁶.

Thus for functionals whose wells are given by sets of two or three matrices our main theorem is of no interest, for in these cases we can calculate the scaling and it can be seen instantly that the energy of functionalI^ataken over a space of function that are piecewise affine on a grid of size√

scales in the same way as the energy of functionalI^a. The point of this paper is that we study functionalI with wells

K=SO(2)A1∪SO(2)A2. . . SO(2)AN

and for these wells the scaling of the energy ofIand the scaling of the energy ofI over the space of piecewise affine functions are completely unknown. In this case our main theorem tells us that these two problem, one discrete and one continuous, are scaling equivalent.

1.1. Background and statement of main result

To state our theorem we need to give some background. Given a polytopal region Ω and some small constant ς ∈ (0,1) we say a collection of disjoint triangles {τi} is an (h, ς)-triangulation of Ω if

iτⁱ = Ω and every triangle τⁱ contains a ball of radiusςh and has diameter less thanς⁻¹h. Givenw∈ S¹ we denote by ^ς_h(w) the set of regular triangulations with respect to axisw,w^⊥ axis, by this we mean every triangleτiof distance ς⁻¹hfrom∂Ω is a right angle triangle with sides parallel tow,w^⊥. Finally we letF_F^ς,h(w) denote the space of functions that are piecewise affine on some triangulation in^ς_h(w) and satisfy the affine boundary condition u=l^F on∂Ω, wherel^F is a fixed affine function withDl^F =F.

Given two connected subsets of matricesM, N ⊂M^2×2we sayM andN arerank-1 connectedif and only if there existsA∈M andB∈N and v∈S¹ such thatAv=Bv. The set ofrank-1 directions connectingM,N are the set of vectorsv∈S¹ satisfyingAv=Bv for someA∈M,B∈N.

1Whose edges are not parallel to the rank-1 connections betweenAandB.

For given triangulation{τi} and functionu∈ FF^ς,h(w) and triangleτⁱ we define theneighbouring gradients by

Ni(u) =

Duτj :τ^j∩τⁱ=∅

forisuch that τⁱ∩∂Ω =∅ Duτj :τ^j∩τⁱ=∅

∪ {F} forisuch that τⁱ∩∂Ω=∅. (1.3) And foru∈ F_F^ς,h we define thejump trianglesby

J(u) :=

i:∃A, B∈Nⁱ(u) such that |A−B|> ς⁻¹

. (1.4)

Let σbe the minimum of the absolute values of the eigenvalues ofA1, . . . , A^N. Letw1 ∈S¹ be such that for some w2 ∈ w^⊥₁ we have that w1, w2,_|^ww¹₁⁻−^ww²₂| are not in the set of rank-1 directions connecting SO(2)Aⁱ to SO(2)A^j for anyi=j, letς∈

0,10⁻¹σ

we define function space D^ς,hF (w1) :=

⎧⎨

⎩v∈ FF^ς,h(w1) :

i∈J(v)

M∈N_i(v)

Dvτi−M²≤ς⁻¹h⁻²

Ω

d^p(Dv, K)

⎫⎬

⎭. (1.5)

When there is no ambiguity we will denote these function spaces just asFF^ς,horD^ς,hF . Clearly inf_v∈F^ς,h

F I₀^p(v)≤ inf_v∈D^ς,h

F I₀^p(v), the main reason for introducing function spaceD^ς,h_F is that with our methods wecan notshow the sharpness of the lower bound

inf

v∈FF^ς,√

I₀^p(v)≤c inf

v∈A_FI(v) (1.6)

(whereAF is the subspace of functions inu∈W^2,2(Ω) withDu=F in the sense of trace). So instead we will prove the stronger lower bound inf_v∈Dς,√

F I₀^p(v)≤cinfu∈A_F I(u) and it turns out that function spaceD^ς,h_F has enough structure to allow us to show the upper bound²

u∈infA_FI^p(u)≤c inf

v∈DF^ς,√

I₀^p(v). (1.7)

Our main theorem is the following.

Theorem 1.1. Let K := SO(2)A1∪SO(2)A2. . . SO(2)A^N where A1, A2, . . . , A^N ∈ M^2×2 are matrices of non-zero determinant. Let σbe the minimum of the absolute values of the eigenvalues ofA1, . . . , A^N.

Letw1∈S¹be such that for somew2∈w^⊥1, the vectorsw1, w2,_|^ww¹₁⁻−^ww²₂| are not in the set of rank-1directions connectingSO(2)Aⁱ toSO(2)A^j for anyi=j. LetΩ⊂R² be a polytopal convex domain. Forp∈[1,2]define

I^p(u) :=

Ω

d^p(Du(z), K) +D²u(z)²dL²z.

Let F ∈ K and let A^F denote the subspace of functions in W^2,2(Ω) that have boundary condition Du = F on ∂Ωin the sense of trace. Forς ∈

0,10⁻¹σ

let function spaceD^ς,hF (w1)be defined by(1.5). If we define αp(h) := inf

v∈DF^ς,h(w₁)I₀^p(v) andm^p := inf

u∈A_FI^p(u) then there are positive constants C1<1<C2 (depending only on σ,ς,p) for which

C1α^p√

≤m^p ≤ C2α^p√

for all >0. (1.8)

2For further explanation as to why function spaceD^ς,h_F allows us to prove (1.7) where as with our methods we can not show the same inequality for the larger function spaceF_F^ς,hsee Section2.2.

The point of introducing parameter ς into the definition of D^ς,hF is that we would like to use the greater flexibility it allows for a potentialfuture improvement of our main result. To explain this further note that the definition ofD^ς,hF gives us the inclusion

L^ς,hF :=

v∈ FF^ς,h:DvL^∞(Ω)≤4⁻¹ς⁻¹

⊂ D^ς,hF ,

so clearly we have the upper bound (1.7) for this function space. Given the results of [28] it seems reasonable to hope that minimisers of a functional equivalent³to I^p for p >1 are Q-Lipschitz (for some possibly large⁴ Q independent of ) inside the whole domain Ω. In [28] this has only been proved for p= 2 in an interior domain. If such a result could be proved the methods of this paper would allow us to show the lower bound inf_v∈L^ς,h

F I^p(v)≤cinfu∈A_F I^p(u) which together with the upper bound would imply forp >1 the equivalence of the scaling ofm^p to the scaling ofI^p over the space of Lipschitz finite elements. This is the principle reason for introducing parameterς.

In truth our main motivation for establishing Theorem 1.1was that we hoped to use it as a tool to under- standing the minimiser ofI^p. To explain this further we will simplify and takeK=SO(2)∪SO(2)H where H is a diagonal matrix of determinant 1 and we takep= 1.

As mentioned, nothing is known about the minimiser of the functional I¹. In particular it is completely unknown if for very small the minimiser is something like the absolute minimiser of I0 provided by CI⁵. In some sense this might seem reasonable, we refer to the D²u²term as the “surface energy” and the

d(Du, K) term as the “bulk energy”, as→0 the surface energy becomes less and less important, the main thing to be minimised is the bulk energy and of course CI solutions have zero bulk energy.

This question is best expressed by considering the scaling ofm¹. An upper bound ofm¹≤c¹⁶ is provided by the standard double laminate which follows from the characterisation of the quasiconvex hull ofSO(2)∪SO(2)H provided by [35], see Figure1.

Ifm∼^16+αforα >0 then the minimiser will have to take a very different form than the double laminate.

On the other hand ifα= 0 then energetically the minimiser does no better than the double laminate.

This question is important because CI solutions are important, many counter examples to natural conjectures in PDE have been achievedviaCI [13,19,31,32]. Minimising functionalIis the simplest problem that constrains oscillation in some slight way where we can hope to see the effect of the existence of exact minimisers of (1.1).

In the proof of Theorem1.1 we have to work quite hard to establish the result forp= 1, we do so because functional I¹ is particularly clean in the sense that it is not necessary to consider laminates with “domain branching” to construct upper bounds (contrast this with the case p = 2 [8,22]) as such the upper bound is given byc¹⁶ and is domain independent.

Let w1 ∈ S¹ be such that for w2 ∈ w₁^⊥ we have w1, w2,_|^ww¹₁⁻−^ww²₂| do not belong to the rank-1 connections betweenSO(2) andSO(2)H. If ˜u∈ FF^ς,h(w1) and τ1, τ2 ∈ ^ς_h(w1) are such that d

Du˜τ₁, SO(2)

≈0 and d

Du˜τ₂, SO(2)H

≈0, it is not too hard to seeτ1 can not touchτ2,i.e. there must be a triangleτ3 between τ1and τ2 for whichd

Duτ₃, K

≥o(1).

For example if we have an interpolant of a laminate, and triangleτⁱcuts through an interface of the laminate the affine map we get from interpolating the laminate on the corners ofτⁱwill have its linear part some distance from the wells. See Figure2.

3In order to apply the result of [28] we need a functional that is quadratic at infinity in a strong sense, but givenW(·)∼d^p(·, K) it is easy to construct a function W such thatW =W in a large ballBR andW−WL^∞ ≤cwhileW(M) =|M|^pfor any M∈B2R. DefiningI^p(v) =

ΩW(Dv) +D²v²we haveI^p(v)−I^p(v)≤cfor anyv∈W^2,2(Ω) so obviously the energy ofI^p

andI^pscale the same way with respect toand for the casep= 2 it is possible to apply the results of [28] to the minimiser ofI^p. 4Foundviaa compactness argument.

5We know it can notbea functionuwithI0(u) = 0 because the result of Dolzmann-M¨uller [16], that anyuwith this property and with the property thatDuis a BV has to be laminate.

SO(2) SO(2)H

R1

Rλ

R2

F G

H

0 0 1 0

0 0 0 0

−1 0 0 0

−1 0

0 −1 −1 0

0 1

Figure (b) Figure (a)

( )

( ) ( ) ( )

( )

Figure 1. Rank-1 connections between sets of matrices.

Figure 2. A finite element approximation to a laminate.

So we can not lower the energy of I0 overFF^ς,h(w1) by simply making a laminate type function with finer layers, there is a competition between the surface energy as given by the error contributed from the interfaces and the bulk energy which in the case of the laminate is the width of the interpolation layer.

As mentioned for functionalI² with wells

−1 0 0 −1

,

−1 0

0 1

,

1 0 0 0

in [23] it was shown the energy ofI²over the spaceF_F^ς,h₁ (whereF1=

0 0 0 0

) scales likeh¹³. From ˇSver´ak’s characterisation [35] we know the exact arrangement of rank-1 connections between the matrices in the setSO(2)∪SO(2)H and a matrix in the interior of the quasiconvex hull ofSO(2)∪SO(2)H, see Figure1a. As we can see from Figures1a and1b, the finite well functionalI² precisely mimics these rank-1 connections.

Conjecture 1.1. Let K = SO(2)∪SO(2)H where H is a diagonal matrix with eigenvalues σ, σ⁻¹. Let w1∈S¹andw2∈w₁^⊥ be such thatw1,w2, _|^w_w^1−w2

1−w₂| are not in the set of rank-1 connections betweenSO(2)and SO(2)H. LetΩbe a polytopal convex region, ς∈

0,10⁻¹σ

. Given F ∈int (K^qc), let function space FF^ς,h(w1) denote the space of functions that are piecewise affine on some regular triangulation {τi} ∈ ^ςh(w1). There existsc0=c0(σ, ς)>0 such that

inf

u∈FF^ς,h

I₀¹(u)≥c0h¹³ for allh >0.

So from Theorem1.1, if Conjecture1.1could be proved it would imply the scalingm¹ ∼¹⁶. Unfortunately even though the minimisation of I₀¹ over FF^ς,h is discrete problem, it appears to be quite hard to prove lower bounds.

2. Sketch of the Proof

Written out in detail, the proof of Theorem1.1 is not short, however the basic ideas are quite simple. We give a sketch of the proof based on two lemmas that are only approximate principles, by this we mean that either we can not prove them, or only a weaker form hold true. This may be a bit unconventional, but it seems to us to be the best way to get to the heart of the matter without being flooded with details.

2.1. Lower bound

We focus on the casep= 1 and take Ω =Q1(0). LetM =

⁻¹²

. We cut the square Ω intoM²sub-squares of side length _M¹, letc1, c2, . . . , cM² be the centres of these squares. SoQ1(0) =M²

i=1Q_M¹ (cⁱ). LetC1=C1(σ) be some small constant we decide on later. Now we define the “bad” squares to beB:=

i:

Q₁

M(ci)D²u²≥ C1

. Approximate principle 1. For anyi ∈

1,2, . . . , M²

\B definevⁱ(z) =u

cⁱ+_M^z

M we have that there exists affine functionLⁱ withDLⁱ∈K such that

vi−LiL^∞(Q₁(0))≤c

Q₁(0)

d(Dvi, K) +D²vi². (2.1)

Approximate principle 2. The minimiseruofI is a Lipschitz.

Let us make it once again clear we can not prove eitherapproximate principle, they are simply a device to show the strategy of the proof. Now we split every sub-square Q_M¹ (cⁱ) into two right angle triangles, denote them τⁱ, τi+M² so the set {τ1, τ2, . . . , τ2M²} is a triangulation of Ω. Let ˜ube the piecewise affine function we obtain fromuby defining ˜uτi to be the affine map we get from interpolatinguon the corners ofτⁱ.

Now for anyi∈B letωⁱ₁, ω₂ⁱ, ω₃ⁱ denotes the corners ofτⁱ, sol, q∈ {1,2,3}

Du˜τ_i

ωlⁱ−ωqⁱ

ωlⁱ−ωqⁱ

!

−DLi

ωlⁱ−ωqⁱ

ωlⁱ−ωqⁱ!

≤Mu ωⁱl

−Li

ωlⁱ

− u

ωⁱq

−Li

ωqⁱ

(2.1)

≤ c

Q₁

M(ci)

M²d(Du, K) +D²u². (2.2)

Since (2.2) holds true for every l, q ∈ {1,2} we have Du˜τi−DLⁱ ≤ c

Q₁

M(c_i)M²d(Du, K) +D²u². In exactly the same wayDu˜τ_i+M2−DLi+M²≤c

Q₁

M(c_i)M²d(Du, K) +D²u². So

i∈{1,2,...,M²}\B

Du˜τ_i−DLⁱL²(τⁱ) +Du˜τ

i+M2−DLi+M²L²(τi+M²)≤cm¹. (2.3) Now for anyi∈B, sinceuis Lipschitz, forl, q∈ {1,2,3}we have

Du˜τi

ωlⁱ−ωⁱq

ωlⁱ−ωⁱq! =

u ωⁱl

−u ωqⁱ

ωlⁱ−ωⁱq

≤c thusd

Du˜τ_i, K

≤c and in the same wayd

Du˜τ_i+M2, K

≤cso

i∈B

Du˜τi−DLⁱL²(τⁱ) +Du˜τ_i+M2−DLi+M²L²(τi+M²)≤ c M²

i∈B

Q₁

M(c_i)

D²u²≤cm¹.(2.4) So as {τi} is a √

,10⁻¹σ

-triangulation and from (2.3), (2.4) we have α(√

) ≤ cm¹ which establishes the lower bound.

It is easy to construct a counter example to the “morally true” Lemma 1, however as a substitute we have Proposition5.1, see Section5. Sincei∈B it should seem reasonable that there existsk0 such that

Q₁(0)

d(Dvⁱ, SO(2)A^k0)≤c

Q₁(0)

d(Dvⁱ, K). (2.5)

This follows from a kind of capacity type argument that is Step 1 of Proposition 5.1. Alternatively imagine we had slightly more integrability ofD²vⁱ and hence that

Q₁(0)D²v^i2+δ_2+δ¹

is “small” (in fact vⁱ satisfies a fourth order elliptic PDE coming from the Euler Lagrange equation of uso we could indeed establish such higher integrabilityvia reverse Holder inequalities), then by Sobolev embedding we would have thatDvⁱ stays in a neighbourhood of some wellSO(2)Ak₀ and so (2.5) trivially follows.

Now if we were considering thed^p(·, K) distance from the wells then we could apply Theorem3.1to obtain sharpL^p control of the distance ofDvi from a matrix in K. For thep= 1 case Theorem3.1 is false [10] and so we need to use the fact that the “tangent space” to the set SO(2) around the identity is the set of skew symmetric matrices. This allows us to apply the Korn type Poincar´e inequality given by Lemma 3.1to gain sharp control of theL¹ distance ofvⁱ from the affine function.

Note that Proposition5.1is not enough since in the argument given in (2.2) we need to control the function exactly at the corners of the triangles. The trick to overcome this is the following. Letv :Q^M(0) →R² be defined by v(z) =u_z

M

M. By the Co-area formula we can find a grid of squares of side length 1, labelled

S1, S2, . . . , SM²−4M such that for eachithere exists affine functionLⁱ withDLⁱ∈K such that c

∂S_i

|v−Li|+D²v²+d(Dv, SO(2) sym (DLi))≤

N₁(S_i)

d(Dv, K) +D²v²=:αi (2.6) (where sym (A) denotes the symmetric part of matrixAwe obtain by polar decomposition). We can splitSⁱinto disjoint trianglesτⁱ,τi+M². Letaⁱ, bⁱ, cⁱ be the corners ofτⁱ where [aⁱ, bⁱ]∪[bⁱ, cⁱ] =∂τⁱ∩∂Sⁱ. The important point is that Dv along [aⁱ, bⁱ] varies by at most √

αⁱ and so its not hard to show Dv(z) ∈ B^c√αi(DLⁱ) for allz∈[aⁱ, bⁱ]. For simplicity let us assume sym (DLⁱ) =Id.

Given ˜bi∈[ai, bi], by trigonometry this allows to conclude v(ai)−v

˜bi≥(1−cαi)ai−˜bi.

And very easily from (2.6) (since we have assumed sym (DLi) =Id) we have v(aⁱ)−v

˜bⁱ≤(1 +cαⁱ)ai−˜bⁱ. The point ˜bⁱ can be easily chosen so that v

˜bⁱ −Lⁱ

˜bⁱ ≤ cαⁱ. In exactly the same way we can find c˜ⁱ ∈ [aⁱ, cⁱ] such that |v(˜cⁱ)−Lⁱ(˜cⁱ)| ≤ cαⁱ and ||v(aⁱ)−v(˜cⁱ)| − |ai−˜cⁱ|| ≤ cαⁱ. Let γ1 = aⁱ−˜bⁱ and γ2=|ai−˜ci|so (definingNδ(A) :={x:d(x, A)< δ}) we have

v(ai)∈Ncαi

∂Bγ₁

˜bi

∩Ncαi(∂Bγ₂(˜ci)). (2.7)

See Figure 4. From (2.7) it is not hard to show v(ai) ∈ Bcαi(Li(ai)). We can control the corners bi, ci in the same way. Therefor if we defineli to be the affine map we get from interpolating v on{ai, bi, ci} we have d(Dli, DLi)≤cαi. Since"

iαi ≤c⁻¹m^p this gives the lower bound.

2.2. Upper bound

To obtain the upper bound we will have to convert a function v that is piecewise affine on a (√ , ς)- triangulation into a function u ∈ W^2,2(Ω) with affine boundary condition Du = F on ∂Ω (in the sense of trace), recall we denote the space of such functions by A^F. The most natural way to do this is to convolvev with a functionψ^√ whereψ^√(z) :=⁻¹ψ

√z

andψ∈C0^∞(B1(0) :R+) withψ= 1 onB¹₂(0).

LetG0:=

i:d

Dvτi, K

≤ ^d(SO(2),SO₈ ^(2)H)

and define E(x) :=

i:τⁱ∩B^√(x)=∅

. Supposex∈Ω is such thatE(x)⊂G0, for simplicity we will assumed

Dvτi, SO(2)

=d

Dvτi, K

for everyi∈E(x). Since for any k, l∈ E(x) withH¹(τ^k∩τ^l)>0 we have that there existsw ∈S¹ such that Dvτ_kw=Dvτ_lw and thusDvτ_k−Dvτ_l≤c

d

Dvτ_k, SO(2) +d

Dvτ_l, SO(2)

because ifDvτ_k∈SO(2) andDvτ_l∈SO(2) the fact that Dvτ_kw = Dvτ_lw would imply Dvτ_k = Dvτ_l, so the difference between Dvτ_k and Dvτ_l is controlled by the distance of these matrices fromSO(2).

A relatively easy generalisation of this is that for anyxwhere E(x)⊂G0

Dvτk−Dvτl≤cmax d

Dvτi, K

:i∈E(x)

for anyk, l∈E(x). (2.8)

NowDu(x) ="

i∈E(x)Dvτi

τ_iψ^√(z−x) dL²z. Let’s picki0∈E(x) we then have Du(x)−Dvτi0

=

i∈E(x)

Dvτi−Dvτi0

τi

ψ^√(z−x) dL²z

(2.8)

≤ cmax d

Dvτ_i, K

:i∈E(x)

. (2.9)

So for any x ∈ Ω such that E(x) ⊂ G0, d(Du(x), K) is comparable to d

Dvτ_i

0, K

with error given by max

d

Dvτi, K

:i∈E(x)

and thus

{x:E(x)⊂G₀}d^p(Du(z), K) dL²z≤c"

id^p

Dvτi, K . Since |Du(x)| ≤ c"

i∈E(x)Dvτi thus d^p(Du(x), K) ≤ c"

i∈E(x)d^p

Dvτi, K + 1

so as L²({x∈Ω :E(x)⊂G0})≤cL²

i∈G₀τⁱ

≤cm^p we have

{x:E(x)⊂G₀}d^p(Du(x), K)≤cm^p. So all that remains is to control the

ΩD²u² term. For x∈Ω such thatE(x)⊂G0 this is relatively easy since

D²u(x) =−

Dv(z)⊗Dψ^√(z−x) dL²z (2.10) and as

Dψ^√(z−x) dL²z= 0 we have

D²u(x) = − Dv(z)−Dvτ_i

0

⊗Dψ^√(z−x) dL²z

≤ c⁻¹²maxDvτ_j−Dvτ_i

0

:j∈E(x)

. So

D²u(x)² ≤ c⁻¹

maxDvτ_j−Dvτ_i

0

:j∈E(x) p (2.8)

≤ c⁻¹max d^p

Dvτ_i, K

:i∈E(x) . Thus

{x:E(x)⊂G₀}

D²u(x)²dL²x≤c⁻¹

i

d^p

Dvτi, K

L²(τⁱ)≤c⁻¹m^p.

So far everything goes well simply by using (2.8), however forx∈Ω such thatE(x)⊂G0 we have a problem because the quantity we are interested in is D²u(x)² and from equation (2.10), if the jump from Dvτ_i to Dvτ_l is much greater than 1 we can not estimateD²u²by anyL¹control of the distance ofDvfromK. Quite simply if we have an arbitrary functionv∈ F(^ς,√)

F and we form functionuby convolving it withψ^√ it could be the case that

Ωd^p(Du, K) +D²u²m^p. In order for the estimate we want to hold true we need some condition that bounds the square of all the jumps of order>1 by the quantity ⁻¹m^p. The way we deal with this problem is by circumventing it: in establishing the lower bound we showed that from a function u∈A^F we can create a function ˜u that is piecewise affine on a (√

, ς) triangulation and

Ωd(Du, K˜ )≤ cm^p, if we were smarter we could show the function ˜uthat we created had even stronger properties. For example ifuwas Lipschitz then ˜uwould also be Lipschitz and our problems would be over. Unfortunately we can not proveuis Lipschitz, however what we have for free is that

ΩD²u²≤⁻¹m^p. It turns out that for sufficiently careful choice of triangulation this is strong enough for us to be able to construct a function ˜usuch that if we define