https://doi.org/10.1051/cocv/2021019 www.esaim-cocv.org
APPROXIMATION OF PLANAR SOBOLEV W
2,1HOMEOMORPHISMS BY PIECEWISE QUADRATIC
HOMEOMORPHISMS AND DIFFEOMORPHISMS
∗Daniel Campbell
1,2,**and Stanislav Hencl
3Abstract. Given a Sobolev homeomorphismf ∈W2,1 in the plane we find a piecewise quadratic homeomorphism that approximates it up to a set ofεmeasure. We show that this piecewise quadratic map can be approximated by diffeomorphisms in theW2,1 norm on this set.
Mathematics Subject Classification.46E35.
Received August 18, 2020. Accepted February 8, 2021.
1. Introduction
In this paper we address the issue approximation of Sobolev homeomorphisms with diffeomorphisms. Let us briefly explain the motivation for this problem that comes from Nonlinear Elasticity. Let Ω⊂Rn be a domain which models a body made out of homogeneous elastic material, and let f : Ω→Rn be a mapping modeling the deformation of this body with prescribed boundary values. In the theory of nonlinear elasticity pioneered by Ball and Ciarlet, see e.g. [2, 3, 12], we study the existence and regularity properties of minimizers of the energy functionals
I(f) = Z
Ω
W(Df) dx,
whereW :Rn×n→Ris the so-called stored-energy functional, andDf is the differential matrix of the mapping f. The physically relevant assumptions on the model include:
(W1) W(A)→+∞as detA→0,i.e.mapping does not compress too much, (W2) W(A) = +∞if detA≤0, which guarantees that the orientation is preserved.
∗The first author was supported by the grant GA ˇCR 20-19018Y. The second author was supported by the grant GA ˇCR P201/18-07996S.
Keywords and phrases:Diffeomorphic approximation, Ball-Evan’s, Sobolev homeomorphism.
1 Department of Mathematics, University of Hradec Kr´alov´e, Rokitansk´eho 62, 500 03 Hradec Kr´alov´e, Czech Republic.
2 Faculty of Economics, University of South Bohemia, Studentsk´a 13, Cesk´e Budejovice, Czech Republic.
3 Department of Mathematical Analysis, Charles University, Sokolovsk´a 83, 186 00 Prague 8, Czech Republic.
**Corresponding author:[email protected]
Article published by EDP Sciences c EDP Sciences, SMAI 2021
In particular, any admissible deformationf satisfies
Jf(x) := detDf(x)>0 for a.e.x∈Ω.
With the help of some growth assumptions onW we can prove that a mapping with finite energy is continuous and one-to-one, which corresponds to the non-impenetrability of the matter. Hence it is natural to study Sobolev homeomorphisms with Jf >0 a.e. that minimize the energy.
As pointed out by Ball [4, 5] (who ascribes the question to Evans [15]), an important issue toward under- standing the regularity of the minimizers in this setting would be to show the existence of minimizing sequences given by piecewise affine homeomorphisms or by diffeomorphisms. This question is called the Ball-Evans approx- imation problem and asks as a first step to approximate any homeomorphism u∈W1,p(Ω;Rn), p∈[1,+∞) by piecewise affine homeomorphisms or by diffeomorphisms inW1,p norm. The motivation is that regularity is typically proven by testing the weak equation or the variation formulation by the solution itself; but without some a priori regularity of the solution, the integrals are not finite. Thus we need to test the equation with a smooth test mapping of finite energy which is close to the given homeomorphism instead. Besides Nonlinear Elasticity, an approximation result of homeomorphisms with diffeomorphisms would be a very useful tool as it allows a number of proofs to be significantly simplified. Let us note that finding diffeomorphisms near a given homeomorphism is not an easy task, as the usual approximation techniques like mollification or Lipschitz extension using the maximal operator destroy, in general, injectivity.
Let us describe the known results about the Ball-Evans approximation problem. The problems of approxi- mation by diffeomorphisms or piecewise affine planar homeomorphisms are in fact equivalent by the result of Mora-Corral and Pratelli [25] (see also [21]). The first positive results on approximation of planar homeomor- phisms smooth outside a point are by Mora-Corral [24]. The celebrated breakthrough result in the area which stimulated much interest in the subject was given by Iwaniec, Kovalev and Onninen in [19], where they found diffeomorphic approximations to any homeomorphism f ∈W1,p(Ω,R2), for any 1< p <∞in the W1,p norm.
The remaining missing case p= 1 in the plane has been solved by Hencl and Pratelli in [17] by a different method. This method was extended to cover other function spaces like Orlicz-Sobolev spaces (see Campbell [8]), BV space (see Pratelli, Radici [26]) orW Xfor nice rearrangement invariant Banach function spaceX (see Campbell et al.[11]). It is possible to approximate also f−1 in the Sobolev norm for p= 1 (see Pratelli [27]) or for 1≤p <∞under the additional assumption that the mapping is bi-Lipschitz (see Daneri and Pratelli in [13]). Moreover, it is possible to characterize all strong limits of Sobolev diffeomorphisms (not only homeomor- phisms) as shown by Iwaniec and Onninen [20] forp≥2 and by De Philippis and Pratelli [14] for 1≤p <2. The higher dimensional casen≥3 is widely open and essentially nothing is known forn= 3. However, forn≥4 and 1≤p <[n2] there exists a Sobolev homeomorphism inW1,pwhich cannot be approximated by diffeomorphisms (see Hencl and Vejnar [18], Campbell, Hencl and Tengvall [10], Campbell, D’Onofrio and Hencl [9]).
Our aim is to find the corresponding planar result for models with second gradient, i.e.we would like to approximate W2,q homeomorphisms by diffeomorphisms. Models with the second gradient
E(f) = Z
Ω
W(Df(x)) +δ0|D2f(x)|q
dx, (1.1)
where q∈[1,∞) andδ0>0, were introduced by Toupin [28], [29] and later considered by many other authors, seee.g.Ball, Curie, Olver [6], Ball, Mora-Corral [7], M¨uller ([23], Sect. 6), Ciarlet ([12], page 93) and references given there. The contribution of the higher gradient is usually connected with interfacial energies and is used to model various phenomena like elastoplasticity or damage. If qis much bigger than the dimension 2, then under some additional assumptions we can actually conclude thatJf ≥σ >0 and we can approximate (see [16]). The more physically relevant assumptions are q= 1 or q= 2. In that case the usual convolution approximation is not useful for approximation as it in general destroys injectivity in places where the Jacobian is close to zero.
In this paper we start to study the case q= 1. We cannot use the approach of [19] as there is no analogy of the key extension procedure, i.e. of Rado-Choquet-Knesser theorem. Indeed, even in the one-dimensional
case the minimizers of theW2,q((0,1)) energy do not need to be injective once we prescribe the boundary data f(0), f(1)> f(0), f0(0)>0 andf0(1)>0 with derivatives much bigger than f(1)−f(0). Instead we use some ideas of [17], we cover Ω by triangles and we divide triangles into good and bad according to the behavior of f like differentiability on them. The total measure of bad triangles is small and we approximate f on good triangles by quadratic polynomials. Then we smoothen this piecewise quadratic mapping along the edges of triangles and we obtain the desired diffeomorphism. Note that piecewise linear approximation on triangles as in [17] is not good as the second derivative on linear pieces is zero and we would not be able to approximate strongly the second derivative.
We call Ω⊂R2a polygonal domain, if we can find triangles{Ti}ki=1 with pairwise disjoint interiors so that Ω =Sk
i=1Ti. We say that f : Ω→R is piecewise quadratic, if it is continuous and there is a triangulation Ω =Sk
i=1Ti such thatf|Ti is a quadratic function in each coordinate,i.e.
f(x, y) = [a1+a2x+a3y+a4x2+a5xy+a6y2, b1+b2x+b3y+b4x2+b5xy+b6y2] onTi.
Note that with our quadratic approximation we cannot achieve the continuity of the derivative in the direction perpendicular to sides of the triangles, but we can achieve that the jumps of the derivative there are small.
Thus our piecewise quadratic mapping does not belong toW2,1 but it belongs toW BV,i.e.its derivative is a BV mapping. ByD2sf we denote the singular part of the second derivative which is supported inSk
i=1∂Tiand corresponds to jump of derivatives between touching triangles.
Our first result is an analogy of [25] for second derivatives, but with no control of derivative of the inverse. It states that given a nice piecewise quadratic approximation (with small jumps of derivatives, see (1.2)) we can find a diffeomorphic approximation. For the definition ofW BV(Ω,R2) see the preliminaries.
Theorem 1.1. Let Ω⊂R2 be a polygonal domain. Let δ, d >0 and assume that f : Ω→R2 is a piecewise quadratic homeomorphism so that
Z
Ω
|D2sf|< δ andJf > da.e. inΩ. (1.2) Then for every ε >0we can find a C∞ diffeomorphismg: Ω→R2 such that
kf−gkW BV(Ω,R2)< ε+Cδ andkf −gkL∞(Ω,R2)< ε.
In our second result we apply the previous result to show a diffeomorphic approximation ofW2,1homeomor- phism up to a set of small measure. This part is more difficult than the corresponding result in [17] as we have to deal also with second derivatives and with piecewise quadratic approximation.
Theorem 1.2. LetΩ⊂R2be a domain of finite measure. Letf ∈W2,1(Ω,R2)be a homeomorphism such that Jf >0 a.e. Then for every ν >0 we can find squares {Qi}∞i=1 which are locally finite (i.e. each compact set K⊂Ωintersects only finitely many of them) with
L2
[∞
i=1
Qi
< ν
and we can find C∞ diffeomorphismg: Ω\S∞
i=1Qi→R2 such that kf −gkW2,1(Ω\S∞
i=1Qi,R2)< ν.
The natural plan for our future research is to obtain some analogy of the key extension result ([17], Thm. 2.1) which will lead to the full approximation result of W2,1 homeomorphisms, i.e.we would be able to deal also
with a set of small measure SQi. Moreover, we could try to obtain an analogy of Theorem 3.1 in [17], which would even remove the assumptionJf >0 a.e., even-though it is quite natural in models of Nonlinear Elasticity.
2. Preliminaries
By [x, y] we denote the point inR2 with coordinatesxandy. The scalar product ofu, v∈R2 is denoted by hu, vi. ByB(c, r) we denote the ball centered atc∈R2with radiusr >0 andQ(c, r) denotes the corresponding square.
Letu, v∈R2 be nonzero. Then we have the following elementary estimate
u
|u| − v
|v|
=
u
|u|− v
|u|+v|v| − |u|
|u| |v|
≤2|u−v|
|u| . (2.1)
We introduce a smooth function which grows from 0 to 1 on [0,1] and plays an important role in our construction.
Notation 2.1. Letη:R→Rbe a fixed smooth function withη(x) = 0,x≤0 andη(x) = 1,x≥1,ηincreasing on [0,1], 0≤η0≤2 and|η00| ≤4.
For f :R2 → R2 we use the notation for first derivatives Dxf = ∂f∂x, Dy = ∂f∂x and similarly for second derivatives Dxxf =Dx(Dxf), Dyyf =Dy(Dyf) and Dxy =Dx(Dyf). Similarly for any vector u∈ R2 we denote byDuf the derivative of f in theudirection.
It is well-known that forC1mapping the classical and distributional derivatives agree. Hence for any domain G⊂R2,f ∈C1(G,R2) and{u, v} and{~u, ~v} a pair of positively oriented orthonormal bases ofR2we have
Jf(x, y) = detDf(x, y) =hDuf(x, y), ~uihDvf(x, y), ~vi − hDuf(x, y), ~vihDvf(x, y), ~ui, (2.2) for almost every [x, y] ∈ G where Jf is the weak Jacobian of f. This is essentially the invariance of the determinant with respect to the choice of a positively oriented orthonormal basis.
2.1. Representation of higher order derivatives
Given f ∈C2(Ω,R2) we can view Df as the the mapping from Ω→C(R2×2) (i.e.eachDf(x) is a matrix) and we can define the symbolD2f as the mapping from Ω→C(R2×2×2) (i.e.eachD2f(x) is the operator on 2×2 matrices). We know that D(f g) = (Df)g+f(Dg) as matrices and similarly we can symbolically write
D2(f g) =D (Df)g+f(Dg)
=D2f g+Df Dg+Df Dg+f D2g
where on the righthand side we see the correct terms of the product. At the end we will just estimate the norm of this by the corresponding product of norms and the exact terms will not be important for us.
2.2. ACL condition
Let Ω⊂R2 be an open set. It is a well-known fact (see e.g.[1], Sect. 3.11) that a mapping u∈L1(Ω,Rm) is in W1,1(Ω,Rm) if and only there is a representative which is an absolutely continuous function on almost all lines parallel to coordinate axes and the variation on these lines is integrable.
Analogously for any given direction v ∈R2 we can fix v⊥⊥v and define hs(t) =u(sv⊥+tv) for the right representative ofu. Then for a.e.sthe function hs is absolutely continuous onLs:={t:sv⊥+tv∈Ω}and
Z ∞
∞
Z
Ls
|h0s(t)|dtds≤ Z
Ω
|Du(x)|dx.
2.3. Spaces with derivatives as measures
Let Ω⊂R2 we call the BV(Ω) the class of functions u: Ω→R such thatu∈L1(Ω) there exists a pair of (signed) Radon measuresµx andµy
Z
Ω
u(x, y)∂xϕ(x, y) dL2(x, y) =− Z
Ω
ϕ(x, y) dµx(x, y) and
Z
Ω
u(x, y)∂yϕ(x, y) dL2(x, y) =− Z
Ω
ϕ(x, y) dµy(x, y)
for allϕ∈ Cc1(Ω). Then we denoteDxu=µxandDyu=µy. We define the following norm onBV(Ω) kukBV =kuk1+ supnZ
Ω
udivϕdL2(x, y) ; ϕ∈ Cc1(Ω),kϕk∞≤1o .
The above is all standard and we refer the reader to [1] for references.
In this paper we refer to the spaceW BV(Ω,R2). We define this space as the class of mappingsu: Ω→R2such that u∈W1,1(Ω,R2) and for each j= 1,2 it holds thatDxuj, Dyuj ∈BV(Ω), where uj is thejth coordinate function ofu. An equivalent norm onW BV(Ω,R2) can be defined simply as
kukW BV(Ω,R2)=kuk1+
2
X
j=1
kDxujkBV +kDyujkBV.
A minor abuse of the notation allows us to writeR
Ω|D2u| for u∈W BV(Ω,R2) which we use as a shorthand for
Z
Ω
|D2u|d(x, y) =
2
X
j=1
Z
Ω
d|D2xxuj|+ Z
Ω
d|D2xyuj|+ Z
Ω
d|Dyx2 uj|+ Z
Ω
d|D2yyuj|(x, y)
where by|µ|we denote the so-called total variation of the measureµ.
2.4. FEM quadratic approximation on triangles
We need to define a quadratic polynomialAthat approximates our mappingf on a triangleT. Without loss of generality letT have verticesv1= [0,0],v2= [r,0] andv3= [0, r] for somer >0 (other triangles we use are just a translation and rotation). We have a mapping f :T →R2and we want to define mapping
A(x, y) = [a1+a2x+a3y+a4x2+a5xy+a6y2, b1+b2x+b3y+b4x2+b5xy+b6y2], where ai, bi∈R,i∈ {1,2,3,4,5,6}. We choose these constants so that forj = 1,2,3
A(vj) =− Z
B(vj,10r)
f, DxA(v1) =− Z
B(v1,10r)
Dxf, −DyA(v3) =− Z
B(v3,10r)
−Dyf and (−Dx+Dy)A(v2) =−
Z
B(v2,10r)
(−Dx+Dy)f,
(2.3)
Figure 1.Triangulation and direction in vertices.
that is values of A in vertices corresponds to values off (in averaged sense) and values of derivatives ofA in vertices along three sides correspond to derivatives of f. Note that these 6 equations in first coordinate (resp.
second coordinates) determine the 6 coefficients ai (resp. bi) uniquely. Moreover, imagine that we have two trianglesT1andT2 with common side and that we defineA1 onT1 andA2 onT2 by procedure (2.3) described above. ThenA1=A2 on∂T1∩∂T2since it is a quadratic polynomial of one variable on this segment (in each coordinate) and it has the same value at two vertices and the same derivative along the segment in one of the vertices (see Fig.1).
2.5. Estimates of piecewise quadratic homeomorphisms around the vertices
Lemma 2.2. Let Q1, Q2, . . . QN :R2→R2 be quadratic mappings with Qi(0,0) = [0,0]. Let 0 ≤ω0 < ω1 <
· · ·< ωN−1 < ωN =ω0+ 2π <4πand let ω˜i = [cosωi,sinωi]∈S1 be angles ordered anti-clockwise around S1. Let R >0andf :B(0, R)→R2 be the map defined by
f(tcosθ, tsinθ) =Qi(tcosθ, tsinθ)for all 0≤t≤R and all ωi−1≤θ≤ωi.
Further assume that thisf is a homeomorphism anddetDQi≥d >0onB(0, R). LetLandM denote positive numbers such that|DQi| ≤L onB(0, R)and|D2Qi| ≤M. Then the map
h(tcosθ, tsinθ) =DQi(0,0)[tcosθ, tsinθ] t∈[0,∞)andωi−1≤θ≤ωi (2.4) is a piecewise linear homeomorphism (see Fig.2). Moreover
h tcosθ, tsinθ
=t D[cosθ,sinθ]h cosθ,sinθ
(2.5) for all 0< t≤R and all θ∈R. Further it holds that
d
L ≤ |Dwh(x, y)| ≤L and d
L ≤ |Dwf(x, y)| ≤L for allw∈S1 (2.6) and
h(x, y)−f(x, y) ≤M
2 [x, y]
2 (2.7)
Figure 2.Mappingf maps rays [0, ρ0)×ω˜i onto quadratic curves (bold on the right side) and hmaps these rays onto touching segments (dashed on the right side).
for all [x, y]∈B(0, R). Moreover for any pairu⊥v∈S1,u clockwise from v, denoting~v= |DDvf(x,y)
vf(x,y)| and~u⊥~v is clockwise from~v, it holds that
d L ≤
Duf(x, y), ~u
(2.8) for all [x, y]∈B(0, R). Finally
|Df(x, y)−Dh(x, y)| ≤M [x, y]
. (2.9)
Similarly if r, ` > 0 and f(x, y) =Q1(x, y) on [−r,0]×[0, `] and f(x, y) =Q2(x, y) on [0, r]×[0, `] is a homeomorphism with |Df| ≤L,Jf ≥dand|D2f| ≤M then
d
L ≤ |Dwf(x, y)| ≤L for allw∈S1. (2.10)
Further, for any pair u⊥v∈S1, uclockwise from v, denoting~v= |DDvf(x,y)
vf(x,y)| and ~u⊥~v is clockwise from ~v, we have
d L ≤
Duf(x, y), ~u
(2.11) for all [x, y]∈[−r, r]×[0, `].
Proof. First we prove that the map h defined by (2.4) is a piecewise linear homeomorphism. Denote ˜θ = [cosθ,sinθ] and note that on eachωi≤θ≤ωi+1 we haveDh(tθ) =˜ DQi(0,0). Since detDQi(0,0)≥d >0 for all 1≤i≤Nwe know thathis a homeomorphism on eachωi≤θ≤ωi+1. Furtherf is continuous on ˜ωi×[0, ρ0] and hence ∂ω˜iQi(0,0) =∂ω˜iQi+1(0,0) which implies that his continuous. By the piecewise linearity ofh, the continuity ofhandJh>0 a.e. it is not difficult to deduce thathis a homeomorphism (see Fig.2).
The equality (2.5) is obvious from the piece-wise linearity ofh. Since|Df| ≤LandJf ≥dwe obtain for any pairu, v∈S1withu⊥vthat
d≤Jf(x, y)≤ |Duf(x, y)| · |Dvf(x, y)| ≤L|Dvf(x, y)|,
which shows (2.6) forf. AnalogouslyJh≥da.e. (as detDQi(0,0)≥dfor alli) and|Dwh| ≤Lfor anyw∈S1 imply (2.6) forh.
For all ωi ≤θ ≤ωi+1 we have DQi(0,0) = lims→0Df(scosθ, ssinθ). For any ωi ≤θ≤ωi+1, calling ˜θ = [cosθ,sinθ] we have usingDθ˜Qi(0,0) =Dθ˜h(sθ)˜
f(tθ)˜ −h(tθ) =˜ Z t
0
Dθ˜f(sθ)˜ − Z t
0
Dθ˜Qi(0,0) ds
= Z t
0
Z s 0
Dθ˜θ˜f(zθ)dz˜ ds
(2.12)
but since|Dθ˜θ˜f| ≤M we have
f(tθ)˜ −h(tθ)˜ ≤ M
2 t2 which is (2.7).
Since~v= |DDvf(x,y)
vf(x,y)| and~u⊥~v we obtainhDvf, ~ui= 0. Thus we can use (2.2) to obtain d≤Jf(x, y) =hDuf(x, y), ~uihDvf(x, y), ~vi
and using (2.6) we get (2.8). The equation (2.9) follows from the fact that |D2f| ≤M andDh(˜θ) =DQi(0,0) forωi< θ < ωi+1. The proof of (2.10) and (2.11) is analogous to that of (2.6) and (2.8).
Lemma 2.3. Let f ∈ W1,∞(B(0, R),R2) and f is C1 smooth except on a finite number of rays
˜
ω1R+,ω˜2R+, . . .ω˜NR+, ω˜i ∈S1,f(0,0) = 0 and |f(x, y)|>0 for [x, y]6= [0,0]. Let R:B(0, R)→[0,∞) and ϕ:B(0, R)→S1 be a pair of functions such that for all[x, y]∈B(0, R)we have
f(x, y) =R(x, y)ϕ(x, y).
Then for anyt∈(0, R)and anyθ∈[0,2π)(callingθ˜= [cosθ,sinθ],θ˜⊥= [−sinθ,cosθ]and callingϕ⊥(tθ)˜ ∈S1 the vector anti-clockwise perpendicular to ϕ(tθ)) it holds that˜
∂
∂θϕ(tcosθ, tsinθ), ϕ⊥(tθ)˜
= t
R(tθ)˜
Dθ˜⊥f(tθ), ϕ˜ ⊥(tθ)˜ .
Proof. Without loss of generality we may assume that ˜θ=e1=ϕ(t,0) and ˜θ⊥ =e2=ϕ⊥(t,0) (just consider suitable rotations). Since ϕ= |f|f , f is Lipschitz and |f(x, y)|>0 for |[x, y]|>0 we obtain that ϕ is locally Lipschitz outside of 0. This and the fact that
[tcosθ, tsinθ]−[t, ttanθ]
≤θ2 for smallθ
implies
h ∂
∂θϕ(tcosθ, tsinθ)i
θ=0= lim
θ→0
ϕ(tcosθ, tsinθ)−ϕ(t,0) θ
= lim
θ→0
ϕ(tcosθ, tsinθ)−ϕ(t, ttanθ) θ
+ lim
θ→0
ttanθ θ
ϕ(t, ttanθ)−ϕ(t,0) ttanθ
=tDθ˜⊥ϕ(t,0)
(2.13)
because ttanθ θ →t. Now
Dθ˜⊥f(t,0) =Dyf(t,0) =DyR(t,0)ϕ(t,0) +R(t,0)Dyϕ(t,0) and hence
Dθ˜⊥f(t,0), ϕ⊥(t,0)
=
R(t,0)Dθ˜⊥ϕ(t,0), ϕ⊥(t,0) and our conclusion follows using (2.13).
3. Approximation of piecewise quadratic homeomorphisms around the edges
Recall that η denotes the function from the Preliminaries, Notation2.1.
Lemma 3.1(Approximation along the edge). LetQ1, Q2:R2→R2be a pair of quadratic mappings coinciding on the line{x= 0}and letρ0, ` >0be such that the mapf =Q1on[−ρ0,0]×[0, `]andf =Q2on[0, ρ0]×[0, `]
is a homeomorphism with
d= min
detDQ1(x, y),detDQ2(x, y); [x, y]∈[−ρ0, ρ0]×[0, `] >0.
Let L andM denote positive numbers such that |DQj| ≤L and |D2Qj| ≤M on [−ρ0, ρ0]×[0, `] for j= 1,2.
Fix N ∈N, N≥4 such that ρ= `
2N <minn
ρ0, d
1000(M+ 1)(L+ 1), 1 320
d2 M L3
o
. (3.1)
Then there exists an r0 (depending on the geometry of f({0} ×[0, `])) such that for every 0 < r <
min{r0,2(L+1)ρ2 ,40ρ,2M ρL2} there exists a diffeomorphismg: [−ρ0, ρ0]×[0, `] satisfying g(x, y) =f(x, y)for all |x|> randy∈[0, `]
and
Z
[r,r]×[0,`]
|D2g| ≤C Z `
0
|Dxxf(0, y)|+CM `r (3.2)
where by |Dxxf(0, y)|=|DxQ2(0, y)−DxQ1(0, y)| we denote the size of the Dirac measure of Dxxf at[0, y].
Further, call ~u(x, y)∈S1 the vector clockwise perpendicular to |DDyg(0,y)
yg(0,y)| for all [x, y]∈[−r, r]×[0, `], then we
have
Dxg(x, y), ~u(x, y)
≥ 9d
10L. (3.3)
Proof.
Step 1. Initial setup and definition of f˜r.
We haveρwhich satisfies (3.1) and 2ρ` =N ∈N. We divide [−r, r]×[0, `] intoN rectangles [−r, r]×[(2i− 2)ρ,2iρ] fori= 1, . . . , N. We denote
~vi= ∂yQ1(0,(2i−1)ρ)
|∂yQ1(0,(2i−1)ρ)| fori= 1,2, . . . , N and we fix~ui∈S1, ~ui⊥~vi (3.4) so that{~ui, ~vi} is a positively oriented basis ofR2. That is~ui is clockwise perpendicular to~vi. Then we define a tentative map ˜fr as an appropriate convex combination ofQ1 andQ2,i.e.
f˜r(x, y) =
(1−η(xr))Q1(x, y) +η(xr)Q2(x, y)
ifa)hDxQ2(0,(2i−1)ρ), ~uii ≥ hDxQ1(0,(2i−1)ρ), ~uii, (1−η(x+rr ))Q1(x, y) +η(x+rr )Q2(x, y)
ifb)hDxQ1(0,(2i−1)ρ), ~uii>hDxQ2(0,(2i−1)ρ), ~uii.
(3.5)
for every [x, y]∈[−r, r]×[(2i−2)ρ,2iρ]. Note that ˜fr=Q1 forx <−rand ˜fr=Q2forx > r.
Step 2. Define g.
The above definition of ˜fr is fine if all rectangles [−r, r]×[(2i−2)ρ,2iρ] are of type a) or if all of them are of type b). Otherwise we have to continuously connect rectangles of type a) to rectangles of type b).
Suppose that we have a pair of neighboring rectangles (−r, r)×((2i−2)ρ,2iρ) of type a) and (−r, r)× (2iρ,(2i+ 2)ρ) of typeb) then we definegas follows
g(x, y) =
1−η xr +η(yρ−1−2i)
Q1(x, y) +η xr +η(yρ−1−2i)
Q2(x, y) (3.6)
for all [x, y]∈(−ρ0, ρ0)×[2iρ,(2i+ 1)ρ]. Note that fory1= 2iρand y2= (2i+ 1)ρwe have η xr+η(y1ρ−1−2i)
=η xr
andη xr +η(y2ρ−1−2i)
=η x+rr and so it agrees with (3.5) there.
Similarly when we have a pair of adjacent rectangles (−r, r)×((2i−2)ρ,2iρ) of typeb) and (−r, r)×(2iρ,(2i+ 2)ρ) of type a) then we definegas follows
g(x, y) =
1−η x+rr −η(yρ−1−2i)
Q1(x, y) +η x+rr −η(yρ−1−2i)
Q2(x, y) (3.7) on (−ρ0, ρ0)×[2iρ,(2i+ 1)ρ]. On the rest of [−ρ0, ρ0]×[0, `] we letg(x, y) = ˜fr(x, y). Immediately we see from the smoothness of Q1, Q2andη that gis smooth on (−ρ0, ρ0)×(0, `).
Step 3. Lower bound for Jg.
We firstly show that Jg>0 which shows thatgis locally a homeomorphism. In the next step we show that g is injective on ∂([−ρ0, ρ0]×[0, `]). Together these two facts imply that the smooth mappingg is in fact a diffeomorphism (see e.g.[22]).
The following calculations are for rectangles wherea)-type transfers intob)-type andg is given by (3.6). The calculations are analogous for (3.7) and are even simpler on rectangles where g≡f˜r. Since Q1, Q2 and η are
smooth we immediately get that ˜fr is smooth on each rectangle (−ρ0, ρ0)×((2i−2)ρ,2iρ). Further Dxf˜r(x, y) = (1−η(xr))DxQ1(x, y) +η(xr)DxQ2(x, y) +1rη0(xr)(Q2(x, y)−Q1(x, y)) andDyf˜r(x, y) = (1−η(xr))DyQ1(x, y) +η(xr)DyQ2(x, y).
(3.8)
If (−r, r)×((2i−2)ρ,2iρ) is ana)-type rectangle and (−r, r)×(2iρ,(2i+ 2)ρ) is a b)-type rectangle then on (−ρ0, ρ0)×(2iρ,(2i+ 1)ρ) we calculate
Dxg(x, y) =
1−η xr +η(yρ−1−2i)
DxQ1(x, y) +η xr +η(yρ−1−2i)
DxQ2(x, y) +1rη0 xr +η(yρ−1−2i)
(Q2(x, y)−Q1(x, y)). (3.9)
Moreover we calculate Dyg(x, y) =
1−η xr+η(yρ−1−2i)
DyQ1(x, y) +η xr +η(yρ−1−2i)
DyQ2(x, y) +1ρη0 xr+η(yρ−1−2i)
η0(yρ−1−2i)(Q2(x, y)−Q1(x, y)). (3.10) Because Q1(0, y) =Q2(0, y) fory∈[0, `] we have
|Q2(x, y)−Q1(x, y)| ≤ Z x
0
|DxQ1(s, y)|+|DxQ2(s, y)| ds≤2Lr.
Utilizing this fact, (3.8), (3.9), (3.10), 0≤η≤1,|η0| ≤2 andr < 40ρ we get that
|Dg| ≤8L. (3.11)
on each (−r, r)×((2i−2)ρ,2iρ). Sinceg is a convex combination ofQ1 andQ2,f is equal either to Q1or Q2 andQ1=Q2on 0×[0, `] we get
kg−fk∞≤ kQ1(x, y)−Q1(0, y)k∞+kQ2(x, y)−Q2(0, y)k∞≤2Lr. (3.12) UsingQ1(0, y) =Q2(0, y) we also have
Q2(x, y)−Q1(x, y), ~ui
= Z x
0
hDxQ2(s, y), ~uii − hDxQ1(s, y), ~uiids.
Use|D2Qj| ≤M forj= 1,2 andr≤40ρ to get DQj(s, y)−DQj(0,(2i−1)ρ)
≤2M ρfors∈[−r, r] andy∈[(2i−2)ρ,2iρ] (3.13) and hence with the help ofρ≤ 1000M Ld
Q2(x, y)−Q1(x, y), ~ui
≥ −4M ρx≥ − dr 250L. From (2.11) at the point [0,(2i−1)ρ] forv= [0,1] andu= [1,0] we obtain
DxQ1(0,(2i−1)ρ), ~ui
≥ d L
and hence we can combine it with the previous inequality to obtain 2
r
Q2(x, y)−Q1(x, y), ~ui
≥ − 1 125
DxQ1(0,(2i−1)ρ), ~ui
. (3.14)
Using (3.8),hDxQ2(0,(2i−1)ρ), ~uii ≥ hDxQ1(0,(2i−1)ρ), ~uii(which holds for a) type rectangles) and (3.13) we obtain (for [x, y] whereg= ˜fr)
hDxg(x, y), ~uii=
(1−η(xr))DxQ1(x, y) +η(xr)DxQ2(x, y) +1rη0(xr)(Q2(x, y)−Q1(x, y)), ~ui
≥
(1−η(xr))DxQ1(0,(2i−1)ρ) +η(xr)DxQ2(0,(2i−1)ρ), ~ui
− |DxQ1(0,(2i−1)ρ)−DxQ1(x, y)| − |DxQ2(0,(2i−1)ρ)−DxQ2(x, y)|
+1rη0(xr)
Q2(x, y)−Q1(x, y), ~ui
≥
DxQ1(0,(2i−1)ρ), ~ui
−4M ρ+1rη0(xr)
Q2(x, y)−Q1(x, y), ~ui ,
≥
DxQ1(0,(2i−1)ρ), ~ui 1− 1
250− 1 125
,
(3.15)
where we have used (2.11) and ρ ≤ 1000M Ld to estimate the term 4M ρ and the term 1rη0(xr)
Q2(x, y)− Q1(x, y), ~ui
is either positive and then we can estimate it by 0 or it is negative and then we use |η0| ≤ 2 and (3.14). Similarly we can use (3.9) and also in this case we obtain
hDxg(x, y), ~uii ≥ 123 125
DxQ1(0,(2i−1)ρ), ~ui
(3.16)
on (−r, r)×(0, `) which together with (2.11) implies
hDxg(x, y), ~uii ≥ 123 125
d
L. (3.17)
Now d ≤ JQ1 ≤ |DxQ1||DyQ1| implies |DyQ1| ≥ Ld. Recall that ~u =~u(y) is the vector in S1 clockwise perpendicular to |DDyQi(0,y)
yQi(0,y)|. Using (3.4), (2.1), (3.11) and (3.1) we obtain hDxg(x, y), ~uii − hDxg(x, y), ~ui
≤ |Dxg(x, y)| |~ui−~u|=|Dxg(x, y)| |~vi−~v|
≤8L|DyQ1(0,(2i−1)ρ)−DyQ1(0, y)|
|DyQ1(0,(2i−1)ρ)| 2
≤8LM ρ
d L
2≤ 1 20
d L
which together with (3.17) imply (3.3) sinceg=Q1 in [0, y] (see (3.5) and (3.6)). In (3.9) we dealt only with thea)-type tob)-type transitions but the calculations easily extend also for theb)-type toa)-type transitions.
The only difference is that we usehDxQ1(0,(2i−1)ρ), ~uii ≥ hDxQ2(0,(2i−1)ρ), ~uiiin (3.15) above and hence we have hDxQ2(0,0), ~uiion the righthand side of (3.16).
By the definition of~vi(3.4),~ui⊥~viandQ1=Q2on{0} ×[0, `] we havehDyQj(0,(2i−1)ρ), ~uii= 0. It follows using (3.13) that
|hDyQj(x, y), ~uii| ≤2M ρforj= 1,2.
With the help ofr < 2M ρL 2 we obtain
|Q1(x, y)−Q2(x, y)| ≤ |Q1(x, y)−Q1(0, y)|+|Q2(0, y)−Q2(x, y)| ≤2Lr≤4M ρ2 and hence
Q2(x, y)−Q1(x, y), ~ui
≤4M ρ2. Applying this in (3.10) we get that
hDyg(x, y), ~uii ≤2M ρ+4
ρ4M ρ2≤18M ρ. (3.18)
We can express the values ofDgwith respect to the basis{~ui, ~vi} as Dg(x, y) =
hDxg(x, y), ~uii, hDxg(x, y), ~vii hDyg(x, y), ~uii, hDyg(x, y), ~vii
=
a1, a2
b1, b2
.
Therefore, applying (3.16), (3.11), (3.18),ρ < 1000LMd , definition ofv~i(3.4), (3.13) and (2.10) (i.e.hDyQ1(0,(2i− 1)ρ), ~vii ≥ Ld) we conclude that
a1> 123 125
DxQ1(0,(2i−1)ρ), ~ui
|a2| ≤8L
|b1| ≤18M ρ≤ d 50L
b2≥ |DyQ1(0,(2i−1)ρ)| −2M ρ≥ 499
500|DyQ1(0,(2i−1)ρ)|
(3.19)
on the entire rectangle (−r, r)×((2i−2)ρ,2iρ). From the definition of~viandu~i⊥~viwe know thathDyQ1(0,(2i− 1)ρ), ~uii= 0 and hence using (2.2)
DxQ1(0,(2i−1)ρ), ~ui
|DyQ1(0,(2i−1)ρ)| ≥detDQ1(0,(2i−1)ρ)≥d.
Therefore simple computation gives
Jg(x, y)≥61377 62500d−8d
50 ≥ 4
5d (3.20)
on (−r, r)×((2i−2)ρ,2iρ).
Step 4. The injectivity of g.
By a combination of (3.16) for i= 0 andi=N and the fact that f is a homeomorphism we get thatg is injective on both segments [−ρ0, ρ0]× {0} and [−ρ0, ρ0]× {`}. Becausef is a homeomorphism we have that
dist
f([−r, r]× {0}), f ∂([−ρ0, ρ0]×[0, `])\ [−ρ0, ρ0]× {0}
>0 and similarly
dist
f([−r, r]× {`}), f ∂([−ρ0, ρ0]×[0, `])\ [−ρ0, ρ0]× {`}
>0.
Therefore, by (3.12) and f(x, y) =g(x, y) for |x| ≥r there exists anr0>0 (this is the r0 of our claim) such that for all 0< r < r0the mappingg constructed from ˜fr satisfies
g([−ρ0, ρ0]× {0})∩g ∂([−ρ0, ρ0]×[0, `])\ [−ρ0, ρ0]× {0}
=∅ and
g([−ρ0, ρ0]× {`})∩g ∂([−ρ0, ρ0]×[0, `])\ [−ρ0, ρ0]× {`}
=∅
for all r≤r0. But together that means that g is injective on ∂([−ρ0, ρ0]×[0, `]). Since (3.20) implies local injectivity this is enough to conclude thatgis injective everywhere in [−ρ0, ρ0]×[0, `] and thus a diffeomorphism (seee.g.[22]).
Step 5. Estimates of |D2g|.
We calculate the estimates of D2g in detail only for the a) to b) type transition given by (3.6). It is not difficult to check that the computation for b) to a) type transition given by (3.7) are essentially the same and the estimates for the set where{f˜r=g} given by (3.5) are even simpler.
We have the following elementary estimates (recall that |Dxxf(0, y)|=|DxQ2(0, y)−DxQ1(0, y)|)
|DxQ2(x, y)−DxQ1(x, y)| ≤ |Dxxf(0, y)|+ 2M|x| ≤ |Dxxf(0, y)|+ 2M r, (3.21) further, sinceDyQ2(0, y) =DyQ1(0, y), we have
|DyQ2(x, y)−DyQ1(x, y)| ≤2M r (3.22) and
|Q2(x, y)−Q1(x, y)| ≤ |Dxxf(0, y)| · |x|+
2
X
j=1
|Qj(x, y)−Qj(0, y)−xDxQj(0, y)|
≤ |Dxxf(0, y)|r+
2
X
j=1
Z x 0
DxQj(s, y)−DxQj(0, y) ds
≤ |Dxxf(0, y)|r+M r2.
(3.23)
The second derivatives of (3.6) are calculated by Dxxg(x, y) = (1−η xr +η(yρ−1−2i)
)DxxQ1+η xr+η(yρ−1−2i) DxxQ2
+ 1
r2η00 xr+η(yρ−1−2i)
(Q2(x, y)−Q1(x, y)) +1
rη0 xr +η(yρ−1−2i)
(DxQ2(x, y)−DxQ1(x, y)).
Using|D2Qj(x, y)| ≤M,|η0| ≤2,|η00| ≤4, (3.21) and (3.23) we get
|Dxxg(x, y)| ≤ C
r|Dxxf(0, y)|+CM.
Further
Dxyg(x, y) = 1−η xr +η(yρ−1−2i)
DxyQ1(x, y) +η xr +η(yρ−1−2i)
DxyQ2(x, y) +1
rη0 xr +η(yρ−1−2i)
(DyQ2(x, y)−DyQ1(x, y)) +1
ρη0 xr +η(yρ−1−2i)
η0(yρ−1−2i)(DxQ2(x, y)−DxQ1(x, y)) + 1
rρη00 xr +η(yρ−1−2i)
η0(yρ−1−2i)(Q2(x, y)−Q1(x, y)) and using |D2Qi(x, y)| ≤M,|η0| ≤2,|η00| ≤4, (3.21), (3.22) and (3.23) we get
|Dxyg(x, y)| ≤CM+C
ρ|Dxxf(0, y)|+CM r ρ and the estimate holds for all [x, y]∈[−r, r]×[0, `] where (3.6) applies. Finally
Dyyg(x, y) =
1−η xr +η(yρ−1−2i)
DyyQ1(x, y) +η xr +η(yρ−1−2i)
DyyQ2(x, y) +ρ1η0 xr+η(yρ−1−2i)
η0(yρ−1−2i)(DyQ2(x, y)−DyQ1(x, y)) +ρ12η00 xr+η(yρ−1−2i)
η0(yρ−1−2i)2
(Q2(x, y)−Q1(x, y)) +ρ12η0 xr+η(yρ−1−2i)
η00(yρ−1−2i)(Q2(x, y)−Q1(x, y)) so
|Dyyg(x, y)| ≤M +CM r ρ +Cr
ρ2|Dxxf(0, y)|+CM r2 ρ2 . Integrating the above over [−r, r]×[0, `] and estimating
|D2g(x, y)| ≤ |Dxxg(x, y)|+ 2|Dxyg(x, y)|+|Dyyg(x, y)|
we get usingr≤40ρ Z
[−r,r]×[0,`]
|D2g(x, y)| ≤Cr Z `
0
|Dxxf(0, y)|dy1 r+1
ρ+ r ρ2
+Cr`h
M +M r
ρ +M r2 ρ2
i
≤C Z `
0
|Dxxf(0, y)| dy+CM `r,
and (3.2) follows.
4. Approximation of piecewise quadratic homeomorphisms around the vertices and proof of Theorem 1.1
Againη denotes the function from the Preliminaries, Notation2.1.
Lemma 4.1(Approximation near vertices). LetQ1, Q2, . . . QN :R2→R2be quadratic mappings. Let0≤ω0<
ω1<· · ·< ωN−1< ωN =ω0+ 2π <4πand letω˜i= [cosωi,sinωi]∈S1 be points ordered anti-clockwise around
Figure 3. The setsOi in blue contained inside red cones around rays parallel to ˜ωi. Outside B(0, R) we use the same approach as Lemma 3.1. InsideB(0,3R/4) we use a linear map. In the annulus we interpolate by first squashing onto rings and then rotating.
S1 and call
ω∗= min{π8, ωi+1−ωi;i= 0, . . . , N−1}.
Callω˜⊥i = [−sinωi,cosωi]∈S1the vector anti-clockwise perpendicular toω˜i. Letf :B(0, ρ0)→R2 be the map defined by
f(tcosθ, tsinθ) =Qi(tcosθ, tsinθ)for all0≤t≤ρ0 and allωi−1≤θ≤ωi.
Further assume that thisf is a homeomorphism anddetDQi≥d >0onB(0, ρ0). LetLandM denote positive numbers such that|DQi| ≤L onB(0, ρ0) and|D2Qi| ≤M. For every ρ1, ρ2. . . , ρN and everyR such that
0< R < 12min{ρi, i= 1, . . . , N}< 12minn
ρ0, min{d, d2}
1000(M + 1)(L+ 1)4, 1 320
d2 M L3,1
8 L M+ 1
o
(4.1) and every
0< ri≤minn d2R 432L4, Rd
1200L2, ρ2i 2(L+ 1),R
2 tanω∗ 3
o
we call r= (R, ρ1, . . . , ρN, r1, . . . , rN).
Then for all such r, the rectangles (see Fig.3)
Oi=
tω˜i+s˜ω⊥i ;t∈[R2, ρi], s∈[−ri, ri]
are pairwise disjoint. Further call ~vi = |DDωi˜ Qi(ρiω˜i)
ωi˜ Qi(ρiω˜i)| and call ~ui ∈S1 the vector clockwise perpendicular to ~vi. Define f˜r as
f˜r(x, y) =
f(x, y)for[x, y]∈/ SN i=1Oi, 1−η r1
ih[x, y],−˜ω⊥i i
Qi(x, y) +η r1
ih[x, y],−˜ωi⊥i
Qi+1(x, y) for[x, y]∈Oi ifhD−ω˜⊥
i Qi+1(ρiω˜i), ~uii ≥ hD−˜ω⊥
i Qi(ρiω˜i), ~uii, 1−η r1
ih[x, y],−˜ω⊥i i+ 1
Qi(x, y) +η r1
ih[x, y],−˜ωi⊥i+ 1
Qi+1(x, y) for[x, y]∈Oi ifhD−ω˜⊥
i Qi+1(ρiω˜i), ~uii<hD−˜ω⊥
i Qi(ρiω˜i), ~uii.
(4.2)
Then there exists aC∞diffeomorphismgrdefined onB(0,2R)withgr(x, y) = ˜fr(x, y)for allR≤ |[x, y]| ≤2R and
Z
B(0,R)
|D2gr|< CR (4.3)
where the constant C depends ond,L,M andN but is independent ofR.
Proof. Without loss of generality we may assume thatf(0,0) = [0,0].
Step 1. Proving that Oi are pair-wise disjoint.
The first claim we prove is that Oi∩Oj =∅ for any 1≤i < j≤N. On the one hand we have that ri ≤
R
2 tanω3∗ and on the other hand we have that
min{|[x, y]|; [x, y]∈Oi}=12R.
Therefore Oi lies inside a cone whose axis goes throughωi and the angle at the apex is 23ω∗. These cones are pairwise disjoint and therefore so areOi (see Fig. 3).
From ri≤ R2 we get qR2
4 +R42 < 3R4 and hence R
2ω˜i+s˜ωi⊥;s∈[−ri, ri] ⊂B(0,3R4 ). (4.4) It follows that this inner edge of Oi (where ˜fr is discontinuous) is a subset of B(0,3R4 ) and thus we can use Lemma 3.1to conclude that ˜fris a diffeomorphism onB(0,2R)\B(0,34R) since (4.2) agrees with rotated and translated version of (3.5) there (our ri and ρi play the role ofr and ρ in Lem.3.1). Note that d, L and M play the same role as in Lemma3.1and that (4.1) verifies (3.1). In the following computation we will use some estimates from Lemma 3.1.
We have shown that ˜fris smooth for 3R4 ≤ |[x, y]| ≤2R <min{ρi;i= 1,2, . . . , N}.
Step 2. Proving h∂θ∂ ϕf(tcosθ,tsinθ),ϕ⊥f(tcosθ,tsinθ)i ≥C >0.
Now we express ˜fr in polar coordinates in the image, i.e.we define the pair of functions Rf :B(0,2R)→ [0,∞) asRf(x, y) =|f˜r(x, y)|andϕf :B(0,2R)\ {[0,0]} →S1⊂R2 asϕf(x, y) = f˜r(x,y)
|f˜r(x,y)|. Then f˜r(x, y) =Rf(x, y)ϕf(x, y) onB(0,2R).
Since ˜fr isC∞ smooth on B(0,2R)\B(0,34R) and|f˜r(x, y)|= 0 if and only if [x, y] = [0,0], we have thatRf
andϕf areC∞smooth there. Further we define ϕ⊥f(x, y) =
−(ϕf(x, y))2,(ϕf(x, y))1
the π2 anti-clockwise rotation ofϕf. For brevity call ˜θ= [cosθ,sinθ] and ˜θ⊥ = [−sinθ,cosθ]. Our aim is to prove that in B(0, R)
Dθ˜⊥ϕf(tθ), ϕ˜ ⊥f(tθ)˜
= ∂
∂θϕf(tθ), ϕ˜ ⊥f(tθ)˜
≥C >0.
Step 2.A. The [x,y]∈/Oi case.
By Lemma2.2the map
h(tcosθ, tsinθ) =DQi(0,0)(tcosθ, tsinθ) fort∈[0,∞) andωi−1≤θ≤ωi is a piecewise linear homeomorphism. From Lemma2.3we have
h∂
∂θϕf(tcosθ, tsinθ), ϕ⊥f(tθ)i˜ = t
Rf(tθ)˜ hDθ˜⊥f˜r(tθ), ϕ˜ ⊥f(tθ)i.˜ (4.5) We call
ϕh(tθ) =˜ h(tθ)˜
|h(tθ)|˜ andϕ⊥h(x, y) =
−(ϕh(x, y))2,(ϕh(x, y))1
the π2 anti-clockwise rotation ofϕh. For brevity we use the notation [x, y] =tθ, where˜ t=|[x, y]|and ˜θ= |[x,y]|[x,y] . By linearity ϕhdepends only on θand nottand henceDθ˜(ϕh(x, y)) = 0 which implies
Dθ˜h(x, y) =Dθ˜ |h(x, y)|
ϕh(x, y) +|h(x, y)|Dθ˜ ϕh(x, y)
=Dθ˜ |h(x, y)|
ϕh(x, y).
It follows that
0<
Dθ˜h(x, y), ϕh(x, y)
≤L and
Dθ˜h(x, y), ϕ⊥h(x, y)
= 0.
Using (2.2) we obtain
d≤Jh(x, y) =
Dθ˜h(x, y), ϕh(x, y)
Dθ˜⊥h(x, y), ϕ⊥h(x, y) and together with|Dwh(x, y)| ≤Lfor all [x, y]∈B(0, R) and allw∈S1 this implies
d
L ≤ hDθ˜h(x, y), ϕh(x, y)i ≤L and d
L ≤ hDθ˜⊥h(x, y), ϕ⊥h(x, y)i ≤L. (4.6) Therefore Ld|[x, y]| ≤ |h(x, y)| ≤L|[x, y]|. We use this estimate together with (2.7) and the fact that |[x, y]|<
R < 18min{ML,LMd } to get that 15d
L16|[x, y]| ≤ |f(x, y)| ≤ 17L
16 |[x, y]|, i.e. 16
17L ≤ |[x, y]|
Rf(x, y) ≤ 16L
15d. (4.7)
