Approximation of planar Sobolev W 2,1 homeomorphisms by Piecewise Quadratic Homeomorphisms and Diffeomorphisms
aa r X i v : . [ m a t h . F A ] A ug APPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMSBY PIECEWISE QUADRATIC HOMEOMORPHISMS ANDDIFFEOMORPHISMS
DANIEL CAMPBELL AND STANISLAV HENCL
Abstract.
Given a Sobolev homeomorphism f ∈ W , in the plane we find a piecewisequadratic homeomorphism that approximates it up to a set of ε measure. We show thatthis piecewise quadratic map can be approximated by diffeomorphisms in the W , normon this set. Introduction
In this paper we address the issue approximation of Sobolev homeomorphisms withdiffeomorphisms. Let us briefly explain the motivation for this problem that comes fromNonlinear Elasticity. Let Ω ⊂ R n be a domain which models a body made out of homoge-neous elastic material, and let f : Ω → R n be a mapping modeling the deformation of thisbody with prescribed boundary values. In the theory of nonlinear elasticity pioneered byBall and Ciarlet, see e.g. [2, 3, 12], we study the existence and regularity properties ofminimizers of the energy functionals I ( f ) = Z Ω W ( Df ) dx, where W : R n × n → R is the so-called stored-energy functional, and Df is the differentialmatrix of the mapping f . The physically relevant assumptions on the model include:(W1) W ( A ) → + ∞ as det A →
0, i.e. mapping does not compress too much,(W2) W ( A ) = + ∞ if det A ≤
0, which guarantees that the orientation is preserved.In particular, any admissible deformation f satisfies J f ( x ) := det Df ( x ) > x ∈ Ω.With the help of some growth assumptions on W we can prove that a mapping with finiteenergy is continuous and one-to-one, which corresponds to the non-impenetrability of thematter. Hence it is natural to study Sobolev homeomorphisms with J f > u ∈ W ,p (Ω; R n ), p ∈ [1 , + ∞ ) bypiecewise affine homeomorphisms or by diffeomorphisms in W ,p norm. The motivation Date : August 14, 2020.The first author was supported by the grant GA ˇCR 20-19018Y. The second author was supported bythe grant GA ˇCR P201/18-07996S. is that regularity is typically proven by testing the weak equation or the variation for-mulation by the solution itself; but without some a priori regularity of the solution, theintegrals are not finite. Thus we need to test the equation with a smooth test mappingof finite energy which is close to the given homeomorphism instead. Besides NonlinearElasticity, an approximation result of homeomorphisms with diffeomorphisms would bea very useful tool as it allows a number of proofs to be significantly simplified. Let usnote that finding diffeomorphisms near a given homeomorphism is not an easy task, asthe usual approximation techniques like mollification or Lipschitz extension using themaximal operator destroy, in general, injectivity.Let us describe the known results about the Ball-Evans approximation problem. Theproblems of approximation by diffeomorphisms or piecewise affine planar homeomor-phisms are in fact equivalent by the result of Mora-Corral and Pratelli [25] (see also [21]).The first positive results on approximation of planar homeomorphisms smooth outside apoint are by Mora-Corral [24]. The celebrated breakthrough result in the area which stim-ulated much interest in the subject was given by Iwaniec, Kovalev and Onninen in [19],where they found diffeomorphic approximations to any homeomorphism f ∈ W ,p (Ω , R ),for any 1 < p < ∞ in the W ,p norm. The remaining missing case p = 1 in the planehas been solved by Hencl and Pratelli in [17] by a different method. This method wasextended to cover other function spaces like Orlicz-Sobolev spaces (see Campbell [8]), BVspace (see Pratelli, Radici [26]) or W X for nice rearrangement invariant Banach functionspace X (see Campbell, Greco, Schiattarella, Soudsk´y [11]). It is possible to approximatealso f − in the Sobolev norm for p = 1 (see Pratelli [27]) or for 1 ≤ p < ∞ under theadditional 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 (notonly homeomorphisms) as shown by Iwaniec and Onninen [20] for p ≥ ≤ p <
2. The higher dimensional case n ≥ n = 3. However, for n ≥ ≤ p < [ n ] there existsa Sobolev homeomorphism in W ,p which cannot be approximated by diffeomorphisms(see Hencl and Vejnar [18], Campbell, Hencl and Tengvall [10], Campbell, D’Onofrio andHencl [9]).Our aim is to find the corresponding planar result for models with second gradient, i.e.we would like to approximate W ,q homeomorphisms by diffeomorphisms. Models withthe second gradient(1.1) E ( f ) = Z Ω (cid:0) W ( Df ( x )) + δ | D f ( x ) | q (cid:1) dx, where q ∈ [1 , ∞ ) and δ >
0, were introduced by Toupin [28], [29] and later consideredby many other authors, see e.g. Ball, Curie, Olver [6], Ball, Mora-Corral [7], M¨uller[23, Section 6], Ciarlet [12, page 93] and references given there. The contribution of thehigher gradient is usually connected with interfacial energies and is used to model variousphenomena like elastoplasticity or damage. If q is much bigger than the dimension 2, thenunder some additional assumptions we can actually conclude that J f ≥ σ > q = 1 or q = 2.In that case the usual convolution approximation is not useful for approximation as it ingeneral 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] asthere is no analogy of the key extension procedure, i.e. of Rado-Choquet-Knesser theorem. PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 3
Indeed, even in the one-dimensional case the minimizers of the W ,q ((0 , f (0) , f (1) > f (0) , f ′ (0) > f ′ (1) > f (1) − f (0). Instead we use some ideasof [17], we cover Ω by triangles and we divide triangles into good and bad accordingto the behavior of f like differentiability on them. The total measure of bad trianglesis small and we approximate f on good triangles by quadratic polynomials. Then wesmoothen this piecewise quadratic mapping along the edges of triangles and we obtainthe 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 ableto approximate strongly the second derivative.We call Ω ⊂ R a polygonal domain, if we can find triangles { T i } ki =1 with pairwisedisjoint interiors so that Ω = S ki =1 T i . We say that f : Ω → R is piecewise quadratic,if it is continuous and there is a triangulation Ω = S ki =1 T i such that f | T i is a quadraticfunction in each coordinate, i.e. f ( x, y ) = [ a + a x + a y + a x + a xy + a y , b + b x + b y + b x + b xy + b y ] on T i . Note that with our quadratic approximation we cannot achieve the continuity of thederivative in the direction perpendicular to sides of the triangles, but we can achieve thatthe jumps of the derivative there are small. Thus our piecewise quadratic mapping doesnot belong to W , but it belongs to W BV , i.e. its derivative is a BV mapping. By D s f we denote the singular part of the second derivative which is supported in S ki =1 ∂T i andcorresponds to jump of derivatives between touching triangles.Our first result is an analogy of [25] for second derivatives, but with no control ofderivative 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. Theorem 1.1.
Let Ω ⊂ R be a polygonal domain. Let δ, d > and assume that f : Ω → R is a piecewise quadratic homeomorphism so that (1.2) Z Ω | D s f | < δ and J f > d a.e. in Ω . Then for every ε > we can find a C ∞ diffeomorphism g : Ω → R such that k f − g k W BV (Ω , R ) < ε + Cδ and k f − g k L ∞ (Ω , R ) < ε. In our second result we apply the previous result to show a diffeomorphic approximationof W , homeomorphism up to a set of small measure. This part is more difficult thanthe corresponding result in [17] as we have to deal also with second derivatives and withpiecewise quadratic approximation. Theorem 1.2.
Let Ω ⊂ R be a domain of finite measure. Let f ∈ W , (Ω , R ) be ahomeomorphism such that J f > a.e. Then for every ν > we can find squares { Q i } ∞ i =1 which are locally finite (i.e. each compact set K ⊂ Ω intersects only finitely many ofthem) with L (cid:16) ∞ [ i =1 Q i (cid:17) < ν and we can find C ∞ diffeomorphism g : Ω \ S ∞ i =1 Q i → R such that k f − g k W , (Ω \ S ∞ i =1 Q i , R ) < ν. DANIEL CAMPBELL AND STANISLAV HENCL
The natural plan for our future research is to obtain some analogy of the key extensionresult [17, Theorem 2.1] which will lead to the full approximation result of W , homeo-morphisms, i.e. we would be able to deal also with a set of small measure S Q i . Moreover,we could try to obtain an analogy of [17, Theorem 3.1], which would even remove theassumption J f > Preliminaries
By [ x, y ] we denote the point in R with coordinates x and y . The scalar product of u, v ∈ R is denoted by h u, v i . By B ( c, r ) we denote the ball centered at c ∈ R withradius r > Q ( c, r ) denotes the corresponding square.Let u, v ∈ R be nonzero. Then we have the following elementary estimate(2.1) (cid:12)(cid:12)(cid:12) u | u | − v | v | (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) u | u | − v | u | + v (cid:16) | v | − | u || u | | v | (cid:17)(cid:12)(cid:12)(cid:12) ≤ | u − v || u | . We introduce a smooth function which grows from 0 to 1 on [0 ,
1] and plays an importantrole in our construction.
Notation 2.1.
Let η : R → R be a fixed smooth function with η ( x ) = 0 , x ≤ and x = 1 , x ≥ , η increasing on [0 , , ≤ η ′ ≤ and | η ′′ | ≤ . For f : R → R we use the notation for first derivatives D x f = ∂f∂x , D y = ∂f∂x andsimilarly for second derivatives D xx f = D x ( D x f ), D yy f = D y ( D y f ) and D xy = D x ( D y f ).Similarly for any vector u ∈ R we denote by D u f the derivative of f in the u direction.It is well-known that for C mapping the classical and distributional derivatives agree.Hence for any domain G ⊂ R , f ∈ C ( G, R ) and { u, v } and { ~u, ~v } a pair of positivelyoriented orthonormal bases of R we have(2.2) J f ( x, y ) = det Df ( x, y ) = h D u f ( x, y ) , ~u ih D v f ( x, y ) , ~v i − h D u f ( x, y ) , ~v ih D v f ( x, y ) , ~u i , for almost every [ x, y ] ∈ G where J f is the weak Jacobian of f . This is essentially the in-variance of the determinant with respect to the choice of a positively oriented orthonormalbasis.2.1. Representation of higher order derivatives.
Given f ∈ C (Ω , R ) we can view Df as the the mapping from R × → C (Ω) (i.e. as a matrix) and we can define thesymbol D f as the mapping from R × × → C (Ω) (i.e. as the operator on 2 × D ( f g ) = ( Df ) g + f ( Dg ) as matrices and similarly we can symbolicallywrite D ( f g ) = D (cid:0) ( Df ) g + f ( Dg ) (cid:1) = D f g + Df Dg + Df Dg + f D g where on the righthand side we see the correct terms of the product. At the end we willjust estimate the norm of this by the corresponding product of norms and the exact termswill not be important for us.2.2. ACL condition.
Let Ω ⊂ R be an open set. It is a well-known fact (see e.g. [1,Section 3.11]) that a mapping u ∈ L (Ω , R m ) is in W , (Ω , R m ) if and only there is arepresentative which is an absolutely continuous function on almost all lines parallel tocoordinate axes and the variation on these lines is integrable. PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 5
Figure 1.
Triangulation and direction in verticesAnalogously for any given direction v ∈ R we can fix v ⊥ ⊥ v and define h s ( t ) = u ( sv ⊥ + tv ) for the right representative of u . Then for a.e. s the function h s is absolutely continuouson L s := { t : sv ⊥ + tv ∈ Ω } and Z ∞∞ Z L s | h ′ s ( t ) | dt ds ≤ Z Ω | Du ( x ) | dx. FEM quadratic approximation on triangles.
We need to define a quadraticpolynomial A that approximates our mapping f on a triangle T . Without loss of generalitylet T have vertices v = [0 , v = [ r,
0] and v = [0 , r ] for some r > f : T → R and we wantto define mapping A ( x, y ) = [ a + a x + a y + a x + a xy + a y , b + b x + b y + b x + b xy + b y ] , where a i , b i ∈ R , i ∈ { , , , , , } . We choose these constants so that for j = 1 , , A ( v j ) = − Z B ( v j , r ) f, D x A ( v ) = − Z B ( v , r ) D x f, − D y A ( v ) = − Z B ( v , r ) − D y f and ( − D x + D y ) A ( v ) = − Z B ( v , r ) ( − D x + D y ) f, that is values of A in vertices corresponds to values of f (in averaged sense) and values ofderivatives of A in vertices along three sides correspond to derivatives of f . Note that these6 equations in first coordinate (resp. second coordinates) determine the 6 coefficients a i (resp. b i ) uniquely. Moreover, imagine that we have two triangles T and T with commonside and that we define A on T and A on T by procedure (2.3) described above. Then A = A on ∂T ∩ ∂T since 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 derivativealong the segment in one of the vertices (see Fig. 1).2.4. Estimates of piecewise quadratic homeomorphisms around the vertices.Lemma 2.2.
Let Q , Q , . . . Q N : R → R be quadratic mappings with Q i (0 ,
0) = [0 , .Let ≤ ω < ω < · · · < ω N − < ω N = ω + 2 π < π and let ˜ ω i = [cos ω i , sin ω i ] ∈ S DANIEL CAMPBELL AND STANISLAV HENCL f, h ˜ ω = ˜ ω ˜ ω ˜ ω ˜ ω ˜ ω [0 ,
0] [0 , Figure 2.
Mapping f maps rays [0 , ρ ) × ˜ ω i onto quadratic curves (boldon the right side) and h maps these rays onto touching segments (dashedon the right side). be angles ordered anti-clockwise around S . Let R > and f : B (0 , R ) → R be the mapdefined by f ( t cos θ, t sin θ ) = Q i ( t cos θ, t sin θ ) for all ≤ t ≤ R and all ω i − ≤ θ ≤ ω i . Further assume that this f is a homeomorphism and det DQ i ≥ d > on B (0 , R ) . Let L and M denote positive numbers such that | DQ i | ≤ L on B (0 , R ) and | D Q i | ≤ M . Thenthe map (2.4) h ( t cos θ, t sin θ ) = DQ i (0 , t cos θ, t sin θ ] 0 ≤ t ≤ R and ω i − ≤ θ ≤ ω i is a piecewise linear homeomorphism (see Fig. 2). Moreover (2.5) h (cid:0) t cos θ, t sin θ (cid:1) = t (cid:0) D [cos θ, sin θ ] h (cid:0) cos θ, sin θ (cid:1)(cid:1) for all < t ≤ R and all θ ∈ R . Further it holds that (2.6) dL ≤ | D w h ( x, y ) | ≤ L and dL ≤ | D w f ( x, y ) | ≤ L for all w ∈ S and (2.7) (cid:12)(cid:12) h ( x, y ) − f ( x, y ) (cid:12)(cid:12) ≤ M (cid:12)(cid:12) [ x, y ] (cid:12)(cid:12) for all [ x, y ] ∈ B (0 , R ) . Moreover for any pair u ⊥ v ∈ S , u clockwise from v , denoting ~v = D v f ( x,y ) | D v f ( x,y ) | and ~u ⊥ ~v is clockwise from ~v , it holds that (2.8) dL ≤ (cid:10) D u f ( x, y ) , ~u (cid:11) for all [ x, y ] ∈ B (0 , R ) . For the mapping f we have (2.9) dL t − M t ≤ (cid:12)(cid:12) f ( t cos θ, t sin θ ) (cid:12)(cid:12) ≤ Lt especially if | [ x, y ] | ≤ dLM we have (2.10) d L (cid:12)(cid:12) [ x, y ] (cid:12)(cid:12) ≤ | f ( x, y ) | ≤ L (cid:12)(cid:12) [ x, y ] (cid:12)(cid:12) . PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 7
Finally (2.11) | Df ( x, y ) − Dh ( x, y ) | ≤ M (cid:12)(cid:12) [ x, y ] (cid:12)(cid:12) . Similarly if r, ℓ > and f ( x, y ) = Q ( x, y ) on [ − r, × [0 , ℓ ] and f ( x, y ) = Q ( x, y ) on [0 , r ] × [0 , ℓ ] is a homeomorphism with | Df | ≤ L , J f ≥ d and | D f | ≤ M then (2.12) dL ≤ | D w f ( x, y ) | ≤ L for all w ∈ S . Further, for any pair u ⊥ v ∈ S , u clockwise from v , denoting ~v = D v f ( x,y ) | D v f ( x,y ) | and ~u ⊥ ~v isclockwise from ~v , we have (2.13) dL ≤ (cid:10) D u f ( x, y ) , ~u (cid:11) for all [ x, y ] ∈ [ − r, r ] × [0 , ℓ ] .Proof. First we prove that the map h defined by (2.4) is a piecewise linear homeo-morphism. Denote ˜ θ = [cos θ, sin θ ] and note that on each ω i ≤ θ ≤ ω i +1 we have Dh ( t ˜ θ ) = DQ i (0 , DQ i (0 , ≥ d > ≤ i ≤ N we know that h isa homeomorphism on each ω i ≤ θ ≤ ω i +1 . Further f is continuous on ˜ ω i × [0 , ρ ] andhence ∂ ˜ ω i Q i (0 ,
0) = ∂ ˜ ω i Q i +1 (0 ,
0) which implies that h is continuous. By the piecewiselinearity of h , the continuity of h and J h > h is ahomeomorphism (see Fig. 2).The equality (2.5) is obvious from the piece-wise linearity of h . Since | Df | ≤ L and J f ≥ d we obtain for any pair u, v ∈ S with u ⊥ v that d ≤ J f ( x, y ) ≤ | D u f ( x, y ) | · | D v f ( x, y ) | ≤ L | D v f ( x, y ) | , which shows (2.6) for f . Analogously J h ≥ d a.e. (as det DQ i (0 , ≥ d for all i ) and | D w h | ≤ L for any w ∈ S imply (2.6) for h .For all ω i ≤ θ ≤ ω i +1 we have DQ i (0 ,
0) = lim s → Df ( s cos θ, s sin θ ). For any ω i ≤ θ ≤ ω i +1 , calling ˜ θ = [cos θ, sin θ ] we have using D ˜ θ Q i (0 ,
0) = D ˜ θ h ( s ˜ θ )(2.14) f ( t ˜ θ ) − h ( t ˜ θ ) = Z t D ˜ θ f ( s ˜ θ ) − Z t D ˜ θ Q i (0 , ds = Z t Z s D ˜ θ ˜ θ f ( z ˜ θ ) dz ds but since | D ˜ θ ˜ θ f | ≤ M we have (cid:12)(cid:12) f ( t ˜ θ ) − h ( t ˜ θ ) (cid:12)(cid:12) ≤ M t which is (2.7).Since ~v = D v f ( x,y ) | D v f ( x,y ) | and ~u ⊥ ~v we obtain h D v f, ~u i = 0. Thus we can use (2.2) to obtain d ≤ J f ( x, y ) = h D u f ( x, y ) , ~u ih D v f ( x, y ) , ~v i and using (2.6) we get (2.8). The equation (2.9) follows from (2.14) with the help of | D f | ≤ M and (2.6). Further (2.10) follows immediately from (2.9). The equation (2.11)follows from the fact that | D f | ≤ M and Dh (˜ θ ) = DQ i (0 ,
0) for ω i < θ < ω i +1 . Theproof of (2.12) and (2.13) is analogous to that of (2.6) and (2.8). (cid:3) DANIEL CAMPBELL AND STANISLAV HENCL
Lemma 2.3.
Let f ∈ W , ∞ ( B (0 , R ) , R ) and f is C smooth except on a finite numberof rays ˜ ω R + , ˜ ω R + , . . . ˜ ω N R + , ˜ ω i ∈ S , f (0 ,
0) = 0 and | f ( x, y ) | > for [ x, y ] = [0 , .Let R : B (0 , R ) → [0 , ∞ ) and ϕ : B (0 , R ) → S 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 any t ∈ (0 , R ) and any θ ∈ [0 , π ) (calling ˜ θ = [cos θ, sin θ ] , ˜ θ ⊥ = [ − sin θ, cos θ ] and calling ϕ ⊥ ( t ˜ θ ) ∈ S the vector anti-clockwise perpendicular to ϕ ( t ˜ θ ) ) it holds that (cid:10) ∂∂θ ϕ ( t cos θ, t sin θ ) , ϕ ⊥ ( t ˜ θ ) (cid:11) = t R ( t ˜ θ ) (cid:10) D ˜ θ ⊥ f ( t ˜ θ ) , ϕ ⊥ ( t ˜ θ ) (cid:11) . Proof.
Without loss of generality we may assume that ˜ θ = e = ϕ ( t,
0) and ˜ θ ⊥ = e = ϕ ⊥ ( t,
0) (just consider suitable rotations). Since ϕ = f | f | , f is Lipschitz and | f ( x, y ) | > | [ x, y ] | > ϕ is locally Lipschitz outside of 0. This and the fact that (cid:12)(cid:12) [ t cos θ, t sin θ ] − [ t, t tan θ ] (cid:12)(cid:12) ≤ θ for small θ implies(2.15) h ∂∂θ ϕ ( t cos θ, t sin θ ) i θ =0 = lim θ → ϕ ( t cos θ, t sin θ ) − ϕ ( t, θ = lim θ → ϕ ( t cos θ, t sin θ ) − ϕ ( t, t tan θ ) θ + lim θ → t tan θθ ϕ ( t, t tan θ ) − ϕ ( t, t tan θ = tD ˜ θ ⊥ ϕ ( t, t tan θθ → t . Now D ˜ θ ⊥ f ( t,
0) = D y f ( t,
0) = D y R ( t, ϕ ( t,
0) + R ( t, D y ϕ ( t, (cid:10) D ˜ θ ⊥ f ( t, , ϕ ⊥ ( t, (cid:11) = (cid:10) R ( t, D ˜ θ ⊥ ϕ ( t, , ϕ ⊥ ( t, (cid:11) and our conclusion follows using (2.15). (cid:3) Approximation of piecewise quadratic homeomorphisms around theedges
Recall that η denotes the function from the Preliminaries, Notation 2.1. Lemma 3.1 (Approximation along the edge) . Let Q , Q : R → R be a pair of quadraticmappings coinciding on the line { x = 0 } and let ρ , ℓ > be such that the map f = Q on [ − ρ , × [0 , ℓ ] and f = Q on [0 , ρ ] × [0 , ℓ ] is a homeomorphism with d = min (cid:8) det DQ ( x, y ) , det DQ ( x, y ); [ x, y ] ∈ [ − ρ , ρ ] × [0 , ℓ ] (cid:9) > . Let L and M denote positive numbers such that | DQ j | ≤ L and | D Q j | ≤ M on [ − ρ , ρ ] × [0 , ℓ ] for j = 1 , . Fix N ∈ N , N ≥ such that (3.1) ρ = ℓ N < min n ρ , d M + 1)( L + 1) , d M L o . PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 9
Then there exists an r (depending on the geometry of f ( { } × [0 , ℓ ]) ) such that for ev-ery < r < min { r , ρ L +1) , ρ , Mρ L } there exists a diffeomorphism g : [ − ρ , ρ ] × [0 , ℓ ] satisfying g ( x, y ) = f ( x, y ) for all | x | > r and y ∈ [0 , ℓ ] and (3.2) Z [ r,r ] × [0 ,ℓ ] | D g | ≤ C Z ℓ | D xx f (0 , y ) | + CM ℓr where by | D xx f (0 , y ) | = | D x Q (0 , y ) − D x Q (0 , y ) | we denote the size of the Dirac measureof D xx f at [0 , y ] . Further, call ~u ( x, y ) ∈ S the vector clockwise perpendicular to D y g (0 ,y ) | D y g (0 ,y ) | for all [ x, y ] ∈ [ − r, r ] × [0 , ℓ ] , then we have (3.3) (cid:10) D x g ( x, y ) , ~u ( x, y ) (cid:11) ≥ d L .
Proof.
Step 1. Initial setup and definition of ˜ f r . We have ρ which satisfies (3.1) and ℓ ρ = N ∈ N . We divide [ − r, r ] × [0 , ℓ ] into N rectangles [ − r, r ] × [(2 i − ρ, iρ ] for i = 1 , . . . , N . We denote(3.4) ~v i = ∂ y Q (0 , (2 i − ρ ) | ∂ y Q (0 , (2 i − ρ ) | for i = 1 , , . . . , N and we fix ~u i ∈ S , ~u i ⊥ ~v i so that { ~u i , ~v i } is a positively oriented basis of R . That is ~u i is clockwise purpendicularto ~v i . Then we define a tentative map ˜ f r as an appropriate convex combination of Q and Q , i.e.(3.5) ˜ f r ( x, y ) = (1 − η ( xr )) Q ( x, y ) + η ( xr ) Q ( x, y )if a ) h D x Q (0 , (2 i − ρ ) , ~u i i ≥ h D x Q (0 , (2 i − ρ ) , ~u i i , (1 − η ( x + rr )) Q ( x, y ) + η ( x + rr ) Q ( x, y )if b ) h D x Q (0 , (2 i − ρ ) , ~u i i > h D x Q (0 , (2 i − ρ ) , ~u i i . for every [ x, y ] ∈ [ − r, r ] × [(2 i − ρ, iρ ]. Note that ˜ f r = Q for x < − r and ˜ f r = Q for x > r . Step 2. Define g . The above definition of ˜ f r is fine if all rectangles [ − r, r ] × [(2 i − ρ, iρ ] are of type a)or if all of them are of type b). Otherwise we have to continuously connect rectangles oftype a) to rectangles of type b).Suppose that we have a pair of neighboring rectangles ( − r, r ) × ((2 i − ρ, iρ ) of type a ) and ( − r, r ) × (2 iρ, (2 i + 2) ρ ) of type b ) then we define g as follows(3.6) g ( x, y ) = (cid:2) − η (cid:0) xr + η ( yρ − − i ) (cid:1)(cid:3) Q ( x, y ) + η (cid:0) xr + η ( yρ − − i ) (cid:1) Q ( x, y )for all [ x, y ] ∈ ( − ρ , ρ ) × [2 iρ, (2 i + 1) ρ ]. Note that for y = 2 iρ and y = (2 i + 1) ρ wehave η (cid:0) xr + η ( yρ − − i ) (cid:1) = η (cid:0) xr (cid:1) and η (cid:0) xr + η ( yρ − − i ) (cid:1) = η (cid:0) x + rr (cid:1) and so it agrees with (3.5) there. Similarly when we have a pair of adjacent rectangles ( − r, r ) × ((2 i − ρ, iρ ) of type b ) and ( − r, r ) × (2 iρ, (2 i + 2) ρ ) of type a ) then we define g as follows(3.7) g ( x, y ) = (cid:2) − η (cid:0) x + rr − η ( yρ − − i ) (cid:1)(cid:3) Q ( x, y ) + η (cid:0) x + rr − η ( yρ − − i ) (cid:1) Q ( x, y )on ( − ρ , ρ ) × [2 iρ, (2 i + 1) ρ ]. On the rest of [ − ρ , ρ ] × [0 , ℓ ] we let g ( x, y ) = ˜ f r ( x, y ).Immediately we see from the smoothness of Q , Q and η that g is smooth on ( − ρ , ρ ) × (0 , ℓ ). Step 3. The injectivity of g . We firstly show that J g > g is locally a homeomorphism. Secondlywe show that g is injective on ∂ ([ − ρ , ρ ] × [0 , ℓ ]). Together these two facts imply thatthe smooth mapping g is in fact a diffeomorphism (see e.g. [22]).The following calculations are for rectangles where a )-type transfers into b )-type and g is given by (3.6). The calculations are analogous for (3.7) and are even simpler onrectangles where g ≡ ˜ f r . Since Q , Q and η are smooth we immediately get that ˜ f r issmooth on each rectangle ( − ρ , ρ ) × ((2 i − ρ, iρ ). Further(3.8) D x ˜ f r ( x, y ) = (1 − η ( xr )) D x Q ( x, y ) + η ( xr ) D x Q ( x, y ) + r η ′ ( xr )( Q ( x, y ) − Q ( x, y ))and D y ˜ f r ( x, y ) = (1 − η ( xr )) D y Q ( x, y ) + η ( xr ) D y Q ( x, y ) . If ( − r, r ) × ((2 i − ρ, iρ ) is an a )-type rectangle and ( − r, r ) × (2 iρ, (2 i + 2) ρ ) is a b )-typerectangle then on ( − ρ , ρ ) × (2 iρ, (2 i + 1) ρ ) we calculate(3.9) D x g ( x, y ) = (cid:2) − η (cid:0) xr + η ( yρ − − i ) (cid:1)(cid:3) D x Q ( x, y ) + η (cid:0) xr + η ( yρ − − i ) (cid:1) D x Q ( x, y )+ r η ′ (cid:0) xr + η ( yρ − − i ) (cid:1) ( Q ( x, y ) − Q ( x, y )) . Moreover we calculate(3.10) D y g ( x, y ) = (cid:2) − η (cid:0) xr + η ( yρ − − i ) (cid:1)(cid:3) D y Q ( x, y ) + η (cid:0) xr + η ( yρ − − i ) (cid:1) D y Q ( x, y )+ ρ η ′ (cid:0) xr + η ( yρ − − i ) (cid:1) η ′ ( yρ − − i )( Q ( x, y ) − Q ( x, y )) . Because Q (0 , y ) = Q (0 , y ) for y ∈ [0 , ℓ ] we have | Q ( x, y ) − Q ( x, y ) | ≤ Z x | D x Q ( s, y ) | + | D x Q ( s, y ) | ds ≤ Lr.
Utilizing this fact, (3.8), (3.9), (3.10), 0 ≤ η ≤ | η ′ | ≤ r < ρ we get that(3.11) | Dg | ≤ L. on each ( − r, r ) × ((2 i − ρ, iρ ). Since g is a convex combination of Q and Q , f isequal either to Q or Q and Q = Q on 0 × [0 , ℓ ] we get(3.12) k g − f k ∞ ≤ k Q ( x, y ) − Q (0 , y ) k ∞ + k Q ( x, y ) − Q (0 , y ) k ∞ ≤ Lr.
Using Q (0 , y ) = Q (0 , y ) we also have (cid:10) Q ( x, y ) − Q ( x, y ) , ~u i (cid:11) = Z x h D x Q ( s, y ) , ~u i i − h D x Q ( s, y ) , ~u i i ds. Use | D Q j | ≤ M for j = 1 , r ≤ ρ to get(3.13) (cid:12)(cid:12) DQ j ( s, y ) − DQ j (0 , (2 i − ρ ) (cid:12)(cid:12) ≤ M ρ for s ∈ [ − r, r ] and y ∈ [(2 i − ρ, iρ ] PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 11 and hence with the help of ρ ≤ d ML (cid:10) Q ( x, y ) − Q ( x, y ) , ~u i (cid:11) ≥ − M ρx ≥ − dr L .
From (2.13) at the point [0 , (2 i − ρ ] for v = [0 ,
1] and u = [1 ,
0] we obtain (cid:10) D x Q (0 , (2 i − ρ ) , ~u i (cid:11) ≥ dL and hence we can combine it with the previous inequality to obtain(3.14) 2 r (cid:10) Q ( x, y ) − Q ( x, y ) , ~u i (cid:11) ≥ − (cid:10) D x Q (0 , (2 i − ρ ) , ~u i (cid:11) . Using (3.8), h D x Q (0 , (2 i − ρ ) , ~u i i ≥ h D x Q (0 , (2 i − ρ ) , ~u i i (which holds for a) typerectangles) and (3.13) we obtain (for [ x, y ] where g = ˜ f r )(3.15) h D x g ( x, y ) , ~u i i = (cid:10) (1 − η ( xr )) D x Q ( x, y ) + η ( xr ) D x Q ( x, y ) + r η ′ ( xr )( Q ( x, y ) − Q ( x, y )) , ~u i (cid:11) ≥ (cid:10) (1 − η ( xr )) D x Q (0 , (2 i − ρ ) + η ( xr ) D x Q (0 , (2 i − ρ ) , ~u i (cid:11) − | D x Q (0 , (2 i − ρ ) − D x Q ( x, y ) | − | D x Q (0 , (2 i − ρ ) − D x Q ( x, y ) | + r η ′ ( xr ) (cid:10) Q ( x, y ) − Q ( x, y ) , ~u i (cid:11) ≥ (cid:10) D x Q (0 , (2 i − ρ ) , ~u i (cid:11) − M ρ + r η ′ ( xr ) (cid:10) Q ( x, y ) − Q ( x, y ) , ~u i (cid:11) , ≥ (cid:10) D x Q (0 , (2 i − ρ ) , ~u i (cid:11)(cid:16) − − (cid:17) , where we have used (2.13) and ρ ≤ d ML to estimate the term 4 M ρ and the term r η ′ ( xr ) (cid:10) Q ( x, y ) − Q ( x, y ) , ~u i (cid:11) is either positive and then we can estimate it by 0 or it isnegative and then we use | η ′ | ≤ h D x g ( x, y ) , ~u i i ≥ (cid:10) D x Q (0 , (2 i − ρ ) , ~u i (cid:11) on ( − r, r ) × (0 , ℓ ) which together with (2.13) implies(3.17) h D x g ( x, y ) , ~u i i ≥ dL . Now d ≤ J Q ≤ | D x Q || D y Q | implies | D x Q | ≥ dL . Recall that ~u = ~u ( y ) is the vector in S clockwise purpendicular to D y Q i (0 ,y ) | D y Q i (0 ,y ) | . Using (3.4), (2.1), (3.11) and (3.1) we obtain (cid:12)(cid:12) h D x g ( x, y ) , ~u i i − h D x g ( x, y ) , ~u i (cid:12)(cid:12) ≤ | D x g ( x, y ) | | ~u i − ~u | = | D x g ( x, y ) | | ~v i − ~v |≤ L | D y Q (0 , (2 i − ρ ) − D y Q (0 , y ) || D y Q (0 , (2 i − ρ ) | ≤ L M ρ dL ≤ dL which together with (3.17) imply (3.3) since g = Q in [0 , y ] (see (3.5) and (3.6)). In(3.9) we dealt only with the a )-type to b )-type transitions but the calculations easilyextend also for the b )-type to a )-type transitions. The only difference is that we use h D x Q (0 , (2 i − ρ ) , ~u i i ≥ h D x Q (0 , (2 i − ρ ) , ~u i i in (3.15) above and hence we have h D x Q (0 , , ~u i i on the righthand side of (3.16). By the definition of ~v i (3.4), ~u i ⊥ ~v i and Q = Q on { } × [0 , ℓ ] we have h D y Q j (0 , (2 i − ρ ) , ~u i i = 0. It follows using (3.13) that |h D y Q j ( x, y ) , ~u i i| ≤ M ρ for j = 1 , . With the help of r < Mρ L we obtain | Q ( x, y ) − Q ( x, y ) | ≤ | Q ( x, y ) − Q (0 , y ) | + | Q (0 , y ) − Q ( x, y ) | ≤ Lr ≤ M ρ and hence (cid:10) Q ( x, y ) − Q ( x, y ) , ~u i (cid:11) ≤ M ρ . Applying this in (3.10) we get that(3.18) h D y g ( x, y ) , ~u i i ≤ M ρ + 4 ρ M ρ ≤ M ρ.
We can express the values of Dg with respect to the basis { ~u i , ~v i } as Dg ( x, y ) = (cid:18) h D x g ( x, y ) , ~u i i , h D x g ( x, y ) , ~v i ih D y g ( x, y ) , ~u i i , h D y g ( x, y ) , ~v i i (cid:19) = (cid:18) a , a b , b (cid:19) . Therefore, applying (3.16), (3.11), (3.18), ρ < d LM , definition of ~v i (3.4), (3.13) and(2.12) (i.e. h D y Q (0 , (2 i − ρ ) , ~v i i ≥ dL ) we conclude that(3.19) a > (cid:10) D x Q (0 , (2 i − ρ ) , ~u i (cid:11) | a | ≤ L | b | ≤ M ρ ≤ d Lb ≥ | D y Q (0 , (2 i − ρ ) | − M ρ ≥ | D y Q (0 , (2 i − ρ ) | on the entire rectangle ( − r, r ) × ((2 i − ρ, iρ ). From the definition of ~v i and ~u i ⊥ ~v i weknow that h D y Q (0 , (2 i − ρ ) , ~u i i = 0 and hence using (2.2) (cid:10) D x Q (0 , (2 i − ρ ) , ~u i (cid:11) | D y Q (0 , (2 i − ρ ) | ≥ det DQ (0 , (2 i − ρ ) ≥ d. Therefore simple computation gives(3.20) J g ( x, y ) ≥ d − d ≥ d on ( − r, r ) × ((2 i − ρ, iρ ). Step 4. The injectivity of g . By a combination of (3.16) for i = 0 and i = N and the fact that f is a homeomorphismwe get that g is injective on both segments [ − ρ , ρ ] × { } and [ − ρ , ρ ] × { ℓ } . Because f is a homeomorphism we have thatdist (cid:16) f ([ − r, r ] × { } ) , f (cid:0) ∂ ([ − ρ , ρ ] × [0 , ℓ ]) \ (cid:0) [ − ρ , ρ ] × { } (cid:1)(cid:1)(cid:17) > (cid:16) f ([ − r, r ] × { ℓ } ) , f (cid:0) ∂ ([ − ρ , ρ ] × [0 , ℓ ]) \ (cid:0) [ − ρ , ρ ] × { ℓ } (cid:1)(cid:1)(cid:17) > . PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 13
Therefore, by (3.12) and f ( x, y ) = g ( x, y ) for | x | ≥ r there exists an r > r of our claim) such that for all 0 < r < r the mapping g constructed from ˜ f r satisfies g ([ − ρ , ρ ] × { } ) ∩ g (cid:0) ∂ ([ − ρ , ρ ] × [0 , ℓ ]) \ (cid:0) [ − ρ , ρ ] × { } (cid:1)(cid:1) = ∅ and g ([ − ρ , ρ ] × { ℓ } ) ∩ g (cid:0) ∂ ([ − ρ , ρ ] × [0 , ℓ ]) \ (cid:0) [ − ρ , ρ ] × { ℓ } (cid:1)(cid:1) = ∅ for all r ≤ r . But together that means that g is injective on ∂ ([ − ρ , ρ ] × [0 , ℓ ]). Since(3.20) implies local injectivity this is enough to conclude that g is injective everywhere in[ − ρ , ρ ] × [0 , ℓ ] and thus a diffeomorphism (see e.g. [22]). Step 5. Estimates of | D g | . We calculate the estimates of D g in detail only for the a) to b) type transition givenby (3.6). It is not difficult to check that the computation for b) to a) type transition givenby (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 | D xx f (0 , y ) | = | D x Q (0 , y ) − D x Q (0 , y ) | )(3.21) | D x Q ( x, y ) − D x Q ( x, y ) | ≤ | D xx f (0 , y ) | + 2 M | x | ≤ | D xx f (0 , y ) | + 2 M r, further, since D y Q (0 , y ) = D y Q (0 , y ), we have(3.22) | D y Q ( x, y ) − D y Q ( x, y ) | ≤ M r and(3.23) | Q ( x, y ) − Q ( x, y ) | ≤ | D xx f (0 , y ) | · | x | + X j =1 | Q j ( x, y ) − Q j (0 , y ) − xD x Q j (0 , y ) |≤ | D xx f (0 , y ) | r + X j =1 (cid:12)(cid:12)(cid:12)Z x (cid:0) D x Q j ( s, y ) − D x Q j (0 , y ) (cid:1) ds (cid:12)(cid:12)(cid:12) ≤ | D xx f (0 , y ) | r + M r . The second derivatives of (3.6) are calculated by D xx g ( x, y ) =(1 − η (cid:0) xr + η ( yρ − − i ) (cid:1) ) D xx Q + η (cid:0) xr + η ( yρ − − i ) (cid:1) D xx Q + 1 r η ′′ (cid:0) xr + η ( yρ − − i ) (cid:1) ( Q ( x, y ) − Q ( x, y ))+ 1 r η ′ (cid:0) xr + η ( yρ − − i ) (cid:1) ( D x Q ( x, y ) − D x Q ( x, y )) . Using | D Q j ( x, y ) | ≤ M , | η ′ | ≤ | η ′′ | ≤
4, (3.21) and (3.23) we get | D xx g ( x, y ) | ≤ Cr | D xx f (0 , y ) | + CM.
Further D xy g ( x, y ) = (cid:0) − η (cid:0) xr + η ( yρ − − i ) (cid:1)(cid:1) D xy Q ( x, y ) + η (cid:0) xr + η ( yρ − − i ) (cid:1) D xy Q ( x, y )+ 1 r η ′ (cid:0) xr + η ( yρ − − i ) (cid:1) ( D y Q ( x, y ) − D y Q ( x, y ))+ 1 ρ η ′ (cid:0) xr + η ( yρ − − i ) (cid:1) η ′ ( yρ − − i )( D x Q ( x, y ) − D x Q ( x, y ))+ 1 rρ η ′′ (cid:0) xr + η ( yρ − − i ) (cid:1) η ′ ( yρ − − i )( Q ( x, y ) − Q ( x, y ))and using | D Q i ( x, y ) | ≤ M , | η ′ | ≤ | η ′′ | ≤
4, (3.21), (3.22) and (3.23) we get | D xy g ( x, y ) | ≤ CM + Cρ | D xx f (0 , y ) | + CM rρ and the estimate holds for all [ x, y ] ∈ [ − r, r ] × [0 , ℓ ] where (3.6) applies. Finally D yy g ( x, y ) = (cid:2) − η (cid:0) xr + η ( yρ − − i ) (cid:1)(cid:3) D yy Q ( x, y ) + η (cid:0) xr + η ( yρ − − i ) (cid:1) D yy Q ( x, y )+ ρ η ′ (cid:0) xr + η ( yρ − − i ) (cid:1) η ′ ( yρ − − i )( D y Q ( x, y ) − D y Q ( x, y ))+ ρ η ′′ (cid:0) xr + η ( yρ − − i ) (cid:1)(cid:2) η ′ ( yρ − − i ) (cid:3) ( Q ( x, y ) − Q ( x, y ))+ ρ η ′ (cid:0) xr + η ( yρ − − i ) (cid:1) η ′′ ( yρ − − i )( Q ( x, y ) − Q ( x, y ))so | D yy g ( x, y ) | ≤ M + CM rρ + Crρ | D xx f (0 , y ) | + CM r ρ . Integrating the above over [ − r, r ] × [0 , ℓ ] and estimating | D g ( x, y ) | ≤ | D xx g ( x, y ) | + 2 | D xy g ( x, y ) | + | D yy g ( x, y ) | we get using r ≤ ρ Z [ − r,r ] × [0 ,ℓ ] | D g ( x, y ) | ≤ Cr Z ℓ | D xx f (0 , y ) | dy (cid:16) r + 1 ρ + rρ (cid:17) + Crℓ h M + M rρ + M r ρ i ≤ C Z ℓ | D xx f (0 , y ) | dy + CM ℓr, and (3.2) follows. (cid:3) Approximation of piecewise quadratic homeomorphisms around thevertices and proof of Theorem 1.1
Again η denotes the function from the Preliminaries, Notation 2.1. Lemma 4.1 (Approximation near vertices) . Let Q , Q , . . . Q N : R → R be quadraticmappings. Let ≤ ω < ω < · · · < ω N − < ω N = ω + 2 π < π and let ˜ ω i =[cos ω i , sin ω i ] ∈ S be angles ordered anti-clockwise around S and call ω ∗ = min { π , ω i +1 − ω i ; i = 0 , . . . N } . Call ˜ ω ⊥ i = [ − sin ω i , cos ω i ] ∈ S the vector anti-clockwise perpendicular to ˜ ω i . Let f : B (0 , ρ ) → R be the map defined by f ( t cos θ, t sin θ ) = Q i ( t cos θ, t sin θ ) for all ≤ t ≤ ρ and all ω i − ≤ θ ≤ ω i . PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 15˜ ω = ˜ ω ˜ ω ˜ ω ˜ ω ˜ ω ˜ ω O = O O O O O O R R ω ∗ Figure 3.
The sets O i in blue contained inside red cones around raysparallel to ˜ ω i . Outside B (0 , R ) we use the same approach as Lemma 3.1.Inside B (0 , R/
4) we use a linear map. In the annulus we interpolate byfirst squashing onto rings and then rotating.
Further assume that this f is a homeomorphism and det DQ i ≥ d > on B (0 , ρ ) . Let L and M denote positive numbers such that | DQ i | ≤ L on B (0 , ρ ) and | D Q i | ≤ M . Forevery ρ , ρ . . . , ρ N and every R such that (4.1)0 < R < min { ρ i , i = 1 , . . . , N } < min n ρ , min { d, d } M + 1)( L + 1) , d M L , LM + 1 o and every < r i ≤ min n d R L , Rd L , ρ i L + 1) , R ω ∗ o we call r = ( R, ρ , . . . , ρ N , r , . . . , r N ) . Then for all such r , the rectangles (see Fig. 3) O i = (cid:8) t ˜ ω i + s ˜ ω ⊥ i ; t ∈ [ R , ρ i ] , s ∈ [ − r i , r i ] (cid:9) are pairwise disjoint. Further call ~v i = D ˜ ωi Q i ( ρ i ˜ ω i ) | D ˜ ωi Q i ( ρ i ˜ ω i ) | and call ~u i ∈ S the vector clockwiseperpendicular to ~v i . Define ˜ f r as (4.2)˜ f r ( x, y ) = f ( x, y ) for [ x, y ] / ∈ S Ni =1 O i , (cid:2) − η (cid:0) r i h [ x, y ] , − ˜ ω ⊥ i i (cid:1)(cid:3) Q i ( x, y ) + η (cid:0) r i h [ x, y ] , − ˜ ω ⊥ i i (cid:1) Q i +1 ( x, y ) for [ x, y ] ∈ O i if h D − ˜ ω ⊥ i Q i +1 ( ρ i ˜ ω i ) , ~u i i ≥ h D − ˜ ω ⊥ i Q i ( ρ i ˜ ω i ) , ~u i i , (cid:2) − η (cid:0) r i h [ x, y ] , − ˜ ω ⊥ i i + 1 (cid:1)(cid:3) Q i ( x, y ) + η (cid:0) r i h [ x, y ] , − ˜ ω ⊥ i i + 1 (cid:1) Q i +1 ( x, y ) for [ x, y ] ∈ O i if h D − ˜ ω ⊥ i Q i +1 ( ρ i ˜ ω i ) , ~u i i < h D − ˜ ω ⊥ i Q i ( ρ i ˜ ω i ) , ~u i i . Then there exists a C ∞ diffeomorphism g r defined on B (0 , R ) with g r ( x, y ) = ˜ f r ( x, y ) for all R ≤ | [ x, y ] | ≤ R and (4.3) Z B (0 ,R ) | D g r | < CR where the constant C depends on d , L , M and N but is independent of R .Proof. Without loss of generality we may assume that f (0 ,
0) = [0 , Step 1. Proving that O i are pair-wise disjoint. The first claim we prove is that O i ∩ O j = ∅ for any 1 ≤ i < j ≤ N . On the one handwe have that r i ≤ R tan ω ∗ and on the other hand we have thatmin {| [ x, y ] | ; [ x, y ] ∈ O i } = R. Therefore O i lies inside a cone whose axis goes through ω i and the angle at the apex is ω ∗ . These cones are pairwise disjoint and therefore so are O i (see Fig. 3).From r i ≤ R we get q R + R < R and hence(4.4) (cid:8) R ˜ ω i + s ˜ ω ⊥ i ; s ∈ [ − r i , r i ] (cid:9) ⊂ B (0 , R ) . It follows that this inner edge of O i (where ˜ f r is discontinuous) is a subset of B (0 , R ) andthus we can use Lemma 3.1 to conclude that ˜ f r is a diffeomorphism on B (0 , R ) \ B (0 , R )since (4.2) agrees with rotated and translated version of (3.5) there (our r i and ρ i play therole of r and ρ in Lemma 3.1). Note that d , L and M play the same role as in Lemma 3.1and that (4.1) verifies (3.1). In the following computation we will use some estimatesfrom Lemma 3.1.We have shown that ˜ f r is smooth for R ≤ | [ x, y ] | ≤ R < min { ρ i ; i = 1 , , . . . , N } . Step 2. Proving h ∂∂θ ϕ f ( t cos θ, t sin θ ) , ϕ ⊥ f ( t cos θ, t sin θ ) i ≥ C > . Now we express ˜ f r in polar coordinates in the image, i.e. we define the pair of functions R f : B (0 , R ) → [0 , ∞ ) as R f ( x, y ) = | ˜ f r ( x, y ) | and ϕ f : B (0 , R ) \ { [0 , } → S ⊂ R as ϕ f ( x, y ) = ˜ f r ( x,y ) | ˜ f r ( x,y ) | . Then ˜ f r ( x, y ) = R f ( x, y ) ϕ f ( x, y ) on B (0 , R ) . Since ˜ f r is C ∞ smooth on B (0 , R ) \ B (0 , R ) and | ˜ f r ( x, y ) | = 0 if and only if [ x, y ] = [0 , R f and ϕ f are C ∞ smooth there. Further we define ϕ ⊥ f ( x, y ) = (cid:2) − ( ϕ f ( x, y )) , ( ϕ f ( x, y )) (cid:3) the π anti-clockwise rotation of ϕ f . For brevity call ˜ θ = [cos θ, sin θ ] and ˜ θ ⊥ = [ − sin θ, cos θ ].Our aim is to prove that in B (0 , R ) (cid:10) D ˜ θ ⊥ ϕ f ( t ˜ θ ) , ϕ ⊥ f ( t ˜ θ ) (cid:11) = (cid:10) ∂∂θ ϕ f ( t ˜ θ ) , ϕ ⊥ f ( t ˜ θ ) (cid:11) ≥ C > . Step 2.A. The [ x, y ] / ∈ O i case. By Lemma 2.2 the map h ( t cos θ, t sin θ ) = DQ i (0 , t cos θ, t sin θ ) for t ∈ [0 , ∞ ) and ω i − ≤ θ ≤ ω i PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 17 is a piecewise linear homeomorphism. From Lemma 2.3 we have(4.5) h ∂∂θ ϕ f ( t cos θ, t sin θ ) , ϕ ⊥ f ( t ˜ θ ) i = t R f ( t ˜ θ ) h D ˜ θ ⊥ ˜ f r ( t ˜ θ ) , ϕ ⊥ f ( t ˜ θ ) i . We call ϕ h ( t ˜ θ ) = h ( t ˜ θ ) | h ( t ˜ θ ) | and ϕ ⊥ h ( x, y ) = (cid:2) − ( ϕ h ( x, y )) , ( ϕ h ( x, y )) (cid:3) the π anti-clockwise rotation of ϕ h . For brevity we use the notation [ x, y ] = t ˜ θ , where t = | [ x, y ] | and ˜ θ = [ x,y ] | [ x,y ] | . By linearity ϕ h depends only on θ and not t and hence D ˜ θ ( ϕ h ( x, y )) = 0 which implies D ˜ θ h ( x, y ) = D ˜ θ (cid:0) | h ( x, y ) | (cid:1) ϕ h ( x, y ) + | h ( x, y ) | D ˜ θ (cid:0) ϕ h ( x, y ) (cid:1) = D ˜ θ (cid:0) | h ( x, y ) | (cid:1) ϕ h ( x, y ) . It follows that0 < (cid:10) D ˜ θ h ( x, y ) , ϕ h ( x, y ) (cid:11) ≤ L and (cid:10) D ˜ θ h ( x, y ) , ϕ ⊥ h ( x, y ) (cid:11) = 0 . Using (2.2) we obtain d ≤ J h ( x, y ) = (cid:10) D ˜ θ h ( x, y ) , ϕ h ( x, y ) (cid:11)(cid:10) D ˜ θ ⊥ h ( x, y ) , ϕ ⊥ h ( x, y ) (cid:11) and together with | D w h ( x, y ) | ≤ L for all [ x, y ] ∈ B (0 , R ) and all w ∈ S this implies(4.6) dL ≤ h D ˜ θ h ( x, y ) , ϕ h ( x, y ) i ≤ L and dL ≤ h D ˜ θ ⊥ h ( x, y ) , ϕ ⊥ h ( x, y ) i ≤ L. Therefore dL | [ x, y ] | ≤ | h ( x, y ) | ≤ L | [ x, y ] | and | [ x, y ] | < R < min { L, dL } gives (see (2.7))that(4.7) 15 dL | [ x, y ] | ≤ | f ( x, y ) | ≤ L | [ x, y ] | , i.e. 1617 L ≤ | [ x, y ] |R f ( x, y ) ≤ L d . Further for all | [ x, y ] | = t ≤ R ≤ min { d,d } M +1)( L +1) we have using (2.1) and (2.7)(4.8) (cid:12)(cid:12) ϕ h ( t ˜ θ ) − ϕ f ( t ˜ θ ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) h ( t ˜ θ ) | h ( t ˜ θ ) | − f ( t ˜ θ ) | f ( t ˜ θ ) | (cid:12)(cid:12)(cid:12) ≤ | f ( t ˜ θ ) − h ( t ˜ θ ) || h ( t ˜ θ ) | ≤ M t dL t < (cid:8) , dL (cid:9) . Therefore, using (4.6), we get99 d L ≤ (cid:10) D ˜ θ ⊥ h ( x, y ) , ϕ ⊥ f ( x, y ) (cid:11) ≤ L. uv [ x, y ] = a ˜ ω i + b ˜ ω ⊥ i = √ a + b ˜ θθ α < u = − ˜ ω ⊥ i v = ˜ ω i − ˜ θ ⊥ i α R r i α max αO i Figure 4.
Position of vectors and points in O i .In this case we estimate for all 0 < | [ x, y ] | = t ≤ R ≤ d M +1)( L +1) using (4.5), (4.7)and (2.11) to get(4.9) (cid:10) ∂∂θ ϕ f ( t cos θ, t sin θ ) , ϕ ⊥ f ( t ˜ θ ) (cid:11) = t R f ( t ˜ θ ) (cid:10) D ˜ θ ⊥ ˜ f r ( t ˜ θ ) , ϕ ⊥ f ( t ˜ θ ) (cid:11) ≥ L h D ˜ θ ⊥ h ( t ˜ θ ) , ϕ ⊥ f ( t ˜ θ ) i − L d (cid:12)(cid:12) h D ˜ θ ⊥ h ( t ˜ θ ) − D ˜ θ ⊥ ˜ f r ( t ˜ θ ) , ϕ ⊥ f ( t ˜ θ ) i (cid:12)(cid:12) ≥ L h D ˜ θ ⊥ h ( t ˜ θ ) , ϕ ⊥ f ( t ˜ θ ) i − L d M t ≥ L d L − d L ≥ d L . Step 2.B. The [ x, y ] ∈ O i case. PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 19
In the case [ x, y ] ∈ O i we calculate as follows. Let u, v ∈ S satisfy u ⊥ v and set w = u cos α + v sin α for some α ∈ [ − π/ , π/ w ∈ S and α is the anti-clockwiseoriented angle between u and w . By linearity we obtain(4.10) h D w ˜ f r , ϕ ⊥ f i = cos α h D u ˜ f r , ϕ ⊥ f i + sin α h D v ˜ f r , ϕ ⊥ f i = cos α h D u ˜ f r , ~u i ih ϕ ⊥ f , ~u i i + cos α h D u ˜ f r , ~v i ih ϕ ⊥ f , ~v i i + sin α h D v ˜ f r , ϕ ⊥ f i . Given that [ x, y ] ∈ O i ∩ B (0 , R ), then we can uniquely express[ x, y ] = a ˜ ω i + b ˜ ω ⊥ i for a ∈ [ R, R ] and b ∈ [ − r i , r i ] . Further there exists a unique θ ∈ [0 , π ) and using our standard notation that ˜ θ =[cos θ, sin θ ] and ˜ θ ⊥ = [ − sin θ, + cos θ ] we have [ x, y ] = √ a + b ˜ θ . We plan to use (4.10),with w = − ˜ θ ⊥ , u = − ˜ ω ⊥ i and v = ˜ ω i . The situation is depicted in Fig. 4. The anglebetween u and w is the same as the angle between v and ˜ θ and using r i ≤ d R L and d ≤ L we calculate that (see Fig 4)(4.11) | sin α | ≤ | tan α | ≤ r iR ≤ d L ≤ d L ≤ α ≥ . Using also (3.11) ( | D ˜ f r | ≤ L ) in (4.10) we get(4.12) h D ˜ θ ⊥ ˜ f r ( t ˜ θ ) , ϕ ⊥ f ( t ˜ θ ) i ≥ h D ˜ ω ⊥ i ˜ f r ( t ˜ θ ) , ~u i ih ϕ ⊥ f ( t ˜ θ ) , ~u i i − (cid:12)(cid:12) h D ˜ ω ⊥ i ˜ f r ( t ˜ θ ) , ~v i i (cid:12)(cid:12) h ϕ ⊥ f ( t ˜ θ ) , ~v i i− d L (cid:12)(cid:12) h D ˜ ω i ˜ f r ( t ˜ θ ) , ϕ ⊥ f ( t ˜ θ ) i (cid:12)(cid:12) ≥ h D ˜ ω ⊥ i ˜ f r ( t ˜ θ ) , ~u i ih ϕ ⊥ f ( t ˜ θ ) , ~u i i − L h ϕ ⊥ f ( t ˜ θ ) , ~v i i − d L L. By (3.3) we have that h D − ˜ ω ⊥ i ˜ f r ( t ˜ θ ) , ~u i i ≥ d L (note that in order to apply (3.3) we take − ˜ ω ⊥ i as the clockwise rotation of ˜ ω i because also [1 ,
0] is the clockwise rotation of [0 . ϕ ⊥ f is anti-clockwise perpendicular to ϕ f but ~u i is clockwise perpendicularto ~v i and hence h ϕ ⊥ f ( t ˜ θ ) , ~u i i is negative. Combining the two previous facts we get that h D ˜ ω ⊥ i ˜ f r ( t ˜ θ ) , ~u i ih ϕ ⊥ f ( t ˜ θ ) , ~u i i ≥ d L |h ϕ ⊥ f ( t ˜ θ ) , ~u i i| . Applying this in (4.12) we get(4.13) h D ˜ θ ⊥ ˜ f r ( t ˜ θ ) , ϕ ⊥ f ( t ˜ θ ) i ≥ d L |h ϕ ⊥ f ( t ˜ θ ) , ~u i i| − L h ϕ ⊥ f ( t ˜ θ ) , ~v i i − d L .
The factors h ϕ ⊥ f , ~u i i and h ϕ ⊥ f , ~v i i are a question of the geometry of ˜ f r ( O i ). We express[ x, y ] = a ˜ ω i + b ˜ ω ⊥ i for R ≤ a ≤ R and − r i ≤ b ≤ r i . We use (3.12) ( | ˜ f r − f | ≤ Lr i on O i ), (2.7), (2.6), r i ≤ dR L , a + b ≤ R and R ≤ d ML and we get(4.14) (cid:12)(cid:12) ˜ f r ( a ˜ ω i + b ˜ ω ⊥ i ) (cid:12)(cid:12) ≥ (cid:12)(cid:12) h ( a ˜ ω i ) (cid:12)(cid:12) − (cid:12)(cid:12) h ( a ˜ ω i + b ˜ ω ⊥ i ) − h ( a ˜ ω i ) (cid:12)(cid:12) − (cid:12)(cid:12) ˜ f r ( a ˜ ω i + b ˜ ω ⊥ i ) − f ( a ˜ ω i + b ˜ ω ⊥ i ) (cid:12)(cid:12) − (cid:12)(cid:12) h ( a ˜ ω i + b ˜ ω ⊥ i ) − f ( a ˜ ω i + b ˜ ω ⊥ i ) (cid:12)(cid:12) ≥| h ( a ˜ ω i ) | − Lr i − Lr i − M a + b ) ≥ a | D ˜ ω i h (˜ ω i ) | − dR L − dR L ≥ d L R.
By (3.11) (cid:12)(cid:12) ˜ f r ( a ˜ ω i + b ˜ ω ⊥ i ) − ˜ f r ( a ˜ ω i ) (cid:12)(cid:12)(cid:12) ≤ Lr i ≤ d L R. Combining these two facts and calling ζ the angle between ϕ f ( a ˜ ω i + b ˜ ω ⊥ i ) and ϕ f ( a ˜ ω i )we get(4.15) | tan ζ | ≤ d L R d L R ≤ d L . On the other hand using ρ < d M +1)( L +1) and | D Q i | ≤ M we have (cid:12)(cid:12) D ˜ ω i Q i ( ρ ˜ ω i ) − D ˜ ω i Q i (0 , (cid:12)(cid:12) ≤ M ρ ≤ d L + 1) . Therefore, because ~v i = D ˜ ωi Q i ( ρ ˜ ω i ) | D ˜ ωi Q i ( ρ ˜ ω i ) | and | D ˜ ω i Q i ( t ˜ ω i ) | ≥ dL (see (2.6)) we have analogouslyto (4.8) that(4.16) (cid:12)(cid:12)(cid:12) ~v i − D ˜ ω i Q i (0 , | D ˜ ω i Q i (0 , | (cid:12)(cid:12)(cid:12) ≤ d L + 1) Ld d L + 1) . From (4.2) we obtain that ˜ f r = f on the ray ˜ ω i R + since for [ x, y ] = t ˜ ω i we have h [ x, y ] , ˜ ω ⊥ i i = 0. Hence ˜ f r is smooth along this ray and ϕ f ( t ˜ ω i ) = ˜ f r ( t ˜ ω i ) | ˜ f r ( t ˜ ω i ) | = R t D ˜ ω i ˜ f r ( s ˜ ω i ) ds | R t D ˜ ω i ˜ f r ( s ˜ ω i ) ds | = t R t D ˜ ω i Q i ( s ˜ ω i ) ds | t R t D ˜ ω i Q i ( s ˜ ω i ) ds | and so using t ≤ R ≤ d M +1)( L +1) using (2.1)(4.17) (cid:12)(cid:12)(cid:12) ϕ f ( t ˜ ω i ) − D ˜ ω i Q i (0 , | D ˜ ω i Q i (0 , | (cid:12)(cid:12)(cid:12) ≤ t R t | D ˜ ω i Q i ( s ˜ ω i ) − D ˜ ω i Q i (0 , ds || D ˜ ω i Q i (0 , | ≤ M t dL ≤ d L + 1) . Combining (4.16) and (4.17) we obtain that the angle between ϕ f ( a ˜ ω i ) and ~v i (call it ζ )satisfies tan ζ ≤ d L + 1) . PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 21
Call ζ the angle between ϕ f ( a ˜ ω i + b ˜ ω ⊥ i ) and ~v i . From the previous inequality and (4.15)we obtain that | ζ | ≤ | ζ | + | ζ | implies | sin( ζ ) | ≤ | sin ζ | + | sin ζ | ≤ | sin ζ | + 2 (cid:12)(cid:12) sin ζ (cid:12)(cid:12) ≤ d L + 2 d L + 1) ≤ d L . Then also |h ϕ ⊥ f , ~v i i| = | sin( ζ ) | ≤ d L and since dL ≤ |h ϕ ⊥ f , ~u i i| = | cos( ζ ) | ≥ . Applying this in (4.13) we get h D ˜ θ ⊥ ˜ f r ( x, y ) , ϕ ⊥ f ( x, y ) i ≥
910 81 d L − d L − d L ≥ d L for all [ x, y ] ∈ O i ∩ B (0 , R ). Because together (4.14) and (3.11) imply that R f ( t ) ≈ t weconclude from the above using (4.5) that h ∂∂θ ϕ f ( t cos θ, t sin θ ) , ϕ ⊥ f ( t ˜ θ ) i ≥ C. Step 3. Proving that ∂∂t R f ( t ˜ θ ) ≥ C > . In this section we show that ∂∂t R f ( t ˜ θ ) > R ≤ t ≤ R . For [ x, y ] = t ˜ θ , where t = | [ x, y ] | and ˜ θ = [ x,y ] | [ x,y ] | we consider firstly t ˜ θ / ∈ S Ni =1 O i using the following facts. Firstly,for all w ∈ S , we have h ( tw ) = tD w h ( w ) and | D w h | ≥ dL . This means that(4.18) (cid:12)(cid:12) D w h ( w ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ∂∂t h ( tw ) (cid:12)(cid:12)(cid:12) ≥ dL and D ∂∂t h ( tw ) , ϕ h ( tw ) E = D ∂∂t (cid:0) tD w h ( w ) (cid:1) , tD w h ( w ) | tD w h ( w ) | E = (cid:12)(cid:12)(cid:12) ∂∂t h ( tw ) (cid:12)(cid:12)(cid:12) for all w ∈ S . Secondly, because t ≤ R ≤ d ML and t ˜ θ / ∈ S Ni =1 O i we have using (2.7), (cid:12)(cid:12) ˜ f r ( t ˜ θ ) − h ( t ˜ θ ) (cid:12)(cid:12) = (cid:12)(cid:12) f ( t ˜ θ ) − h ( t ˜ θ ) (cid:12)(cid:12) ≤ M t ≤ dtL ≤ | h ( t ˜ θ ) | . This implies (using the fact that | ϕ h − ϕ f | is less than the arclength between them on S )that | ϕ h − ϕ f | ≤ arctan 11000 ≤ . Finally we obtain using (2.11) and t ≤ R ≤ d ML | ∂∂t ˜ f r ( t ˜ θ ) − ∂∂t h ( t ˜ θ ) | ≤ M t ≤ d L ≤ | ∂∂t h ( t ˜ θ ) | . We estimate with the help of (4.18) h ∂∂t ˜ f r ( t ˜ θ ) , ϕ f ( t ˜ θ ) i ≥h ∂∂t h ( t ˜ θ ) , ϕ h ( t ˜ θ ) i − |h ∂∂t ˜ f r ( t ˜ θ ) − ∂∂t h ( t ˜ θ ) , ϕ f ( t ˜ θ ) i|− |h ∂∂t h ( t ˜ θ ) , ϕ f ( t ˜ θ ) − ϕ h ( t ˜ θ ) i|≥| ∂∂t h ( t ˜ θ ) | − | ∂∂t h ( t ˜ θ ) | − | ∂∂t h ( t ˜ θ ) | ≥ d L and using ˜ f r ( t ˜ θ ) = | ˜ f r ( t ˜ θ ) | ϕ f ( t ˜ θ ) and ∂∂t h ϕ f , ϕ f i = 0 we obtain(4.19) h ∂∂t ˜ f r ( t ˜ θ ) , ϕ f ( t ˜ θ ) i = ∂∂t (cid:12)(cid:12) ˜ f r ( t ˜ θ ) (cid:12)(cid:12) h ϕ f ( t ˜ θ ) , ϕ f ( t ˜ θ ) i + (cid:12)(cid:12) ˜ f r ( t ˜ θ ) (cid:12)(cid:12) h ∂∂t ϕ f ( t ˜ θ ) , ϕ f ( t ˜ θ ) i = ∂∂t R f ( t ˜ θ )and hence ∂∂t R f ( t ˜ θ ) > C .When t ˜ θ ∈ S Ni =1 O i we use (4.19) and calculate similarly as in (4.10) and (4.12) (again α denotes the angle between ˜ θ and ˜ ω i ) ∂∂t R f ( t ˜ θ ) = h ∂∂t ˜ f r ( t ˜ θ ) , ϕ f ( t ˜ θ ) i = h D ˜ θ ˜ f r ( t ˜ θ ) , ϕ f ( t ˜ θ ) i≥ cos α (cid:10) D ˜ ω i ˜ f r , ϕ f ( t ˜ θ ) (cid:11) − (cid:12)(cid:12) sin α (cid:10) D ˜ ω ⊥ i ˜ f r , ϕ f ( t ˜ θ ) (cid:11)(cid:12)(cid:12) ≥ cos α D D ˜ ω i ˜ f r , D ˜ ω i f ( ρ i ˜ ω i ) | D ˜ ω i f ( ρ i ˜ ω i ) | E h ϕ f ( t ˜ θ ) , ϕ f ( t ˜ ω i ) i− (cid:12)(cid:12)(cid:12)D D ˜ ω i ˜ f r , (cid:16) D ˜ ω i f ( ρ i ˜ ω i ) | D ˜ ω i f ( ρ i ˜ ω i ) | (cid:17) ⊥ E(cid:12)(cid:12)(cid:12) − | sin α | | D ˜ ω ⊥ i ˜ f r | . In (3.19) (term corresponding to b ) we estimated that D D ˜ ω i ˜ f r , D ˜ ω i f ( ρ i ˜ ω i ) | D ˜ ω i f ( ρ i ˜ ω i ) | E ≥ | D ˜ ω i f ( ρ i ˜ ω i ) | ≥ d L and (term corresponding to b ) that (cid:12)(cid:12)(cid:12)D D ˜ ω i ˜ f r , (cid:16) D ˜ ω i f ( ρ i ˜ ω i ) | D ˜ ω i f ( ρ i ˜ ω i ) | (cid:17) ⊥ E(cid:12)(cid:12)(cid:12) ≤ d L .
Now computing similarly as in (4.12) we obtain that sin α ≤ d L , cos α ≥ (see (4.11))and applying the previous to the above estimate we get(4.20) ∂∂t R f ( t ˜ θ ) ≥
910 499 d L h ϕ f ( t ˜ θ ) , ϕ f ( t ˜ ω i ) i − d L − L d L . Call t ˜ θ = a ˜ ω i + b ˜ ω ⊥ i . Then we obtain using (2.1), (3.11), (4.14) and r i ≤ Rd L that | ϕ f ( t ˜ θ ) − ϕ f ( a ˜ ω i ) | ≤ | ˜ f r ( t ˜ θ ) − ˜ f r ( a ˜ ω i ) || ˜ f r ( t ˜ θ ) | ≤ Lr i d L R ≤ . Similarly using (2.1), (2.7) and (2.6) (obviously ϕ h ( ρ i ˜ ω i ) = ϕ h ( a ˜ ω i )) we have | ϕ f ( ρ i ˜ ω i ) − ϕ f ( a ˜ ω i ) | ≤| ϕ f ( ρ i ˜ ω i ) − ϕ h ( ρ i ˜ ω i ) | + | ϕ f ( a ˜ ω i ) − ϕ h ( a ˜ ω i ) |≤ M ρ i Ldρ i M a Lda ≤ M Ld ρ i ≤ . Since | ϕ f ( t ˜ θ ) − ϕ f ( ρ i ˜ ω i ) | < we obtain h ϕ f ( t ˜ θ ) , ϕ f ( t ˜ ω i ) i ≥ and so continuing theestimate (4.20) ∂∂t R f ( t ˜ θ ) ≥
910 499 d L − d L − d L ≥ C > . Step 4. Proving that g r is a diffeomorphism. PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 23
Call λ = d L . We need to redefine our mapping close to the origin so it is smooth there.We define it as a proper interpolation between a linear mapping [ x, y ] → λ [ x, y ] and ourmapping ˜ f r . We define it as g r ( x, y ) = R g ( x, y ) ϕ g ( x, y ) , where R g ( t ˜ θ ) = (cid:0) − η (cid:0) t − RR (cid:1)(cid:1) λt + η (cid:0) t − RR (cid:1) R f ( t ˜ θ (cid:1) and ϕ g ( t ˜ θ ) = (cid:0) − η (cid:0) t − RR (cid:1)(cid:1) ˜ θ + η (cid:0) t − RR (cid:1) ϕ f ( t ˜ θ (cid:1) . Note that this is equal to λ [ x, y ] on B (0 , R ) and it is equal to ˜ f r outside of B (0 , R ). It ischanging the angle on B (0 , R ) \ B (0 , R ) while keeping the distance from the origin ofa map λ [ x, y ] and it is changing the distance from the origin on B (0 , R ) \ B (0 , R ) whilekeeping the angle of ˜ f r .Immediately from the definition of g r it is obvious that it is smooth since [ x, y ] → λ · [ x, y ]is smooth and η, R f and ϕ f are all smooth away from the origin. Let us define ˆ ϕ f ∈ [0 , π )(resp. ˆ ϕ g ) as the corresponding angle of ϕ f ∈ S (resp. ϕ g ) modulo 2 π . From Step 2 weknow (cid:10) ∂∂θ ϕ f ( t cos θ, t sin θ ) , ϕ ⊥ f ( t cos θ, t sin θ ) (cid:11) ≥ C for all t ∈ [ R, R ] and θ. Using derivative of composed mapping for ϕ f ( t cos θ, t sin θ ) = (cid:2) cos ˆ ϕ f ( t cos θ, t sin θ ) , sin ˆ ϕ f ( t cos θ, t sin θ ) (cid:3) in the above inequality and ϕ ⊥ f ( t cos θ, t sin θ ) = (cid:2) − sin ˆ ϕ f ( t cos θ, t sin θ ) , cos ˆ ϕ f ( t cos θ, t sin θ ) (cid:3) this implies that ∂∂θ ˆ ϕ f ( t cos θ, t sin θ ) ≥ C . It follows that ∂∂θ ˆ ϕ g ( t cos θ, t sin θ ) = (cid:0) − η (cid:0) t − RR (cid:1)(cid:1) ∂∂θ ( θ ) + η (cid:0) t − RR (cid:1) ∂∂θ ˆ ϕ f ( t cos θ, t sin θ ) ≥ C > . Further, because (see (4.7) and (4.14)) | ˜ f r ( t cos θ, t sin θ ) | ≥ dR L > λt for all 34 R ≤ t ≤ R we have that ∂∂t R g ( t ˜ θ ) = (cid:0) − η (cid:0) t − RR (cid:1)(cid:1) λ + η (cid:0) t − RR (cid:1) ∂∂t R f + 8 R η ′ (cid:0) t − RR (cid:1) ( R f ( t ˜ θ ) − λt )but as shown above each of the above terms is positive. Because ∂∂t R g ( t cos θ, t sin θ ) ≥ C > ∂∂θ ˆ ϕ g ( t cos θ, t sin θ ) ≥ C > < t ≤ R and all θ we easily conclude that g r is a diffeomorphism on B (0 , R ) by considering the three parts B (0 , R ), B (0 , R ) \ B (0 , R ) and B (0 , R ) \ B (0 , R ) separately. Further, because g r coincides with the diffeomorphism ˜ f r on B (0 , R ) \ B (0 , R ) it must be a diffeomorphismon B (0 , R ). Step 5. Estimating R B (0 ,R ) | D g r | . Clearly D g r = D ( λ [ x, y ]) = 0 for [ x, y ] ∈ B (0 , R ) so it remains to estimate it for[ x, y ] ∈ B (0 , R ) \ B (0 , R ). We have | D g r | = | D ( R g ϕ g ) | . We calculate D R g = (cid:0) − η (cid:0) | [ x,y ] |− RR (cid:1)(cid:1) λD | [ x, y ] | + η (cid:0) | [ x,y ] |− RR (cid:1) D | ˜ f r ( x, y ) | + D [ x, y ] 8 R η ′ ( | [ x,y ] |− RR ) (cid:0) | ˜ f r ( x, y ) | − λ | [ x, y ] | (cid:1) and (see Section 2.1) D R g = (cid:0) − η (cid:0) | [ x,y ] |− RR (cid:1)(cid:1) λD | [ x, y ] | + 8 R η ′ (cid:0) | [ x,y ] |− RR (cid:1) λD [ x, y ] D | [ x, y ] | + η (cid:0) | [ x,y ] |− RR (cid:1) D | ˜ f r ( x, y ) | + 8 R η ′ (cid:0) | [ x,y ] |− RR (cid:1) D [ x, y ] D | ˜ f r ( x, y ) | + D [ x, y ] 8 R η ′ (cid:0) | [ x,y ] |− RR (cid:1)(cid:0) | ˜ f r ( x, y ) | − λ [ x, y ] (cid:1) + 64 R D [ x, y ] D [ x, y ] η ′′ (cid:0) | [ x,y ] |− RR (cid:1)(cid:0) | ˜ f r ( x, y ) | − λ | [ x, y ] | (cid:1) + 8 R D [ x, y ] η ′ (cid:0) | [ x,y ] |− RR (cid:1)(cid:0) D | ˜ f r ( x, y ) | − λD | [ x, y ] | (cid:1) . We now separate B (0 , R ) \ B (0 , R ) into parts B (0 , R ) \ [ B (0 , R ) ∪ S Ni =1 O i ] and theparts S Ni =1 O i . For [ x, y ] / ∈ S Ni =1 O i we know that ˜ f r = f and hence | ˜ f r ( x, y ) | ≤ CLR, D | ˜ f r | ≤ L and D | ˜ f r | ≤ M. By elementary computation D | [ x, y ] | ≤ , (cid:12)(cid:12) D | [ x, y ] | (cid:12)(cid:12) ≤ C | [ x, y ] | , | D [ x, y ] | ≤ C and D [ x, y ] = 0 . Therefore |R g ( x, y ) | ≤ CR, | D R g ( x, y ) | ≤ C and | D R g ( x, y ) | ≤ CR for all [ x, y ] ∈ B (0 , R ) \ [ B (0 , R ) ∪ S Ni =1 O i ]. Further Dϕ g = (cid:0) − η (cid:0) | [ x,y ] |− RR (cid:1)(cid:1) D [ x, y ] | [ x, y ] | + 8 R D [ x, y ] η ′ (cid:0) | [ x,y ] |− RR (cid:1)(cid:16) ˜ f r ( x, y ) | ˜ f r ( x, y ) | − [ x, y ] | [ x, y ] | (cid:17) + η (cid:0) | ( x,y ) |− RR (cid:1) D ˜ f r ( x, y ) | ˜ f r ( x, y ) | , PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 25 and (see Section 2.1) D ϕ g = (cid:0) − η (cid:0) | [ x,y ] |− RR (cid:1)(cid:1) D [ x, y ] | [ x, y ] | − R D [ x, y ] η ′ (cid:0) | [ x,y ] |− RR (cid:1) D [ x, y ] | [ x, y ] | + 8 R D [ x, y ] η ′ (cid:0) | [ x,y ] |− RR (cid:1)(cid:16) ˜ f r ( x, y ) | ˜ f r ( x, y ) | − [ x, y ] | [ x, y ] | (cid:17) + 64 R D [ x, y ] D [ x, y ] η ′′ (cid:0) | [ x,y ] |− RR (cid:1)(cid:16) ˜ f r ( x, y ) | ˜ f r ( x, y ) | − [ x, y ] | [ x, y ] | (cid:17) + 8 R D [ x, y ] η ′ (cid:0) | [ x,y ] |− RR (cid:1)(cid:16) D ˜ f r ( x, y ) | ˜ f r ( x, y ) | − D [ x, y ] | [ x, y ] | (cid:17) + 8 R D [ x, y ] η ′ (cid:0) | [ x,y ] |− RR (cid:1) D ˜ f r ( x, y ) | ˜ f r ( x, y ) | + η (cid:0) | [ x,y ] |− RR (cid:1) D ˜ f r ( x, y ) | ˜ f r ( x, y ) | . It is easy to calculate that (cid:12)(cid:12)(cid:12) D [ x, y ] | [ x, y ] | (cid:12)(cid:12)(cid:12) ≤ CR and (cid:12)(cid:12)(cid:12) D [ x, y ] | [ x, y ] | (cid:12)(cid:12)(cid:12) ≤ CR . Further basic calculus (and | ˜ f r ( x, y ) | ≈ | [ x, y ] | ) gives (cid:12)(cid:12)(cid:12) D ˜ f r ( x, y ) | ˜ f r ( x, y ) | (cid:12)(cid:12)(cid:12) ≤ CR and (cid:12)(cid:12)(cid:12) D ˜ f r ( x, y ) | ˜ f r ( x, y ) | (cid:12)(cid:12)(cid:12) ≤ CR for all [ x, y ] ∈ B (0 , R ) \ [ B (0 , R ) ∪ N [ i =1 O i ] . Therefore | ϕ g ( x, y ) | ≤ C, | Dϕ g ( x, y ) | ≤ CR , | D ϕ g ( x, y ) | ≤ CR for all [ x, y ] ∈ B (0 , R ) \ [ B (0 , R ) ∪ S Ni =1 O i ]. Therefore | D g r | = | D ( R g ϕ g ) | ≤ C (cid:0) | D R g | · | ϕ g | + | D R g | · | Dϕ g | + |R g | · | D ϕ g | (cid:1) ≤ CR and by integrating(4.21) Z B (0 ,R ) \ [ B (0 , R ) ∪ S Ni =1 O i ] | D g r | < CR. Now we continue with the case [ x, y ] ∈ S Ni =1 O i . We work in each O i separately. Use K i to denote K i = max (cid:8) | D s f | ( t ˜ ω i ); t ∈ [0 , R ] (cid:9) ≤ L. It still holds that | ˜ f r ( x, y ) | ≤ CR and D | ˜ f r ( x, y ) | ≤ C but the difference is that D | ˜ f r ( x, y ) | ≤ C + CK i r i (see estimates in step 5 of Lemma 3.1, most importantly the D xx term). Therefore |R g ( x, y ) | ≤ CR, | D R g ( x, y ) | ≤ C and | D R g ( x, y ) | ≤ CR + CK i r i for all [ x, y ] ∈ O i Similarly as before basic calculus with | f ( x, y ) | ≈ | [ x, y ] | gives (cid:12)(cid:12)(cid:12) D ˜ f r ( x, y ) | ˜ f r ( x, y ) | (cid:12)(cid:12)(cid:12) ≤ CR and (cid:12)(cid:12)(cid:12) D ˜ f r ( x, y ) | ˜ f r ( x, y ) | (cid:12)(cid:12)(cid:12) ≤ CR + CRr i K i , where the constants C depend on d, M and L etc., but not R . Therefore | ϕ g ( x, y ) | ≤ C, | Dϕ g ( x, y ) | ≤ CR , | D ϕ g ( x, y ) | ≤ CR + CK i Rr i for | [ x, y ] | ∈ O i . Calculating as before | D g r | = | D ( R g ϕ g ) | ≤ C (cid:0) | D R g | · | ϕ g | + | D R g | · | Dϕ g | + |R g | · | D ϕ g | (cid:1) ≤ CR + CK i r i Integrating the above estimates over O i we get Z O i | D g r | ≤ CR Rr i + CK i r i Rr i ≤ Cr i + CLR ≤ CR.
Summing the above over i ∈ { , , . . . , N } and adding to (4.21) we get (4.3). (cid:3) Proof of Theorem 1.1.
Let A denote the finite set A = { a , a , . . . , a I } of vertices and S denote the finite set S = { s , s , . . . , s J } of sides of triangles of the definition of polyg-onal domain. We know that our quadratic mappings Q j defined on triangles T j satisfydet DQ j ≥ d > L >
M > | DQ j | ≤ L and | D Q j | ≤ M for all j . Thus we can apply Lemma 4.1 to a translation of f in the imageand preimage at each vertex a i ∈ A . We find ρ > B ( a i , ρ ) are pairwise disjoint.We find N ∈ N , N ≥ ℓ s length of s ∈ S ) we havemax { ℓ s : s ∈ S } N < min n ρ , min { d, d } M + 1)( L + 1) o . Then we call ρ s = ℓ s N for each s ∈ S .Having chosen a ρ s for every s ending at a i we choose R i ≤
12 min n εI ( M + 1) , ρ s o <
14 min n ρ , min { d, d } M + 1)( L + 1) , LM + 1 o . For each s j ∈ S we choose an r s j > r s j is smaller thanthe corresponding r ( s j ), the number from Lemma 3.1. Further we require that r s j ≤ min n d R i L , R i d L , ρ s j L + 1) , R i ω ∗ i , J ε ( M + 1) ℓ s o for both endpoints a i = a s j , and a i = a s j , . For each a i we call r i = ( R i , ρ , . . . ρ n i , r s ji, , . . . , r s ji,ni ).Having made the above choices, we have satisfied the hypothesis of Lemma 3.1 (up toappropriate rotations and translations) for each side s j ∈ S by the choice of r = r s j andhence we can construct a smooth g = g s on a small rectangular neighborhood of each side s . Similarly we satisfy the hypothesis of Lemma 4.1 and because of the same choice ofparameters the smooth map g = g a i is equal to g s ( a i ∈ s ) as soon as the argument is R i distant from a i . Both of the maps equal the original homeomorphism f as soon as we are R i distant from a i and r s distant from s , which is C ∞ smooth on that set. Therefore themap g ( x, y ) = g a i ( x, y ) | [ x, y ] − a i | ≤ R i g s ( x, y ) dist([ x, y ] , s ) ≤ r s and | [ x, y ] − a i | ≥ R i f ( x, y ) otherwiseis a C ∞ -diffeomorphism. Notice that the balls B ( a i , R i ) are pairwise disjoint and so thedefinition is correct. PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 27
For each s j ∈ S we call O j the 2 r s j -wide rectangular neighborhood of the line s j asin Lemma 3.1. Now we use g ( x, y ) = f ( x, y ) for all [ x, y ] / ∈ S i B ( a i , R i ) ∪ S j O j , whichimplies that Z Ω | D f − D g | ≤ I X i =1 Z B ( a i ,R i ) ( | D f | + | D g | ) + J X j =1 Z O j ( | D f | + | D g | ) . We estimate by summing (4.3) over a i ∈ A (recall R i < εI ( M +1) ) and using | D f | ≤ M toobtain I X i =1 Z B ( a i ,R i ) ( | D f | + | D g | ) ≤ I X i =1 ( M πR i + CR i ) ≤ Cε.
Finally we sum (3.2) over s j ∈ S (recall r s j ≤ J ε ( M +1) ℓ s ) and we get using Theorem 1.1 J X j =1 Z O j ( | D f | + | D g | ) ≤ C J X j =1 ( l s j r s j M + | D s f | ( s )) ≤ C ( δ + ε ) . By (3.11) and the estimates in Lemma 4.1 Step 5 it is immediately obvious that k f − g k ∞ < max i CR i + max s Lr s , which is as small as we like. (cid:3) Piecewise quadratic approximation on good squares - Proof ofTheorem 1.2
In this section we first show that the quadratic polynomials constructed in subsection2.3 approximate our homeomorphism f well on some good squares and then we showTheorem 1.2. As noted in subsection 2.3 we know that the two quadratic polynomials onadjacent triangles have the same values on T ∩ T but the derivatives (in the orthogonaldirection) are not necessarily the same. The key observation (5.5) ( ii ) below shows thatthey do not differ too much. Theorem 5.1.
Let T be a triangle with vertices v = [0 , , v = [ r, and v = [0 , r ] forsome r > . Let T be an adjacent triangle, i.e. either with vertices { [ r, , [0 , r ] , [ r, r ] } or { [0 , , [ r, , [ r, − r ] } or { [0 , , [0 , r ] , [ − r, r ] } . Let us assume that we have a homeomor-phism f ∈ W , ( Q ([0 , , r ) , R ) . Let < δ < and assume that (5.1) J f (0 , > δ, k Df (0 , k < δ and for ε > we have (5.2) | f ( z ) − f (0 , − Df (0 , z | < ε | z | for z ∈ Q ([0 , , r ) , (5.3) − Z Q ([0 , , r ) | Df ( z ) − Df (0 , | dz < ε and (5.4) − Z Q ([0 , , r ) | D f ( z ) − D f (0 , | dz < ε. Then there are absolute constant C > and quadratic mappings A , A : Q ([0 , , r ) → R so that (5.5)( i ) D A i is constant and | D A i (0 , − D f (0 , | < Cε, ( ii ) | DA ( z ) − DA ( z ) | < εr for every z ∈ T ∩ T , ( iii ) A = ( A on T A on T is homeomorphism with det A > δ on T ∪ T , if ε < C δ , ( iv ) | f ( z ) − A ( z ) | < C rε for every z ∈ T . Further the map A is independent of the choice of T .Proof. We define A on T by (2.3) and A on T by a similar procedure, i.e. values of A in corners of T are determined by the average values of f nearby and a derivative at eachvertex along a given side is determined by the average of the corresponding derivativeof f . We just make sure that on the side T ∩ T both A and A use the same vertexfor the definition of derivative along that side. In fact we can divide the whole R intosquares of sidelength r , divide them into two triangles (by segment in direction [ − , T and T above (see Fig. 1). Part ( i ) : We have(5.6) A ([ r, − A ([0 , − rD x A ([0 , Z r D x A ([ t, dt − Z r D x A ([0 , dt = Z r ( r − a ) D xx A ([ a, da. In preparation for (5.8) we define a function w ( z , z ) = Z min (cid:8) √ r / − z ,z (cid:9) max (cid:8) − √ r / − z ,z − r (cid:9) r + s − z L ( B (0 , r )) ds on the set ([0 , r ] × { } ) + B (0 , r ). Becausemin { p r / − y , z } − max {− p r / − y , z − r } ≤ r ≤ | r + s − z | ≤ Cr we have a geometric constant C such that(5.7) 0 ≤ w ( z ) ≤ C. PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 29
By the definition of A (see (2.3)), the ACL condition and straight forward Fubini theorem(5.6) is equal to(5.8) − Z B (0 , r ) (cid:2) f ([ r,
0] + z ) − f ([0 ,
0] + z ) − rD x f ([0 ,
0] + z ) (cid:3) dz == − Z B (0 , r ) Z r ( r − a ) D xx f ([ a,
0] + z ) da dz = Z r − r Z q r − z − q r − z Z z + rz D xx f ( t, z ) r + z − t L ( B (0 , r/ dt dz dz = Z r − r Z r + q r − z − q r − z D xx f ( t, y ) Z min { √ r / − z ,t } max {− √ r / − z ,t − r } r + s − t L ( B (0 , r/ ds dt dz = Z ([0 ,r ] ×{ } )+ B (0 , r ) w ( z ) D xx f ( z ) dz. Further(5.9) Z ([0 ,r ] ×{ } )+ B (0 , r ) w ( z ) dz = Z r ( r − a ) da = r , can be easily deduced by considering the special case D xx f ≡
1. Since D xx A is constantwe can use equality of (5.6) and (5.8) together with (5.9), (5.7), and (5.4) to obtain(5.10) (cid:12)(cid:12)(cid:12) D xx A ([0 , − D xx f ([0 , (cid:12)(cid:12)(cid:12) = 1 R r ( r − a ) da (cid:12)(cid:12)(cid:12)Z r ( r − a ) (cid:0) D xx A ([ a, − D xx f ([0 , (cid:1) da (cid:12)(cid:12)(cid:12) = 2 r (cid:12)(cid:12)(cid:12)Z ([0 ,r ] ×{ } )+ B (0 , r ) w ( z ) (cid:0) D xx f ( z ) − D xx f ([0 , (cid:1) dz (cid:12)(cid:12)(cid:12) ≤ Cr Z ([0 ,r ] ×{ } )+ B (0 , r ) (cid:12)(cid:12) D xx f ( z ) − D xx f ([0 , (cid:12)(cid:12) dz< Cε. By similar reasoning on side [0 , , r ] with the help of D y A ([0 , r ]) we obtain that(5.11) (cid:12)(cid:12) D yy A ([0 , − D yy f ([0 , (cid:12)(cid:12) < Cε. It remains to consider D xy . We use A ([ r, A ([0 , r ]), ( − D x + D y ) A ([ r, h ( t ) = f ([ r,
0] + t [ − , h ′ ( t ) = − D x f ([ r − t, t ]) + D y f ([ r − t, t ]) and h ′′ ( t ) = D xx f ([ r − t, t ]) − D yx f ([ r − t, t ]) − D xy f ([ r − t, t ]) + D yy f ([ r − t, t ]) . Now D xy f = D yx f as distributional derivatives are always interchangeable. An analogyof the inequality (5.10) above together with the fact that we already know (5.10) and(5.11) for D xx and D yy implies that (cid:12)(cid:12) D xy A ([0 , − D xy f ([0 , (cid:12)(cid:12) < Cε. The proof for | D A (0 , − D f (0 , | < Cε on T is similar. Therefore (5.5) i ) has beenproved. Part ( ii ) : We know that T has vertices [0 , r,
0] and [0 , r ]. We assume that T hasvertices [0 , r,
0] and [ r, − r ] as other cases can be treated similarly. Our A on T isdefined by (2.3) and A on T is defined using (average) values at vertices and derivativesalong sides D x A ([0 , − Z B ([0 , , r ) D x f, − D y A ([ r, − Z B ([ r, , r ) − D y f and ( − D x + D y ) A ([ r, − r ]) = − Z B ([ r, − r ] , r ) ( − D x + D y ) f. In this way we have D x A ([0 , D x A ([0 , x ∈ [0 , r ] we have with the help of D x A ([0 , D x A ([0 , D A i is constantand (5.5) ( i ) (which was proved in part (i))(5.12) (cid:12)(cid:12) D x ( A − A )([ x, (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) D x ( A − A )([0 , Z x D xx ( A − A )([ a, da (cid:12)(cid:12)(cid:12) ≤ r (cid:16) | D xx A ([0 , − D xx f ([0 , | + | D xx A ([0 , − D xx f ([0 , | (cid:17) ≤ Crε.
It remains to show that D y ( A − A ) along T ∩ T is small. By the definition of A i D y A ([ r, − Z B ([ r, , r ) D y f and ( − D x + D y ) A ([ r, − Z B ([ r, , r ) ( − D x + D y ) f. and hence (cid:12)(cid:12) D y ( A − A )([ r, (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) D x A ([ r, − − Z B ([ r, , r ) D x f (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) D x A ([ r, − D x A ([0 , − rD xx A ([0 , (cid:12)(cid:12)(cid:12) ++ r (cid:12)(cid:12)(cid:12) D xx f ([0 , − D xx A ([0 , (cid:12)(cid:12)(cid:12) ++ (cid:12)(cid:12)(cid:12) − Z B ([ r, , r ) D x f − − Z B ([0 , , r ) D x f − rD xx f ([0 , (cid:12)(cid:12)(cid:12) . The first expression on the righthand is zero by the fundamental theorem of calculus for D x as D xx A is constant and the second one is bounded by Crε by ( i ). It remains toestimate the last term using ACL condition, fundamental theorem of calculus and (5.4) (cid:12)(cid:12)(cid:12) − Z B ([ r, , r ) D x f − − Z B ([0 , , r ) D x f − rD xx f ([0 , (cid:12)(cid:12)(cid:12) == (cid:12)(cid:12)(cid:12) − Z B ([0 , , r ) Z r (cid:2) D xx f ([ t,
0] + z ) − D xx f ([0 , (cid:3) dt dz (cid:12)(cid:12)(cid:12) ≤ Cr − Z Q ([0 , , r ) (cid:12)(cid:12) D xx f ( z ) − D xx f ([0 , (cid:12)(cid:12) dz ≤ Crε.
It follows that | D ( A − A )([ r, | ≤ Crε . PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 31
Similarly to (5.12) we obtain for a ∈ [0 , r ](5.13) (cid:12)(cid:12) D y ( A − A )([0 , a ]) − D y ( A − A )([0 , (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)Z a D yy ( A − A )([0 , t ]) dt (cid:12)(cid:12)(cid:12) ≤ Crε and with the help of | D ( A − A )([ r, | ≤ Crε also for t ∈ [0 , r ](5.14) (cid:12)(cid:12) ( − D x + D y )( A − A )([ t, r − t ]) (cid:12)(cid:12) ≤ (cid:12)(cid:12) ( − D x + D y )( A − A )([ r, (cid:12)(cid:12) + Crε ≤ Crε.
The integral of the derivative along the closed curve is zero and thus(5.15)0 = Z r D x ( A − A )([ a, da + Z r ( − D x + D y )( A − A )([ r − a, a ]) da + Z r ( − D y )([0 , r − a ]) da. It follows using (5.13), (5.12) and (5.14) that r (cid:12)(cid:12) D y ( A − A )([0 , (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)Z r D y ( A − A )([0 , a ]) da (cid:12)(cid:12)(cid:12) + Cr ε ≤ (cid:12)(cid:12)(cid:12)Z r D x ( A − A )([ a, da (cid:12)(cid:12)(cid:12) ++ (cid:12)(cid:12)(cid:12)Z r ( − D x + D y )( A − A )([ r − a, a ]) da (cid:12)(cid:12)(cid:12) + Cr ε ≤ Cr ε. We have just shown that | D ( A − A )([0 , | ≤ Crε . Similar reasoning can estimate thederivative of A − A at other points [ x, ∈ T ∩ T , we just use a triangle with vertices[ x, r,
0] and [ x, r − x ] in an analogy of (5.15). Part ( iii ) : We know by the definition of T and T that A = A on T ∩ T (seesubsection 2.3) and hence A is continuous. It remains to show that det A > δ and that A is 1 − T ∪ T .The definition (2.3) of A in fact means that to determine the coefficients of quadraticfunction A we solve the equation M a = c , where c determines the averaged values of f and Df along the sides (in vertices), a is the vector of coefficients of A and(5.16) M = r r r r − − r r
00 0 − − r and M − = − r r − r r − r r − r − r r − r r − r − r . If fact we solve this for a = [ a , a , a , a , a , a ] where c is determined by the first coor-dinate function of f and we solve it for a = [ b , b , b , b , b , b ] where c is determined bythe second coordinate function of f .We know that (5.2) and (5.3) hold for f and thus we can divide it into linear part L ( z ) = f (0 ,
0) + Df (0 , z plus E := f − L and | f − L | ≤ ε | z | on Q ([0 , , r ). Thus wecan divide the right-hand side c into two terms c L + c E , c L corresponding to the linearpart L of f and c E corresponding to the remaining ( f − L )-term. Our equation is linear and the unique solution to the linear part L is the same linear function (with determinant > δ ). Let us estimate the derivative of E = f − L . From | f − L | ≤ εr on Q ([0 , , r ) (see(5.2)), definition of A (2.3) and L ( B ( v i , r )) ≥ Cr we see that | ( c E ) | ≤ Cεr, | ( c E ) | ≤ Cεr and | ( c E ) | ≤ Cεr.
Similarly we obtain from (5.3) and (2.3) that | ( c E ) | ≤ Cε, | ( c E ) | ≤ Cε and | ( c E ) | ≤ Cε.
Given the form of M − (5.16) it is now easy to see that the solution a E := M − c E satisfies | ( a E ) i | ≤ Cε for i = 1 , , | ( a E ) i | ≤ C εr for i = 4 , , . Now for every z ∈ Q ([0 , , r ) we have(5.17) | DE ( z ) | ≤ C ( a + a + a r + a r + a r ) ≤ Cε.
Thus we have a quadratic function A = L + E , where (see (5.1)) det DL > δ , | DL | < δ and | DE | ≤ Cε . Assume that 0 < ε < C δ . Now det D ( L + E ) contains det DL plusother terms whose sum is smaller than C | DE | ( | DL | + | DE | ) ≤ Cε (cid:0) δ + ε (cid:1) ≤ CC δ. Now it is easy to see that we can choose an absolute constant C so thatdet DA ( z ) = det( L + E )( z ) > δ z ∈ Q ([0 , , r ) . Now we prove that A is 1 − T . Let us denote by λ , λ the eigenvalues of thematrix Df (0 , λ λ > δ and max {| λ | , | λ |} < δ and hence min {| λ || , | λ |} > δ . It follows that the linear function L ( z ) = f (0 ,
0) + Df (0 , z satisfies(5.18) | L ( z ) − L ( w ) | ≥ δ | z − w | for every z, w ∈ Q ([0 , , r ) . From A = L + E , | DE | ≤ Cε and ε < C δ we obtain for z, w ∈ Q ([0 , , r ) | A ( z ) − A ( w ) | ≥ | L ( z ) − L ( w ) | − | E ( z ) − E ( w ) | ≥ δ | z − w | − Cε | z − w | ≥ δ | z − w | once C is chosen sufficiently small. It follows that A is 1 − T and similarly we canshow that A is 1 − T .It remains to show that we cannot have A ( z ) = A ( w ) for z ∈ T and w ∈ T . Wefind v ∈ T ∩ T on the line segment between z and w . We know that A = L + E and A = L + E with | DE | ≤ Cε and | DE | ≤ Cε . Analogously as above we use A ( v ) = A ( v ) to obtain(5.19) | A ( z ) − A ( w ) | ≥ | L ( z ) − L ( w ) | − | E ( z ) − E ( v ) | − | E ( v ) − E ( w ) |≥ δ | z − w | − Cε | z − w | − Cε | z − w |≥ δ | z − w | once C is chosen sufficiently small. Hence A is 1 − T ∪ T . PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 33
Moreover, we can divide Q ([0 , , r ) into 32 triangles, define quadratic functions A oneach of them by (translated and rotated version of) (2.3). Similarly to (5.19) we can evenshow that A is a homeomorphism on the whole Q ([0 , , r ) (once C is sufficiently smallbut fixed absolute constant) since we subtract only bounded number of terms Cε | z − w | in analogy of (5.19). Part ( iv ) : We know that A = L + E where L ( z ) = f (0 , Df (0 , z and | DE | ≤ Cε (see (5.17)). It follows using (5.2) and | DE | ≤ Cε that for z ∈ T (5.20) | f ( z ) − A ( z ) | ≤ | f ( z ) − L ( z ) | + | A ( z ) − L ( z ) |≤ ε | z | + | E ( z ) − E (0 , | + | E (0 , |≤ εr + Cεr + | E (0 , | . Clearly E (0 ,
0) = A (0 , − f (0 ,
0) = [ a , b ] − f (0 ,
0) and the coefficients [ a , b ] are givenby (see (2.3)) [ a , b ] = − Z B ([0 , , r ) f ( z ) dz. Hence we obtain using (5.2)(5.21) | E (0 , | = (cid:12)(cid:12)(cid:12) − Z B ([0 , , r ) (cid:0) f ( z ) − L ( z ) (cid:1) dz (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) − Z B ([0 , , r ) (cid:0) f ( z ) − f (0 , − Df (0 , z (cid:1) dz (cid:12)(cid:12)(cid:12) ≤ εr. Our conclusion for C := 2 + C follows from (5.20) and (5.21). (cid:3) Proof of Theorem 1.2.
Let us recall that we have a W , homeomorphism so that J f > η > η := { z ∈ Ω : dist( z, ∂ Ω) > η } satisfies L (Ω \ Ω η ) < ν . Since J f > δ > δ := n z ∈ Ω : J f ( z ) > δ, k Df ( z ) k < δ o satisfies L (Ω \ Ω δ ) < ν . We know that f is differentiable a.e. and that a.e. point is a Lebesgue point for both Df and D f . It follows that for a.e. z ∈ Ω we havelim w → z | f ( w ) − f ( z ) − Df ( z )( w − z ) || w − z | = 0 , lim r → − Z Q ( z,r ) | Df ( w ) − Df ( z ) | dw = 0 and lim r → − Z Q ( z,r ) | D f ( w ) − D f ( z ) | dw = 0 . We fix 0 < ε < min { C δ , η, δ , δ C } , where C and C are constants from Theorem 5.1( iii ) and ( iv ). From previous limits we know that for a.e. z there is r z > < r ≤ r z we have(5.22) (cid:12)(cid:12) f ( w ) − f ( z ) − Df ( z )( w − z ) (cid:12)(cid:12) < ε | z − w | for w ∈ Q ( z, r ) , (5.23) − Z Q ( z, r ) | Df ( w ) − Df ( z ) | dw < ε and − Z Q ( z, r ) | D f ( w ) − D f ( z ) | dw < ε. Now we fix 0 < r < η small enough so that the good set G := { z ∈ Ω δ : r z > r } satisfies L (Ω \ G ) < ν . Now we would like to cover Ω η by squares of sidelength 2 r so that most corners ofthose squares belong to G . That is for z ∈ Q (0 , r ) we consider Q z := n Q ( z + 2 kr , r ) : k ∈ Z , Q ( z + kr , r ) ∩ Ω η = ∅ o . Since L (Ω \ G ) < ν we can find and fix z so that the number of good vertices (with z + 2 kr ∈ G ) is bigger than the average and we have that(5.24) Q := n Q ( z + 2 kr , r ) ∈ Q z : z + 2 kr ∈ G o satisfies L (cid:16) [ Q ∈Q z \Q Q (cid:17) < ν . Now we choose a Whitney type covering of Ω \ S Q ∈Q z Q and our set of squares { Q i } ∞ i =1 for the statement consists ofsquares in Q z \ Q together with all cubes covering Ω \ [ Q ∈Q z Q. It is clear that these squares are locally finite and (5.24) and L (Ω \ Ω η ) < ν imply that L (cid:16) ∞ [ i =1 Q i (cid:17) < ν. It remains to define an approximation of f on S Q ∈Q Q . We first divide each such Q intotwo triangles T Q , ˜ T Q by joining the lower-right corner with upper-left corner. We denote T := [ Q ∈Q { T Q , ˜ T Q } . As a first step we use Theorem 5.1 for each T ∈ T to obtain a piecewise quadraticapproximation A T there. The assumption (5.1) is verified by the definition of Ω δ and G above (recall that corners of Q ∈ Q belong to G ) and assumptions (5.2), (5.3) and (5.4)are verified by (5.22) and (5.23). We define A ( z ) = A T ( z ) for z ∈ T and T ∈ T . We know that A is a homeomorphism on each T , T ∈ T , by Theorem 5.1 ( iii ) and more-over it is a homeomorphism on each Q ( z T , r ) ∩ S Q ∈Q Q , where z T is the correspondingvertex of T ∈ T , as we have discussed at the end of proof of Theorem 5.1 ( iii ).We claim that it is a homeomorphism on the whole S Q ∈Q Q . Assume for contrary that A ( z ) = A ( w ) for some z, w ∈ S Q ∈Q Q , z = w . We find z a vertex of some triangle T ∈ T so that z ∈ G , z ∈ T and (5.22) holds for z . Since A is a homeomorphism on B ( z , r ) ∩ S Q ∈Q Q we obtain that w / ∈ B ( z , r ). From Theorem 5.1 ( iv ) and A ( z ) = A ( w )we obtain(5.25) | f ( z ) − f ( w ) | ≤ | f ( z ) − A ( z ) | + | A ( w ) − f ( w ) | ≤ C rε. PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 35
For every v ∈ ∂B ( z , r ) we obtain from analogy of (5.18), (5.22) for z and ε < δ | f ( v ) − f ( z ) | ≥| Df ( z )( v − z ) | − | f ( v ) − f ( z ) − Df ( z )( v − z ) |−− | f ( z ) − f ( z ) − Df ( z )( z − z ) |≥ δ | v − z | − ε ( | v − z | + | z − z | ) ≥ δ r − ε r > δ r. Since f is a homeomorphism and w / ∈ B ( z , r ) we obtain now that | f ( z ) − f ( w ) | ≥ inf {| f ( z ) − f ( v ) | : v ∈ ∂B ( z , r ) } ≥ δ r. This is a contradiction with (5.25) by our choice of ε < δ C .We know that Ω has bounded measure and triangles in T have sidelength r and thus T ≤ Cr . Now clearly A ∈ W BV and singular part of second derivative D s A is supportedon S T ∈T ∂T and corresponds to jump of the derivative there. We estimate it with thehelp of Theorem 5.1 ( ii ) as(5.26) Z S T ∈T T | D s A | ≤ T C max T ,T ∈T | D s A | ( T ∩ T ) ≤ T Crε H ( ∂T ) ≤ Cr CrεCr = Cε.
Moreover, the absolutely continuous part D a A satisfies by Theorem 5.1 ( i ) and (5.23)(call v T the corresponding vertex of T ) Z S T ∈T T | D a A − D f | ≤ X T ∈T Z T (cid:0) | D A − D f ( v T ) | + | D f − D f ( v T ) | (cid:1) ≤ X T ∈T Cε L ( T ) ≤ Cε L (Ω) . Finally we use Theorem 1.1 for our mapping A to obtain a C ∞ diffeomorphism g on S T ∈T T such that k f − g k L ∞ < ν and using (5.26) Z S T ∈T T | D f − D g | ≤ Z S T ∈T T (cid:0) | D f − D a A | + | D g − D a A | (cid:1) ≤ Cε L (Ω) + ε + Cε ≤ Cν.
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PPROXIMATION OF PLANAR SOBOLEV W , HOMEOMORPHISMS 37
Faculty of Economics, University of South Bohemia, Studentsk´a 13, Cesk´e Budejovice,Czech Republic
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