Deformations of Bi-conformal Energy and a new Characterization of Quasiconformality
DDEFORMATIONS OF BI-CONFORMAL ENERGY ANDA NEW CHARACTERIZATION OFQUASICONFORMALITY
TADEUSZ IWANIEC, JANI ONNINEN, AND ZHENG ZHU
Abstract.
The concept of hyperelastic deformations of bi-conformalenergy is developed as an extension of quasiconformality. These arehomeomorphisms h : X onto −−→ Y between domains X , Y ⊂ R n of theSobolev class W ,n loc ( X , Y ) whose inverse f def == h − : Y onto −−→ X also be-longs to W ,n loc ( Y , X ) . Thus the paper opens new topics in GeometricFunction Theory (GFT) with connections to mathematical models ofNonlinear Elasticity (NE). In seeking differences and similarities withquasiconformal mappings we examine closely the modulus of continuityof deformations of bi-conformal energy. This leads us to a new charac-terization of quasiconformality. Specifically, it is observed that quasi-conformal mappings behave locally at every point like radial stretchings.Without going into detail, if a quasiconformal map h admits a function φ as its optimal modulus of continuity at a point x ◦ , then f = h − admits the inverse function ψ = φ − as its modulus of continuity at y ◦ = h ( x ◦ ) . That is to say; a poor (possibly harmful) continuity of h at a given point x ◦ is always compensated by a better continuity of f at y ◦ , and vice versa. Such a gain/loss property, seemingly overlookedby many authors, is actually characteristic of quasiconformal mappings.It turns out that the elastic deformations of bi-conformal energy arevery different in this respect. Unexpectedly, such a map may have thesame optimal modulus of continuity as its inverse deformation. In linewith Hooke’s Law, when trying to restore the original shape of the body(by the inverse transformation) the modulus of continuity may neitherbe improved nor become worse. However, examples to confirm thisphenomenon are far from being obvious; indeed, elaborate computationsare on the way. We eventually hope that our examples will gain aninterest in the materials science, particularly in mathematical models ofhyperelasticity.2010 Mathematics Subject Classification.
Primary 30C65; Secondary 46E35, 58C07.
Key words and phrases.
Quasiconformality, bi-conformal energy, mapping of integrabledistortion, modulus of continuity.T. Iwaniec was supported by the NSF grant DMS-1802107. J. Onninen was supportedby the NSF grant DMS-1700274. This research was done while Z. Zhu was visiting Math-ematics Department at Syracuse University. He wishes to thank SU for the hospitality. a r X i v : . [ m a t h . C A ] A p r T. IWANIEC, J. ONNINEN, AND Z. ZHU Introduction
We study Sobolev homeomorphisms h : X onto −−→ Y between domains X , Y ⊂ R n , together with their inverse mappings denoted by f def == h − : Y onto −−→ X .We impose two standing conditions on these mappings: • The conformal energy of h (stored in X ) is finite; that is,(1.1) E X [ h ] def == (cid:90) X | Dh ( x ) | n d x < ∞• The conformal energy of f (stored in Y ) is also finite;(1.2) E Y [ f ] def == (cid:90) Y | Df ( y ) | n d y < ∞ Hereafter, | A | stands for the Hilbert-Schmidt norm of a linear map A , de-fined by the rule | A | = Tr ( A t A ). It should be noted that the above energyintegrals are invariant under conformal change of variables in their domainsof definition ( X and Y , respectively). This motivates us calling such home-omorphisms Deformations of Bi-conformal Energy
Clearly, such deformations include quasiconformal mappings. A Sobolevhomeomorphisms h : X onto −−→ Y is said to be a quasiconformal mapping ifthere exists a constant K such that(1.3) | Dh ( x ) | n (cid:54) K J ( x, h ) , J ( x, h ) = det Dh ( x ) . The conformal energy integral (1.1), an n -dimensional alternative to theclassical Dirichlet integral, has drawn the attention of researchers in themultidimensional GFT [7, 19, 20, 27, 46, 47, 51]. In Geometric Analysis theSobolev space W ,n ( X , R n ) plays a special role for several reasons. First,this space is on the edge of the continuity properties of Sobolev’s mappings.Second, just the fact that h is a homeomorphism allows us to establishuniform bounds of its modulus of continuity. Precisely, given a compactsubset X (cid:98) X , there exists a constant C ( X , X ) so that for all distinctpoints x , x ∈ X , we have:(1.4) | h ( x ) − h ( x ) | (cid:54) C ( X , X ) n (cid:112) E X [ h ]log n (cid:16) diam X | x − x | (cid:17) For a historical account and more details concerning this estimate we referthe reader to Section 7.4, Section 7.5 and Corollary 7.5.1 in the monograph[27].For the same reasons, to every compact Y (cid:98) Y there corresponds aconstant C ( Y , Y ) such that for all distinct points y , y ∈ Y , we have: I-CONFORMAL ENERGY AND QUASICONFORMALITY 3 (1.5) | f ( y ) − f ( y ) | (cid:54) C ( Y , Y ) n (cid:112) E Y [ f ]log n (cid:16) diam Y | y − y | (cid:17) In other words, h and f admit the same function ω = ω ( t ) ≈ log − n (1 + 1 /t )as a modulus of continuity. Shortly, h and f are ω -continuous. There isstill a slight improvement to these estimates; namely,(1.6) lim | x − x |→ | h ( x ) − h ( x ) | log n (cid:18) X | x − x | (cid:19) = 0The question whether the modulus of continuity ω = ω ( t ) ≈ log − n (1 + 1 /t )is the best and universal for all bi-conformal energy mappings remains un-clear. We shall not enter this issue here. The optimal modulus of continuity of h : X onto −−→ Y at a given point x ◦ ∈ X is defined by(1.7) ω h ( x ◦ ; t ) def == max | x − x ◦ | = t | h ( x ) − h ( x ◦ ) | for 0 (cid:54) t < dist( x ◦ , ∂ X ) . Nevertheless, it is easy to see, via examples of radial stretchings, that inthe class of functions that are powers of logarithms the exponent α = n issharp; meaning that for α > n it is not generally true that(1.8) | h ( x ) − h ( x ) | (cid:52) log − α (cid:18) X | x − x | (cid:19) To this end, we take a quick look at the radial homeomorphism h : B n onto −−→ B n of the unit ball B n ⊂ R n onto itself,(1.9) h ( x ) = x | x | (cid:0) − log | x | (cid:1) n (cid:2) log( e − log | x | ) (cid:3) β , where β > n It is often seen that the inverse map f def == h − : Y → X admits better mod-ulus of continuity than h , or vice versa. Just for h defined in (1.9), itsinverse is even C ∞ -smooth. Such a gain/loss rule about the moduli of con-tinuity for a map and its inverse is typical of the radial stretching/squeezing.It turns out that the gain/loss rule gives a new characterization for a widelystudied class of quasiconformal mappings. Theorem 1.1.
Let h : X onto −−→ Y be a homeomorphism between domains X , Y ⊂ R n and let f : Y onto −−→ X denote its inverse. Then h is quasiconformalif and only if for every pair ( x ◦ , y ◦ ) ∈ X × Y , y ◦ = h ( x ◦ ) , the optimal modulusof continuity functions ω h = ω h ( x ◦ ; t ) and ω f = ω f ( y ◦ ; s ) are quasi-inverseto each other; that is, there is a constant K (cid:62) (independent of ( x ◦ , y ◦ ) )such that K − s (cid:54) ( ω h ◦ ω f )( s ) (cid:54) K s Hereafter the notation A (cid:52) B stands for the inequality A (cid:54) c B in which c > T. IWANIEC, J. ONNINEN, AND Z. ZHU for sufficiently small s > . See Section 3 for fuller discussion. It should be noted that for a radialstretching/squeezing homeomorphism h ( x ) = H ( | x | ) x | x | , H (0) = 0, we al-ways have ( ω h ◦ ω f )( s ) ≡ s Thus it amounts to saying that
Quasiconformal mappings are characterized by being comparatively radialstrectching/squeezing at every point.
At the first glance, the gain/loss rule seems to generalize to deformationsof bi-conformal energy. Here we refute this view, by constructing examplesin which both h and f admit the same modulus of continuity. Theseexamples work well regardless of whether or not the modulus of continuity(given upfront) is close to the borderline case ω = ω ( t ) ≈ log − n (1 + 1 /t ) .Without additional preliminaries, we now can illustrate this instance with arepresentative case of Theorem 14.1. Theorem 1.2 (A Representative Example) . Consider a modulus of conti-nuity function φ : [0 , ∞ ) onto −−→ [0 , ∞ ) defined by the rule (1.10) φ ( s ) = s = 0 (cid:2) log (cid:0) es (cid:1)(cid:3) − n (cid:2) log log (cid:0) e e s (cid:1) (cid:3) − if 0 < s (cid:54) s if s (cid:62) Then there exists a deformation of bi-conformal energy H : R n onto −−→ R n suchthat • H (0) = 0 , H ( x ) ≡ x , for | x | (cid:62) • | H ( x ) − H ( x ) | (cid:52) φ ( | x − x | ) , for all x , x ∈ R n Its inverse F def == H − : R n onto −−→ R n also admits φ as a modulus of continu-ity, • | F ( y ) − F ( y ) | (cid:52) φ ( | y − y | ) , for all y , y ∈ R n Furthermore, φ represents the optimal modulus of continuity at the originfor both H and F ; that is, for every (cid:54) s < ∞ we have (1.11) ω H (0 , s ) = φ ( s ) = ω F (0 , s ) . Remark . More specifically, letting ψ : [0 , ∞ ) onto −−→ [0 , ∞ ) denote theinverse of φ , the maxima in (1.11) are attained on the vertical axes, wherewe have(1.12) H (0 , ..., , x n ) = (cid:26) (0 , ..., , φ ( x n ) ) if x n (cid:62) , ..., , ψ ( x n ) ) if x n (cid:54) F (0 , ..., , y n ) = (cid:26) (0 , ..., , ψ ( y n ) ) if y n (cid:62) , ..., , φ ( y n ) ) if y n (cid:54) In the above estimates the implied constants depend only on n . I-CONFORMAL ENERGY AND QUASICONFORMALITY 5
It is worth noting here that in our representative examples the inverse func-tion ψ : [0 , ∞ ) onto −−→ [0 , ∞ ) will be even C ∞ -smooth near 0 .There are many more reasons for studying deformations of bi-conformalenergy. First, a homeomorphism h : X → Y in W ,n ( X , Y ) whose inverse f def == h − : Y → X also lies in W ,n ( Y , X ) include ones with integrableinner distortion , see (1.15). From this point of view our study not onlyexpands the theory of quasiconformal mappings but also mappings of finitedistortion. The latter can be traced back to the early paper by Goldsteinand Vodop’yanov [17] (1976) who established continuity of such mappings.However, a systematic study of mappings of finite distortion has begun in1993 with planar mappings of integrable distortion [34] (Stoilow factoriza-tion), see also the monographs [3, 27, 20]. The optimal modulus of continuityfor mappings of finite distortion and their inverse deformations have beenstudied in numerous publications [9, 11, 21, 22, 24, 38, 44, 45]. In all of theseresults, except in [44], the sharp modulus of continuity is obtained amongthe class of radially symmetric mapping.In a different direction, the essence of elasticity is reversibility. All ma-terials have limits of the admissible distortions. Exceeding such a limit onebreaks the internal structure of the material (permanent damage). Here wetake on stage the materials of bi-conformal stored-energy (1.14) E XY [ h, f ] def == E X [ h ] + E Y [ f ] = (cid:90) X | Dh ( x ) | n d x + (cid:90) Y | Df ( y ) | n d y The bi-conformal energy reduces to an integral functional defined solely overthe domain X by the rule:(1.15) E XY [ h, f ] = E X [ h ] def == (cid:90) X (cid:110) (cid:12)(cid:12) Dh ( x ) (cid:12)(cid:12) n + (cid:12)(cid:12) D (cid:93) h ( x ) (cid:12)(cid:12) n [ J h ( x ) ] n − (cid:111) d x where the ratio term represents the inner distortion of h . For more detailswe refer the reader to [4]. Examples abound in which one can return thedeformed body to its original shape with conformal energy, but not neces-sarily via the inverse mapping f = h − : Y onto −−→ X , because f need noteven belong to W ,n ( Y , R n ) . This typically occurs when the boundary ofthe deformed configuration (like a ball with a straight line slit cut) differstopologically from the boundary of the reference configuration (like a ballwithout a cut) [29, 30, 31]. We believe that the geometric/topological ob-structions for reversibility of elastic deformations might be of interest inmathematical models of nonlinear elasticity (NE) [1, 5, 10, 41]. In our set-ting, by virtue of the Hooke’s Law, it is naturally to study deformations ofbi-conformal energy. One of the important problems in nonlinear elasticityis whether or not a radially symmetric solution of a rotationally invariantminimization problem is indeed the absolute minimizer. In the case of bi-conformal energy this is proven to be the case in low dimension models( n = 2 ,
3) [32]. The radial symmetric solutions, however, may fail to beabsolute minimizers if n (cid:62) T. IWANIEC, J. ONNINEN, AND Z. ZHU of NE and GFT, are devoted to understand the expected radial symmetricproperties [2, 6, 12, 18, 23, 25, 26, 28, 33, 35, 36, 39, 42, 43, 48, 49, 50].2.
Quick review of the modulus of continuity
Let us recall the concept of modulus of continuity , also known as modulusof oscillation ; the concept introduced by H. Lebesgue [40] in 1909.We are dealing with continuous mappings h : X → Y between subsets X ⊂ X and Y ⊂ Y of normed spaces ( X , | · | ) and ( Y , || · || ) .A modulus of continuity is any continuous function ω : [0 , ∞ ) → [0 , ∞ )that is strictly increasing and ω (0) = 0 . Definition 2.1.
A continuous mapping h : X → Y is said to admit ω asits (local) modulus of continuity at the point x ◦ ∈ X if(2.1) || h ( x ) − h ( x ◦ ) || (cid:52) ω ( | x − x ◦ | ) , for all x ∈ X Here the implied constant may depend on x ◦ , but not on x . In short, h is ω -continuous at the point x ◦ . If this inequality holds for all x, x ◦ ∈ X with an implied constant independent of x and x ◦ then h is said to admit ω as its (global) modulus of continuity in X . Definition 2.2 (Optimal Modulus of Continuity) . Every uniformly contin-uous function h : X → Y admits the optimal modulus of continuity at agiven point x ◦ ∈ X , given by the rule:(2.2) ω h ( x ◦ ; t ) def == sup { || h ( x ) − h ( x ◦ ) || : x ∈ X , | x − x ◦ | (cid:54) t } No implied constant is involved in this definition. Similarly, the function(2.3) Ω h ( t ) def == sup { || h ( x ) − h ( x ◦ ) || : x, x ◦ ∈ X , | x − x ◦ | (cid:54) t } is referred to as (globally) optimal modulus of continuity of h in X . Definition 2.3 (Bi-modulus of Continuity) . The term bi-modulus of conti-nuity of a homeomorphism h : X onto −−→ Y refers to a pair ( φ, ψ ) of continu-ously increasing functions φ : [0 , ∞ ) onto −−→ [0 , ∞ ) and ψ : [0 , ∞ ) onto −−→ [0 , ∞ )in which φ is a modulus of continuity of h and ψ is a modulus of conti-nuity of the inverse map f def == h − : Y onto −−→ X . Such a pair is said to be theoptimal bi-modulus of continuity at the point ( x ◦ , y ◦ ) ∈ X × Y , y ◦ = h ( x ◦ ) ,if φ ( t ) = ω h ( x ◦ ; t ) and ψ ( s ) = ω f ( y ◦ ; s )3. Quasiconformal Mappings
Let us take a quick look at the radial stretching/squeezing homeomor-phism h : R n onto −−→ R n defined by:(3.1) h ( x ) = H ( | x | ) x | x | , for x ∈ R n I-CONFORMAL ENERGY AND QUASICONFORMALITY 7 where the function H : [0 , ∞ ) onto −−→ [0 , ∞ ) (interpreted as radial stress func-tion) is continuous and strictly increasing. Its inverse f def == h − : R n onto −−→ R n becomes a squeezing/stretching homeomorphism of the form:(3.2) f ( y ) = F ( | y | ) y | y | , for y ∈ R n where F : [0 , ∞ ) onto −−→ [0 , ∞ ) stands for the inverse function of H . Thesetwo radial stress functions are exactly the optimal moduli of continuity at0 ∈ R n of h and f , respectively. By the definition, ω h ( t ) def == ω h (0 , t ) = max | x | = t | h ( x ) | = H ( t ) ω f ( s ) def == ω f (0 , s ) = max | y | = s | f ( y ) | = F ( s ) . Therefore(3.3) ω f ( ω h ( t )) ≡ t for all t (cid:62) , and ω h ( ω f ( s )) ≡ s for all s (cid:62) . The above identities admit of a simple interpretation:
The better is the optimal modulus of continuity of h , the worse isthe optimal modulus of continuity of its inverse map f, and vice versa. Look at the power type stretching h ( x ) = | x | N x | x | and f ( y ) = | y | N y | y | . To an extent, this interpretation pertains to all quasiconformal homeomor-phisms. There are three main equivalent definitions for quasiconformal map-pings: metric, geometric, and analytic. The analytic definition (1.3) wasfirst considered by Lavrentiev in connection with elliptic systems of partialdifferential equations. Here we will relay on the metric definition , whichsays that “infinitesimal balls are transformed to infinitesimal ellipsoids ofbounded eccentricity.” The interested reader is referred to [3, Chapter 3.]to find more about the foundations of quasicoformal mappings.
Definition 3.1.
Let X and Y be domains in R n , n (cid:62) h : X onto −−→ Y a homeomorphism. For every point x ◦ ∈ X we define.(3.4) H h ( x ◦ , r ) def == max | x − x ◦ | = r | h ( x ) − h ( x ◦ ) | min | x − x ◦ | = r | h ( x ) − h ( x ◦ ) | whenever 0 < r < dist( x ◦ , ∂ X ) . Also define(3.5) 1 (cid:54) H h ( x ◦ ) def == lim sup r → H h ( x ◦ , r ) (cid:54) ∞ and call it the linear dilatation of h at x ◦ . If, furthermore,(3.6) K h def == sup x ◦ ∈ X H h ( x ◦ ) < ∞ T. IWANIEC, J. ONNINEN, AND Z. ZHU then we call K h the maximal linear dilatation of h in X and h a quasi-conformal mapping. Finally, h is K -quasiconformal, 1 (cid:54) K < ∞ if(3.7) ess-sup x ◦ ∈ X H h ( x ◦ ) (cid:54) K It should be noted that the inverse map f def == h − : Y onto −−→ X is also K -quasiconformal.Next, we invoke the optimal modulus of continuity at a point x ◦ ∈ X : ω h ( t ) def == ω h ( x ◦ ; t ) = max | x − x ◦ | = t | h ( x ) − h ( x ◦ ) | , for 0 (cid:54) t < t ◦ def == dist( x ◦ ; ∂ X ) . This defines a continuous strictly increasing function ω h : [0 , t ◦ ) onto −−→ [0 , s ◦ ) ,where s ◦ def == dist( y ◦ ; ∂ Y ) . Similar definitions apply to the inverse map f : Y onto −−→ X which is also K -quasiconformal. Its optimal modulus of continuityat the image point y ◦ = h ( x ◦ ) is given by ω f ( s ) def == ω f ( y ◦ ; s ) = max | y − y ◦ | = s | f ( y ) − f ( y ◦ ) | , for 0 (cid:54) s < s ◦ Therefore, both compositions ω f ( ω h ( t )) and ω h ( ω f ( s )) are well defined for0 (cid:54) t < t ◦ and 0 (cid:54) s < s ◦ , respectively. Unlike the radial stretchings,the function ω f ( s ) is generally not the inverse of ω h ( t ) , but very close toit. Namely, the optimal modulus of continuity of h and that of f are quasi-inverse to each other. Let us make this statement more precise by thefollowing theorem. Theorem 3.2 (Local quasi-inversion) . Let a map h : X onto −−→ Y be K -quasiconformal and f : Y onto −−→ X denote its inverse. Then there is a constant K = K ( n, K ) (cid:62) such that for every point x ◦ ∈ X and its image y ◦ = h ( x ◦ ) ∈ Y it holds (3.8) K − s (cid:54) ω h ( ω f ( s )) (cid:54) K s and K − t (cid:54) ω f ( ω h ( t )) (cid:54) K t whenever (cid:54) t (cid:54) t ( x ◦ ) and (cid:54) s (cid:54) s ( y ◦ ) . Here the upper bounds positivenumbers t ( x ◦ ) and s ( y ◦ ) , depend only on dist( x ◦ ; ∂ X ) and dist( y ◦ ; ∂ Y ) ,respectively. Before proceeding to the proof, we recall a very useful Extension Theoremby F. W. Gehring [14], see also the book by J. V¨ais¨al¨a [52] (Theorem 41.6).This theorem allows us to reduce a local quasiconformal problem to ananalogous problem for mappings defined in the entire space R n . Lemma 3.3 (F. W. Gehring) . Every quasiconformal map h : B ( x ◦ , r ) into −→ R n defined in a ball B ( x ◦ , r ) ⊂ R n admits a quasiconformal mapping h (cid:48) : R n onto −−→ R n which equals h on B ( x ◦ , r ) . The dilatation of h (cid:48) depends onlythat of h and the dimension n . Accordingly, we may (and do) assume that X = Y = R n . This will giveus a more precise information about the constant K = K ( n, K ) . I-CONFORMAL ENERGY AND QUASICONFORMALITY 9
Theorem 3.4 (Global quasi-inversion) . Let a map h : R n onto −−→ R n be K -quasiconformal and f : R n onto −−→ R n denote its inverse. Then there is aconstant K = K ( n, K ) (cid:62) such that for every point x ◦ ∈ R n and itsimage y ◦ = h ( x ◦ ) it holds (3.9) K − s (cid:54) ω h ( ω f ( s )) (cid:54) K s and K − t (cid:54) ω f ( ω h ( t )) (cid:54) K t for all s (cid:62) and t (cid:62) . Rather than using the original definition we will appeal to Gehring’s char-acterization of quasiconformal mappings, see Inequality (3.3) in [16] andsome related articles [13, 15, 51, 52, 37, 53]. The interested reader is re-ferred to a book by P. Caraman [8] on various definitions and extensiveearly literature on the subject.
Proposition 3.5 (Three points condition) . To every λ (cid:62) there corre-sponds a constant (cid:54) K λ = K λ ( n, K ) such that:Whenever three distinct points x ◦ , x , x ∈ R n satisfy the ratio condition (3.10) | x − x ◦ || x − x ◦ | (cid:54) λ, the image points under h : R n onto −−→ R n satisfy analogous condition (3.11) | h ( x ) − h ( x ◦ ) || h ( x ) − h ( x ◦ ) | (cid:54) K λ = K λ ( n, K )In particular, Proposition 3.6.
Let h : R n onto −−→ R n be K -quasiconformal. Then forevery point x ◦ ∈ X and < r < ∞ we have (3.12) H h ( x ◦ , r ) def == max | x − x ◦ | = r | h ( x ) − h ( x ◦ ) | min | x − x ◦ | = r | h ( x ) − h ( x ◦ ) | (cid:54) K = K ( n, K ) Proof. (of Theorem 3.4) It is clearly sufficient to make the computationwhen x ◦ = 0 and y ◦ = 0 . In this case the condition (3.12) takes the form(3.13) 1 K | h ( x ) | (cid:54) | h ( x ) | (cid:54) K | h ( x ) | , whenever | x | = | x | (cid:54) = 0By the definition of the optimal modulus of continuity at the origin, wehave: • ω h ( ω f ( s )) = | h ( x ) | for some x ∈ R n with | x | = ω f ( s ) • ω f ( s ) = | f ( y ) | for some y ∈ R n with | y | = s • Therefore, ω h ( ω f ( s )) = | h ( x ) | , for some | x | = | f ( y ) | Now, the right hand side of inequality at (3.13) gives the desired upperbound ω h ( ω f ( s )) = | h ( x ) | (cid:54) K | h ( f ( y )) | = K | y | = K s , whereas the lefthand side gives the lower bound ω h ( ω f ( s )) = | h ( x ) | (cid:62) K − | h ( f ( y )) | = K − | y | = K − s . The analogous bounds for ω f ( ω h ( t )) at (3.8) follow by interchanging the roles of h and f ; as they are both K -quasiconformal.This completes the proof of Theorem 3.4. (cid:3) The converse statement to Theorem 3.2 reads as:
Theorem 3.7.
Consider a homeomorphism h : X onto −−→ Y , its inverse map-ping f : Y onto −−→ X , and their optimal moduli of continuity at a point x ◦ ∈ X and y ◦ = h ( x ◦ ) , respectively: ω h ( t ) def == max | x − x ◦ | = t | h ( x ) − h ( x ◦ ) | and ω f ( s ) def == max | y − y ◦ | = s | f ( y ) − f ( y ◦ ) | for (cid:54) t < dist( x ◦ , ∂ X ) and (cid:54) s < dist( y ◦ , ∂ Y ) . Assume the followingone-sided quasi-inverse condition at every point x ◦ ∈ X , with a constant K (cid:62) . (3.14) ω h ( ω f ( r )) (cid:54) K r for all sufficiently small r > x ◦ ) Then h is K -quasiconformal. Here is a simple geometric proof.
Proof.
We shall actually show that Condition (3.14) at the given point x ◦ ∈ X implies(3.15) H h ( x ◦ , t ) = max | x − x ◦ | = t | h ( x ) − h ( x ◦ ) | min | x − x ◦ | = t | h ( x ) − h ( x ◦ ) | (cid:54) K , for t > x ◦ ∈ X it holds that:(3.16) lim sup t → H h ( x ◦ , t ) (cid:54) K , as required.A sufficient upper bound of t at (3.15) depends on dist( x ◦ , ∂ X ) , but weshall not enter into this issue. It simplifies the writing, and causes no lossof generality, to assume that x ◦ = y ◦ = 0 . Thus we are reduced to showingthat(3.17) max | x | = t | h ( x ) | (cid:54) K min | x | = t | h ( x ) | , for all sufficiently small t > . To this end, consider the ball B ( x ◦ , t ) ⊂ X centered at x ◦ = 0 and withsmall radius t > h , denoted by Ω = h ( B ( x ◦ , t )) ⊂ Y ,contains the origin y ◦ = 0 . Let r > B r ⊂ Ω , centered at y ◦ = 0 . Thusmin | x | = t | h ( x ) | = r Similarly, denote by R the smallest radius of a ball B R ⊃ Ω centered at y ◦ = 0 , see Figure 1. Thus R = max | x | = t | h ( x ) | def == ω h ( t ) I-CONFORMAL ENERGY AND QUASICONFORMALITY 11
Figure 1.
The ratio Rr (cid:54) K Now the inverse map f : Y onto −−→ X takes Ω onto B ( x ◦ , t ) . In particular,it takes the common point of ∂ B r and ∂ Ω into a point of ∂ B ( x ◦ , t ) . Thismeans that t = max | y | = r | f ( y ) | def == ω f ( r )The proof is completed by invoking the quasi-inverse condition at (3.14), R = ω h ( t ) = ω h ( ω f ( r )) (cid:54) K r (cid:3) Doubling Property.
It is worth discussing another special property ofquasiconformal mappings in relation to their bi-modulus of continuity. Tosimplify matters we confine ourselves to quasiconformal mappings definedon the entire space, h : R n onto −−→ R n and its inverse f : R n onto −−→ R n . Itturns out that at every point x ◦ ∈ R n the optimal modulus of continuity φ ( t ) def == ω h ( x ◦ ; t ) , as well as its inverse function φ − : [0 , ∞ ) onto −−→ [0 , ∞ )have a doubling property. Observe that φ − is not exactly the optimalmodulus of continuity of the inverse map f = h − , the latter is only quasi-inverse to φ − . It should be emphasized at this point that doubling propertyof the modulus of continuity is rather rare, see our representative examplesin Section 6. Proposition 3.8.
Consider all K -quasiconformal mappings h : R n onto −−→ R n . To every λ (cid:62) there corresponds a constant K λ (actually the onespecified in (3.11) ), and there is a constant C λ = C λ ( n, K ) (independentof h ) such that at every point x ◦ ∈ R n we have (3.18) ω h ( x ◦ ; λ t ) (cid:54) K λ ω h ( x ◦ ; t ) and (3.19) ω − h ( x ◦ ; λ s ) (cid:54) C λ ω − h ( x ◦ ; s ) for all (cid:54) t < ∞ and (cid:54) s < ∞ .Proof. We may again assume that x ◦ = 0 and h ( x ◦ ) = 0 . This simplifiesthe notation ω h ( x ◦ ; t ) def == ω h ( t ) . The proof of the first inequality is imme-diate from the three points ratio condition in Proposition 3.5 , which givesus exactly the constant K λ from this condition. Indeed, we have • ω h ( λ t ) = | h ( x ) | , for some x ∈ R n with | x | = λ t • ω h ( t ) = | h ( x ) | , for some x ∈ R n with | x | = t • Hence, | x || x | (cid:54) λ . • Consequently | h ( x ) || h ( x ) | (cid:54) K λ , which is the desired estimate. (cid:3) Clearly, for every y ◦ ∈ R n we also have(3.20) ω f ( y ◦ ; λ s ) (cid:54) K λ ω f ( y ◦ ; s ) for all 0 (cid:54) s < ∞ , simply by interchanging the roles of h and f .We precede the proof of the doubling condition for ω − h , with a quicklemma.3.0.2. A quick lemma on doubling condition.
Consider an arbitrary con-tinuously increasing function φ : [0 , ∞ ) onto −−→ [0 , ∞ ) (in our application, φ ( t ) = ω h ( t ) ). It is commonly said that φ satisfies doubling condition ifthere is a constant C φ (cid:62) φ (2 t ) (cid:54) C φ φ ( t ) for all t (cid:62) generalized doubling condition ,which reads as:(3.21) φ ( λ t ) (cid:54) C φ ( λ ) φ ( t ) , for all t (cid:62) λ - constant C φ ( λ ) (cid:62) φ (2 t ) (cid:54) C φ φ ( t ) .Associated with φ is its quasi-inverse function . This term pertains to anycontinuous and strictly increasing function ψ : [0 , ∞ ) onto −−→ [0 , ∞ ) such that(3.22) m t (cid:54) ψ ( φ ( t )) (cid:54) M t , for all t (cid:62) < m (cid:54) (cid:54) M < ∞ are constants. In general, ψ does not satisfydoubling condition, but its inverse ψ − : [0 , ∞ ) onto −−→ [0 , ∞ ) does. Lemma 3.9.
To every factor λ (cid:62) there corresponds a generalized doublingconstant for ψ − . For all t (cid:62) we have (3.23) ψ − ( λ t ) (cid:54) C ψ − ( λ ) ψ − ( t ) . Explicitly C ψ − ( λ ) def == C φ ( M λ/m ) . Proof.
Choose and fix λ (cid:62) ψ − ( m t ) (cid:54) φ ( t ) (cid:54) ψ − ( M t ) , for all t (cid:62) . I-CONFORMAL ENERGY AND QUASICONFORMALITY 13
Upon substitution t (cid:32) λtm in the left hand side, we obtain ψ − ( λ t ) (cid:54) φ ( λ tm ) = φ ( Mλm · tM ) (cid:54) C φ ( Mλm ) · φ ( tM )The proof of the lemma is completed by invoking the right hand side ofinequality (3.24) which, upon substitution t (cid:32) tM , gives us the desiredestimate φ ( tM ) (cid:54) ψ − ( t ) . (cid:3) We summarize this section with the following theorem, which is an ex-panded version of Theorem 1.1:
Theorem 3.10.
Let h : R n onto −−→ R n be a K -quasiconformal mapping and f : R n onto −−→ R n its inverse. Choose and fix an arbitrary point x ◦ ∈ R n an itsimage point y ◦ = h ( x ◦ ) . Denote by φ ( t ) = ω h ( x ◦ ; t ) the optimal modulusof continuity of h at x ◦ and by ψ ( s ) = ω f ( y ◦ ; s ) the optimal modulus ofcontinuity of f at y ◦ . Then the following statements hold true. (Q1) The functions φ and ψ are quasi-inverse to each other. Precisely,there is a constant K = K ( n, K ) such that (3.25) K − t (cid:54) ψ ( φ ( t )) (cid:54) K t and K − s (cid:54) φ ( ψ ( s )) (cid:54) K s for all t, s ∈ [0 , ∞ ) . (Q2) Both φ and ψ satisfy the general doubling condition; that is, forevery λ (cid:62) there is a constant K λ such that (3.26) φ ( λ t ) (cid:54) K λ φ ( t ) and ψ ( λ s ) (cid:54) K λ ψ ( s ) for all t, s ∈ [0 , ∞ ) . (Q3) As a consequence of Conditions ( Q and ( Q , the inverse func-tions ψ − and φ − also satisfy a general doubling conditions; namely, (3.27) φ − ( λ s ) (cid:54) C λ φ − ( s ) and ψ − ( λ t ) (cid:54) C λ ψ − ( s ) for all t, s ∈ [0 , ∞ ) , where the constant C λ = K λ ( λ K ) . Let us now proceed to more general mappings of bi-conformal energy.4.
A handy metric in R n (cid:39) R n − × R It will be convenient to consider the space R n as Cartesian product R n − × R , with the purpose of using cylindrical coordinates. Accordingly, R n = R n − × R = (cid:8) X = ( x, t ); x = ( x , ..., x n − ) ∈ R n − and t ∈ R (cid:9) Hereafter, we change the notation of the variables; the lowercase letter x designates a point ( x , ...x n − ) ∈ R n − while the uppercase letter X =( x, t ) is reserved for points in R n . The Euclidean norm of x ∈ R n − isdenoted by | x | def == (cid:113) x + · · · + x n − . The space R n − × R is furnishedwith the norm || X || def == | x | + | t | , for X = ( x, t ) = ( x , ..., x n − , t ) ∈ R n − × R In this metric the closed unit ball in R n − × R becomes the Euclidean doublecone C = { ( x, t ) ∈ R n ; | x | + | t | (cid:54) } = C + ∪ C − where we split C into the upper and lower cones: C + = { ( x, t ); | x | + t (cid:54) , t (cid:62) } , C − = { ( x, t ); | x | − t (cid:54) , t (cid:54) } The idea of the construction of H : C onto −−→ C Our construction of a bi-conformal energy map H : C onto −−→ C , whose op-timal modulus of continuity at the origin coincides with that of the inversemap, will be carried out in two steps. First we construct a homeomor-phism H : C + onto −−→ C + of finite conformal-energy which equals the identityon ∂ C + . Its inverse map F def == H − : C + onto −−→ C + will also have finiteconformal-energy. The substance of the matter is that their optimal mod-uli of continuity ( ω H and ω F , respectively) are inverse to each other; thusgenerally not equal. In fact ω H will be stronger that ω F . In the secondstep we adopt the modulus of continuity of F : C + onto −−→ C + to an extensionof H to C − , simply by reflecting F twice about R n − . Let the reflec-tion r : R n onto −−→ R n be defined by r ( x, t ) = ( x, − t ) . This gives rise to amap r ◦ F ◦ r : C − onto −−→ C − , which we glue to H : C + onto −−→ C + along thecommon base ∂ C + ∩ ∂ C − ⊂ R n − . Precisely, the desired homeomorphism H : C onto −−→ C , still denoted by H , will be defined by the rule(5.1) H def == (cid:26) H : C + onto −−→ C + r ◦ F ◦ r : C − onto −−→ C − Its inverse, also denoted by F : C onto −−→ C , is defined analogously byinterchanging the roles of F and H .(5.2) F def == (cid:26) F : C + onto −−→ C + r ◦ H ◦ r : C − onto −−→ C − As a result, the optimal modulus of continuity of H will be attained in theupper cone C + , whereas the optimal modulus of continuity of F will beattained in the lower cone C − . Clearly, they are the same for the doublecone C = C + ∪ C − , and this is the essence of our construction.Explicit formula for H can easily be stated, see Definition 7.1 in Section 7.Since H : C onto −−→ C and its inverse F def == H − : C onto −−→ C are both equal tothe identity on ∂ C we can extend them to R n as the identity outside C .Whenever it is convenient, we shall speak of H : R n onto −−→ R n and its inverse F : R n onto −−→ R n as homeomorphisms of the entire space R n onto itself. I-CONFORMAL ENERGY AND QUASICONFORMALITY 15
Figure 2.
A mapping H : C onto −−→ C and its inverse F : C onto −−→ C , will have the same optimal modulus of continuityat the center of C .6. Preconditions on the modulus of continuityand the representative examples
Let us introduce a fairly general class of moduli of continuity to be con-sidered. These classes are intended to unify the proofs. It will also give usan aesthetic appearance of the inequalities. On that account, our moduli ofcontinuity, will be made of functions φ : [0 , onto −−→ [0 ,
1] in C [0 , ∩ C (0 , C ) φ (0) = 0 , φ (1) = 1 ( can be extended by φ ( s ) = s for s (cid:62) C )(6.1) φ (cid:48) ( s ) (cid:54) φ ( s ) s (cid:54) M [ φ (cid:48) ( s )] , for some contant 1 (cid:54) M < ∞ ( C ) Finite Energy Condition :(6.2) E [ φ ] def == (cid:90) | φ ( s ) | n d ss < ∞ . As a consequence of Conditions ( C ) and ( C ) we have: • (6.3) λ ( s ) def == φ ( s ) s (cid:62) φ (cid:48) ( s ) (cid:62) M for all 0 < s (cid:54) φ ( s ) ≡ s )we have even stronger property; namely, lim s → φ (cid:48) ( s ) = ∞ . • The function λ ( s ) is non-increasing. This follows from(6.4) λ (cid:48) ( s ) = φ (cid:48) ( s ) s − φ ( s ) s (cid:54) . • Thus in fact,(6.5) λ ( s ) = φ ( s ) s (cid:62) φ (1)1 = 1 , for all 0 < s (cid:54) Representative examples. ( E ) For 0 < ε (cid:54) φ ( s ) = s ε . In the borderline case, φ ( s ) = s ( E ) For n < α (cid:54) φ ( s ) = log − α (cid:16) es (cid:17) = (cid:18) a log 1 s (cid:19) − α ( E ) For n < α (cid:54) φ ( s ) = (cid:18) a log 1 s (cid:19) − n (cid:16) a log log es (cid:17) − α ( E ) For n < α (cid:54) φ ( s ) = (cid:18) a log 1 s (cid:19) − n (cid:16) a log log es (cid:17) − n (cid:18) a log log log e e s (cid:19) − α Continuing in this fashion, we define a sequence of functions φ k , k =0 , , , ... in which the last product-term in the round parantheses involves k -times iterated logarithm and ( k −
1) -times iterated power of e . All theabove functions can be extended by setting φ k ( s ) ≡ s, for s (cid:62) Remark . The coefficients a k in the above formulas are adjusted to ensurethe inequality φ (cid:48) ( s ) (cid:54) φ ( s ) s , which is required by Condition ( C ) . Thisworks well with a k def == (1 − n ) k − . Indeed, the reader may wish to verifythat the expression sφ (cid:48) k ( s ) φ k ( s ) is increasing, thus assumes its maximum valueat s = 1 . It is then readily seen that its maximum value is not exceeding n ( a + a + ... + a k − ) + α a k = 1 − (cid:0) − α (cid:1) (cid:0) − n (cid:1) k − (cid:54) The definition of H : C + onto −−→ C + First we set H on the vertical axis of the upper cone by the rule. H ( , t ) = ( , φ ( t )) . Here and below ( , t ) def == (0 , ..., , t ) ∈ R n − × R . We wish H to be theidentity map on the base of the cone, which consists of points ( x, ∈ R n − × R with | x | (cid:54) x,
0) with the point( , | x | ) by a straight line segment and map it linearly onto the straightlinesegment with endpoints at ( x,
0) and ( , φ ( | x | ) ) . Explicitly, I-CONFORMAL ENERGY AND QUASICONFORMALITY 17
Definition 7.1.
The map H : C + onto −−→ C + ⊂ R n − × R is given by theformula(7.1) H ( x, t ) def == ( x, t λ ( t + | x | ) ) , for 0 (cid:54) t (cid:54) | x | + t (cid:54) λ ( s ) def == φ ( s ) s for 0 < s (cid:54) . Indeed, for α, β (cid:62) α + β = 1 , we have H (cid:2) α ( x,
0) + β ( , | x | ) (cid:3) = α ( x,
0) + β ( , φ ( | x | ) , which means that H is a linear transformationbetween the above-mentioned segments. A formula for the inverse map F : C + onto −−→ C + is not that explicit. Figure 3.
Diagonals of rectangles built on the curve t = φ ( s ) . The map F is linear on each such diagonal, as well ason their rotations.8. The Jacobian matrix of H and its inverse A straightforward computation of the Jacobian matrix of H , at the point X def == ( x, t ) = ( x , ..., x n − , t ) ∈ R n − × R shows that(8.1) DH ( x, t ) = . . . . . . . . . D D D . . . D n − D where D i = (cid:2) t λ (cid:48) ( t + | x | ) (cid:3) x i | x | and D = λ ( t + | x | ) + t λ (cid:48) ( t + | x | ) is theJacobian determinant, later also denoted by J H ( X ) . Then the inversematrix ( DH ) − takes the form (8.2) ( DH ) − = 1 D D . . . D . . . . . . D − D − D − D . . . − D n − Square of the Hilbert Schmidt norm of a matrix is the sum of squares of itsentries. Accordingly,(8.3) | DH | = n − (cid:2) t λ (cid:48) ( t + | x | ) (cid:3) + (cid:2) λ ( t + | x | ) + t λ (cid:48) ( t + | x | ) (cid:3) and | ( DH ) − | = 1 (cid:2) λ ( t + | x | ) + t λ (cid:48) ( t + | x | ) (cid:3) + (cid:2) t λ (cid:48) ( t + | x | ) (cid:3) (cid:2) λ ( t + | x | ) + t λ (cid:48) ( t + | x | ) (cid:3) + n − The Jacobian determinant D = J H ( X )We have the following bounds of the Jacobian determinant, including auniform lower bound for all X = ( x, t ) ∈ C + .(9.1) λ ( || X || ) (cid:62) J H ( X ) (cid:62) φ (cid:48) ( || X || ) (cid:62) M Proof.
Using the notation || X || = s = t + | x | (cid:54) D = dd t (cid:16) t λ ( t + | x | ) (cid:17) = dd t (cid:16) t φ ( t + | x | ) t + | x | (cid:17) = φ ( s ) s + t (cid:18) φ (cid:48) ( s ) s − φ ( s ) s (cid:19) = φ ( s ) s (cid:18) − ts (cid:19) + t φ (cid:48) ( s ) s (9.2)Now, since φ (cid:48) ( s ) (cid:54) φ ( s ) s = λ ( s ) , it follows that D (cid:54) λ ( s ) . On the otherhand φ (cid:48) ( s ) (cid:62) M and φ ( s ) s (cid:62) (cid:62) M , whence D (cid:62) M . (cid:3) Conformal-energy of H : C + onto −−→ C + In the forthcoming computation the ”implied constants” depend only onthe dimension n (cid:62) Lemma 10.1.
We have (10.1) (cid:90) C + | DH ( x, t ) | n d x d t (cid:52) E [ φ ] I-CONFORMAL ENERGY AND QUASICONFORMALITY 19
Proof.
Formula (8.3) yields the inequality: (cid:90) C + | DH ( x, t ) | n d x d t (cid:52) (cid:90) t + | x | (cid:54) | λ ( t + | x | ) | n d x d t + (cid:90) t + | x | (cid:54) t n | λ (cid:48) ( t + | x | ) | n d x d t (10.2)Obviously, for the constant term we have 1 (cid:52) E [ φ ] . For the first integralin the right hand side we make the substitution t = s − | x | and use Fubini’sformula to obtain (cid:90) t + | x | | λ ( t + | x | ) | n d x d t = (cid:90) | x | (cid:54) s (cid:54) | λ ( s ) | n d x d s == (cid:90) | λ ( s ) | n (cid:32)(cid:90) | x | (cid:54) s d x (cid:33) d s = ω n − n − (cid:90) s n − | λ ( s ) | n d s == ω n − n − (cid:90) | φ ( s ) | n d ss (cid:52) E [ φ ]For the second integral in (10.2), we make the same substitution t = s − | x | and proceed as follows (cid:90) t + | x | (cid:54) t n | λ (cid:48) ( t + | x | ) | n d x d t = (cid:90) | x | (cid:54) s (cid:54) ( s − | x | ) n | λ (cid:48) ( s ) | n d x d s = (cid:90) | λ (cid:48) ( s ) | n (cid:32)(cid:90) | x | (cid:54) s ( s − | x | ) n d x (cid:33) d s = c n (cid:90) s n − | λ (cid:48) ( s ) | n d s (cid:52) (cid:90) | φ ( s ) | n d ss = E [ φ ]Here, we used the inequality | λ (cid:48) ( s ) | = φ ( s ) s − φ (cid:48) ( s ) s (cid:54) φ ( s ) s .The proof is complete. (cid:3) Conformal-energy of the inverse map
This brings us back to the seminal work [4] on extremal mappings offinite distortion. Going into this in detail would take us too far afield, so weconfine ourselves to a simplified variant.Consider a homeomorphism H : X onto −−→ Y between bounded domains ofSobolev class W ,n loc ( X , Y ) and assume (just to make it easier) that the Ja-cobian J H def == det [DH] is positive almost everywhere, as in (9.1). Definition 11.1.
The differential expression(11.1) K H ( X ) def == (cid:12)(cid:12) [ DH ( X )] − (cid:12)(cid:12) n J H ( X ) = (cid:12)(cid:12) D (cid:93) H ( X ) (cid:12)(cid:12) n [ J H ( X ) ] n − , is called the inner distortion function of H . Here the symbol D (cid:93) H standsfor the cofactor matrix of DH , defined by Cramer’s rule.The following identity was first observed with a complete proof of it in[4] , see Theorem 9.1 therein. Proposition 11.2.
Under the assumptions above, if K H ∈ L ( X ) thenthe inverse map F : Y onto −−→ X belongs to W ,n ( Y , X ) and (11.2) (cid:90) Y (cid:12)(cid:12) DF ( Y ) (cid:12)(cid:12) n d Y = (cid:90) X K H ( X ) d X .
In our case, since H is locally Lipschitz on C + , the derivation of thisidentity is straightforward. Simply, the differential matrix DF ( Y ) at thepoint Y = H ( X ) equals [ DH ( X )] − . We may change variables Y = H ( X )in the energy integral for F , to obtain (cid:90) C + (cid:12)(cid:12) DF ( Y ) (cid:12)(cid:12) n d Y = (cid:90) C + (cid:12)(cid:12) [ DH ( X )] − (cid:12)(cid:12) n J H ( X ) d X def == (cid:90) C + K H ( X ) d X Now, by (8.1) and (8.2), we have a point wise inequality J H ( X ) (cid:12)(cid:12) [ DH ( X )] − (cid:12)(cid:12) (cid:54) √ n − (cid:12)(cid:12) DH ( X ) (cid:12)(cid:12) , which yields:(11.3) K H (cid:54) ( n − n | DH | n ( J H ) n − (cid:54) ( n − n M n − | DH | n ∈ L ( C + )because J H ( X ) (cid:62) M , by (9.1).12. Modulus of continuity of H : C + onto −−→ C + We start with the straightforward estimates of the modulus of continuityat ( , ∈ R n − × R . In consequence of λ ( || X || ) (cid:62) || X || = | x | + t (cid:54) | x | + t λ ( t + | x | ) (cid:54) | x | λ ( || X || ) + t λ ( || X || ) = φ ( || X || )Here the middle term | x | + tλ ( t + | x | ) = || H ( X ) || . Therefore,(12.1) || X || (cid:54) || H ( X ) || (cid:54) φ ( || X || ) Corollary 12.1.
The function φ is the optimal modulus of continuity ofthe map H : C + onto −−→ C + at ( , ∈ R n − × R + ; that is, (12.2) sup || X || = s || H ( X ) || = φ ( s ) , whenever X ∈ C + and 0 (cid:54) s (cid:54) X = ( , s ) on the verticalaxis of the cone C + , because H ( , s ) = ( , φ ( s ) ) . Remark . It is perhaps worth remarking in advance that both inequal-ities at (12.1) remain valid in terms of the Euclidean norm of R n as well, I-CONFORMAL ENERGY AND QUASICONFORMALITY 21 where | X | = | ( x, t ) | = (cid:112) | x | + t (cid:54) || X || . To this end, since λ is decreas-ing to its minimum value λ (1) = 1 , for X ∈ C + we can write | X | (cid:54) | x | + t λ ( || X || ) = | H ( X ) | (cid:54) | x | λ ( | X | ) + t λ ( | X | ) = φ ( | X | ) , Let us record this fact as:(12.3) | X | (cid:54) | H ( X ) | (cid:54) φ ( | X | )For the inverse map F = F ( Y ) , these inequalities take the form(12.4) ψ ( | Y | ) (cid:54) | F ( Y ) | (cid:54) | Y | (cid:54) φ ( | Y | ) for all Y ∈ C + because s (cid:54) φ ( s )where ψ ; [0 , onto −−→ [0 ,
1] denotes the inverse function of φ . This, however,does not necessarily imply that F is Lipschitz continuous, as shown by ourrepresentative examples.We shall now prove that H admits φ as global modulus of continuity;that is, everywhere in C + . Precisely, we have Proposition 12.3.
For X = ( x, t ) ∈ C + and X (cid:48) = ( x (cid:48) , t (cid:48) ) ∈ C + it holdsthat (12.5) || H ( X ) − H ( X (cid:48) ) || (cid:54) φ ( || X − X (cid:48) || ) Thus, according to (2.3) , Ω H ( t ) (cid:54) φ ( t ) . . Proof.
Recall that || X || def == | x | + | t | and H ( x, t ) def == ( x, tλ ( || X || ) ) . Thus(12.6) || H ( X ) − H ( X (cid:48) ) || (cid:54) | x − x (cid:48) | + | tλ ( || X || ) − t (cid:48) λ ( || X (cid:48) || ) | The first term is easily estimated as | x − x (cid:48) | (cid:54) φ ( | x − x (cid:48) | ) (cid:54) φ ( || X − X (cid:48) || ) ,because s (cid:54) φ ( s ) and φ is increasing in s ∈ [0 ,
1] . The second term needsmore work. First observe that for 0 < A (cid:54) B (cid:54) < λ ( A ) − λ ( B ) (cid:54) A − φ ( B − A )Indeed, λ ( A ) − λ ( B ) = n φ ( A ) − φ ( B ) A + B − AA λ ( B ) (cid:54) B − AA λ ( B − A ) = A − φ ( B − A )In the above formula, the first term is negative because φ is increasing.In the second term we have used the inequality λ ( B ) (cid:54) λ ( B − A ) , because λ is nonincreasing.In Inequality (12.5) we may (and do) assume that || X (cid:48) || (cid:54) || X || , forotherwise we can interchange X with X (cid:48) . This yields || X − X (cid:48) || (cid:54) || X || and, consequently, λ ( || X || ) (cid:54) λ ( || X − X (cid:48) || ) = φ ( || X − X (cid:48) || ) / || X − X (cid:48) || (cid:54) φ ( || X − X (cid:48) || ) / || X − X (cid:48) || . Having this and (12.7) at hand, weconclude with the desired estimate | tλ ( || X || ) − t (cid:48) λ ( || X (cid:48) || ) | (cid:54) | t − t (cid:48) | λ ( || X || ) + t (cid:48) | λ ( || X || ) − λ ( || X (cid:48) || ) | (cid:54) | t − t (cid:48) ||| X − X (cid:48) || φ ( || X − X (cid:48) || )+ t (cid:48) || X (cid:48) || φ ( || X || − || X (cid:48) || ) (cid:54) φ ( || X − X (cid:48) || ) + φ ( || X − X (cid:48) || )= 3 φ ( || X − X (cid:48) || ) (cid:3) Modulus of continuity of F : C + onto −−→ C + All representative functions φ = φ k , k = 0 , , . . . that are listed in( E ) . . . ( E k ) . . . are concave ( φ (cid:48)(cid:48) k (cid:54) ,
1] . Actually, upon minor modifications awayfrom the origin all the above functions can be made concave in the entireinterval [0 ,
1] . But their aesthetic appearance will be lost. Thus, ratherthan modifying those examples, in the first step we restrict our attentionto a neighborhood of the origin. Outside such a neighborhood the mapping F : C + onto −−→ C + is Lipschitz continuous. This will take care of the globalestimate.The additional condition imposed on φ reads as follows:( C ) There is an interval (0 , r ] ⊂ (0 ,
1] in which φ is C -smooth andconcave; that is,(13.1) φ (cid:48)(cid:48) ( s ) (cid:54) , for 0 < s (cid:54) r We shall now prove that F admits φ as global modulus of continuity in C + . Proposition 13.1.
For arbitrary two points Y = ( y, τ ) ∈ C + and Y (cid:48) =( y (cid:48) , τ (cid:48) ) ∈ C + it holds: (13.2) || F ( Y ) − F ( Y (cid:48) ) || (cid:52) φ ( || Y − Y (cid:48) || ) The implied constant depends on the conditions imposed through ( C ) − ( C ) . Proof.
A seemingly routine proof below, actually took an effort to accom-plish all details. Let us begin with the definition of the map H : C + onto −−→ C + and some new related notation. For X = ( x, t ) ∈ C + ⊂ R n − × R , we recallthat || X || = | x | + t and H ( X ) = (cid:0) x, t λ ( t + | x | ) (cid:1) def == (cid:0) y, τ (cid:1) = Y ∈ R n − × R I-CONFORMAL ENERGY AND QUASICONFORMALITY 23
For the inverse map F = H − we write || Y || = | y | + τ and F ( Y ) = (cid:0) y, T (cid:1) ∈ C + ⊂ R n − × R where the vertical coordinate T = T ( τ, | y | ) is determined uniquely from theequation(13.3) T λ ( T + | y | ) = τ In much the same way as in (9.2) we find that the function T (cid:32) T λ ( T + | y | )is strictly increasing. We actually haved T λ ( T + | y | )d T = λ ( T + | y | ) + T λ (cid:48) ( T + | y | ) (cid:62) M Even more can be said about the above expression. Indeed, denoting by s def == T + | y | (cid:54) λ ( s ) + T λ (cid:48) ( s ) = φ ( s ) s + T · (cid:18) φ (cid:48) ( s ) s − φ ( s ) s (cid:19) = (cid:18) − Ts (cid:19) φ ( s ) s + Ts · φ (cid:48) ( s )which, in view of Condition ( C ) at (6.1), also yields a useful upper bound,(13.4) φ ( s ) s (cid:62) λ ( s ) + T λ (cid:48) ( s ) (cid:62) φ (cid:48) ( s ) (cid:62) M The latter follows from the Condition ( C ) at (6.1) as well.Now, implicit differentiation in (13.3) with respect to τ -variable gives(13.5) 0 (cid:54) ∂T ( τ, | y | ) ∂τ = 1 λ ( T + | y | ) + T λ (cid:48) ( T + | y | ) (cid:54) M On the other hand, differentiation with respect to the | y | -variable gives(13.6) 0 (cid:54) ∂T ( τ, | y | ) ∂ | y | = − T λ (cid:48) ( T + | y | ) λ ( T + | y | ) + T λ (cid:48) ( T + | y | ) = − T λ (cid:48) ( s ) λ ( s ) + T λ (cid:48) ( s )It should be noted that λ (cid:48) ( s ) (cid:54) s def == T + | y | (cid:54) (cid:54) − λ (cid:48) ( s ) = − φ (cid:48) ( s ) s + φ ( s ) s (cid:54) φ ( s ) s From this and the lower bound in (13.4) we infer that0 (cid:54) ∂T ( τ, | y | ) ∂ | y | (cid:54) T φ ( s ) s φ (cid:48) ( s ) (cid:54) φ ( s ) s φ (cid:48) ( s ) (cid:54) M φ (cid:48) ( s ) = M φ (cid:48) ( T + | y | ) , the latter being guaranteed by the right hand side of inequality (6.1).It is at this point that we are going to use the additional assumption that φ is concave near the origin; namely, φ (cid:48) is non-increasing in (0 , r ] ⊂ (0 ,
1] .Examine an arbitrary point Y = ( y, τ ) ∈ C + of lengths || Y || def == τ + | y | (cid:54) rM to show that T + | y | (cid:54) r . Recall that T is determined by the equation T λ ( T + | y | ) = τ . Thus, we have TM (cid:54) φ (cid:48) ( T + | y | ) T (cid:54) φ ( T + | y | ) T + | y | T = τ ,by Condition (6.1). Hence T + | y | (cid:54) M τ + | y | (cid:54) M ( τ + | y | ) (cid:54) r . Since s = T + | y | (cid:62) | y | and φ (cid:48) is non-increasing in (0 , r ] , we infer that(13.8) 0 (cid:54) ∂T ( τ, | y | ) ∂ | y | (cid:54) M φ (cid:48) ( | y | ) , whenever τ + | y | def == || Y || (cid:54) rM We are now ready to formulate an estimate of the modulus of continuity of F within the neighborhood of the origin that is determined by || Y || (cid:54) rM . Proposition 13.2.
Let Y = ( y, τ ) ∈ R n − × R and Y (cid:48) = ( y (cid:48) , τ (cid:48) ) ∈ R n − × R be points in C + such that || Y || (cid:54) rM and || Y (cid:48) || (cid:54) rM . Then (13.9) || F ( Y ) − F ( Y (cid:48) ) || (cid:54) M φ ( || Y − Y (cid:48) || ) Proof.
With the notation for F ( Y ) = ( y , T ( τ, | y | )) and F ( Y (cid:48) ) = ( y (cid:48) , T ( τ (cid:48) , | y (cid:48) | ))we begin with the computation, || F ( Y ) − F ( Y (cid:48) ) || = | y − y (cid:48) | + | T ( τ, | y | ) − T ( τ (cid:48) , | y (cid:48) | ) | (cid:54) | y − y (cid:48) | + | T ( τ, | y | ) − T ( τ, | y (cid:48) | ) | + | T ( τ, | y (cid:48) | ) − T ( τ (cid:48) , | y (cid:48) | ) | (cid:54) (in view of (13 . (cid:54) | y − y (cid:48) | + | T ( τ, | y | ) − T ( τ, | y (cid:48) | ) | + M | τ − τ (cid:48) | (cid:54) | T ( τ, | y | ) − T ( τ, | y (cid:48) | ) | + M || Y − Y (cid:48) || (cid:54) | T ( τ, | y | ) − T ( τ, | y (cid:48) | ) | + M φ ( || Y − Y (cid:48) || )The latter is obtained by the inequality s (cid:54) φ ( s ) , see (6.5). It remains toestablish the following estimates, say when 0 < | y (cid:48) | (cid:54) | y | (cid:54) r . (13.10) | T ( τ, | y | ) − T ( τ, | y (cid:48) | ) | (cid:54) M φ ( | y − y (cid:48) | ) (cid:54) M φ ( || Y − Y (cid:48) || )To that end, we begin with the following expression: T ( τ, | y | ) − T ( τ, | y (cid:48) | ) = (cid:90) dd γ (cid:104) T ( τ, | γ y + (1 − γ ) y (cid:48) | ) (cid:105) d γ = (cid:90) T ξ ( τ, | γ y + (1 − γ ) y (cid:48) | ) (cid:28) γ y + (1 − γ ) y (cid:48) | γ y + (1 − γ ) y (cid:48) | (cid:12)(cid:12)(cid:12) y − y (cid:48) (cid:29) d γ where T ξ ( τ, ξ ) def == ∂T ( τ,ξ ) ∂ ξ . In view of (13.8) , we obtain(13.11) | T ( τ, | y | ) − T ( τ, | y (cid:48) | ) | (cid:54) M | y − y (cid:48) | (cid:90) φ (cid:48) ( | γ y + (1 − γ ) y (cid:48) | ) d γ It is important to notice that | γ y + (1 − γ ) y (cid:48) | (cid:54) r , which enables us toinvoke Condition ( C ) at (13.1); that is, φ (cid:48) is non-increasing in the interval(0 , r ] . The following interesting lemma comes into play. Lemma 13.3.
Let
Φ : (0 , r ] → (0 , ∞ ) be continuous non-increasing andintegrable, (cid:90) r Φ( s ) d s < ∞ . I-CONFORMAL ENERGY AND QUASICONFORMALITY 25
Then for every vectors a , b in a normed space ( N ; | . | ) , such that < | a | (cid:54) r and < | b | (cid:54) r , it holds: (13.12) (cid:90) Φ( | γ a + (1 − γ ) b | ) d γ (cid:54) | a | + | b | (cid:32)(cid:90) | a | Φ( s ) d s + (cid:90) | b | Φ( s ) d s (cid:33) Equality occurs if a is a negative multiple of b .Proof. Since φ is non-increasing, by triangle inequality it follows that (cid:90) Φ( | γ a + (1 − γ ) b | ) d γ (cid:54) (cid:90) Φ (cid:0)(cid:12)(cid:12) (1 − γ ) | b | − γ | a | (cid:12)(cid:12)(cid:1) d γ = (cid:90) | b || a | + | b | Φ (cid:0) (1 − γ ) | b | − γ | a | (cid:1) d γ + (cid:90) | b || a | + | b | Φ (cid:0) γ | a | − (1 − γ ) | b | (cid:1) d γ In the first integral we make a substitution s = (1 − γ ) | b | − γ | a | , whichplaces s in the interval (0 , | b | ) and | d s | = ( | a | + | b | ) d λ . This gives us thesecond integral-term of the right hand side of (13.12), and similarly for thefirst integral-term. (cid:3) Since φ (cid:48) is non-increasing in the interval (0 , r ] (by inequality (13.1) atCondition ( C ) ), we may apply Estimate (13.12) to Φ = φ (cid:48) . Now, return-ing to (13.11), the inequality (13.10) is readily inferred as follows: | T ( τ, | y | ) − T ( τ, | y (cid:48) | ) | (cid:54) M | y − y (cid:48) | φ ( | y | ) + φ ( | y (cid:48) | ) | y | + | y (cid:48) | == M φ ( | y − y (cid:48) | ) | y − y (cid:48) | φ ( | y − y (cid:48) | ) φ ( | y | ) + φ ( | y (cid:48) | ) | y | + | y (cid:48) | (cid:54)(cid:54) M φ ( | y − y (cid:48) | ) | y | + | y (cid:48) | φ ( | y | + | y (cid:48) | ) φ ( | y | ) + φ ( | y (cid:48) | ) | y | + | y (cid:48) | (cid:54) M φ ( | y − y (cid:48) | ) , because sφ ( s ) = λ ( s ) is non-decreasing, see (6.4) , and φ ( s ) is increasing.The proof of Proposition 13.2 is complete. (cid:3) Finally, the global estimate (13.2) in Proposition 13.1 follows from Propo-sition 13.2, whenever || Y || (cid:54) rM and || Y (cid:48) || (cid:54) rM . Whereas its extensionto all points Y and Y (cid:48) is fairly straightforward by invoking Lipschitz con-tinuity of F away from the origin. (cid:3) Conclusion
Choose an arbitrary modulus of continuity function φ : [0 , ∞ ) onto −−→ [0 , ∞ )that satisfies conditions ( C ) ( C ) ( C ) and ( C ) . Then consider a bi-conformal energy map H : C + onto −−→ C + defined in (7.1) together with itsinverse map F : C + onto −−→ C + . Extend H and F to the double cone C = C + ∪ C − by the reflection rule at (5.1). Afterwards extend H and F to Figure 4.
Bi-conformal energy mapping H and its inverse F exhibit the same optimal modulus of continuity at theorigin of the double cone C .the entire space R n by setting H = Id : R n \ C onto −−→ R n \ C and F = Id : R n \ C onto −−→ R n \ C . Then we obtain: Theorem 14.1.
For every modulus of continuity function φ : [0 , ∞ ) onto −−→ [0 , ∞ ) satisfying conditions ( C ) ( C ) ( C ) and ( C ) , there exists a home-omorphism H : R n onto −−→ R n of Sobolev class W ,n loc ( R n , R n ) , whose inverse F = H − : R n onto −−→ R n also lies in the Sobolev space W ,n loc ( R n , R n ) . More-over • H (0) = 0 , H ( X ) ≡ X , for || X || (cid:62) X = ( x , ..., x n − , . • H : R n onto −−→ R n admits φ as its global modulus of continuity; thatis, (14.1) || H ( X ) − H ( X ) || (cid:52) φ ( || X − X || ) , for all X , X ∈ R n . • The inverse map F : R n onto −−→ R n satisfies the same condition: (14.2) || F ( Y ) − F ( Y ) || (cid:52) φ ( || Y − Y || ) , for all Y , Y ∈ R n . • H and F share the same optimal moduli of continuity at the origin;namely (14.3) ω H (0 , r ) = max || X || = r | H ( X ) | = φ ( r ) = max || Y || = r | F ( Y ) | = ω F (0 , r ) for all (cid:54) r < ∞ . References [1] S. S. Antman,
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Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA
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