Bilipschitz mappings with derivatives of bounded variation

Publ. Mat. 52 (2008), 91–99 BILIPSCHITZ MAPPINGS WITH DERIVATIVES OF BOUNDED VARIATION Stanislav Hencl Abstract Let Ω ⊂Rnbe open and suppose that f: Ω →Rnis a bilipschitz mapping such that Df ∈BVloc(Ω,Rn2). We show that under these assumptions the inverse satisfies Df−1∈BVloc(f(Ω),Rn2). 1. Introduction Suppose that Ω ⊂Rnis an open set and let f: Ω →f(Ω) ⊂Rnbe a homeomorphism. In this paper we address the issue of the regularity of f−1under regularity assumptions on f. The starting point for us is the following very recent result from [4] (see Preliminaries for the definition of the space BV ). Theorem 1.1. Let Ω,Ω′⊂R2be open and suppose that f: Ω →Ω′is a homeomorphism. Then f∈BVloc(Ω; R2)if and only if f−1 ∈BVloc(Ω′;R2). Moreover, both fand f−1are differentiable almost everywhere. In the same paper we also studied the conditions that guarantee that f−1∈BVloc(Ω,Rn) in higher dimensions. With some additional assumptions (namely that fis a mapping of finite distortion) it is moreover possible to prove that f−1∈W1,1 loc (Ω,Rn) (see [2], [3] and [5]). In this paper we want to address the issue of regularity of the second derivative of f−1. The classical inverse function theorem states that if fis C2and Jf(x0)6= 0 then there is a small neighborhood of x0 where fis homeomorphism and f−1is C2. We will assume that fis a bilipschitz mapping and show that Df−1∈BVloc provided that Df ∈ BVloc. This resembles the result from [6] that the inverse of bilipschitz Delta-convex mapping is again Delta-convex. 2000 Mathematics Subject Classification. 26B30. Key words. Functions of bounded variation, inverse. The author was supported in part by GAˇ CR 201/06/P100 and in part by MSM 0021620839. 92 S. Hencl Theorem 1.2. Let Ω,Ω′⊂Rnbe open and suppose that f: Ω →Ω′ is a bilipschitz mapping such that Df ∈BVloc(Ω; Rn2). Then Df−1∈ BVloc(Ω′;Rn2). It is moreover possible to show that Df−1belongs to the Sobolev space W1,p loc if Df ∈W1,p loc . Theorem 1.3. Let Ω,Ω′⊂Rnbe open, p≥1and suppose that f: Ω → Ω′is a bilipschitz mapping such that Df ∈W1,p loc (Ω; Rn2). Then Df−1∈ W1,p loc (Ω′;Rn2). Let us make a comment on our assumptions. Let α > 1 and consider the function f: (−1,1) →Rdefined as f(x) = |x|αsgn x. Then it is easy to check that fis Lipschitz, homeomorphism, Df ∈W1,1((−1,1)), but Df−1/∈BVloc((−1,1)). Thus the assumption that f−1is Lipschitz cannot be omitted. In Section 4 we give an example which shows that Theorem 1.2 is not valid in dimension n≥4 without the assumption that fis Lipschitz. If n= 1, then Df ∈BV implies that Df is bounded and that fis Lipschitz, and thus this assumption is redundant. We would like to know if a homeomorphism f: Ω →Ω′such that f−1is Lipschitz and Df ∈BVloc(Ω; Rn2) must satisfy Df−1∈BVloc(Ω′;Rn2) in dimensions n= 2 and n= 3. Unfortunately our method of the proof and our counterexample do not provide an answer to this question. 2. Preliminaries By e1,...,enwe denote the canonical basis in Rn. For x∈Rnwe write x1,...,xnfor its coordinates, i.e. x=Pn i=1 xiei. The euclidean distance of x, y ∈Rnis denoted by |x−y|and the norm of the ntimes n matrix Ais denoted by kAk. In the whole paper Ω will denote an open subset of Rn. We say that F: Ω →Rnis a Lipschitz map if there is a constant K > 0 such that |F(x)−F(y)| ≤ K|x−y| for every x, y ∈Ω. Further Fis said to be bilipschitz if it is an invertible mapping and both F: Ω →Rnand F−1:F(Ω) →Rnare Lipschitz. The Lebesgue measure of a set A⊂Rnis denoted by Ln(A). A mapping f: Ω →Rnis said to satisfy the Lusin condition (N) if Ln(f(A))=0 for every A⊂Ω such that Ln(A) = 0. Let Ω ⊂Rnbe open and m∈N. A function h∈L1(Ω) is of bounded variation, h∈BV (Ω),if the distributional partial derivatives of hare Mappings with Derivatives in BV 93 measures with finite total variation in Ω: there are Radon (signed) measures µ1,...,µndefined in Ω so that for i= 1,...,n, |µi|(Ω) <∞and ZΩ hDiϕ dx =−ZΩ ϕ dµi for all ϕ∈C∞ 0(Ω). We say that f∈L1(Ω,Rm) belongs to BV (Ω,Rm) if the coordinate functions of fbelong to BV (Ω). Analogously we define the Sobolev space: f∈W1,1(Ω,Rm) if f∈L1(Ω,Rm) and the distributional derivatives of the coordinate functions are in L1(Ω,Rn). Further, f∈BVloc(Ω,Rm) (or f∈W1,1 loc (Ω,Rm)) requires that f∈BV (Ω′,Rm) (or f∈W1,1(Ω′,Rm)) for each open Ω′⋐Ω. For an introduction to the theory of BV and W1,1spaces see [1], [7]. The function h: Ω →Rmis said to be a representative of g: Ω →Rmif h=galmost everywhere with respect to Lebesgue measure. For function f: (a, b)→Rmwe define Vf, (a, b):= sup (k X i=1 |f(ai)−f(bi)|: (ai, bi) are pairwise disjoint intervals in (a, b)). The function fis said to have finite variation if Vf, (a, b)<∞. It is a well-known fact (see e.g. [1, Section 3.11]) that a mapping u∈ L1 loc(Ω,Rm) is in BVloc(Ω,Rm) (or in W1,1 loc (Ω,Rm)) if and only if there is a representative which has bounded variation (or is an absolutely continuous function) on almost all lines parallel to coordinate axes and the variation on these lines is integrable. More precisely, let i∈ {1,2,...,n} and denote by πithe projection on the hyperplane perpendicular to the xi-axis. Suppose that Q(c, r) := (c1−r, c1+r)×· · ·×(cn−r, cn+r)⊂Ω for some c∈Rn,r > 0 and set Qi(c, r) = πi(Q(c, r)). Let y∈Qi(c, r) and denote ui,y(t) = u(y+tei) for t∈(ci−r, ci+r). Theorem 2.1. Let Ω⊂Rnbe open and let u∈L1 loc(Ω,Rm). (i) Then u∈W1,1 loc (Ω,Rm)if and only if the following happens. For every cube Q(c, r)⋐Ωand for every i∈ {1,...,n}there is a representative ˜uof usuch that the function ˜ui,y(t)is absolutely continuous on (ci−r, ci+r)(i.e. each coordinate function is absolutely 94 S. Hencl continuous) for Ln−1almost every y∈Qi(c, r)and moreover (2.1) ZQi(c,r)Zci+r ci−r |∇˜ui,y(t)|dt dy < ∞. (ii) Then u∈BVloc(Ω,Rm)if and only if the following happens. For every cube Q(c, r)⋐Ωand for every i∈ {1,...,n}there is a representative ˜uof usuch that the function ˜ui,y(t)has bounded variation on (ci−r, ci+r)for Ln−1almost every y∈Qi(c, r)and moreover (2.2) ZQi(c,r) V˜ui,y,(ci−r, ci+r)dy < ∞. We shall also need that the composition of BV function and a homeomorphism with Lipschitz inverse is in BV (see [1, Theorem 3.16 and Corollary 3.19]). Theorem 2.2. Let Ω,Ω′⊂Rnbe open and let u: Ω →Rm. Suppose that F: Ω →Ω′is Lipschitz and homeomorphism. (i) If u∈BVloc(Ω,Rm), then u◦F−1∈BV (Ω′,Rm). (ii) If u∈W1,1 loc (Ω,Rm)and F−1is Lipschitz, then u◦F−1∈W1,1 loc (Ω′ ,Rm) and Du ◦F−1(y) = Du(F−1(y))DF −1(y)for almost every y∈Ω′. 3. Regularity of the inverse Proof of Theorem 1.2: We want to show that Df−1has bounded variation on almost all lines parallel to coordinate axes and therefore Df−1∈ BVloc (see Theorem 2.1 (ii)). Fix Q(c, r)⋐f(Ω) and i∈ {1,...,n}. From Theorem 2.2 we know that Df ◦f−1∈BVloc. Denote by ha good representative of Df ◦f−1from Theorem 2.1 (ii) and set hi,y(t) := h(y+tei) for y∈Qi(c, r). From (2.2) we have (3.1) ZQi(c,r) Vhi,y,(ci−r, ci+r)dy < ∞. Denote A=x∈Q(c, r) : h(x) = Df ◦f−1(x), f−1is differentiable at x and fis differentiable at f−1(x). Lipschitz functions are differentiable almost everywhere and map Lebesgue null sets to Lebesgue null sets and therefore Ln(A) = Ln(Q(c, r)). From the definition of Awe have (3.2) Df−1(x)Df(f−1(x)) = Ifor every x∈A. Mappings with Derivatives in BV 95 Fix y∈Qi(c, r) and let {(aj, bj)}k j=1 be a system of pairwise disjoint subintervals of (ci−r, ci+r) such that Aj:= y+ajei∈Aand Bj:= y+bjei∈Afor every j. Plainly kDf−1(x)k ≤ Kwhere Kdenotes the Lipschitz constant of f−1. Together with (3.2) this imply k X j=1 kDf−1(Aj)−Df−1(Bj)k = k X j=1 kDf−1(Aj)Df(f−1(Bj))−Df(f−1(Aj))Df−1(Bj)k ≤K2 k X j=1 kDf(f−1(Bj)) −Df(f−1(Aj))k ≤CV hi,y,(ci−r, ci+r). (3.3) From (3.1) and Ln(A) = Ln(Q(c, r)) we know that Vhi,y,(ci−r, ci+ r)<∞and L1π−1 i(y)∩A= 2rfor Ln−1almost every y. Fix such ay∈Qi(c, r). From (3.3) and elementary properties of functions of bounded variation we obtain that there is a function ˜ui,y : (ci−r, ci+r)→ Rn2such that Df−1(y+tei) = ˜ui,y(t) for every t∈(ci−r, ci+r)∩A and (3.4) V˜ui,y,(ci−r, ci+r)≤CV hi,y,(ci−r, ci+r). It follows that there is a function ˜usuch that ˜u(x) = Df−1(x) almost everywhere and this new representative has bounded variation on Q(c, r)∩π−1 i(y) for Ln−1almost every y∈Qi(c, r). Now (3.4) and (3.1) yields (3.5) ZQi(c,r) V˜ui,y,(ci−r, ci+r)dy < ∞ which verifies (2.2) for Df−1. Proof of Theorem 1.3: First let us prove the theorem in the case p= 1. The proof of this case is analogous to the previous proof and therefore we only sketch it and point out the differences. From Theorem 2.2 (ii) we know that Df ◦f−1∈W1,1 loc . Fix y∈Qi(c, r) such that hi,y is absolutely continuous on (ci−r, ci+r). Given ε > 0 find δ > 0 from the absolute continuity of hi,y. Choose Ajand Bjas before and moreover assume 96 S. Hencl that Pk j=1 |Aj−Bj|< δ. Analogously to (3.3) we obtain k X j=1 kDf−1(Aj)−Df−1(Bj)k< Cε. Reasoning analogously to the previous proof we conclude that Df−1has a representative which is absolutely continuous on almost all lines parallel to coordinate axes. On those lines we have V˜ui,y,(ci−r, ci+r)=Zci+r ci−r |∇˜ui,y(t)|dt. From Theorem 1.2 and Theorem 2.1 (ii) we already know (2.2) and thus we obtain (2.1). Now let us return to the case p > 1. We already know that Df−1∈ W1,1 loc . Therefore we can use Theorem 2.2 (ii) and differentiate twice the identity f◦f−1(y) = yto obtain (3.6) D2f(f−1(y))Df−1(y)Df−1(y) + Df(f−1(y))D2f−1(y) = 0. Here and in the sequel we identify the second derivative with an linear operator from Rn2to Rn2. Clearly kDf(f−1(y))−1k ≤ C, kDf−1(y)k ≤ Cand |Jf−1(y)| ≥ C at almost every point since fis bilipschitz. From (3.6) and substitution formula we now obtain ZA kD2f−1(y)kpdy ≤CZA kD2f(f−1(y))kp|Jf−1(y)|dy =CZf−1(A) kD2f(x)kpdx for every open set A⋐f(Ω) and the claim follows. 4. Necessity of the Lipschitz condition for ffor n≥4 Example 4.1. Let n≥4. There is a homeomorphism f: (−1,1)n→Rn such that Df ∈W1,1((−1,1)n,Rn2) and f−1is Lipschitz, but Df−1/∈ BVlocf((−1,1)n),Rn2. Proof: Given x∈Rnwe denote ˜x= [x1,...,xn−1]∈Rn−1and k˜xk= qx2 1+···+x2 n−1. Mappings with Derivatives in BV 97 Let α=1 2n,β=3 4and set f(x) = n−1 X i=1 eixik˜xkα−1+enxn+k˜xksin(k˜xk−β) if k˜xk>0 and f(x) = enxnif k˜xk= 0. Our mapping fis clearly continuous and it is easy to check that fis a one-to-one map since xik˜xkα−1=zik˜zkα−1for every i∈ {1,...,n−1} ⇒ ⇒xi=zifor every i∈ {1,...,n−1}. Therefore fis a homeomorphism. A direct computation shows that the second partial derivatives of fi, i∈ {1,...,n−1}, are smaller than Ck˜xkα−2and therefore integrable. Moreover, ∂fn(x) ∂x1 =x1k˜xk−1sin(k˜xk−β)− k˜xkβx1 k˜xkβ+2 cos(k˜xk−β). It is not difficult to compute that we can bound each partial derivative of this expression by Ck˜xk1−2(β+1). Clearly 2(β+1)−1< n−1 and therefore these second partial derivatives are integrable. Analogously we can estimate other second partial derivatives of fn. Since the second derivatives of fare smooth outside the segment {[0,...,0, t] : t∈(−1,1)}and |D2f| ∈ L1((−1,1)n) it is easy to see that Df ∈W1,1((−1,1)n,Rn2). The inverse of fis given by f−1(y) = n−1 X i=1 eiyik˜yk1 α−1+enyn− k˜yk1 αsin(k˜yk−β α) if k˜yk>0 and f−1(y) = enynif k˜yk= 0. The derivative of the function φ(t) = t1 αsin(t−β α) is bounded on (−1,1) and therefore φis Lipschitz. Thus it is not difficult to see that f−1is Lipschitz. The second derivative of f−1is clearly continuous outside the segment {[0,...,0, t] : t∈R}. Elementary computation gives us ∂(f−1)n(y) ∂y1 =−y1 αk˜yk1 α−2sin(k˜yk−β α) + k˜yk1 αβ α y1 k˜ykβ α+2 cos(k˜yk−β α) and therefore the second derivative ∂2(f−1)n(y) ∂y2 1 contains some integrable terms and (4.1) −k˜yk1 αβ2 α2 y2 1 k˜yk2β α+4 sin(k˜yk−β α). 98 S. Hencl Since 2β α+ 1−1 α> n −1 we obtain that the integral of the absolute value of (4.1) over the set S=y∈f((−1,1)n) : y1>1 2||˜y||,sin(k˜yk−β α)>1 2 is infinite. Hence |D2f−1|/∈L1 loc and it is not difficult to deduce that Df−1/∈BVloc. Acknowledgements. The author would like to thank to Professor L. Zaj´ıˇcek for suggesting the problem. The author wishes to express his thanks to Professor J. Mal´y for many stimulating conversations and for helpful suggestions. References [1] L. Ambrosio, N. Fusco, and D. Pallara,“Functions of bounded variation and free discontinuity problems”, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. [2] S. Hencl and P. Koskela, Regularity of the inverse of a planar Sobolev homeomorphism, Arch. Ration. Mech. Anal. 180(1) (2006), 75–95. [3] S. Hencl, P. Koskela, and J. Mal´ y, Regularity of the inverse of a Sobolev homeomorphism in space, Proc. Roy. Soc. Edinburgh Sect. A 136(6) (2006), 1267–1285. [4] S. Hencl, P. Koskela, and J. Onninen, Homeomorphisms of bounded variation, Arch. Ration. Mech. Anal. (to appear). [5] J. Onninen, Regularity of the inverse of spatial mappings with finite distortion, Calc. Var. Partial Differential Equations 26(3) (2006), 331–341. [6] L. Vesel´ y and L. Zaj´ ıˇ cek, Delta-convex mappings between Banach spaces and applications, Dissertationes Math. (Rozprawy Mat.) 289 (1989), 52 pp. [7] W. P. Ziemer,“Weakly differentiable functions. Sobolev spaces and functions of bounded variation”, Graduate Texts in Mathematics 120, Springer-Verlag, New York, 1989. Mappings with Derivatives in BV 99 Department of Mathematical Analysis Charles University Sokolovsk´a 83 186 00 Prague 8 Czech Republic E-mail address:[email protected] Primera versi´o rebuda el 26 de setembre de 2006, darrera versi´o rebuda el 21 de mar¸c de 2007.