HOMEOMORPHISMS OF BOUNDED VARIATION

Transcription

1 HOMEOMORPHISMS OF BOUNDED VARIATION STANISLAV HENCL, PEKKA KOSKELA AND JANI ONNINEN Abstract. We show that the inverse of a planar homeomorphism of bounded variation is also of bounded variation. In higher dimensions we show that f 1 is of bounded variation provided that f W 1,1 (Ω; R n ) is a homeomorphism with Df in the Lorentz space L,1 (Ω). 1. Introduction Let Ω R n, n 1, be a domain. Recall that a mapping f L 1 (Ω; R n ) is of bounded variation, f BV (Ω; R n ), if the coordinate functions of f belong to the space BV (Ω). This means that the distributional partial derivatives of each coordinate function h of f are measures with finite total variation in Ω : there are Radon (signed) measures µ 1,, µ n defined in Ω so that for i = 1,, n, µ i (Ω) < and hd i ϕ dx = ϕ dµ i Ω Ω for all ϕ C (Ω). The gradient of h is then a vector-valued measure with finite total variation { Dh = sup h div v dx : v = (v 1,, v n ) C (Ω; R n ), Ω } v(x) 1 for x Ω <. The total variation of Dh can be considered as a Radon measure: given A Ω we set { Dh (A) = sup h div v dx : v = (v 1,, v n ) C (Ω; R n ), Ω } v(x) χ A (x) for x Ω. 2 Mathematics Subject Classification. Primary 3C6; Secondary 35J15, 35J7. Key words and phrases. Sobolev mapping, Jacobian, inverse. Hencl and Koskela were supported in part by the Academy of Finland. Onninen by the National Science Foundation grant DMS

2 2 STANISLAV HENCL, PEKKA KOSKELA AND JANI ONNINEN If h W 1,1 (Ω), then Dh (A) = u dx. For all this see [1], [1]. A Given f BV (Ω; R n ), we write n Df (A) := Df i (A). i=1 Further, f BV loc (Ω; R n ) requires that f BV (Ω ; R n ) for each subdomain Ω Ω. In the one-dimensional setting, each monotone function f L 1 (Ω; R) belongs to BV (Ω; R). Thus, given a homeomorphic f BV loc (Ω; R), also f 1 BV loc (f(ω); R). We show that, surprisingly, this also holds in dimension two. Theorem 1.1. Let Ω, Ω R 2 be open and suppose that f : Ω Ω is a homeomorphism. Then f BV loc (Ω; R 2 ) if and only if f 1 BV loc (Ω ; R 2 ). Moreover, both f and f 1 are differentiable almost everywhere. One cannot replace the space BV with the Sobolev class W 1,1 in Theorem 1.1. However, if the Jacobian J f of f is strictly positive a.e. in Ω for a homeomorphic f W 1,1 (Ω; R 2 ), then f 1 W 1,1 loc (f(ω); R2 ). For this see [2]. The assumption that n = 2 in Theorem 1.1 is essential. Indeed, given n 3 and < ε < 1, we produce below an example of homeomorphic f W 1, ε (( 1, 1) n ; R n ) for which f 1 / BV loc (f(ω); R n ). The next theorem tells us that f 1 has bounded variation in space if we require more regularity from f. Theorem 1.2. Let Ω, Ω R n be open and suppose that f : Ω Ω is a homeomorphism. Suppose that f W 1,1 loc (Ω; Rn ) and Df L,1 loc (Ω). Then both f and f 1 are differentiable almost everywhere and f 1 BV loc (f(ω); R n ). Here L,1 (Ω) is a Lorentz space and L 1,1 (Ω) = L 1 (Ω). To illustrate the meaning of Theorem 1.2 for a reader not familiar with Lorentz spaces we give the following example. Example 1.3. Let f : Ω Ω be a homeomorphic mapping of the class W 1,1 loc (Ω; Rn ) where Ω, Ω R n are open. Suppose that Df log α (e + Df ) < Ω for some α > n 2. Then f 1 BV loc (Ω ; R n ) and both f and f 1 are differentiable almost everywhere. We would like to know if Theorem 1.2 holds under the assumption that Df L loc (Ω) when n 3. Regarding the W 1,1 -setting in higher dimensions we refer the reader to [3] and [6].

3 HOMEOMORPHISMS OF BOUNDED VARIATION 3 2. Regularity of the inverse mapping By B(c, r) we denote the n-dimensional open ball centered at c R n with radius r >. Given sets E, F R n we write E + F = {x + y : x E, y F }. Given an open set A R n and a mapping f : A R n we write f A = 1 f A A Rn for the integral average of f. The closure of a set A is denoted by A. We will need the following generalization of [2, Lemma 3.1] which may be of independent interest. Lemma 2.1. Let Ω R 2 be a domain. Suppose that f BV loc (Ω; R 2 ) is a homeomorphism and let B(y, 2r) f(ω). Then (2.1) r diam f 1 (B(y, r)) C Df ( f 1 (B(y, 2r)) ), where C is an absolute constant. Proof. Set d = diam f 1 (B(y, r)) and pick a, b f 1 (B(y, r)) such that a b = d. Without loss of generality, we will suppose that a = (, ) and b = (d, ). We can clearly find k N large enough such that U := f 1 (B(y, 2r)) + B (, 2 d k ) Ω, (2.2) Df (U) 2 Df ( f 1 (B(y, 2r)) ) and (2.3) x 1, x 2 U, x 1 x 2 < 2d k f(x 1) f(x 2 ) < r 1. For i {1,..., k} we denote S i = { [s, t] R 2 : s [ i 1 k d, i k d], [s, t] f 1 (B(y, 2r)) } and U i = { [s, t] U : s [ i 2 k d, i + 1 k d]}. For a fixed i {1,..., k} we pick balls B 1,..., B l of radius d such k that l l S i B j U i, χ Bj 2 and B j B j+1 > 1 1 B j j=1 j=1 for every j {1,..., l 1}. Clearly, there is x 1 B 1 such that x 1 / f 1 (B(y, 2r)) and it is not difficult to find out that there is j {1,..., l} and x 2 B j such that x 2 f 1 (B(y, r)). Therefore (2.3) implies f B1 y 2r r 1 and f B j y r + r 1.

4 4 STANISLAV HENCL, PEKKA KOSKELA AND JANI ONNINEN From the triangle inequality and the Poincare inequality for BV functions (see e.g. page 23 in [1]) we obtain l r f Bj f Bj+1 C C j=1 l 1 j=1 l 1 j=1 1 f f Bj B B j j+1 + C f f Bl 1 B B j B l l B l k d Df (B j B j+1 ) C k d Df (U i ). By summing these inequalities over i and using (2.2) conclude that rd C k i=1 Df (U i ) C Df (U) C Df ( f 1 (B(y, 2r)) ). Proof of Theorem 1.1. Suppose that f BV loc (Ω; R 2 ). Our first step is to show that f 1 is differentiable almost everywhere. For that we recall a differentiability result for Radon measures. If µ and ν are Radon measures on R n, then the derivative ( ) µ B(x, r) (2.4) D(µ, ν, x) = lim r ν ( B(x, r) ) exists and is finite ν-almost everywhere. By the Rademacher-Stepanov Theorem, see [9, Theorem 29.1], f 1 is differentiable almost everywhere provided that (2.5) lim sup r diam f 1( B(y, r) ) r < a.e. y f(ω). This condition follows immediately from Lemma 2.1 by defining µ(b(y, r)) = Df ( f 1 (B(y, r)) ) and ν(b(y, r)) = B(y, r). Now we will prove that the inverse has bounded variation. Let A f(ω) be a fixed domain. First, we will construct approximations to f 1. Fix ε > such that A + B(, 1ε ) Ω. Let < ε ε.

5 HOMEOMORPHISMS OF BOUNDED VARIATION 5 We denote the ε-grid in R 2 by G ε := (εz) (εz). Pick a partition of unity {φ z } z Gε such that (2.6) Now we set each φ z : R 2 R is continuously differentiable; supp φ z B(z, 2ε) and φ z C ε ; z G ε φ z (y) = 1 for every y R 2. g ε (y) = z G ε φ z (y)f 1 (z) for every y A. This approximation to f 1 clearly satisfies g ε C 1 (A, R 2 ). Then Dg ε (y) C ε diam f 1 (B(y, 2ε)). Indeed, for a fixed y, choose z so that y B(z, 2ε). Then Dg ε (y) = D φ z (y)(f 1 (z) f 1 (z )), z G ε because of (2.6), and the asserted estimate follows. Lemma 2.1 this implies that for every y A we have Together with (2.7) Dg ε (y) C ε 2 Df ( f 1 (B(y, 4ε)) ). It follows that Dg ε (y) dy A z G ε (A+B(,2ε)) B(z,2ε) C Df ( f 1 (A + B(, 8ε)) ). C ε 2 Df ( f 1 (B(z, 6ε)) ) dy Let now ε i = ε /i for i 1. By the previous paragraph, the sequence { Dg εi } i is bounded in L 1 (A). Moreover, g εi f 1 uniformly on A; in particular in L 1 (A). It follows that f 1 BV (A), see [1]. 3. Counterexample in higher dimensions We show that Theorem 1.1 does not extend as such to higher dimensions by constructing the following example. Example 3.1. Let < ε < 1 and n 3. There is a homeomorphism f W 1, ε (( 1, 1) n ; R n ) such that f 1 is continuously differentiable at every point of f(( 1, 1) n ) \ {[,...,, t] R n : t R} and Df 1 / L 1 loc (f(( 1, 1)n )), where Df 1 denotes the pointwise differential of f 1. Consequently, f 1 / BV loc (f(ω); R n ).

6 6 STANISLAV HENCL, PEKKA KOSKELA AND JANI ONNINEN We write e i for the i-th unit vector in R n, i.e. the vector with 1 on the i-th place and everywhere else. Given x = [x 1,..., x n ] R n we denote x = [x 1,..., x ] R and x = x x 2. Let α = ε n 2, β = 1 + ε () and set ( f(x) = e i x i x α 1 + e n xn + x sin( x β ) ) i=1 if x > and f(x) = e n x n if x =. Our mapping f is clearly continuous and it is easy to check that f is a one-to-one map since x i x α 1 = z i z α 1 for every i {1,..., n 1} x i = z i for every i {1,..., n 1}. Therefore f is a homeomorphism. A direct computation shows that the partial derivatives of f i, i {1,..., }, are smaller than C x α 1 and therefore integrable with the exponent p = n 1 ε. Moreover, f n (x) x 1 = x 1 x 1 sin( x β ) x β x 1 x β+2 cos( x β ). The first term is clearly integrable with the exponent n 1 ε. For this degree of integrability of the second one, we need (β )(n 1 ε) < n 1 which is quaranteed by our choice of β. Analogously fn x i is integrable for i = 2,, n 1. Finally, n f x n is bounded. Since f is C 1 - smooth outside the segment {[,,, t] : t ( 1, 1)} and f L ε (( 1, 1) n ) it is easy to see that f W 1, ε (( 1, 1) n ; R n ). The inverse of f is given by ( ) f 1 (y) = e i y i ỹ 1 α 1 + e n yn ỹ 1 α sin( ỹ β α ) i=1 if ỹ > and f 1 (y) = e n y n if ỹ =. The differential of f 1 is clearly continuous outside the segment {[,,, t] : t R}. Computations as above show us that for Df 1 L 1 loc we need that β α α 1 < n 1. This is not satisfied and therefore Df 1 is not locally integrable. 4. Regularity of the inverse mapping in higher dimensions Our starting point is to prove the following auxiliary inequality.

7 HOMEOMORPHISMS OF BOUNDED VARIATION 7 Lemma 4.1. Let Ω R n be open and suppose that a homeomorphism f belongs to the Sobolev class W 1,1 loc (Ω; Rn ) and that Df L,1 loc (Ω). Then there exists Θ L 1 loc (Ω) such that (4.1) diam ( f 1 (B r ) ) C(n)r 1 n Θ(x) dx f 1 (B 2r ) for all balls B r = B(y, r) such that B 3r = B(y, 3r) f(ω). Before giving the proof of this lemma, we recall the definition of a Lorentz space and an oscillation lemma critical for us. The Lorentz space L p,1 (Ω), 1 p <, is defined as the class of all measurable functions f : Ω R for which the norm u L p,1 (Ω) := p { x Ω ; u(x) > t } 1 p dt is finite. The notion of a Lorentz space L p,q (Ω), 1 p <, 1 q, was first introduced in [5]. For further details about Lorentz spaces, see e.g. [8] or [1]. It is a well-known fact that a function in the Sobolev class W 1,p loc (Ω), where Ω R n, is continuous provided p > n. It is also known that this condition can be sharpened to a very precise integrability condition. Namely, if u W 1,1 (Ω) is a function whose weak partial derivatives belong to the Lorentz space L n,1 (Ω), then u is continuous. More precisely, we have the following oscillation lemma, see [7]. Lemma 4.2. Let u be a function belonging to the Sobolev class W 1,1 loc (Ω). If Du L n,1 (Ω), then u(x) u(y) C(n) Du L n,1 (B (x, x y )) for almost every x, y Ω such that B(x, x y ) Ω. Proof of Lemma 4.1. Our first step is to prove a slightly weaker estimate, namely (4.2) diam ( f 1 (B r ) ) C(n)r 1 n Df L,1 (f 1 (B 2r )). For this we may assume that diam f 1 (B r ) = d and that the set f 1 (B r ) contains the origin and the point (,,...,, d). For < t < d, we write L t = { x f 1 (B 2r ) ; x n = t }. Fix t (, d) and a point z L t f 1 (B r ). We consider a (n 1)- dimensional ball B (z, s) L t such that B (z, s) f 1 (B 2r ) is non-empty. Since f is a homeomorphism we have s >. For shorter notation we write B t = B (z, s). Applying the oscillation lemma 4.2, we have r osc(f, B t ) C(n) Df L,1 (B t ) [ = C(n) H ({x B t ; Df(x) > s}) ] 1 ds.

8 8 STANISLAV HENCL, PEKKA KOSKELA AND JANI ONNINEN Integrating this estimate from to d with respect to t we find that d r C(n) d d [ H ({x B t ; Df(x) > s}) ] 1 ds dt ( ) 1 C(n) χ { Df >s} (x) dh (x) dt ds. B t Here, as usually, the notation χ E stands for the characteristic function of a set E. Combining the previous estimate with Hölder s inequality we have d r C(n) C(n) d n 2 d n 2 ( d ( B t χ { Df >s} (x) dh (x) dt χ { Df >s} (z) dz f 1 (B 2r ) ) 1 ds. ) 1 The last inequality follows from the choice of the balls B t, namely from the fact that B t L t. Finally, we conclude that d C(n)r 1 n Df L,1 (f 1 (B 2r )), as desired in (4.2). Our second step is to find Θ. For that we shorten our notation and write ω(s) = { x f 1 (B 2r ) ; Df(x) > s } for all s >. Employing Theorem 2.3 in [4] we find that (4.3) ( Df L,1 (f 1 (B 2r )) C(n) f 1 (B 2r ) [ω(s)] 1 ds ) n 2 Φ( Df(x) ) dx where Φ Df is locally integrable in Ω, see [4, Theorem 2.2]. We choose a a compact set K such that f 1 (B 2r ) K Ω. Now, we are ready to define ( (4.4) Θ(x) = {z K ; Df(z) > s} 1 ds ds ) n 2 Φ( Df(x) ). Employing the definition of the norm of L,1 (K), we conclude that {z K ; Df(z) > s} 1 ds = 1 n 1 Df L,1 (K) <. Our claim follows from (4.2), (4.3), (4.4) and Φ Df L 1 loc (Ω). Proof of Theorem 1.2. The almost everywhere differentiability of f follows from [7] because f is homeomorphic (and thus monotone) and Df L,1 loc (Ω). Our next step is to show that f 1 is differentiable almost everywhere. Analogously to the proof of Theorem 1.1 it is

9 HOMEOMORPHISMS OF BOUNDED VARIATION 9 enough to show (2.5). This follows immediately from (4.1) and (2.4) by defining µ(b r ) = Θ(x) dx and f 1 (B r) ν(b r ) = B r. Here and in what follows we use the notation B s = B(y, s), for s >. Analogously to the proof of Theorem 1.1 we will prove now that the inverse mapping has bounded variation. Let A f(ω) be a fixed domain. Fix ε > such that A + B(, 4nε ) f(ω) and let < ε ε. We denote the ε-grid in R n by G ε := (εz)... (εz). Pick a partition of unity {φ z } z Gε such that Now we set each φ z : R n R is continuously differentiable; supp φ z B(z, nε) and φ z C(n) ; ε φ z (y) = 1 for every y R n. z G ε g ε (y) = z G ε φ z (y)f 1 (z) for every y A. Again g ε C 1 (A, R n ) and with the help of Lemma 4.1 it is easy to check that Dg ε (y) C(n) diam f 1( B(y, nε) ) ε C(n) Θ(x) dx. ε n f 1 (B(y,2nε)) For s > we denote A s := A + B(, s). Now we may use the fact that {f 1 (B(z, 3nε))} z Gε have bounded overlap to obtain Dg ε (y) dy Dg ε (y) dy A z G ε A nε C(n) C(n) B(z,nε) z G ε A nε f 1 (A 4nε ) f 1 (B(z,3nε)) Θ(x) dx <. Θ(x) dx As in the proof of Theorem 1.1, this implies f 1 BV loc.

10 1 STANISLAV HENCL, PEKKA KOSKELA AND JANI ONNINEN 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, 2. [2] S. Hencl and P. Koskela, Regularity of the inverse of a planar Sobolev homeomorphism, Arch. Rational Mech. Anal. 18 (26), [3] S. Hencl, P. Koskela and J. Malý, Regularity of the inverse of a Sobolev homeomorphism in space, Proc. Roy. Soc. Edinburgh Sect. A., to appear. [4] J. Kauhanen, P. Koskela and J. Malý, On functions with derivatives in a Lorentz space, Manuscripta Math. 1,1 (1999), [5] G. G. Lorentz, Some new functional spaces, Ann. of Math. 51 (195), [6] J. Onninen, Regularity of the inverse of spatial mapping with finite distortion, Calc. Var. Partial Differential Equations, to appear. [7] J. Onninen, Differentiability of monotone Sobolev functions, Real Anal. Exchange 26,2 (2/21), [8] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, [9] J. Väisälä, Lectures on n-dimensional quasiconformal mappings, Springer. [1] W. P. Ziemer, Weakly differentiable functions. Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics, 12. Springer-Verlag, New York, Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 Prague 8, Czech Republic address: hencl@karlin.mff.cuni.cz Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FIN-414, Jyväskylä, Finland address: pkoskela@maths.jyu.fi Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA address: jkonnine@syr.edu