A bi-lipschitz continuous, volume preserving map from the unit ball onto a cube

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1 Note i Matematica Note Mat. 8, ISSN 3-536, e-issn DOI.85/i5993v8np77 Note i Matematica 8, n., 8, A bi-lipschitz continuous, volume preserving map from the unit ball onto a cube Jens Anré Griepentrog Weierstrass Institute for Applie Analysis an Stochastics, Mohrenstrasse 39, 7 Berlin, Germany griepent@wias-berlin.e Wolfgang Höppner Weierstrass Institute for Applie Analysis an Stochastics, Mohrenstrasse 39, 7 Berlin, Germany hoeppner@wias-berlin.e Hans-Christoph Kaiser Weierstrass Institute for Applie Analysis an Stochastics, Mohrenstrasse 39, 7 Berlin, Germany kaiser@wias-berlin.e Joachim Rehberg Weierstrass Institute for Applie Analysis an Stochastics, Mohrenstrasse 39, 7 Berlin, Germany rehberg@wias-berlin.e Receive: //6; accepte: /5/7. Abstract. We construct two bi-lipschitz continuous, volume preserving maps from Eucliean space onto itself which map the unit ball onto a cyliner an onto a cube, respectively. Moreover, we characterize invariant sets of these mappings. Keywors: Lipschitz homeomorphism, invariant sets, measure preserving maps MSC classification: primary 53A55, seconary 6B, 57R5 Introuction This paper is eicate to a special case of the general question whether two manifols with measure can be mappe onto each other by a measure preserving map which possesses, aitionally, some continuity properties. This problem has a long history, for instance, in cartography, where surface preserving mappings from the two imensional sphere onto the plane are require. The problem also appears in the investigation of partial ifferential equations on Lipschitz omains, an our work is motivate thereby. Following Grisvar [], we say that a omain is Lipschitzian, if for every point of its bounary

2 78 J. A. Griepentrog, W. Höppner, H.-Chr. Kaiser, J. Rehberg there is a neighbourhoo U an a bi-lipschitz continuous mapping Ψ such that the Ψ-image of the intersection of U with the omain equals the unit half ball. For many purposes these half balls are aequate moel sets: after localization, transformation an reflection one often ens up with an analogous ifferential operator on the unit ball, which is mostly well known. However, some methos use in interpolation theory of function spaces an parabolic regularity theory see [9] an [4] require that the local charts are not only bi-lipschitzian but, aitionally, possess a constant Jacobian. The crucial point is the following: if one wants after suitable localization to transform problems which involve spaces of istributions an function spaces via the Lipschitz iffeomorphism Ψ onto the unit ball, then the way to efine the mapping for functions u is T Ψ u ef = u Ψ, while for istributions the corresponing mapping is TΨ. Hence, to have a common retraction/coretraction for the function spaces an the spaces of istributions the Jacobian of Ψ must be smooth. As the smoothness of this Jacobian is not easy to control, here we are looking for mappings Ψ the Jacobian of which is actually constant. More precisely, we eplicitly construct bi-lipschitz continuous mappings from the ball onto the prototypical nonsmooth objects cube an cyliner. A variant of this problem has been treate by Moser [7], who emonstrate that on a close, smooth manifol any two smooth volume elements are iffeomorphic. Etensions of this result to noncompact manifols are ue to Greene an Shiokama [6], while Banyaga [] establishe it for smooth manifols with bounary. Gromov [5] investigate these questions for real analytic manifols with real analytic volume forms. Zehner [8] prove that certain Höler an Lipschitz continuous volume elements can be mappe onto each other by means of C -mappings with Höler an Lipschitz continuous first erivatives, respectively. For boune omains with Höler continuous volume forms an bounaries of the class C 3,α Dacorogna an Moser [3] emonstrate the eistence of auto iffeomorphisms from the class C,α which provie the equivalence of the volume forms an coincie with the ientity map on the bounary of the manifol. Base upon this result Fonseca an Parry [4, Ch. 5, Thm. 5.4] prove that for any two elements from a class of star shape omains in Eucliean space, there is a Lipschitz homeomorphism, with constant Jacobian, mapping these two omains onto each other. Fonseca an Parry s class contains in particular the ball, the cube, an the cyliner. Our aim is to give a comprehensive proof of Fonseca an Parry s result for the special case of a ball an a cyliner by eplicitly constructing the Lipschitz homeomorphism with constant Jacobian between the two omains. This special Lipschitz homeomorphism has, aitionally, a variety of invariant sets an fie

3 A bi-lipschitz continuous, volume preserving map 79 points, which we characterize. Actually, our investigation has been spurre by just this aitional eman on the mapping. Our construction comes to bear in applications of the concept of Gröger-regular sets in the theory of partial ifferential equations, see [3] an [7] for the concept itself, an [8], [], [], [4], an [6] for applications. Results We investigate bi-lipschitz continuous mappings with constant Jacobian of a ball onto a cyliner an of a ball onto a cube. The special geometric situation allows to reuce the number of spatial variables by making use of rotational symmetry. Thus, we can formulate the constancy of the Jacobian by means of ifferential equations which can be eplicitly solve. Finally, one obtains the sought-after mapping as a rational epression. Theorem. For any integer there is a bi-lipschitz continuous mapping Λ + from R + onto itself with the following properties:. Λ + maps the unit ball B + of R + ontoacylinerwithheight an raius : Λ + : B + onto {, y R + B, y < }.. Λ + maps the halfspace {,,..., + R + j } onto itself for each of the integers l {,,...,+}. 3. Λ + maps each hyperplane containing the rotation ais {, y R + =,y R } onto itself. 4. Λ + is the ientity map on the equatorial hyperplane {, y R + R,y= }. 5. Both poles,...,, an,...,, R + are fie points of Λ The map Λ + is homogeneous of orer : Λ + r,r,...,r + =rλ +,,..., + for all,,..., + R +, r.

4 8 J. A. Griepentrog, W. Höppner, H.-Chr. Kaiser, J. Rehberg 7. The Jacobian of Λ + is constant almost everywhere. We prove the theorem in Section 3. Remark.Inthe two-imensional case the mapping Λ efine by, if = y =, + y, 4 π + y arctan y if y, >, Λ, y ef = + y, 4 π + y arctan y if y, <, 4 π + y arctan y, + y if y, y>, + y arctan y, + y if y, y<, with the inverse 4 π Λ ξ, =, if = ξ =, ξ cos π 4 ξ,ξsin π 4 ξ if ξ, ξ, sin π ξ 4,cos π ξ 4 if ξ,, meets the requirements of Theorem, see also Figure. Λ Λ Figure. The map Λ from R onto R. Theorem implies another special case of Fonseca an Parry s result. 3 Corollary. There is a map from R,, onto itself which is bi- Lipschitz continuous, has an almost everywhere constant Jacobian, an maps the unit ball onto the unit cube.

5 A bi-lipschitz continuous, volume preserving map 8 Proof. In the one-imensional case one can choose the ientity map. In the two-imensional case the mapping Λ from Remark is the right one. Now one euces the statement by inuction over the space imension, thereby making use of Theorem. QED 4 Corollary. There is a bi-lipschitz continuous, volume preserving map from R,, onto itself which maps the unit ball onto a cube. Actually, Corollary 4 an Corollary 3 are equivalent. This follows from the fact that a homothecy has a constant Jacobian. 5Remark. Due to Brouwer s invariance of omain theorem [] the bounaries of the ball an of the cyliner as well as the bounaries of the ball an of the cube are mappe onto each other by the mappings from Theorem an the corollaries, respectively. 3 Proof of the theorem In the following we prove Theorem ; we write the coorinates in R + as,...,,y, for short, y, R, y R, an abbreviate the Eucliean norm R by. For = the mapping Λ from Remark satisfies the assertions of Theorem. Now, we regar the problem in R + with an make the following ansatz for Λ + :, y g, y,..., g, y,h, y for all =,..., R an y R. Moreover, we eman g, y,..., g, y,h, y = g, y,..., g, y, h, y. As a consequence of this ansatz, hyperplanes which contain zero an whose normal vectors are orthogonal to the vector,,...,, are mappe into themselves. If, aitionally, h, y for all y, the halfspace R + + ef = {, y R + y } is mappe into itself, an the hyperplane efine by y = is mappe into itself. Thus, it suffices to efine Λ + on the upper halfspace R + +.Inorertooso, we partition R + + : C γ C γ ef = {, y R + y γ }, 3 ef = {, y R + y γ }, 4

6 8 J. A. Griepentrog, W. Höppner, H.-Chr. Kaiser, J. Rehberg an efine Λ +, thereby observing,, by the bi-lipschitz continuous mappings Λ γ : C γ C, Λ γ : Cγ C, 5 such that the restrictions of these mappings coincie on the common bounary {, y R + y = γ }, 6 of Cγ an C γ. Here, γ>isaconstant, which we specify in Step 3 of the proof.. First, we construct a mapping on the set Cγ, see 3. We efine h, y ef = + y. Now, we are looking for a function g, y =v /y such that the Jacobian satisfies g, y, h, y, y = forall R, y>. 7 This eterminant can be evaluate as follows v + y v y v... y v y v v + y v... y v y v y v... v+ y v... +y +y +y v y y v y v y +y. Aing suitable multiples of the first row to the others we get the eterminant v + y v y v... y v v v y v... v... y v y,

7 A bi-lipschitz continuous, volume preserving map 83 which can be simplifie by aing multiples of the last column to the others v... v v y v... v + y + y... + y y y v Finally, aing suitable multiples of all columns to the first one we en up with + y v... v v + y... + y y + y y v Developing the last eterminant with respect to the first column, conition 7 leas to the following orinary ifferential equation for v: + y v + y y + y +.. v v =, y y which transforms uner the substitution v = w, y = ζ, equivalently into w + ζζ + w = The general solution of this equation is ζ ζ + / ζ ζ ζ ζ +. α α α + c + +/ = ζ + / ζ arctan ζ sin α α + c,

8 84 J. A. Griepentrog, W. Höppner, H.-Chr. Kaiser, J. Rehberg where c is an arbitrary real constant. As one has to avoi a singularity in ζ =, one chooses c =. Thus, one obtains for g : y / arctan /y g, y = + sin α α if, 8 if =. Please note that y arctan /y / lim + sin α α =. It shoul be note that both h an g are rational transformations.. Net, we construct corresponing functions on Cγ, see 4. Because spheres have to pass into cyliner surfaces we efine g, y ef = +y /. Now we are looking for a function h, y =u,y such that the Jacobian satisfies g, y, h, y, y = forall R,,y. 9 It turns out that this conition on the Jacobian together with the requirement that u shoul vanish on the set {, y R + y = } etermines u uniquely. Using the substitution = θ the Jacobian can be evaluate as follows g y y... y y g 4 g 4 g 4 g y g g 4 y... y y g 4 g 4 g y y... g g 4 g 4 y y g 4 g u u θ θ... u u θ y Aing suitable multiples of the first row to the others we get the eterminant g y y... y y g 4 g 4 g 4 g g g , g... g u u θ θ... u u θ y

9 A bi-lipschitz continuous, volume preserving map 85 which can be simplifie by aing multiples of the last column to the others: g... y g g g g... g u θ + y u u y θ + y u y... u θ + y u u y y Finally, aing suitable multiples of all columns to the first one we en up with g... y g g g u θ + y u u y θ + y u y... u θ + y u u y y Developing the last eterminant with respect to the first column, one easily obtains from 9 the conition / + y u y y u =. θ This yiels the partial ifferential equation y uθ, y θ with the bounary conition + θ uθ, y y = θ θ + y / uθ, = for θ<+. By the metho of characteristics, see for instance [5], one fins the solution θ, y arctany/θ θ + y cos α α, an ens up with arctany/ h, y = + y cos α α.

10 86 J. A. Griepentrog, W. Höppner, H.-Chr. Kaiser, J. Rehberg Again it shoul be note that both h an g are rational transformations. 3. Up to now we have constructe two volume preserving mappings g,..., g,h :R + + g,..., g,h :R + + onto R + +, onto R + +, which are homogeneous of orer. These mappings o not epen on γ, see5. We are now going to moify both mappings such that they coincie on the common bounary of the sets Cγ an C γ for some γ>, see 6. To that en we introuce the functions on, + an τ : λ ϱ : λ arctan/λ arctan λ on [, + an efine mappings 5 by Λ g, y g, y γ, y ef =,..., τγ τγ Λ ef γ, y = sin α α / cos α α,,h, y g, y,..., g, y, h, y ϱγ for, y C γ, 3 for, y C γ, 4 for all γ>. The Jacobians of these mappings are Λ, y = an Λ τγ, y = ϱγ, 5 for all, y R + with,y>. If arctan/γ sin α α = arctan γ cos α α, 6 then the values of the Jacobians 5 are equal. There is eactly one γ > which satisfies 6, an in the sequel γ shall be this solution of 6. From the monotonicity properties of τ an ϱ one euces that, in accorance with 5, Λ γ maps the set C γ that τ λ = arctan/λ onto C an that Λ γ maps C γ onto C. Please note / sin α α sin arctan λ +λ = τλ + λ +/ 7

11 A bi-lipschitz continuous, volume preserving map 87 an ϱ λ =cos arctan λ +λ = + λ. 8 +/ The inverse mappings to Λ γ Λ γ ξ, = in the interior of C,an ξ Λ γ ξ, = an Λ γ are given by ξ,...,ξ ξ,τ τγ ξ 9 + τ τγ ξ ξ,...,ξ, ξ ϱ ϱγ + ξ ϱ ϱγ ξ in the interior of C plus continuous etension to C an C, respectively. From the monotonicity properties of τ an ϱ follows that Λ γ maps C onto Cγ an Λ γ maps C onto Cγ. With respect to the solution γ of 6 we now efine the sought-after mapping Λ γ, y if, y Cγ, SΛ Λ +, y ef γ = S, y if S, y C γ, Λ γ, y if, y Cγ, S, y if S, y C SΛ γ where S : R + R + is the reflection at the equatorial plane, given by S, y ef =, y for, y R Finally, we prove the Lipschitz properties of Λ +. First we make sure, that Λ γ C, Cγ an Λ γ C, Cγ, see Step 5. Then we can estimate γ, Λ +, y Λ +, ỹ + ma { Λ γ C, C γ, Λ γ C, C γ }, y, ỹ + for all, y,, ỹ R +. Please note that the segment connecting, y an, ỹ can be split up into finitely many parts in such a way, that each part belongs to one of the sets Cγ, S[Cγ ], Cγ, S[Cγ ]. Λ + Analogously one can prove the Lipschitz continuity of the inverse mapping, see also Step 6.

12 88 J. A. Griepentrog, W. Höppner, H.-Chr. Kaiser, J. Rehberg 5. In the sequel we show that the transforming functions Λ γ an Λ γ are Lipschitz continuous. First, we regar Λ γ on C γ, see 3: The function h is Lipschitz continuous ue to the triangle inequality. Net we prove that the partial erivatives of, y k g, y = k are boune. We substitute λ = bouneness of the function y + arctan /y sin α α / y. The cornerstone of the argument is the λ τλ +λ on, + from below an from above by strictly positive constants. Inee, using the relation α/ sin α α, weget +λ arctan λ +λ +λ arctan λ arctan/λ sin α α / for all λ, +. Hence, it remains to show that the following terms in the partial erivatives of are boune: λ τλ +λ λ = j = jy j λ τλ +λ +λ λτλ + τ λ +λ +λ λ τλ +λ +λ an τλ λ +λ λ y = λτλ +λ + τ λ +λ = λτλ +λ τλ +λ +λ. In the calculations we have use 7. Owing to the bouneness of τ an the function, the epressions on the right-han sie are uniformly boune for all λ, +. Net, we regar Λ γ on Cγ, see 4: The function h, see, has boune partial erivatives. Inee, the secon factor in is boune as well as the

13 A bi-lipschitz continuous, volume preserving map 89 partial erivatives of the first factor. Hence, it remains to show that the following terms in the partial erivatives of h are boune: + y ϱ λ λ y = ϱ λ +λ = + λ / an + y ϱ λ λ = jy j ϱ λ +λ = jy + λ /. Here we have use 8. For y γ these terms are boune. Finally, we prove that the partial erivatives of the function, y k g, y = k + y, 3 are boune. Because the first factor in 3 is boune as well as the partial erivatives of the secon factor, it suffices to note that + k j 3 + y if k = l, y = j k j 3 + y if k l. These terms are uniformly boune on the set {, y R + } y γ. 6. In the sequel we show that the transforming functions Λ γ an Λ γ are Lipschitz continuous. First, we regar Λ γ on C, see 9, an efine s = τγ ξ.inorerto make sure that the partial erivatives of the function τ τγ ξ ξ, + τ τγ ξ are boune, it suffices to consier the terms s τ s +τ s s ξ j = τγξ j ξ τ s +τ s 3/ an s τ s s +τ s = τγ ξ τ s +τ s 3/.

14 9 J. A. Griepentrog, W. Höppner, H.-Chr. Kaiser, J. Rehberg These terms are boune for ξ, since the function s τ s +τ s = +τ s τ τ s 4 is boune on,τγ]. Inee, using 7, the right-han sie equals to +τ s τ τ s = s +τ s which is boune ue to the bouneness of the function. Now, we investigate the partial erivatives of the function ξ, + ξ ξ τ τγ ξ = τγ s +τ s ξ. Please note that the fraction in front of ξ is boune by the positive bouns of the function. Hence, it remains to treat the terms an ξ s τγ s +τ s s = +τ s + sτ sτ s +τ s 3/ ξ s τγ s +τ s s ξ j = τγξ j ξ s +τ s + τ sτ s +τ s. 3/ Both epressions are boune for ξ, thanks to the bouneness of the functions an 4. Finally, we regar Λ γ on C ϱγ, see, an efine t = ξ.first,we investigate the partial erivatives of the function ξ, ξ. + ϱ ϱγ ξ The critical terms are ξ t t +ϱ t = ϱγ ϱ tϱ t +ϱ t 3/

15 A bi-lipschitz continuous, volume preserving map 9 an ξ t +ϱ t t = ϱγ ξ j ξ j ξ ϱ tϱ t +ϱ t 3/. For ξ the bouneness of the right-han sie epressions is a consequence of the bouneness of the function t ϱ t +ϱ t = +ϱ t ϱ ϱ t 5 on the interval [,ϱγ]. Using 8, this follows from +ϱ t ϱ ϱ t = +ϱ t / an the fact that ϱ t [,γ]. Net, we investigate the partial erivatives of the function ξ ϱ ϱγ ξ ξ,. + ϱ ϱγ ξ The critical terms are ξ ϱ t t +ϱ t t = ϱγξ j ξ j ξ ϱ t +ϱ t 3/ an ξ ϱ t t t +ϱ t = ϱγϱ t +ϱ t 3/. For ξ the bouneness of the right-han sie epressions follows from the bouneness of the function 5. 6Remark. In the three-imensional case the solution of 6 an the corresponing values of τ an ϱ, see,, are γ =, τγ =, ϱγ = The mappings g an h are etermine by g, y 3, y =, y 3 + y an h, y =y.

16 9 J. A. Griepentrog, W. Höppner, H.-Chr. Kaiser, J. Rehberg Thus, we get,, if =,y =,,y 3,y, 3, 3 y if 5 y, Λ 3, y = 3,y 3,y 3 + y, 3,y 3,y 3 + y,, y 3 if 5 y, 3,y 3,y 3 + y, 3,y 3,y 3 + y,, y 3 if 5 y. The inverse of Λ 3 is given by Λ 3 ξ, = see also Figure.,, if ξ =, =, ξ 4 9,ξ ξ 4 9, ξ 3 if ξ, ξ 3 ξ,ξ 9 3 ξ, ξ 9 3 if ξ, Λ 3 Λ 3 Figure. The map Λ 3 from R 3 onto R 3. 7Remark. In a way our investigation also is a contribution to the general knowlege about the unit cube, see Zong [9]. 8Remark.Are there other geometrical objects than the cyliner an the ball, such that a mapping of this object onto the unit cube eists an has the properties specifie in our Theorem? Is there a complete geometrical characterization of these objects?

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