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Triangulation of Diffeomorphisms Math. Ann. (2017) 368:1133–1169 DOI 10.1007/s00208-016-1426-x Mathematische Annalen Triangulation of diffeomorphisms Tadeusz Iwaniec1,2 · Jani Onninen1,3 Received: 10 August 2015 / Revised: 26 April 2016 / Published online: 23 May 2016 © Springer-Verlag Berlin Heidelberg 2016 Abstract Is it possible to approximate a diffeomorphism of Euclidean domains with piecewise affine homeomorphisms, locally uniformly up to the first order derivatives? The answer is yes. However, any effort to provide a rigorous and clear proof reveals the complexity of this question, especially in higher dimensions. It is the objective of the present paper to formulate this question in its greatest generality, as well as to provide all details for the affirmative answer, Theorem 1.1. A novelty, which has broader applications, is the construction of selfsimilar isotropic triangulation of the Euclidean domains, Theorem 1.2. Mathematics Subject Classification Primary 41A45; Secondary 57R05 Communicated by Y. Giga. T. Iwaniec was supported by the United States NSF Grant DMS-1301558. J. Onninen was supported by the United States NSF Grant DMS-1301570. B Tadeusz Iwaniec [email protected] Jani Onninen [email protected] 1 Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA 2 Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland 3 Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, 40014 Jyväskylä, Finland 123 1134 T. Iwaniec, J. Onninen Contents 1 Introduction ............................................ 1134 1.1 Two theorems ......................................... 1134 1.2 A brief historical account and overview ........................... 1135 2 Notation and definitions ...................................... 1137 2.1 Model simplexes ....................................... 1138 2.1.1 Model triangulation ................................... 1138 2.1.2 Resizing model simplexes ............................... 1138 2.2 Isotropic dyadic grid G = G (X) ............................... 1139 2.3 Selfsimilar triangulation S (X) ................................ 1139 3 Rectangles and dyadic cubes .................................... 1141 3.1 Hyper-rectangles ....................................... 1141 3.2 Faces .............................................. 1141 3.2.1 Interior and the boundary of a face ........................... 1142 3.3 Bisects of a rectangle ..................................... 1142 3.4 Dyadic cubes in Rn ...................................... 1143 3.5 Dyadic faces .......................................... 1145 4 The dyadic grid (the art of bisecting) ............................... 1147 4.1 Dyadigons ........................................... 1147 4.2 Refining a dyadigon near its boundary ............................ 1147 4.3 Dyadic refinement of the open cover U ........................... 1149 4.4 Exhaustion of X with homogeneous dyadigons ........................ 1150 4.5 Matching the boundary cubes ................................ 1151 4.5.1 Truncating the dyadigons ................................ 1152 4.6 Isotropic dyadic grid G = G (X) ............................... 1152 4.7 Faces of an isotropic grid ................................... 1154 5 Triangulation of the dyadic grid .................................. 1156 5.1 Simplexes (as cones with apex and base) ........................... 1156 5.2 Triangulation of a face F ∈ F d (X) .............................. 1157 5.2.1 Naturally inherited triangulation ............................ 1157 5.3 Simplicial partition of F d (X) ................................ 1157 5.3.1 Simplicial partition of F 1(X) ............................. 1158 5.4 The induction step (construction of S d (X), 1 d n) .................. 1158 5.5 S d (X) is a simplicial partition ................................ 1160 5.5.1 The case (ii) ....................................... 1161 5.6 S (X) is selfsimilar ...................................... 1161 6 Approximation of f ........................................ 1162 6.1 An open cover by small balls ................................. 1163 6.2 Proof of inequality (1) ..................................... 1164 7 Injectivity of f◦ .......................................... 1165 7.1 Distortion of f◦ ........................................ 1165 7.2 Injectivity of f◦ ........................................ 1165 8 Approximation of the inverse map ................................. 1166 References ............................................... 1167 1 Introduction 1.1 Two theorems The primary goal of the present paper is to establish a piecewise affine Lagrange interpolation of a diffeomorphism. Theorem 1.1 (Piecewise affine approximation) Let f : X −onto→ Y be a C 1-smooth diffeomorphism of domains in Rn,n 1, and g : Y −onto→ X denote its inverse. Suppose 123 Triangulation of diffeomorphisms 1135 we are given real continuous functions: ε = ε(x)>0 defined on X and δ = δ(y)>0 onto defined on Y. Then there exists a piecewise affine homeomorphism f◦ : X −→ Y (actually Lagrange interpolation of f ) which satisfies the inequality: || Df(x) − Df◦(x) || ε(x), for almost every x ∈ X. (1) onto Its inverse, denoted by g◦ : Y −→ X, satisfies analogous inequality: || Dg(y) − Dg◦(y) || δ(y), for almost every y ∈ Y . (2) The key ingredient in the proof, which might be of independent interest, is the so-called isotropic selfsimilar triangulation of X. Theorem 1.2 (Isotropic selfsimilar triangulation) Given an open cover U = n {Uα}α∈A of a domain X ⊂ R , there is a locally finite isotropic selfsimilar trian- gulation of X which is a refinement of U . The explanation for Theorem 1.2 is that there is a finite number of so-called model sim- plexes; the others are obtained by rescaling/resizing them. Here the resized simplexes are the only building blocks allowed for the construction of the simplicial partition of X (also called triangulation). They posses an important geometric property of being similar by means of common degree of flatness. 1.2 A brief historical account and overview Consider a homeomorphism f : X −onto→ Y between domains X, Y ⊂ Rn together with a simplicial partition of X. The nodal points of the partition are the vertices of into its simplexes. A Lagrange interpolant is a map f◦ : X −→ Rn which is affine on each simplex and agrees with f at the nodal points. In dimension n = 1, the Lagrange onto interpolation of f results in a homeomorphism f◦ : X −→ Y. In higher dimensions this technique does not work so well. Usually, injectivity is lost. It became apparent that some specific simplicial partitions of X must be administered. Concerning the type of convergence, in case of a general homeomorphism f : X −onto→ Y without any differentiability assumptions, the uniform convergence is the only possible way to approximate f with piecewise affine homeomorphisms. This is what topologists have done. • For dimension n = 2, see Radó [42]. • For dimension n = 3, see Moise [38, Theorem 2], Moise [37] and Bing [10, Theorem 2]. • For dimension n = 4, Donaldson and Sullivan [17, Corollary on page 183] con- structed a homeomorphism f : B4 → R4 which cannot be approximated locally uniformly by piecewise affine homeomorphisms. • For dimensions n 5, a patient reader may find some affirmative answers to the question of uniform approximation of homeomorphisms with piecewise affine maps in [11,15,30,31]. 123 1136 T. Iwaniec, J. Onninen • It should be stressed in this occurrence that the paper [17] does not address any question of diffeomorphic approximation of mappings in the Sobolev class W 1,p(X, Y ),1< p < ∞. In fact this question in dimension n 3 still remains open. Nowadays Sobolev homeomorphisms are of great interest, largely in connection with mathematical models of Nonlinear Elasticity (NE) [3,5,14,35,47,48] and ana- lytical foundations of Geometric Function Theory (GFT) [1,4,25,45,46,49,50]. In most recent years [6,7] and [19] the problem of approximating a Sobolev home- omorphism h ∈ W 1,p(X, Y), 1 < p < ∞, (in the strong Sobolev topology) with diffeomorphisms became ever more important. Let us state the problem in its greatest generality. : X −onto→ Y W 1,p(X, Y) Problem 1.3 Given a homeomorphism h of Sobolev class loc , onto 1 < p < ∞. Does there exist a sequence of diffeomorphisms h j : X −→ Y such that: • − ∈ W 1,p(X, Rn) = , ,... h h j 0 for j 1 2 • h j ⇒ h uniformly •||Dh j − Dh ||L p(X) → 0? In dimension n = 2 this question has been completely solved, in the affirmative [23, 24], by using nonlinear elliptic PDEs (p-harmonic replacements). A piecewise affine variant of this result is straightforward from the present Theorem 1.1. Earlier efforts [9, 39] to approximate h directly with piecewise affine mappings h j turned out to be ineffective. The point is that knowing h j at the vertices of a simplicial partition (no matter how fine a partition is) is insufficient to ensure even local injectivity. The novelty of the approach in [23,24] can be outlined briefly as follows. We consider partition of X by cells (curved squares) as opposed to (straight) triangles that are used in the affine approximations. In fact the cells are the preimages under h of straight squares in the target domain Y. As a first step we replace h on every cell with a diffeomorphism, say h∗, that agrees with h on the boundary of the cell. Naturally, h∗remains a homeomorphism of X −onto→ Y. However, in order to maintain h∗ in close W 1,p-proximity to h the p-harmonic energy of h∗ must be as small as possible on every cell. That is why we choose h∗ to be the p-harmonic
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