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Pacific Journal of Mathematics Vol 230 Issue 2, Apr 2007 PACIFIC JOURNAL OF MATHEMATICS Pacific Journal of Mathematics Volume 230 No. 2 April 2007 Pacific Journal of Mathematics 2007 Vol. 230, No. 2 Volume 230 No. 2 April 2007 PACIFIC JOURNAL OF MATHEMATICS http://www.pjmath.org Founded in 1951 by E. F. Beckenbach (1906–1982) F. Wolf (1904–1989) EDITORS V. S. Varadarajan (Managing Editor) Department of Mathematics University of California Los Angeles, CA 90095-1555 pacifi[email protected] Vyjayanthi Chari Darren Long Sorin Popa Department of Mathematics Department of Mathematics Department of Mathematics University of California University of California University of California Riverside, CA 92521-0135 Santa Barbara, CA 93106-3080 Los Angeles, CA 90095-1555 [email protected] [email protected] [email protected] Robert Finn Jiang-Hua Lu Jie Qing Department of Mathematics Department of Mathematics Department of Mathematics Stanford University The University of Hong Kong University of California Stanford, CA 94305-2125 Pokfulam Rd., Hong Kong Santa Cruz, CA 95064 fi[email protected] [email protected] [email protected] Kefeng Liu Alexander Merkurjev Jonathan Rogawski Department of Mathematics Department of Mathematics Department of Mathematics University of California University of California University of California Los Angeles, CA 90095-1555 Los Angeles, CA 90095-1555 Los Angeles, CA 90095-1555 [email protected] [email protected] [email protected] PRODUCTION pacifi[email protected] Paulo Ney de Souza, Production Manager Silvio Levy, Senior Production Editor Alexandru Scorpan, Production Editor SUPPORTING INSTITUTIONS ACADEMIA SINICA, TAIPEI UNIVERSIDAD DE LOS ANDES UNIV. OF CALIF., SANTA CRUZ CALIFORNIA INST. OF TECHNOLOGY UNIV. OF ARIZONA UNIV. OF HAWAII INST. DE MATEMÁTICA PURA E APLICADA UNIV. OF BRITISH COLUMBIA UNIV. OF MONTANA KEIO UNIVERSITY UNIV. OF CALIFORNIA, BERKELEY UNIV. OF NEVADA, RENO MATH. SCIENCES RESEARCH INSTITUTE UNIV. OF CALIFORNIA, DAVIS UNIV. OF OREGON NEW MEXICO STATE UNIV. UNIV. OF CALIFORNIA, IRVINE UNIV. OF SOUTHERN CALIFORNIA OREGON STATE UNIV. UNIV. OF CALIFORNIA, LOS ANGELES UNIV. OF UTAH PEKING UNIVERSITY UNIV. OF CALIFORNIA, RIVERSIDE UNIV. OF WASHINGTON STANFORD UNIVERSITY UNIV. OF CALIFORNIA, SAN DIEGO WASHINGTON STATE UNIVERSITY UNIV. OF CALIF., SANTA BARBARA These supporting institutions contribute to the cost of publication of this Journal, but they are not owners or publishers and have no respon- sibility for its contents or policies. See inside back cover or www.pjmath.org for submission instructions. Regular subscription rate for 2007: $425.00 a year (10 issues). Special rate: $212.50 a year to individual members of supporting institutions. Subscriptions, requests for back issues from the last three years and changes of subscribers address should be sent to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163, U.S.A. Prior back issues are obtainable from Periodicals Service Company, 11 Main Street, Germantown, NY 12526-5635. The Pacific Journal of Mathematics is indexed by Mathematical Reviews, Zentralblatt MATH, PASCAL CNRS Index, Referativnyi Zhurnal, Current Mathematical Publications and the Science Citation Index. The Pacific Journal of Mathematics (ISSN 0030-8730) at the University of California, c/o Department of Mathematics, 969 Evans Hall, Berkeley, CA 94720-3840 is published monthly except July and August. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices. POSTMASTER: send address changes to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163. PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS at the University of California, Berkeley 94720-3840 A NON-PROFIT CORPORATION Typeset in LATEX Copyright ©2007 by Pacific Journal of Mathematics PACIFIC JOURNAL OF MATHEMATICS Vol. 230, No. 2, 2007 A RESULT ABOUT C3-RECTIFIABILITY OF LIPSCHITZ CURVES SILVANO DELLADIO 1+k Let γ0 :[a, b] → ޒ be Lipschitz. Our main result provides a sufficient condition, expressed in terms of further accessory Lipschitz maps, for the 3 C -rectifiability of γ0([a, b]). 1. Introduction A set in ޒn is C3-rectifiable if Ᏼ1-almost all of it can be covered by countably many curves of class C3 embedded in ޒn. The main goal of this paper is to prove the following result. Theorem 1.1. Let there be given Lipschitz maps 1+k 1+k 1+k γ0, γ1 :[a, b] → ޒ and γ2 = (γ2>, γ2⊥) :[a, b] → ޒ × ޒ and a function ω :[a, b] → {±1} such that 0 0 (1-1) γ0(t) = ω(t) kγ0(t)k γ1(t), 0 0 0 0 (1-2) (γ0(t), γ1(t)) = ω(t) k(γ0(t), γ1(t))k γ2(t) 3 for almost every t ∈ [a, b]. Then γ0([a, b]) is a C -rectifiable set. Remark. In the special case when ω := 1 while γ0 is regular and at least of class 2 C , the conditions (1-1) and (1-2) say that γ1(t) and γ2(t) are, respectively, the unit tangent vector of γ0 at t and the unit tangent vector of (γ0, γ1) at t. This remark is at the root of the applications to geometric variational problems mentioned below. Theorem 1.1 should be considered a step forward in a project, stated in [Delladio 2005], aimed at providing sufficient conditions for the C H -rectifiability of a n- dimensional rectifiable set. Results concerning the case H = 2 were first obtained in [Anzellotti and Serapioni 1994; Delladio 2003; Fu 1998], but subtle mistakes seriously invalidating their proofs were discovered later [Delladio 2004; Fu 2004]. Then the paper [Delladio 2005], cleaning-up the simplest case n = 1 and H = 2, followed. Our future efforts will be aimed at extending the theory to any value of n MSC2000: primary 49Q15, 53A04; secondary 26A12, 26A16, 28A75, 28A78, 54C20. Keywords: rectifiable sets, geometric measure theory, Whitney extension theorem. 257 258 SILVANO DELLADIO and H. Joint work with Joseph Fu on C2-rectifiability in all dimensions (invoking slicing in order to reduce to dimension one) is in progress. Further work is in progress to apply these results to geometric variational prob- lems via geometric measure theory and more precisely through the notion, first in- troduced in [Anzellotti et al. 1990], of a generalized Gauss graph. Former achieve- ments in this direction include [Delladio 2001] (a somehow surprising application to differential geometry context), [Anzellotti and Delladio 1995] (an application to Willmore problem) and [Delladio 1997] (an application to a problem introduced in [Bellettini et al. 1993]). These last two papers followed the idea by De Giorgi of relaxing the functional with respect to L1-convergence of the domains of in- tegration. Now we expect that our results can be applied to handle functionals with integrands involving curvatures with their derivatives and, in particular, to get explicit representation formulas after relaxation. 2 The proof of Theorem 1.1 starts from the C -rectifiability of γ0([a, b]), which is guaranteed by condition (1-1), as shown in [Delladio 2005]. The problem is 2 reduced in Section 2 to proving that γ0([a, b]) intersects the graph of any C map f : ޒ → (ޒu)⊥ (u ∈ ޒ1+k, kuk = 1) in a C3-rectifiable set. From the first and second derivatives of f expressed in terms of the γi , we obtain in Section 3 a second order Taylor-type formula for f with the remainder in terms of the γi . Theorem 1.1 then follows by the Whitney Extension Theorem, also involving a Lusin-type argument (Section 4). Finally, the absolute curvature for a one-dimensional C2-rectifiable set P is defined and proved to be approximately differentiable almost everywhere whenever P is C3-rectifiable (Section 5). 2. Reduction to graphs By virtue of the main result stated in [Delladio 2005], the equality (1-1) implies 2 that γ0([a, b]) is C -rectifiable. As a consequence, there must be countably many unit vectors 1+k u j ∈ ޒ and maps of class C2 ⊥ f j : ޒ → (ޒu j ) such that 1 [ ] \ S = Ᏼ γ0( a, b ) j G f j 0 where := + | ∈ G f j xu j f j (x) x ޒ . A RESULT ABOUT C3-RECTIFIABILITYOFLIPSCHITZCURVES 259 [ ] ∩ 3 Hence we need only show that the sets γ0( a, b ) G f j are C -rectifiable. In other words, Theorem 1.1 becomes an immediate corollary of the following result. Theorem 2.1. Let γ0, γ1, γ2 be as in Theorem 1.1. Consider a map f : ޒ → (ޒu)⊥ (u ∈ ޒ1+k, kuk = 1) of class C2 and define G f := {xu + f (x) | x ∈ ޒ}. 3 Then the set G f ∩ γ0([a, b]) is C -rectifiable. In this section we take the first step toward the proof of Theorem 2.1, which will be concluded later in Section 4. Define := −1 ∩ ∈ [ ] 0 0 0 6= L γ0 (G f ) t a, b γ0(t), γ1(t) exist, γ0(t) 0, (1-1) and (1-2) hold . By Lusin’s Theorem, for any given real number ε > 0, there exists a closed subset Lε of L such that 0 1 (2-1) γ0|Lε and ω|Lε are continuous and ᏸ (L\Lε) ≤ ε. ∗ If Lε denotes the set of the density points of Lε, then ∗ (2-2) Lε ⊂ Lε since Lε is closed. The equality 1 ∗ (2-3) ᏸ (Lε\Lε ) = 0 also holds by a celebrated result of Lebesgue. In the special case that L has measure ∗ zero, we take Lε := ∅, hence Lε := ∅. Now observe that ∩ [ ] \ ∗ ⊂ −1 ∩ [ ]\ ∗ G f γ0( a, b ) γ0(Lε ) γ0 γ0 (G f ) a, b Lε hence 1 ∩ [ ] \ ∗ ≤ 1 −1 ∩ [ ]\ ∗ Ᏼ G f γ0( a, b ) γ0(Lε ) Ᏼ γ0 γ0 (G f ) a, b Lε Z Z 0 0 ≤ kγ0k = kγ0k ≤ ε Lip(γ0), −1 ∩[ ]\ ∗ \ ∗ γ0 (G f ) a,b Lε L Lε which implies 1 ∩ [ ] \ S∞ ∗ = Ᏼ G f γ0( a, b ) j=1 γ0(L1/j ) 0.
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