Representation operators of metric and Euclidian charges Philippe Bouafia To cite this version: Philippe Bouafia. Representation operators of metric and Euclidian charges. General Mathematics [math.GM]. Université Paris Sud - Paris XI, 2014. English. NNT : 2014PA112004. tel-01015943 HAL Id: tel-01015943 https://tel.archives-ouvertes.fr/tel-01015943 Submitted on 27 Jun 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. No d’ordre: THÈSE Présentée pour obtenir LE GRADE DE DOCTEUR EN SCIENCES DE L’UNIVERSITÉ PARIS-SUD XI Spécialité: Mathématiques par Philippe Bouafia École doctorale de Mathématiques de Paris Sud Laboratoire d’Analyse Harmonique Blow-up analysis of multiple valued stationary functions Soutenue le 7 janvier 2014 devant la Commission d’examen: M. Petru Mironescu (Rapporteur, Université Lyon I) M. Camillo De Lellis (Rapporteur, Universität Zürich) M. Guy David (Directeur de thèse, Université Paris Sud) M. Thierry De Pauw (Promoteur de thèse, IMJ) M. Gilles Godefroy (Professeur d’université, IMJ) M. Stefan Wenger (Full professor, University of Fribourg) Thèse préparée au Département de Mathématiques d’Orsay Laboratoire de Mathématiques (UMR 8628), Bât. 425 Université Paris-Sud 11 91 405 Orsay CEDEX Abstract On étudie les fonctions multivaluées vers un espace de Hilbert. Après avoir introduit une bonne notion de p énergie, on donne une définition possible d’espace de Sobolev et on prouve un théorème d’existence des p minimiseurs. Puis on considère les fonctions bivaluées de deux variables, stationnaires pour les déformations au départ et à l’arrivée. On démontre qu’elles sont localement lipschitziennes et on utilise cette régularité pour montrer la convergence forte dans W 1,2 vers leur unique éclatement en un point. L’ensemble de branchement d’une telle fonction est la réunion localement finie de courbes analytiques qui se rencontrent en faisant des angles égaux. Nous donnons aussi un exemple de fonction discontinue et stationnaire seulement pour les déformations au départ. Dans un deuxième temps, on prouve qu’il n’existe pas de rétraction uniformément continue de l’espace des champs vectoriels continus vers le sous-espace de ceux dont la divergence est nulle en un sens distributionnel. On généralise ce résultat en toute codimension en utilisant la notion de m charge et à tout ensemble X Rn vérifiant une hypothèse géométrique mineure. ⊂ Keywords : Stationary multiple valued function, branch set, normal currents, charges, metric calibrations. Abstract We study multiple valued functions with values in a Hilbert space. We introduce a possible definition of Sobolev spaces and the rightful notion of p energy. We prove the existence of p minimizers. Then we consider two-valued real functions of two variables which are station- ary with respect to both domain and range transformations. We prove their local Lipschitz continuity and use it to establish strong convergence in W 1,2 to their unique blow-up at any point. We claim that the branch set of any such function consists of finitely many real analytic curves meeting at nod points with equal angles. We also provide an example showing that stationarity with respect to domain transformations only does not imply continuity. In a second part, we prove that there does not exist a uniformly continuous retraction from the space of continuous vector fields onto the subspace of vector fields whose divergence vanishes in the distributional sense. We then generalise this result using the concept of m charges on any subset X Rn satisfying a mild geometric condition, there is no uniformly ⊂ continuous representation operator for mcharges in X. Keywords : Stationary multiple valued function, branch set, normal currents, charges, metric calibrations. Contents Introduction 8 I Almgren’s multiple valued functions 15 1 Preliminaries 16 1.1 Symmetric powers . 16 1.2 Concatenation and splitting . 18 1.3 Measurability . 19 1.4 Lipschitz extensions . 21 1.5 Differentiability . 28 2 Embeddings 37 n 2.1 Whitney bi-Hölder embedding — The case Y = ℓ2 (K) . 37 2.2 Splitting in case Y = R ............................ 38 n 2.3 Almgren-White locally isometric embedding — The case Y = ℓ2 (R) . 39 ∗ 2.4 Lipeomorphic embedding into Lipy0 (Y ) . 45 3 Sobolev classes 48 3.1 Definition of Lp(X, QQ(Y )) . 48 3.2 Analog of the Fréchet-Kolmogorov compactness Theorem . 49 1 3.3 Definition of Wp (U; QQ(Y )) . 52 3.4 The p energy . 56 3.5 Extension . 62 3.6 Poincaré inequality and approximate differentiability almost everywhere . 64 3.7 Trace . 70 3.8 Analog of the Rellich compactness Theorem . 72 3.9 Existence Theorem . 74 4 Blowup analysis of stationary multiple valued functions 75 4.1 Squeeze and squash variations . 75 4.2 Regularity results . 79 4.3 Blowing up . 84 4.4 Branch set . 90 4.5 Stationary surfaces in R3 ........................... 96 II Representation of charges 98 5 Charges and cohomology 99 5.1 Preleminaries on Federer-Fleming currents . 99 5.2 Topologies on the space of normal currents . 101 5.3 Charges and duality with normal currents . 103 5.4 A representation theorem for charges . 104 5.5 Charges vanishing at infinity . 106 6 Functional analytical tools 108 6.1 Duality of vector lattices . 108 6.2 Representation of abstract M spaces . 110 6.3 Representation of abstract L spaces . 112 6.4 p absolutely summing operators . 114 6.5 Lp spaces . 116 6.6 Grothendieck theorem . 118 6.7 A splitting theorem . 122 7 Continuous representation operators 124 7.1 Main theorem . 124 7.2 Adaptation to charges of positive codimension . 126 8 Appendix to Part II 128 References 132 8 Introduction The Plateau problem When we observe a soap film spanning a wire frame, we can be interested in the principles that enable it to exist in certain geometric configurations and not others. The Plateau problem amounts to asking what are the possible shapes it can assume. The liquid surface acts as an elastic membrane. In a first time, we can say that the total surface energy of such a soap film is proportionnal to its area, though such a statement leaves behind situations with multiple sheets. A shape is a physical solution of the Plateau problem only if it cannot change to a configuration with less energy. Therefore, we will be interested in certain generalized surfaces of R3 that are stable, i.e stable critical points of the area functional. The first physicist to have studied the geometry of soap films appears to be the Belgian Joseph Plateau. In his honor, various mathematical questions dealing with the geometry of soap-film-like surfaces are referred to as the Plateau problem. The first difficulty is that there is no universal agreement on what is meant by a “surface” and its “area”. Here is a list of different formulations, to see how this problem led to a bushy mathematical literature 9 • immersed surfaces: the first progress in this direction was independently achieved in the thirties by Tibor Radó and Jesse Douglas. Both relied on setting up mini- mization problems and complex analysis. However, their work does not extend in higher dimensions. • sets with finite perimeter, introduced by De Giorgi. • rectifiable currents: introduced by Federer and Flemming. They generalize sets with finite perimeter to codimensions greater than 1. • varifolds: they can useful for the study of stationary surfaces which are not nec- essarily minimizing. However, they lack a boundary operator. • Almgren’s (M, 0, δ) minimal sets: they are accurate models of physical solutions, but the lack of a compactness theorem makes it difficult to prove the existence of an area minimizer. Establishing the existence and regularity of solutions to the Plateau problem in general dimensions and codimensions is one of the most challenging problem in geometric measure theory. In the eighties, Frederick J. Almgren proved the following theorem: Theorem 1. Let T be an m dimensional mass minimizing current in Rn. Consider the set reg(T ) of all points x supp T supp ∂T for which there exists an open neighborhood U of x such that supp T ∈ supp ∂T\ U is an C∞ manifold. Then the set sing(T ) := supp T (supp ∂T reg(T\)) has Hausdorff∩ dimension less than m 2. \ ∪ − This theorem is optimal, as shown by the case of complex varieties. Its proof has required development of several new geometric and analytic techniques, central among which is the utilization of multiple valued functions to study branching phenomena. We believe stationary varifolds can be well approximated by stationary multiple valued functions, and that one can deduce from this approximation some regularity results. The following question is open, even in the case Q = 2. Conjecture 1. Suppose (A) V is a 2 dimensional stationary varifold in R3 with integer multiplicities, (B) ε [0, 1), Q N∗, ∈ ∈ (C) V B3(0, 1) π(Q + ε) k k ≤ Then there exists γ (0, 1) such that H 2(sing V B3(0, γ)) = 0. ∈ ∩ Multiple valued functions Let Y be a metric space and Q be an integer. We define QQ(Y ) to be the subset of measures in Y : Q Q (Y ) := y : y , . , y Y . Q i 1 Q ∈ Xi=1 J K 10 We will adopt the compact notation Q y1, . , yQ = yi . Xi=1 J K J K Elements of QQ(Y ) are just unordered Q tuples of points in Y .