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Anisotropic Energies in Geometric Measure Theory

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Anisotropic Energies in Geometric Measure Theory A NISOTROPIC E NERGIESIN G EOMETRIC M EASURE T HEORY Dissertation zur Erlangung der naturwissenschaftlichen Doktorwürde (Dr. sc. nat.) vorgelegt der Mathematisch-naturwissenschaftlichen Fakultät der Universität Zürich von antonio de rosa aus Italien Promotionskommission Prof. Dr. Camillo De Lellis (Vorsitz) Prof. Dr. Guido De Philippis Prof. Dr. Thomas Kappeler Prof. Dr. Benjamin Schlein Zürich, 2017 ABSTRACT In this thesis we focus on different problems in the Calculus of Variations and Geometric Measure Theory, with the common peculiarity of dealing with anisotropic energies. We can group them in two big topics: 1. The anisotropic Plateau problem: Recently in [37], De Lellis, Maggi and Ghiraldin have proposed a direct approach to the isotropic Plateau problem in codimension one, based on the “elementary” theory of Radon measures and on a deep result of Preiss concerning rectifiable measures. In the joint works [44],[38],[43] we extend the results of [37] respectively to any codimension, to the anisotropic setting in codimension one and to the anisotropic setting in any codimension. For the latter result, we exploit the anisotropic counterpart of Allard’s rectifiability Theorem, [2], which we prove in [42]. It asserts that every d-varifold in Rn with locally bounded anisotropic first variation is d-rectifiable when restricted to the set of points in Rn with positive lower d-dimensional density. In particular we identify a necessary and sufficient condition on the Lagrangian for the validity of the Allard type rectifiability result. We are also able to prove that in codimension one this condition is equivalent to the strict convexity of the integrand with respect to the tangent plane. In the paper [45], we apply the main theorem of [42] to the minimization of anisotropic energies in classes of rectifiable varifolds. We prove that the limit of a minimizing sequence of varifolds with density uniformly bounded from below is rectifiable. Moreover, with the further assumption that all the elements of the minimizing sequence are integral varifolds with uniformly locally bounded anisotropic first variation, we show that the limiting varifold is also integral. 2. Stability in branched transport: Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. Given a flow (traffic path) that transports a given measure µ- onto a target measure µ+, along a 1-dimensional network, the transportation cost per unit length is supposed in these models to be proportional to a concave power α 2 (0, 1) of the intensity of the flow. The transportation cost is called α-mass. In the paper [27] we address an open problem in the book [15] and we improve the stability for optimal traffic paths in the Euclidean space Rn with respect to variations - + 1 of the given measures (µ , µ ), which was known up to now only for α > 1 - n . We 1 prove it for exponents α > 1 - n-1 (in particular, for every α 2 (0, 1) when n = 2), for a fairly large class of measures µ+ and µ-. The α-mass is a particular case of more general energies induced by even, subadditive, and lower semicontinuous functions H : R ! [0, ) satisfying H(0) = 0. In the paper [28], we prove that the lower semicontinuous envelope of these energy functionals defined on polyhedral chains coincides on rectifiable1 currents with the H-mass. ZUSAMMENFASSUNG In dieser Arbeit behandeln wir verschiedene Probleme der Variationsrechnung und der Geometrischen Masstheorie, wobei wir jeweils die anisotropen Energien betrachten. Wir können die Probleme in zwei grosse Themen unterteilen: 1. Das anisotrope Plateau Problem: Vor kurzer Zeit haben De Lellis, Maggi und Ghiraldin in [37] eine direkte Betra- chtungsweise des isotropen Plateau Problems in der Kodimension eins eingeführt, basierend auf der “elementaren” Theorie der Radonmasse und einem tiefgreifenden Ergebnis von Preiss über rektifizierbare Masse. In den gemeinsamen Arbeiten [44],[38],[43] erweitern wir die Ergebnisse von [37] jeweils auf eine beliebige Kodimension, auf den anisotropen Fall in der Kodimension eins und auf den anisotropen Fall in jeder Kodimension. Für das letztere Resultat nutzen wir das anisotrope Gegenstück von Allards Rekti- fizierbarkeitstheorem, [2], das wir in [42] beweisen. Es sagt aus, dass jeder d-Varifold in Rn mit lokal beschränkter anisotroper erster Variation d-rektifizierbar ist, wenn er auf die Menge von Punkten in Rn mit positiver niedriger d-dimensionaler Dichte beschränkt wird. Insbesondere identifizieren wir eine notwendige und hinreichende Bedingung an die Anisotropie für die Gültigkeit des Rektifizierbarkeitstheorem heraus. Wir beweisen ausserdem, dass diese Bedingung in Kodimension eins gleichbedeutend zu der strengen Konvexität des Integrands in Bezug auf die Tangentialebene ist. In der Arbeit [45] wenden wir den Hauptsatz von [42] auf die Minimierung anisotroper Energien in Klassen von rektifizierbaren Varifolds an. Wir beweisen, dass der Gren- zwert einer Minimierungsfolge von Varifolds mit gleichmässig von unten begrenzter Dichte rektifizierbar ist. Darüber hinaus zeigen wir unter der weiteren Annahme, dass alle Elemente der Minimierungsfolge integrale Varifolds mit gleichmässig lokal beschränkter anisotroper erster Variation sind, und wir zeigen, dass der Grenzwert der Folge auch integral ist. 2. Stabilität im verzweigten Transport: Modelle mit verzweigten Strukturen werden eingesetzt, um mehrere Supply-Demand- Systeme wie die Struktur der Nerven eines Blattes, das System der Wurzeln eines Baumes oder Nerven- oder Herz-Kreislauf-Systeme zu beschreiben. Bei gegebenem Fluss, der ein bestimmtes Mass µ- über ein 1-dimensionales Strom auf ein Zielmass µ+ transportiert, werden die Transportkosten pro Längeneinheit in diesen Modellen proportional zu einer konkaven Potenzfunktion der Potenz α 2 (0, 1) der Intensität des Flusses sein. Die Transportkosten nennen wir α-Masse. In der Arbeit [27] beziehen wir uns auf ein offenes Problem im Buch [15] und verbessern die Stabilität für optimale Transporte im euklidischen Raum Rn, bezo- - + 1 gen auf Variationen der gegebene Masse (µ , µ ), die bisher nur für α > 1 - n 1 bekannt war. Wir beweisen es für Exponenten α > 1 - n-1 (insbesondere für jedes α 2 (0, 1) wenn n = 2) und für eine grosse Klasse von Massen µ+ und µ-. Die α-Masse ist ein besonderer Fall von allgemeineren Energien, die durch gerade, subadditive und unterhalbstetige Funktionen H : R ! [0, ) mit H(0) = 0 generiert werden. In der Arbeit [28] beweisen wir, dass die unterhalbstetige Hülle dieser auf polyedrischen Strömen definierten Energiefunktionen auf1 rektifizierbaren Strömen dasselbe ist wie die H-Masse. To strive, to seek, to find, and not to yield — A. L. Tennyson, Ulysses ACKNOWLEDGMENTS In the first place, I would like to thank my two advisors Prof. Dr. Camillo De Lellis and Prof. Dr. Guido De Philippis, great examples and sources of inspiration for my research. They have been brilliant teachers and mentors, but especially friends. I have enjoyed the time spent together, the discussions on several topics, the running training sessions, the basketball matches and the great number of actvities done together. I thank all the friends and colleagues who accompanied me along these three years. I have always felt at home with their friendship and support. Particular affection goes to the friends of my research group, who gave me a great time in Zurich and contributed to my mathematical growth with many stimulating discussions. I wish to thank my girlfriend Marika, who has filled up my days with love, bearing the distance with patience. My best gratitude goes to my family. My brother Giovanni has been a constant support, a friend and an inspiration. My parents Giuseppe and Concita made everything possible with their scarifices and their never-ending love. viii CONTENTS 1 introduction1 1.1 The anisotropic Plateau problem 1 1.2 Stability in branched transport 4 i the anisotropic plateau problem7 2 notation of part i9 3 existence and regularity results for the anisotropic plateau prob- lem 15 3.1 Introduction 15 3.2 Setting and preliminaries 15 3.3 Isotropic Plateau problem in higher codimension 16 3.4 Anisotropic Plateau problem in codimension one 31 3.5 Anisotropic Plateau problem in higher codimension 42 3.6 Application 1: Solution to Harrison-Pugh formulation 51 3.7 Application 2: Solution to David formulation 54 3.8 Application 3: Solution to Reifenberg formulation 56 4 the anisotropic counterpart of allard’s rectifiability theorem 61 4.1 Introduction 61 4.2 Preliminary results 63 4.3 Intermediate lemmata 66 4.4 Proof of the main theorem 72 4.5 Proof of the equivalence of the ellipticity definitions in codimension one 73 4.6 Appendix: First variation with respect to anisotropic integrands 76 4.7 Appendix: Proof of the rectifiability Lemma 77 5 compactness for integral varifolds: the anisotropic setting 81 5.1 Introduction 81 5.2 Setting 82 5.3 Preliminary results 83 5.4 An integrality theorem 86 5.5 Proof of the main theorem 90 ii stability in branched transport 97 6 notation of part ii 99 7 improved stability of optimal traffic paths 103 7.1 Introduction 103 7.2 Setting and preliminaries 106 7.3 Known results on optimal traffic paths 108 7.4 Lower semi-continuity of the α-mass 113 7.5 Ideas for the proof of the main theorem 117 7.6 Preliminaries for the proof of the main theorem 122 7.7 Proof of the main theorem 131 8 relaxation of functionals on polyhedral chains 143 8.1 Introduction 143 8.2 Setting and Main Result 143 8.3 Proof of the lower semicontinuity 145 8.4 Proof of the polyhedral approximation 149 8.5 Proof of the representation on all flat chains with finite mass 155 bibliography 159 INTRODUCTION 1 1.1 the anisotropic plateau problem The famous Plateau Problem can be roughly stated as follows: Given a (d - 1)-dimensional boundary in Rn, find a d-dimensional surface with least area among all the surfaces spanning the given boundary.
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