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Regularity Properties of Two-Phase Free Boundary Problems

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Regularity Properties of Two-Phase Free Boundary Problems Regularity properties of two-phase free boundary problems ERIK LINDGREN Doctoral Thesis Stockholm, Sweden 2009 TRITA-MAT-09-MA-07 KTH ISSN 1401-2278 Institutionen för Matematik ISRN KTH/MAT/DA 09/05-SE 100 44 Stockholm ISBN 978-91-7415-288-3 SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan fram- lägges till offentlig granskning för avläggande av teknologie doktorsexamen i matematik fredagen den 5 juni 2009 kl 14.00 i sal F3, Kungl Tekniska högskolan, Lindstedtsvägen 26, Stockholm. c Erik Lindgren, 2009 Tryck: Universitetsservice US AB iii Abstract This thesis consists of four papers which are all related to the regularity prop- erties of free boundary problems. The problems considered have in common that they have some sort of two-phase behaviour. In papers I-III we study the interior regularity of different two-phase free bound- ary problems. Paper I is mainly concerned with the regularity properties of the free boundary, while in papers II and III we devote our study to the regularity of the function, but as a by-product we obtain some partial regularity of the free boundary. The problem considered in paper IV has a somewhat different nature. Here we are interested in certain approximations of the obstacle problem. Two major differences are that we study regularity properties close to the fixed boundary and that the problem converges to a one-phase free boundary problem. iv Sammanfattning Denna avhandling består av fyra artiklar i vilka vi studerar problem relaterade till frirandsproblem med två faser. Gemensamt för alla artiklar är att den nivåyta som studeras utgör varken ett minimum eller ett maximum för funktionen i fråga. Därför kan vi säga att de alla har ett slags två-fas-beteende. I de tre första artiklarna studerar vi inre regularitetsegenskaper hos olika fri- randsproblem med två faser. Artikel I handlar om regularitetsegenskaperna hos den fria randen, medan artiklarna II och III mestdels fokuserar på funktionens regula- ritet. Den fjärde artikeln skiljer sig avsevärt från de tre första. Vi studerar här en approximation av det så kallade hinderproblemet. Skillnaden mot de tre första artiklarna består dels i att vi intresserar oss för regulariteten nära den fixa randen, dels i att problemet konvergerar mot ett en-fas problem. Acknowledgements During my time at KTH I have been fortunate to get financial support not only for employment, but also for travelling. Therefore, I would like to ex- press my gratitude to the ESF program GLOBAL, grant KAW 2005.0098 from the Knut and Alice Wallenberg Foundation, the Royal Swedish Academy of Science and the STINT foundation. While being a PhD student I have encountered many different people and personalities. Some of them I am more grateful to than others. In particular, there are certain persons that need to be mentioned. Obviously, I am deeply indebted to my supervisor Henrik Shahgholian, a born optimist. He has always encouraged me, even at times when I refused to listen to what he said, something that happened every now and then. Many of the ideas I have used originate from him, and without his support, and sometimes also his pressure, this thesis would not exist. In addition, I would like to thank Henrik’s former student Arshak Pet- rosyan whom I visited in the autumn 2005. Arshak was the best host and tutor you could imagine. He spent an enormous amount of time explaining and discussing things with me. I really learned a lot from him. Thanks to Arshak the visit resulted in a paper contained in this thesis. Without my fellow students Anders Edquist and Farid Bozorgnia I would not be where I am today. Thank you guys for not making me work. I think both me and Anders can agree on that if we hadn’t had the same office for the last three years we probably would have finished a long time ago. Regarding Farid, I believe that if he did not have tea 35 times a day we might actually have gotten some work done. Furthermore, I wish to thank those who have read my thesis and given me useful suggestions. This has been important for me during the preparation of my thesis. Evidently, I wouldn’t be where I am now without my family. You have v vi ACKNOWLEDGEMENTS always encouraged me to do (almost) whatever I want. For this and for all the other things you have done for me I am most thankful. Last but not least, thank you Erica. I am very grateful for everything, in particular that you can stand me even when I am as annoying as I sometimes can be. Erik Lindgren Stockholm, February 2009 Contents Acknowledgements v Contents vii Introduction and summary 1 Free boundary problems 1 1.1 The obstacle problem . 1 1.2 Temperature control: the two-phase obstacle problem . 4 1.3 Ice melting: the Stefan problem . 6 1.4 A more mathematical description of FBPs . 7 1.5 The focus of the studies . 8 1.6 Two important monotonicity formulae . 8 1.7 Interior regularity of the function . 9 1.8 Interior regularity of the free boundary . 12 1.9 Regularity properties near the fixed boundary . 14 2 Overview of Paper I 17 2.1 Introduction and basic properties . 17 2.2 Blow-ups . 18 2.3 Differentiability and C1 regularity . 19 3 Overview of Paper II 21 3.1 Introduction and general properties . 21 3.2 Known results . 22 3.3 General idea in the proof of the optimal growth . 22 3.4 (n − 1)-dimensional Hausdorff measure estimates . 23 vii viii CONTENTS 4 Overview of Paper III 25 4.1 Introduction and some basic properties . 25 4.2 The scaling argument . 26 4.3 Estimates of the flatness module . 27 5 Overview of Paper IV 29 5.1 Introduction . 29 5.2 C1,1-estimates up to the fixed boundary . 30 5.3 C1 regularity close to the fixed boundary . 31 5.4 Uniform C1 regularity close to the fixed boundary . 31 References 33 Scientific papers Paper I Regularity of the free boundary in a two-phase semilinear problem in two dimensions (joint with Arshak Petrosyan) Indiana Univ. Math. J. 57 (2008), p. 3397–3418 Paper II A two-phase obstacle-type problem for the p-Laplacian (joint with Anders Edquist) Calculus of Variations, DOI 10.1007/s00526-008-0212-3 Paper III On the two-phase membrane problem with coefficients below the Lip- schitz threshold (joint with Anders Edquist and Henrik Shahgholian) Accepted for publication in Annales de l’Institut Henri Poincaré - Analyse non linéaire Paper IV On the penalized obstacle problem in the unit half ball Preprint Chapter 1 Introduction to free boundary problems In this chapter, the reader will be given a brief introduction to free boundary problems (FBPs). To begin with, some more or less famous examples of FBPs will be given. This is followed by a more mathematical introduction to a large class of FBPs, where the main issues and the usual problems that arise in this area are described. Regarding the references, I do not in any way claim to present a complete list, but rather a selection of what can be found in the vast literature. 1.1 The obstacle problem Imagine an elastic string attached to two points and assume that we can neglect the influence of the gravity. Clearly, it will assume the form of a straight line. Suppose now that we push up the string with some kind of object that we call the obstacle. Then, at some points the string and the obstacle will touch. Moreover, if the obstacle is not flat, somewhere the string will stay above the obstacle. Thus, there it will still be ”free”, which means that it will still be a straight line there. See Figures 1.1 and 1.2. This is a typical free boundary problem. Here the free boundary is the set of points where the string leaves the obstacle. Mathematically, we can formulate this as follows. Suppose that the string is given by the graph of the function f on the interval [0, 1] and that the obstacle is given by the graph of φ on [0, 1]. Then we fix the end points by saying f(0) = a and f(1) = b. The string will 1 2 CHAPTER 1. f(x) f(0) f(1) φ(x) Figure 1.1: When the string does not touch the obstacle. f(x) f(0) f(1) φ(x) Figure 1.2: When the string is pushed up by the obstacle. 1.1. THE OBSTACLE PROBLEM 3 try to minimize the tension energy, which will be proportional to the length of the string, which in turn is given by Z b p L = 1 + |f 0(x)|2 dx. a The minimization will be under the constraints that f(0) = a, f(1) = b and f ≥ φ. If we assume that we can linearize this, we obtain the approximation 1 Z b L ≈ 1 + |f 0(x)|2 dx. 2 a Then f will instead minimize Z b |f 0(x)|2 dx a under the constraints that f(0) = a, f(1) = b and f ≥ φ. In fact, in one dimension this gives the exact same solution as we would get without the approximation. From standard methods in the calculus of variations it follows that f 00 = 0 in the set {f > φ}, i.e., in the set where the string stays above the obsta- cle. This means that the graph of f is a line on every interval in this set. Moreover, it is not hard to prove that f 00 ≤ 0 everywhere.
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