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Conjecture of De Giorgi Nikola Kamburov

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A free boundary problem inspired by a Conjecture of De Giorgi :1hASSACHU6SETTS NSTITIJTF by OF TEC iC1LO0 Y Nikola Kamburov A.B., Princeton University (2007) L B23RAR 1 Submitted to the Department of Mathematics ARCHIVES in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2012 @ Massachusetts Institute of Technology 2012. All rights reserved. Author .............................. ............................ Department of Mathematics April 18, 2012 ~ A Certified by....... ........... David Jerison Professor of Mathematics Thesis Supervisor i '- Aii / / /I Accepted by.. y1 L/ L/ Gigliola Staffilani Abby Rockefeller Mauze Professor of Mathematics Chair, Committee on Graduate Students 2 A free boundary problem inspired by a Conjecture of De Giorgi by Nikola Kamburov Submitted to the Department of Mathematics on April 18, 2012, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract We study global monotone solutions of the free boundary problem that arises from minimizing the energy functional I(u) = f |Vu12 + V(U), where V(u) is the charac- teristic function of the interval (-1, 1). This functional is a close relative of the scalar Ginzburg-Landau functional J(u) = f IVu12 + W(u), where W(u) = (1 - u 2 )2 /2 is a standard double-well potential. According to a famous conjecture of De Giorgi, global critical points of J that are bounded and monotone in one direction have level sets that are hyperplanes, at least up to dimension 8. Recently, Del Pino, Kowalczyk and Wei gave an intricate fixed-point-argument construction of a counterexample in dimension 9, whose level sets "follow" the entire minimal non-planar graph, built by Bombieri, De Giorgi and Giusti (BdGG). In this thesis, we turn to the free boundary variant of the problem and we construct the analogous example; the advantage here is that of geometric transparency as the interphase { ul < 1} will be contained within a unit-width band around the BdGG graph. Furthermore, we avoid the technicalities of Del Pino, Kowalczyk and Wei's fixed-point argument by using barriers only. Thesis Supervisor: David Jerison Title: Professor of Mathematics 3 4 Acknowledgments The hundred odd pages of this thesis will have to serve as compact evidence of the knowledge, understanding, ideas and attitudes that I have internalized over the last five years. To the people who either showed them to me, or helped me form them, appreciate them, develop them and sustain them, here are my words of deepest grat- itude. First, I would like to thank my advisor, David Jerison, for his generosity, un- derstood in all of its aspects. In addition to his time and patience, his unwavering support and enthusiasm, I am grateful for all the knowledge and keen intuition that he has shared with me, communicated in that exquisite, trademark manner: sharp and eloquent to the point where ideas became so concrete that I could almost physi- cally touch them. It has been an honour to be his student and I hope to be a good disciple of his vision that "mathematics is the art of the possible." I would like to thank Tobias Colding and Gigliola Staffilani who served on my thesis committee and who taught me a great deal of geometric analysis and PDE's at MIT. Many thanks to all my teachers in graduate school and in college, especially Denis Auroux, Victor Guillemin, Richard Melrose, Sergiu Klainerman, Igor Rodnian- ski, Yakov Sinai and Elias Stein, whose courses exposed me to beautiful mathematics that shaped my mathematical world. I would also like to mention Donka Stefanova, my physics teacher from high school; through her efforts I caught a first real glimpse of what science is. I am grateful to Michael Eichmair, Mahir Hadzic and Christine Breiner for dis- cussing topics related to my research and for sharing their experience about life after grad school. Special thanks to Michael for teaching a terrific course on minimal surfaces, for all his support, serene counsel and encouragement. I am very fortunate to have been surrounded by an amazing group of colleagues and friends, who made life at MIT and in Boston bright, interesting, dynamic, cheer- ful, cosy and good for the soul. Andrei Negut, Bhairav Singh, Alejandro Morales, Nick Sheridan, Vedran Sohinger, Martina Balagovic, Joel Lewis, Hoeskuldur Hall- 5 dorsson, Niels Moeller, Olga Stroilova, Peter Speh, Ramis Movassagh, Sheel Ganatra, Anatoly Preygel, Nikhil Savale, Angelica Osorno, David Jordan, Martin Frankland, Ana Rita Pires, Dorian Croitoru, Rosalie Belanger-Rioux, Roberto Svaldi, Daniele Valeri - thank you! I would like to thank all the members of the "Perverse Sheaves," our glorious soccer team, for playing together in cold, rain or heat, dreaming the common dream of "lifting the soft-fibre cottonware" one day as league champions (nicely put, Alejandro). I am very grateful to Jennifer Park, Sasha Tsymbaliuk, Giorgia Fortuna, Uhi Rinn Suh and Jiayong Li, who treated my haphazard visits to the office with good humour and chocolate bars. Thanks to my Bulgarian friends Nikolai Slavov, Ivana Dimitrova, Katrina Evtimova, Kaloyan Slavov, Greta Panova, Stefanie Stantcheva, Plamen Nenov and Antoni Rangachev, who kept me culturally connected with the motherland and up to date with many other important things (special thanks to Antoni and Plamen for those late evening meetings that produced some of the most hilarious conversations known to man). To my friends Ana Caraiani, Silviu Pufu and Steven Sivek (who invented the sophisticated game "Three, I win" and beat me soundly at it, I admit); to my friends in California - Vaclav Cvicek, Diana Negoescu and Dzhelil Rufat; to my "notorious" roommate Aaron Silberstein (the person with the most generous heart) - thank you, folks, for sharing this grand adventure with me. Words fail me to describe how much I owe you. Finally, I would like to thank my family: my mother Vanya, my father Angel, my brother Vladi and my uncle Rumen. If it is true that one follows the guiding light of the flame within, then they provided the spark and their love has sustained the flame. 6 Contents 1 Introduction. 9 1.1 Background. .. ... .... .... ... .... ... .... .. 9 1.2 Statement of the problem. .... ... .... ... .... .. 12 1.3 Outline of strategy. .... .... .... ..... .... .... 15 1.4 Organization of the Chapters. .... ... .... ... .... 20 2 Construction of the barriers. 23 2.1 The Bombieri-De Giorgi-Giusti graph F and its approximation. 23 2.1.1 Geometry of the unit-width band around F. .. ... .. 24 2.1.2 Proximity between F and the model graph. .... ... 33 2.2 Construction of the super and subsolution. .. .... ... ... 36 2.2.1 The ansatz. .... ... ... ... .... ... ... 36 2.2.2 Supersolutions for the Jacobi operator. .... .... .. 42 2.2.3 The free boundary super and subsolution. ... ... .. 58 3 The solution. Existence and regularity. 63 3.1 Notation. .. .. ... .. ... .. .. ... ... ... ... 64 3.2 Existence of a local minimizer. .. .. .. .. .. ... ... 64 3.2.1 Lipschitz continuity of local minimizers. ... ... .. 67 3.2.2 Construction of a global minimizer. .. ... ... ... 72 3.3 Regularity of global minimizers. .. .. .. ... ... ... 74 3.3.1 The proof of Theorem 1.2.1. .. .. .... ... 77 7 A Supersolutions for Jr..- Computation of Ho,3. 81 A.1 The relation between the Jacobi operator and the linearized mean cur- vature operator. .. ... .... ... ... .... ... ... .... 82 A.1.1 Computation of H[Fo] and IV(H[F.])I. ... ... ... ... 84 A.1.2 Supersolutions for Jr. .. ... .. .. ... .. .. ... .. 85 A.2 Computation of Ho, 3 . ... .. ... ... ... ... .. ... ... 87 B Bounds on the gradient and hessian of h(r, 0) on I's. 91 8 Chapter 1 Introduction. 1.1 Background. In 1978 Ennio De Giorgi formulated a celebrated conjecture concerning the symmetry properties of global solutions (Q = R') of the semilinear partial differential equation Au = U3 - u inn. (1.1) This PDE arises as the Euler-Lagrange equation for the functional J(u, Q) = jVu2 + W(u) dx, where Q C R' is a bounded domain and W(u) := (1 - u 2)2/4 is a standard "double- well" potential. The functional J appears in several physical applications, e.g. it is used to model the energy of a binary fluid in phase transition phenomena (see [11]): in this model, u represents the concentration of one phase of the fluid or the other and the minima u = ±1 of the potential W correspond to the two stable phases of the fluid. De Giorgi's Conjecture ([13]). If u E C 2 (Rn) is a global solution of (1.1) such that lul < 1 and onu > 0, then the level sets {u = A} are hyperplanes, at least for dimensions n < 8. 9 The conjecture is often considered along with the natural assumption lim u(V', x) = ±1. (1.2) xz-4oo By a theorem of Alberti, Ambrosio and Cabr6 [1], the combination of the monotonicity assumption with (1.2) ensures that one is not just dealing with a critical point of the functional J, but with an honest global minimizer. In the simple statement of De Giorgi's conjecture lurks a fascinating, deep connec- tion with the theory of minimal surfaces. The connection was first elaborated in the works of Modica [22] and Modica-Mortola [23] within the framework of 17-convergence. Note that if u minimizes J(-, B11 ) (B, denotes a ball of radius r, centered at the ori- gin), then the blow-down u,(x) u(x/E) minimizes J,(-, B 1 ), where J, denotes the rescaled energy functional J,(u,/Q) = f V 2 dx±+ W(u) dx. Modica's theorem asserts: Theorem 1.1.1 ([22]). Let u, minimize J,,(-, B1 ) and Jk (u,) C are uniformly bounded. Then as Ek -+ 0, up to a subsequence, {u,,} converges to UEk -+ 1E - 1Bi\E in Ll(Bi), (1.3) where E C B1 is such that aE is a minimal hypersurface.
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