On Some Rigidity Properties in PDEs Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat) der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn vorgelegt von Angkana R¨uland aus Chiang Mai Bonn, 2013 Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms Universit¨at Bonn 1. Gutachter: Prof. Dr. Herbert Koch 2. Gutachter: Prof. Dr. Felix Otto Tag der Promotion: 28.03.2014 Erscheinungsjahr: 2014 Contents I Rigidity Properties in Inverse Problems 1 1 Introduction 2 2 Backward Uniqueness Properties 8 2.1 Introduction................................ 8 2.2 Review: Definitions, Basic Properties and Examples . 11 2.3 HeuristicsfortheBUP.......................... 20 2.4 StatementoftheMainResults . 27 2.5 Elliptic Carleman Estimates and Consequences . 29 2.6 ParabolicCarlemanEstimates . 34 3 The Fractional Laplacian and the UCP 46 3.1 Introduction................................ 46 3.2 Review: Unique Continuation and Carleman Estimates . 48 3.3 TheFractionalLaplacian . .. .. .. .. .. .. .. .. .. .. 52 3.4 TheMainResults............................. 56 3.5 TheWeakUniqueContinuationProperty . 59 3.6 SymmetricCarlemanEstimates. 62 3.7 DoublingandtheWUCP ........................ 76 3.8 TheOne-DimensionalSituation . 80 3.9 Improved Integrability: The Half-Laplacian . 84 3.10VariableCoefficients ........................... 85 II The Cubic-to-Orthorhombic Phase Transition 94 4 Introduction 95 4.1 TheModel ................................ 98 4.2 Heuristics ................................. 101 5 Non-Rigidity 105 5.1 Introduction................................ 105 5.2 TheMainResults............................. 109 5.3 Preliminaries . 112 5.4 Constructions............................... 114 5.5 ControlledConvergenceLemma. 122 5.6 ProofofthePropositions . 123 6 Rigidity 129 6.1 Introduction................................ 129 6.2 TheMainResults............................. 131 6.3 Preliminaries . 134 6.4 ARegularityResult ........................... 136 i ii CONTENTS 6.5 CombinatoricsontheSphere . 140 6.6 PiecewiseAffineStrains .. .. .. .. .. .. .. .. .. .. .. 146 6.7 Appendix ................................. 159 Acknowledgments During the last three years I have had the opportunity and freedom to learn a lot, develop new mathematical interests and to deepen my knowledge on topics which have already previously intrigued me. This would not have been possible without the support of a number of persons to whom I am very grateful. First and foremost, I am greatly indebted to Prof. Koch. His thesis supervision provided just the right mixture of guidance and freedom. I could always rely on his support, his patience and his thoughtful advice. His broad overview and his many heuristic arguments have been an invaluable source of inspiration and moti- vation. Moreover, I would like to thank Prof. Otto for his unwavering interest in and support of my work on phase transitions. In particular, I would like to thank him for introducing me to this fascinating topic, for the many inspiring discussions and invitations to Leipzig. Further thanks go to the Bonn International Graduate School in Mathematics and the Deutsche Telekom Stiftung for their very generous non-monetary and financial support. Bonn and BIGS have provided a very stimulating mathematical environ- ment. Also, I have greatly enjoyed the various seminars and meetings hosted by the Deutsche Telekom Stiftung. It provided me the opportunity to meet other young, enthusiastic researchers from various fields in the natural sciences. This has led to many exciting discussions and new friendships. I also owe great gratitude to those with whom I could discuss numerous mathe- matical and non-mathematical issues on a daily basis. The whole research group including Shaoming Guo, Dominik John, Habiba Kalantarova, Clemens Kienzler, Diogo Oliveira e Silva, Irene Paniccia, Tobias Schottdorf, Stefan Steinerberger and Christian Zillinger have always listened to the latest ideas and problems. I owe them many thoughtful comments and thank them for the wonderful work atmo- sphere they have created. Last but not least, I would like to thank my family and friends – most of all my parents and Christian. Without their infinite encouragement, support and love this would not have been possible. Introduction This thesis is dedicated to two types of “rigidity properties” occurring in certain PDEs. These rigidity notions are rather complementary – the first originating from the study of controllability, the second appearing in the context of material sciences. While the first notion of rigidity is typical of elliptic (and parabolic) equations, the second one is mainly associated with hyperbolic equations and systems. Before describing precisely the setting of our problems, we recall two prototypes of the rigidity properties we have in mind: The first rigidity property we deal with is associated with the unique continu- • ation principle. Here the model operator is given by the Laplacian. Due to its analyticity, a solution which vanishes of infinite order at a point must already vanish globally. Thus, a naturally arising question is whether this extends to more general operators and, in the case of a positive answer, to which ones. In this thesis we deal with two problems of such a flavour: The first is concerned with a parabolic “unique continuation problem at infinity”, while the second treats the unique continuation problem for the fractional Laplacian. In the second problem we put a particular emphasis on requiring as little regularity as pos- sible. The second rigidity problem we investigate concerns a system of PDEs and • is related to the notion of characteristics (in first order equations). Although we are confronted with a system, this type of “rigidity property” is already present in scalar (hyperbolic) equations: A toy problem would, for example, be the transport equation for which the characteristics of the system are straight lines. A more elaborate (toy) model is, for instance, given by the following two-dimensional gradient inclusion problem: 1 0 1 0 u , − + Skew(2). ∇ ∈ 0 1 0 1 − Using the discreteness of the symmetrized gradient and the compatibility con- ditions, one finds that solutions, u, either satisfy e( u) := 1 ( u + ( u)t)= ∇ 2 ∇ ∇ f(x x ) or e( u)= g(x +x ). Thus in both examples, the scalar transport 1 − 2 ∇ 1 2 iv CONTENTS v equation and the differential inclusion, the solutions are necessarily of the form of waves propagating along certain characteristics. In this sense the solutions are very rigid. We remark that although solving a PDE with the method of characteristics is not uncommon when dealing with scalar equations, it often poses problems in the context of systems as the resulting equations are not closed. In this thesis we deal with the classification of all possible solutions of a certain (vector-valued) differential inclusion which arises in the study of phase transi- tions in certain shape-memory materials such as CuAlNi. For this transition we prove two complementary results: On the one hand, one cannot hope for rigidity for a too weak notion of a solution. On the other hand, adding regular- ity constraints, the problem becomes rigid and only very specific, essentially two-dimensional patterns occur. Keeping this brief description of the different notions of rigidity in mind, we present the problems which are discussed in this thesis in greater detail: The backward uniqueness property in conical domains. This problem deals with the controllability of the heat equation (and perturbations thereof): ∂ u ∆u = W u + W u in Ω (0,T ), t − 1 2 · ∇ θ × u = u in Ω 0 , 0 θ ×{ } Here Ωθ is a cone with opening angle θ. We aim at understanding the interplay of the strong diffusivity and the unbounded underlying geometry. As is known since, for example, the work of Zuazua and Micu [MZ01a], [MZ01b], there is a major discrepancy between bounded and unbounded domains. While the heat equation is null-controllable, i.e. by choosing adapted boundary data it is possible to drive any L2 (initial) datum to zero in an arbitrarily short time interval, in bounded domains, this is no longer the case in unbounded do- mains. On top of that depending on the “degree of unboundedness of the domain”, the heat equation is not only not null-controllable but even displays the backward uniqueness property, i.e. in conical domains with sufficiently large opening angles the only solution which can be driven to zero is the trivial solution. In the first part of the thesis we provide a quantitative description of the large angle regime in two spatial dimensions. In this context, it is known that the backward uniqueness property can only hold in angles larger than 90◦ which is a consequence of the Phragmen-Lindel¨of principle [ESS03],ˇ [SS02].ˇ Furthermore, it is conjectured that the backward uniqueness property actually holds in all angles larger than 90◦, reflecting the fact that the diffusivity is not strong enough to drive any nontrivial L2 datum to zero. However, the furthest previous result in this direction only shows that the backward uniqueness property holds in all angles down to approximately 109◦ [LS10].ˇ vi CONTENTS Motivated by understanding a related elliptic “unique continuation problem at infin- ity”, we aim at improving this bound in two spatial dimensions via a more detailed phase space analysis. As in the paper by Sver´akˇ and Li [LS10],ˇ the core of our approach relies on Carleman estimates, i.e. exponentially weighted