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The application of techniques from nonlinear analysis to materials science has seen great developments in the recent years and it has really been a driving force for substantial mathematical research in the area of partial differential equations and the multi-dimensional calculus of variations. This thesis has been motivated by two recent and remarkable experimental observations of H. Seiner in shape-memory alloys which we attempt to interpret mathematically. Much of the work is original and has given rise to deep problems in the calculus of variations. Firstly, we study the formation of non-classical austenite-martensite interfaces. Ball & Carstensen (1997, 1999) theoretically investigated the possibility of the occurrence of such interfaces and studied the cubic-to-tetragonal case extensively. In this thesis, we present an analysis of non-classical austenite-martensite interfaces recently observed by Seiner et al.~in a single crystal of a CuAlNi shape-memory alloy, undergoing a cubic-to-orthorhombic transition. We show that these can be described by the general nonlinear elasticity model and we make some predictions regarding the admissible volume fractions of the martensitic variants involved, as well as the habit plane normals. Interestingly, in the above experimental observations, the interface between the austenite and the martensitic configuration is never exactly planar, but rather slightly curved, resulting from the pattern of martensite not being exactly homogeneous. However, it is not clear how one can reconstruct the inhomogeneous configuration as a stress-free microstructure and, instead, a theoretical approach is followed. In this approach, a general method is provided for the construction of a compatible curved austenite-martensite interface and, by exploiting the structure of quasiconvex hulls, the existence of curved interfaces is shown in two and three dimensions. As far as the author is aware of, this is the first construction of such a curved austenite-martensite interface. Secondly, we study the nucleation of austenite in a single crystal of a CuAlNi shape-memory alloy consisting of a single variant of stabilized 2H martensite. The nucleation process is induced by localized heating and it is observed that, regardless of where the localized heating is applied, the nucleation points are always located at one of the corners of the sample - a rectangular parallelepiped in the austenite. Using a simplified nonlinear elasticity model, we propose an explanation for the location of the nucleation points by showing that the martensite is a local minimizer of the energy with respect to localized variations in the interior, on faces and edges of the sample, but not at some corners, where a localized microstructure can lower the energy. The result for the interior, faces and edges is established by showing that the free-energy function satisfies a set of quasiconvexity conditions at the stabilized variant throughout the specimen, provided this is suitably cut. The proofs of quasiconvexity are based on a rigidity argument and are specific to the change of symmetry in the phase transformation. To the best of the author's knowledge, quasiconvexity conditions at edges and corners have not been considered before.

This thesis concerns the regularity of holonomic minimisers of variational integrals in the context of direct methods in the calculus of variations. Specifically, we consider Sobolev mappings from a bounded domain into a connected compact Riemannian manifold without boundary, to which such mappings are said to be holonomically constrained. For a general class of strictly quasiconvex integral functionals, we give a direct proof of local C^1,α-Hölder continuity, for some 0 < α < 1, of holonomic minimisers off a relatively closed 'singular set' of Lebesgue measure zero. Crucially, the proof constructs comparison maps using the universal covering of the target manifold, the lifting of Sobolev mappings to the covering space and the connectedness of the covering space. A certain tangential A-harmonic approximation lemma obtained directly using a Lipschitz approximation argument is also given. In the context of holonomic minimisers of regular variational integrals, we also provide bounds on the Hausdorff dimension of the singular set by generalising a variational difference quotient method to the holonomically constrained case with critical growth. The results are analogous to energy-minimising harmonic maps into compact manifolds, however in this case the proof does not use a monotonicity formula. We discuss several applications to variational problems in condensed matter physics, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals. In these problems, the class of mappings are constrained to an orbit of 'broken symmetries' or 'manifold of internal states', which correspond to a sub-group of residual symmetries.

Cette thèse est consacrée à l'analyse variationnelle des problèmes d'équilibre avec contraintes d'équilibre (problème d'optimisation bi-niveaux). Ce travail tire sa motivation de modèles de marchés de l'électricité issus de la théorie des jeux non coopératifs. Dans de tels marchés, un régulateur, appelé ISO (Independant System Operator), gère le clearing et les flux d'électricité entre zones d'enchère. Nous avons développé cette analyse variationnelle selon différents axes. Tout d'abord et sur un modèle spécifique, nous avons analysé de façon exhaustive la notion de meilleure réponse d'un producteur, grâce à la détermination d'une formule explicite pour l'unique solution du problème de bas niveau de l'ISO. Puis, pour un modèle plus général de marché, la stabilité des points M(ordukhovitch)-stationnaires a été étudiée via la notion de codérivée limiting de second ordre des opérateurs multivoques. En fin, le concept d'opérateur normal limiting a été introduit et des règles de calcul ont été obtenues, fournissant ainsi un nouvel outil performant pour l'analyse quasiconvexe. L'idée de base a été l'utilisation, pour des cônes normaux à des sous-niveaux d'une fonction, de la construction limiting classiqueen analyse variationnelle moderne. Cette approche est motivée par l'hypothèse de la quasiconvexité des fonctions de coût généralement faite dans de nombreux jeux non-coopératifs. / This thesis is focused on nonsmooth variational analysis of equilibrium problems with equilibrium constraints. Such an eff ort is directly motivated by a model of electricity markets encountered in non-cooperative game theory. In such a marketthere is the so-called Independent System Operator (ISO), a regulator entity that manages the market clearing and the electricity dispatch. This market structure makes the problem of electricity markets challenging from the mathematicalpoint of view. In this area, we discovered several possibilities for further development. First, the best responses of producers in a speci fic variant of a model are fully analysed. This progress was due to an analytical formula for a uniquesolution to the lower level ISO problem. Then, for a more general model of the market, stability of the so-called M(ordukhovich)-stationarity points is provided based on the concept of coderivatives. To this end, the respective second order limiting coderivative was computed. Finally, the concept of limiting normal operator is proposed, a new tool for quasiconvex analysis exhibiting workable calculus rules. The basic idea is to employ the same limiting construction that is used in modern variational analysis in connection with normal cones to sets. This topic is motivated by the classical assumption in many non-cooperative games where the loss function of players is often assumed to be quasiconvex.

Die vorliegende Arbeit beschäftigt sich mit drei Klassen ausgewählter nichtlinearer Probleme, die Forschungsgegenstand der angewandten Mathematik sind. Diese Probleme behandeln die Minimierung von Integralen in der Variationsrechnung (Kapitel 3), das Lösen partieller Differentialgleichungen (Kapitel 4) und das Lösen nichtlinearer Optimierungsaufgaben (Kapitel 5). Mit deren Hilfe lassen sich unterschiedlichste Phänomene der Natur- und Ingenieurwissenschaften sowie der Ökonomie mathematisch modellieren. Als konkretes Beispiel werden mathematische Modelle der Theorie elastischer Festkörper betrachtet. Das Ziel der vorliegenden Arbeit besteht darin, ein gegebenes nichtlineares Problem durch polynomiale Probleme zu approximieren. Um dieses Ziel zu erreichen, beschäftigt sich ein großer Teil der vorliegenden Arbeit mit der polynomialen Approximation von nichtlinearen Funktionen. Den Ausgangspunkt dafür bildet der Weierstraßsche Approximationssatz. Auf der Basis dieses bekannten Satzes und eigener Sätze wird als Hauptresultat der vorliegenden Arbeit gezeigt, dass im Übergang von einer gegebenen Funktion zum approximierenden Polynom wesentliche Eigenschaften der gegebenen Funktion erhalten werden können. Die wichtigsten Eigenschaften, für die dies bisher nicht bekannt war, sind: Quasikonvexität im Sinne der Variationsrechnung, Quasimonotonie im Zusammenhang mit partiellen Differentialgleichungen sowie Quasikonvexität im Sinne der nichtlinearen Optimierung (Theoreme 3.16, 4.10 und 5.5). Schließlich wird gezeigt, dass die zu den untersuchten Klassen gehörenden nichtlinearen Probleme durch polynomiale Probleme approximiert werden können (Theoreme 3.26, 4.16 und 5.8). Die dieser Approximation zugrunde liegende Konvergenz garantiert sowohl eine Approximation im Parameterraum als auch eine Approximation im Lösungsraum. Für letztere werden die Konzepte der Gamma-Konvergenz (Epi-Konvergenz) und der G-Konvergenz verwendet. / In this thesis, we are concerned with three classes of non-linear problems that appear naturally in various fields of science, engineering and economics. In order to cover many different applications, we study problems in the calculus of variation (Chapter 3), partial differential equations (Chapter 4) as well as non-linear programming problems (Chapter 5). As an example of possible applications, we consider models of non-linear elasticity theory. The aim of this thesis is to approximate a given non-linear problem by polynomial problems. In order to achieve the desired polynomial approximation of problems, a large part of this thesis is dedicated to the polynomial approximation of non-linear functions. The Weierstraß approximation theorem forms the starting point. Based on this well-known theorem, we prove theorems that eventually lead to our main result: A given non-linear function can be approximated by polynomials so that essential properties of the function are preserved. This result is new for three properties that are important in the context of the considered non-linear problems. These properties are: quasiconvexity in the sense of the calculus of variation, quasimonotonicity in the context of partial differential equations and quasiconvexity in the sense of non-linear programming (Theorems 3.16, 4.10 and 5.5). Finally, we show the following: Every non-linear problem that belongs to one of the three considered classes of problems can be approximated by polynomial problems (Theorems 3.26, 4.16 and 5.8). The underlying convergence guarantees both the approximation in the parameter space and the approximation in the solution space. In this context, we use the concepts of Gamma-convergence (epi-convergence) and of G-convergence.

This thesis concerns the mathematical modelling of phase transformations with a special emphasis on martensitic phase transformations and their application to the modelling of steels. In Chapter 1, we develop a framework that determines the optimal transformation strain between any two Bravais lattices and use it to give a rigorous proof of a conjecture by E.C. Bain in 1924 on the optimality of the so-called Bain strain. In Chapter 2, we review the Ball-James model and related concepts. We present some simplification of existing results. In Chapter 3, we pose a conjecture for the explicit form of the quasiconvex hull of the three tetragonal wells, known as the three-well problem. We present a new approach to finding inner and outer bounds. In Chapter 4, we focus on highly compatible, so called self-accommodating, martensitic structures and present new results on their fine properties such as estimates on their minimum complexity and bounds on the relative proportion of each martensitic variant in them. In Chapter 5, we investigate the contrary situation when self-accommodating microstructures do not exist. We determine, whether in this situation, it is still energetically favourable to nucleate martensite within austenite. By constructing different types of inclusions, we find that the optimal shape of an inclusion is flat and thin which is in agreement with experimental observation. In Chapter 6, we introduce a mechanism that identifies transformation strains with orientation relationships. This mechanism allows us to develop a simpler, strain-based approach to phase transformation models in steels. One novelty of this approach is the derivation of an explicit dependence of the orientation relationships on the ratio of tetragonality of the product phase. In Chapter 7, we establish a correspondence between common phenomenological models for steels and the Ball-James model. This correspondence is then used to develop a new theory for the (5 5 7) lath transformation in low-carbon steels. Compared to existing theories, this new approach requires a significantly smaller number of input parameters. Furthermore, it predicts a microstructure morphology which differs from what is conventionally believed.