This thesis concerns problems from the calculus of variations, from energy driven pattern formation in thin elastic sheets, to the mean field theory of spin glasses. In Part I, we consider two model problems from the study of compressed thin elastic sheets. In chapter one, we consider the axial compression of a thin elastic cylinder about a hard cylindrical core. We prove upper and lower bounds on the minimum energy of the cylinder that depend on its thickness and the magnitude of compression. We consider two cases: the "large mandrel" case, where the radius of the core exceeds that of the cylinder, and the "neutral mandrel" case, where the radii are the same. In the large mandrel case, our bounds match in their scaling with respect to the external parameters. We find three regimes, corresponding to the presence of many wrinkles, few wrinkles, or no wrinkles at all. In the neutral mandrel case, our bounds match in a certain regime where compression is small as compared to thickness; there, the minimum energy scales as that of the unbuckled configuration. We achieve these results in both the von Kármán-Donnell model and a geometrically nonlinear model of elasticity.
In chapter two, we study symmetry breaking in indented elastic cones. Motivated by simulations of carbon nanocones (see Jordan and Crespi, Phys. Rev. Lett., 2004), we consider a variational model for an elastic cone under compression in the direction of the cone symmetry axis. Assuming radial symmetry, and modeling the compression by imposing Dirichlet conditions at the center and the boundary of the sheet, we identify the energy scaling law in the Föppl-von Kármán model. We find three regimes: for small indentations, the energetic cost of the logarithmic singularity dominates; for moderate indentations, the cone buckles in a thin boundary layer; and for large indentations, a localized inversion takes place in the central region. Then we prove that, for large indentations, energy minimizers are not radially symmetric. We do so by an explicit construction that achieves lower energy than the best radially symmetric deformation.
In Part II, we study problems arising in the mean field theory of spin glasses. In chapters three and four, we study the Parisi variational problem, a non-local and convex minimization problem whose minimizer is thought of as the order parameter for mixed p-spin glass systems. In chapter three, we present a simple proof of the strict convexity of the Parisi functional, a result due originally to Auffinger and Chen (see Auffinger and Chen, Commun. Math. Physics, 2015). In chapter four, we study the question of the phase diagram of general mixed p-spin glasses. We derive self-consistency conditions that generalize the Replica Symmetry (RS) condition of de Almeida and Thouless to all levels of Replica Symmetry Breaking (RSB) and all models. We conjecture that for all models, the RS phase is the region determined by the natural analogue of the de Almeida-Thouless condition. We prove that for all models, the complement of this region is in the RSB phase. Furthermore, we prove that the conjectured phase boundary is exactly the phase boundary in the plane less a bounded set. In the case of the Sherrington-Kirkpatrick model, this bounded set does not contain the critical point at zero external field.
In chapter five, we turn to the Crisanti-Sommers variational problem, which comes from the study of spherical mixed p-spin glasses. We identify the zero-temperature Gamma-limit of the Crisanti-Sommers functionals, thereby establishing a rigorous variational problem for the ground state energy of the spherical mixed p-spin glass. We prove that this problem is dual to an obstacle-type problem. This leads to a simple method for constructing a finite dimensional space in which the optimizers live. As a consequence, we unify predictions of Crisanti-Leuzzi and Auffinger-Ben Arous regarding the 1RSB phase in this limit. We find that the "positive replicon eigenvalue" and "pure-like" conditions are together necessary for optimality, but that neither are themselves sufficient. We prove that these conditions completely characterize the 1RSB phase in 2+p-spin models.
Chapter two in Part I is joint work with S. Conti and H. Olbermann. Part II is joint work with A. Jagannath.
|Advisor:||Kohn, Robert V.|
|Commitee:||Ben Arous, Gerard, Francfort, Gilles, Holmes-Cerfon, Miranda, Lin, Fang-Hua|
|School:||New York University|
|School Location:||United States -- New York|
|Source:||DAI-B 78/04(E), Dissertation Abstracts International|
|Subjects:||Mechanics, Mathematics, Condensed matter physics|
|Keywords:||Calculus of variations, Nonlinear elasticity, Partial differential equations, Spin glasses|
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