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The bending of bilayer plates is a mechanism that allows for large deformations via small externally induced lattice mismatches of the underlying materials. Its mathematical modeling, discussed herein, consists of a nonlinear fourth-order problem with a pointwise isometry constraint. A discretization based on Kirchhoff quadrilaterals is devised and its Γ-convergence is proved. An iterative method that decreases the energy is proposed, and its convergence to stationary configurations is investigated. Its performance, as well as reduced model capabilities, are explored via several insightful numerical experiments involving large (geometrically nonlinear) deformations.© 2015 Wiley Periodicals, Inc.

We study an evolution equation proposed by Chapman, Rubinstein, and Schatzman as a mean-field model for the evolution of the vortex density in a superconductor. We treat the case of a bounded domain where vortices can exit or enter the domain. We show that the equation can be derived rigorously as the gradient flow of some specific energy for the Riemannian structure induced by the Wasserstein distance on probability measures. This leads us to some existence and uniqueness results and energy-dissipation identities. We also exhibit some “entropies” that decrease through the flow and allow us to get regularity results (solutions starting in Lp, p > 1, remain in Lp). © 2007 Wiley Periodicals, Inc.

The J-flow of S. K. Donaldson and X. X. Chen is a parabolic flow on Kahler manifolds with two Kahler metrics. It is the gradient flow of the J-functional that appears in Chen's formula for the Mabuchi energy. We find a positivity condition in terms of the two metrics that is both necessary and sufficient for the convergence of the J-flow to a critical metric. We use this result to show that on manifolds with ample canonical bundle, the Mabuchi energy is proper on all Kahler classes in an open neighborhood of the canonical class defined by a positivity condition. This improves previous results of Chen and of the second author. We discuss the implications of this for the problem of the existence of constant-scalar-curvature Kahler metrics. We also study the singularities of the J-flow and, under certain conditions (which always hold for dimension 2) derive some estimates away from a subvariety. We discuss the conjectural remark of Donaldson that if the J-flow does not converge on a Kahler surface, then it should blow up over some curves of negative self-intersection. © 2007 Wiley Periodicals, Inc.

We study the action minimization problem which is formally associated to phase transformation in the stochastically perturbed Allen-Cahn equation. The sharp-interface limit is related to (but di erent from) the sharp-interface limits of the related energy functional and deterministic gradient flows. In the sharp-interface limit of the action minimization problem, we find distinct “most likely switching pathways,” depending on the relative costs of nucleation and propagation of interfaces. This competition is captured by the limit of the action functional, which we derive formally and propose as the natural candidate for the -limit. Guided by the reduced functional, we prove upper and lower bounds for the minimal action which agree on the level of scaling. (This is a preprint of an article accepted for publication in Comm. Pure App. Math, October 2005.)

We present a method to prove convergence of gradient flows of families of energies that Γ-converge to a limiting energy. It provides lower-bound criteria to obtain the convergence that correspond to a sort of C1-order Γ-convergence of functionals. We then apply this method to establish the limiting dynamical law of a finite number of vortices for the heat flow of the Ginzburg-Landau energy in dimension 2, retrieving in a different way the existing results for the case without magnetic field and obtaining new results for the case with magnetic field. © 2004 Wiley Periodicals, Inc.

In a non-compact context the first natural step in the search for periodic orbits of a hamiltonian flow is to detect bounded ones. In this paper we show that, in a non-compact setting, certain algebraic topological constraints imposed to a gradient flow of the hamiltonian function $f$ imply the existence of bounded orbits for the hamiltonian flow of $f$. Once the existence of bounded orbits is established, under favorable circumstances, application of the $C^{1}$-closing lemma leads to periodic ones., Comment: 54 pages

By studying a negative gradient flow of certain Hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous Hessian equations. To achieve this we derive uniform, first- and second-order a priori estimates for the elliptic and parabolic Hessian equations. Our results generalize well-known results for semilinear elliptic equations and the Monge-Ampere equation. c 2001 John Wiley & Sons, Inc.