The seminar meets regularly on wednesday at 4pm in Eckhart 202. However, we have several special seminars during other days.
I will show how the Gibbs-Thomson law forces Lipschitz interfaces in the Stefan problem to regularize instantaneously (to $C^{2,\alpha}$). This is in contrast with what happens with the classical two-phase Stefan problem where even Lipschitz free boundaries remain just Lipschitz for some time. The proof relies on modified De Giorgi-Nash-Moser estimates and (in anology with quasilinear equations like mean curvature) suggests how one might understand the singular set of the flow in general.
In the early 20th century, S. Bose and A. Einstein predicted the existence of a state of matter composed of weakly interacting bosons (integer spin particles). Today, this is know as the Bose-Einstein condensate. The BEC was first experimentally realized in 1995 by E. Cornell and C. Wieman (U. of Colorado at Boulder) and W. Ketterle (MIT). These experiments have generated a plethora of research concerning both theory and experiment in this area. The work presented in this talk focuses on understanding solitary waves in spinor system. This system is motivated by the spinor BEC which can be described by a quasi-one-dimensional model. Here, we discuss a three-component dynamical lattice which contains a mean-field nonlinearity. Our analysis of solitary waves involves (i) an examination of the anti-continuum limit for our model of interest, (ii) the existence and stability of these solitary waves via a perturbative approach and (iii) understanding the structure of these waves in excited sites of the lattice.
The effect of perturbations of small volume on the Lamé Coefficients of an elastic body can be used to detect imperfections of small volume. In the analysis of such perturbations, an elastic moment tensor appears. This quantity also appears naturally in the theory of composites, when one investigates inclusion shapes of minimal energy. In the context of inverse problems, it is relevant to know if for a given volume, one can distinguish thick inclusions from thin ones. This question can be addressed either as variational minimization problem, a free boundary problem, or a Newtonian potential problem. We will review recen progresses on the characterization of inclusion shapes corresponding to extremal polarization tensors and elastic moment tensor.
In this talk we will describe the method of convex integration applied to construct ”wild” solutions to a general class of non-dissipative active scalar equations. Specifically, we find bounded solutions in which the scalar takes value zero outside a finite time interval, and absolute value one a.e. inside the interval. This shows non-uniqueness in the class of bounded solutions, when no smoothness is imposed. Such solutions have been previously constructed for the Euler equation in a groundbreaking work of De Lellis and Szekelihidy and later for the 2D porous media equation by Cordoba, Gancedo and Faraco. This present work is a direct generalization of the latter and includes other equations such as one arising in magnetostrophic turbulence in the Earths fluid core.
In recent years, methods for differential operators of second order have been applied successfully to integro-differential operators of fractional order. In this context we concentrate on Harnack's inequality and Hölder regularity estimates. We show that one can formulate Harnack's inequality from 1887 in a way such that it holds for the Laplacian and fractional powers of the Laplacian at the same time. This approach generalizes to nonlocal Dirichlet forms. As a result, Hölder regularity estimates are obtained as a consequence of Harnack's inequality. We also discuss some counterexamples.
We start from a lattice based finite range particle potential which is invariant under rigid motion.Depending on the boundary condition that defines a finite volume Gibbs measure.In the elasticity scaling i.e. keeping the interaction strength fixed while letting the volume tend to infinity (or the lattice spacing to zero) we prove a two scale limit result.The macroscopic deformation will be the minimizer of a an elstic free energy,whereas the local fluctuations are described by a mixture of DLR i.e. infinite volume gradient Gibbs measures,something that we called GYGs -Gradient Young Gibbs measures.
The challenge of modeling the complexity of nematic liquid crystals through a model that is both comprehensive and simple enough to manipulate efficiently has led to the existence of several major competing theories. One of the most popular (among physicists) theories was proposed by Pierre Gilles de Gennes in the 70s and was a major reason for awarding him a Nobel prize in 1991. The theory models liquid crystals as functions defined on a two or three dimensional domains with values in the space of Q-tensors (that is symmetric, traceless, three-by-three matrices). Despite its popularity with physicists the theory has received little attention from mathematicians until a few years ago when John Ball initiated its study. Nowadays it is a fast developing area, combining in a fascinating manner topological, geometrical and analytical aspects. The aim of this talk is to survey this development.
I will discuss the regularity theory of the infinity Laplace equation. In particular, I will explain the proof of everywhere differentiability of infinity harmonic functions in all space dimensions.
We investigate the existence of generalized transition waves for a reaction-diffusion equation with a monostable nonlinearity f depending in a general way on time. It is well known that in the homogeneous case the family of speeds associated with traveling fronts is given by a right half-line. We extend this result by introducing a suitable notion of mean. As an application, we obtain the existence of random traveling waves when f is random stationary ergodic. We further present some spreading properties for solutions of the Cauchy problem associated with compactly supported initial data. This is a joint work in progress with G. Nadin (Paris 6).
