Bump solutions for the mesoscopic Allen�CCahn equation in periodic media

Given a double-well potential F, a ${\mathbb{Z}^n}$ -periodic function H, small and with zero average, and ???>?0, we find a large R, a small ?? and a function H _?? which is ??-close to H for which the following two problems have solutions:

1. Find a set E _?? ,R whose boundary is uniformly close to ? B _R and has mean curvature equal to ?H _?? at any point,
2. Find u = u _?? ,R,?? solving $$ -\delta\,\Delta u + \frac{F'(u)}{\delta} +\frac{c_0}{2} H_\varepsilon = 0, $$ such that u _??,R,?? goes from a ??-neighborhood of +?1 in B _R to a ??-neighborhood of ?1 outside B _R .

     

Singularity Formation in Fractional Burgers’ Equations

The formation of singularities in finite time in nonlocal Burgers’ equations, with time-fractional derivative, is studied in detail. The occurrence of finite-time singularity is proved, revealing the underlying mechanism, and precise estimates on the blowup time are provided. The employment of the present equation to model a problem arising in job market is also analyzed.

     

Some connections between results and problems of De Giorgi, Moser and Bangert

Using theorems of Bangert, we prove a rigidity result which shows how a question raised by Bangert for elliptic integrands of Moser type is connected, in the case of minimal solutions without self-intersections, to a famous conjecture of De Giorgi for phase transitions. The work of Enrico Valdinoci was supported by MIUR Variational Methods and Nonlinear Differential Equations. Diese Zusammenarbeit wurde bei einem sehr angenehmen Besuch von EV in Freiburg begonnen.     

Flatness of Bernoulli jets

Level sets of minimizers of a possibly singular or degenerate elliptic variational problem related with fluid jets are shown to possess a Harnack-type inequality. A few flatness and one-dimensional results thence follow. This work has been partially supported by MIUR Variational Methods and Nonlinear Differential Equations. ``...e diedi un volto a quelle mie chimere le navi costruii di forma ardita, concavi navi dalle vele nere...'(Fancesco Guccini, Odysseus.)     

A generalization of Aubry–Mather theory to partial differential equations and pseudo-differential equations

We discuss an Aubry–Mather-type theory for solutions of non-linear, possibly degenerate, elliptic PDEs and other pseudo-differential operators.We show that for certain PDEs and ΨDEs with periodic coefficients and a variational structure it is possible to find quasi-periodic solutions for all frequencies. This results also hold under a generalized definition of periodicity that makes it possible to consider problems in covers of several manifolds, including manifolds with non-commutative fundamental groups.An abstract result will be provided, from which an Aubry–Mather-type theory for concrete models will be derived.     

Critical Points Inside the Gaps of Ground State Laminations for Some Models in Statistical Mechanics

We consider models of interacting particles situated in the points of a discrete set Λ. The state of each particle is determined by a real variable. The particles are interacting with each other and we are interested in ground states and other critical points of the energy (metastable states). Under the assumption that the set Λ and the interaction are symmetric under the action of a group G—which satisfies some mild assumptions—, that the interaction is ferromagnetic, as well as periodic under addition of integers, and that it decays with the distance fast enough, it was shown in a previous paper that there are many ground states that satisfy an order property called self-conforming or Birkhoff. Under some slightly stronger assumptions all ground states satisfy this order property. Under the assumption that the interaction decays fast enough with the distance, we show that either the ground states form a one dimensional family or that there are other Birkhoff critical points which are not ground states, but lying inside the gaps left by ground states. This alternative happens if and only if a Peierls–Nabarro barrier vanishes. The main tool we use is a renormalized energy. In the particular case that the set Λ is a one dimensional lattice and that the interaction is just nearest neighbor, our result establishes Mather’s criterion for the existence of invariant circles in twist mappings in terms of the vanishing of the Peierls–Nabarro barrier. The work of RdlL was supported by NSF grants. The work of EV was supported by GNAMPA and MIUR Variational Methods and Nonlinear Differential Equations.