Entire Minimizers of Allen–Cahn Systems with Sub-Quadratic Potentials

We study entire minimizers of the Allen–Cahn systems. The specific feature of our systems are potentials having a finite number of global minima, with sub-quadratic behaviour locally near their minima. The corresponding formal Euler–Lagrange equations are supplemented with free boundaries. We do not study regularity issues but focus on qualitative aspects. We show the existence of entire solutions in an equivariant setting connecting the minima of W at infinity, thus modeling many coexisting phases, possessing free boundaries and minimizing energy in the symmetry class. We also present a very modest result of existence of free boundaries under no symmetry hypotheses. The existence of a free boundary can be related to the existence of a specific sub-quadratic feature, a dead core, whose size is also quantified

On the structure of phase transition maps for three or more coexisting phases

This paper is partly based on a lecture delivered by the author at the ERC workshop "Geometric Partial Differential Equations" held in Pisa in September 2012. What is presented here is an expanded version of that lecture.Comment: 23 pages, 6 figure

Gradient flows and instantons at a Lifshitz point

I provide a broad framework to embed gradient flow equations in non-relativistic field theory models that exhibit anisotropic scaling. The prime example is the heat equation arising from a Lifshitz scalar field theory; other examples include the Allen-Cahn equation that models the evolution of phase boundaries. Then, I review recent results reported in arXiv:1002.0062 describing instantons of Horava-Lifshitz gravity as eternal solutions of certain geometric flow equations on 3-manifolds. These instanton solutions are in general chiral when the anisotropic scaling exponent is z=3. Some general connections with the Onsager-Machlup theory of non-equilibrium processes are also briefly discussed in this context. Thus, theories of Lifshitz type in d+1 dimensions can be used as off-shell toy models for dynamical vacuum selection of relativistic field theories in d dimensions.Comment: 19 pages, 1 figure, contribution to conference proceedings (NEB14); minor typos corrected in v

B and I-band optical micro-variability observations of the BL Lac objects S5 2007+777 and 3C371

We have observed S5 2007+777 and 3C371 in the B and I bands for 13 and 8 nights, respectively, during various observing runs in 2001, 2002 and 2004. The observations resulted in almost evenly sampled light curves, 6-9 hours long. We do not detect any flares within the observed light curves, but we do observe small amplitude, significant variations, in both bands, on time scales of hours and days. The average variability amplitude on time scales of minutes/hours is 2.5% and 1-1.5% in the case of S5 2007+777 and 3C371, respectively. The average amplitudes increase to 5-12% and 4-6%, respectively, on time scales of days. We find that the B and I band variations are highly correlated, on both short and long time scales. During the 2004 observations, which resulted in the longest light curves, we observe two well defined flux-decay and rising trends in the light curves of both objects. When the flux decays, we observe significant delays, with the B band flux decaying faster than the flux in the I band. As a result, we also observe significant, flux related spectral variations as well. The flux-spectral relation is rather complicated, with loop-like structures forming during the flux evolution. The presence of spectral variations imply that the observed variability is not caused by geometric effects. On the other hand, our results are fully consistent with the hypothesis that the observed variations are caused by perturbations which affect different regions in the jet of the sources.Comment: Accepted for publication in Astronomy and Astrophysic

Three-Fold Diffraction Symmetry in Epitaxial Graphene and the SiC Substrate

The crystallographic symmetries and spatial distribution of stacking domains in graphene films on SiC have been studied by low energy electron diffraction (LEED) and dark field imaging in a low energy electron microscope (LEEM). We find that the graphene diffraction spots from 2 and 3 atomic layers of graphene have 3-fold symmetry consistent with AB (Bernal) stacking of the layers. On the contrary, graphene diffraction spots from the buffer layer and monolayer graphene have apparent 6-fold symmetry, although the 3-fold nature of the satellite spots indicates a more complex periodicity in the graphene sheets.Comment: An addendum has been added for the arXiv version only, including one figure with five panels. Published paper can be found at http://link.aps.org/doi/10.1103/PhysRevB.80.24140

Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension

The qualitative behavior of a thermodynamically consistent two-phase Stefan problem with surface tension and with or without kinetic undercooling is studied. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities in a sense made precise below, it is proved that it exists globally in time and its orbit is relatively compact. In addition, stability and instability of equilibria is studied. In particular, it is shown that multiple spheres of the same radius are unstable, reminiscent of the onset of Ostwald ripening.Comment: 56 pages. Expanded introduction, added references. This revised version is published in Arch. Ration. Mech. Anal. (207) (2013), 611-66

Critical behavior of collapsing surfaces

We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the existence of a critical initial surface that develops a degenerate neckpinch. The limiting flow of the Type II singularity is accurately modeled by the rotationally symmetric translating soliton.Comment: 23 pages, 10 figure