Front propagation in geometric and phase field models of stratified media

We study front propagation problems for forced mean curvature flows and their phase field variants that take place in stratified media, i.e., heterogeneous media whose characteristics do not vary in one direction. We consider phase change fronts in infinite cylinders whose axis coincides with the symmetry axis of the medium. Using the recently developed variational approaches, we provide a convergence result relating asymptotic in time front propagation in the diffuse interface case to that in the sharp interface case, for suitably balanced nonlinearities of Allen-Cahn type. The result is established by using arguments in the spirit of $\Gamma$-convergence, to obtain a correspondence between the minimizers of an exponentially weighted Ginzburg-Landau type functional and the minimizers of an exponentially weighted area type functional. These minimizers yield the fastest traveling waves invading a given stable equilibrium in the respective models and determine the asymptotic propagation speeds for front-like initial data. We further show that generically these fronts are the exponentially stable global attractors for this kind of initial data and give sufficient conditions under which complete phase change occurs via the formation of the considered fronts

Connected surfaces with boundary minimizing the Willmore energy

For a given family of smooth closed curves gamma(1),...,gamma(alpha) subset of R-3 we consider the problem of finding an elastic connected compact surface M with boundary gamma = gamma(1) boolean OR ... boolean OR gamma(alpha). This is realized by minimizing the Willmore energy W on a suitable class of competitors. While the direct minimization of the Area functional may lead to limits that are disconnected, we prove that, if the infimum of the problem is < 4 pi, there exists a connected compact minimizer of W in the class of integer rectifiable curvature varifolds with the assigned boundary conditions. This is done by proving that varifold convergence of bounded varifolds with boundary with uniformly bounded Willmore energy implies the convergence of their supports in Hausdorff distance. Hence, in the cases in which a small perturbation of the boundary conditions causes the non-existence of Area-minimizing connected surfaces, our minimization process models the existence of optimal elastic connected compact generalized surfaces with such boundary data. We also study the asymptotic regime in which the diameter of the optimal connected surfaces is arbitrarily large. Under suitable boundedness assumptions, we show that rescalings of such surfaces converge to round spheres. The study of both the perturbative and the asymptotic regime is motivated by the remarkable case of elastic surfaces connecting two parallel circles located at any possible distance one from the other. The main tool we use is the monotonicity formula for curvature varifolds ([15, 31]) that we extend to varifolds with boundary, together with its consequences on the structure of varifolds with bounded Willmore energy

REIFENBERG FLATNESS FOR ALMOST MINIMIZERS OF THE PERIMETER UNDER MINIMAL ASSUMPTIONS

The aim of this note is to prove that almost minimizers of the perimeter are Reifenberg flat, for a very weak notion of minimality. The main observation is that smallness of the excess at some scale implies smallness of the excess at all smaller scales

Nonlocal minimal clusters in the plane

We prove existence of partitions of an open set \u3a9 with a given number of phases, which minimize the sum of the fractional perimeters of all the phases, with Dirichlet boundary conditions. In two dimensions we show that, if the fractional parameter s is sufficiently close to 1, the only singular minimal cone, that is, the only minimal partition invariant by dilations and with a singular point, is given by three half-lines meeting at 120 degrees. In the case of a weighted sum of fractional perimeters, we show that there exists a unique minimal cone with three phases

Quantitative estimates for bending energies and applications to non-local variational problems

We discuss a variational model, given by a weighted sum of perimeter, bending and Riesz interaction energies, that could be considered as a toy model for charged elastic drops. The different contributions have competing preferences for strongly localized and maximally dispersed structures. We investigate the energy landscape in dependence of the size of the 'charge', that is, the weight of the Riesz interaction energy. In the two-dimensional case, we first prove that for simply connected sets of small elastica energy, the elastica deficit controls the isoperimetric deficit. Building on this result, we show that for small charge the only minimizers of the full variational model are either balls or centred annuli. We complement these statements by a non-existence result for large charge. In three dimensions, we prove area and diameter bounds for configurations with small Willmore energy and show that balls are the unique minimizers of our variational model for sufficiently small charge