The Ginzburg-Landau equation in the Heisenberg group

We consider a functional related with phase transition models in the Heisenberg group framework. We prove that level sets of local minimizers satisfy some density estimates, that is, they behave as "codimension one" sets. We thus deduce a uniform convergence property of these level sets to interfaces with minimal area. These results are then applied in the construction of (quasi)periodic, plane-like minimizers, i.e., minimizers of our functional whose level sets are contained in a spacial slab of universal size in a prescribed direction. As a limiting case, we obtain the existence of hypersurfaces contained in such a slab which minimize the surface area with respect to a given periodic metric.Comment: 49 page

Social and Political Dimensions of Identity

We study the interior regularity of solutions to the Dirichlet problem Lu = g in Omega, u = 0 in R-nOmega, for anisotropic operators of fractional type Lu(x) = integral(+infinity)(0) dp integral(Sn-1) da(w) 2u(x) - u(x + rho w) - u(x - rho w)/rho(1+2s). Here, a is any measure on Sn-1 (a prototype example for L is given by the sum of one-dimensional fractional Laplacians in fixed, given directions). When a is an element of C-infinity(Sn-1) and g is c(infinity)(Omega), solutions are known to be C-infinity inside Omega (but not up to the boundary). However, when a is a general measure, or even when a is L-infinity(s(n-1)), solutions are only known to be C-3s inside Omega. We prove here that, for general measures a, solutions are C1+3s-epsilon inside Omega for all epsilon > 0 whenever Omega is convex. When a is an element of L-infinity(Sn-1), we show that the same holds in all C-1,C-1 domains. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the measure a is singular, we construct an explicit counterexample for which u is not C3s+epsilon for any epsilon > 0 - even if g and Omega are C-infinity

Mean curvature properties for pp-Laplace phase transitions

This paper deals with phase transitions corresponding to an energy which is the sum of a kinetic part of p-Laplacian type and a double well potential h(0) with suitable growth conditions. We prove that level sets of solutions of Delta(p)u=h(0)'(u) possessing a certain decay property satisfy a mean curvature equation in a suitable weak viscosity sense. From this, we show that, if the above level sets approach uniformly a hypersurface, the latter has zero mean curvature

Optimizing the fractional power in a model with stochastic PDE constraints

We study an optimization problem with SPDE constraints, which has the peculiarity that the control parameter s is the s-th power of the diffusion operator in the state equation. Well-posedness of the state equation and differentiability properties with respect to the fractional parameter s are established. We show that under certain conditions on the noise, optimality conditions for the control problem can be established

The effect on fisher-kpp propagation in a cylinder with fast diffusion on the boundary

In this paper we consider a reaction-diffusion equation of Fisher-KPP type inside an infinite cylindrical domain in RN+1, coupled with a reaction-diffusion equation on the boundary of the domain, where potentially fast diffusion is allowed. We will study the existence of an asymptotic speed of propagation for solutions of the Cauchy problem associated with such a system, as well as the dependence of this speed on the diffusivity at the boundary and the amplitude of the cylinder. When N = 1 the domain reduces to a strip between two straight lines. This models the effect of two roads with fast diffusion on a strip-shaped field bounded by them

Nonlocal quantitative isoperimetric inequalities

We show a quantitative-type isoperimetric inequality for fractional perimeters where the deficit of the t-perimeter, up to multiplicative constants, controls from above that of the s-perimeter, with s smaller than t. To do this we consider a problem of independent interest: we characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of the t-perimeter and the s-perimeter. In particular, we show that balls are the unique minimizers if the volume is sufficiently small, depending on t 12 s, while the existence vs. nonexistence of minimizers for large volumes remains open. We also consider the corresponding isoperimetric problem and prove existence and regularity of minimizers for all s, t. When s = 0 this problem reduces to the fractional isoperimetric problem, for which it is well known that balls are the only minimizers

Rigidity results for elliptic PDEs with uniform limits: an abstract framework with applications

We provide an abstract framework for a symmetry result arising in a conjecture of G.W. Gibbons and we apply it to the fractional Laplace operator, to the elliptic operators with constant coefficients, to the quasilinear operators, and to elliptic fully nonlinear operators with possible gradient dependence

Density estimates for a degenerate/singular phase-transition model

We consider a Ginzburg-Landau type phase-transition model driven by a p-Laplacian type equation. We prove density estimates for absolute minimizers and we deduce the uniform convergence of level sets and the existence of plane-like minimizers in periodic media

1D symmetry for semilinear PDEs from the limit interface of the solution

We study bounded, monotone solutions of u=W(u) in the whole of (n), where W is a double-well potential. We prove that under suitable assumptions on the limit interface and on the energy growth, u is 1D.In particular, differently from the previous literature, the solution is not assumed to have minimal properties and the cases studied lie outside the range of -convergence methods.We think that this approach could be fruitful in concrete situations, where one can observe the phase separation at a large scale and wishes to deduce the values of the state parameter in the vicinity of the interface.As a simple example of the results obtained with this point of view, we mention that monotone solutions with energy bounds, whose limit interface does not contain a vertical line through the origin, are 1D, at least up to dimension 4