Interpolation inequalities in pattern formation

We prove some interpolation inequalities which arise in the analysis of pattern formation in physics. They are the strong version of some already known estimates in weak form that are used to give a lower bound of the energy in many contexts (coarsening and branching in micromagnetics and superconductors). The main ingredient in the proof of our inequalities is a geometric construction which was first used by Choksi, Conti, Kohn, and one of the authors in the study of branching in superconductors

Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian

We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation $(-\Delta)^{1/2} u=f(u)$ in \re^n. Our energy estimates hold for every nonlinearity $f$ and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable. As a consequence, in dimension $n=3$, we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation $-\Delta u=f(u)$ in \re^n

A nonlocal supercritical Neumann problem

We establish existence of positive non-decreasing radial solutions for a nonlocal nonlinear Neumann problem both in the ball and in the annulus. The nonlinearity that we consider is rather general, allowing for supercritical growth (in the sense of Sobolev embedding). The consequent lack of compactness can be overcome, by working in the cone of non-negative and non-decreasing radial functions. Within this cone, we establish some a priori estimates which allow, via a truncation argument, to use variational methods for proving existence of solutions. As a side result, we prove a strong maximum principle for nonlocal Neumann problems, which is of independent interest.Comment: 32 pages, 0 figure

Sharp energy estimates for nonlinear fractional diffusion equations

We study the nonlinear fractional equation $(-\Delta)^s u = f(u)$ in $\mathbb{R}^n$, for all fractions $0<s<1$ and all nonlinearities $f$. For every fractional power $s \in (0,1)$, we obtain sharp energy estimates for bounded global minimizers and for bounded monotone solutions. They are sharp since they are optimal for solutions depending only on one Euclidian variable. As a consequence, we deduce the one-dimensional symmetry of bounded global minimizers and of bounded monotone solutions in dimension $n=3$ whenever $1/2 \leq s < 1$. This result is the analogue of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation $-\Delta u = f(u)$ in $\mathbb{R}^n$. It remains open for $n=3$ and $s<1/2$, and also for $n \geq 4$ and all $s$.Comment: arXiv admin note: text overlap with arXiv:1004.286

A nonlinear Liouville theorem for fractional equations in the Heisenberg group

We establish a Liouville-type theorem for a subcritical nonlinear problem, involving a fractional power of the sub-Laplacian in the Heisenberg group. To prove our result we will use the local realization of fractional CR covariant operators, which can be constructed as the Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre, as established in \cite{FGMT}. The main tools in our proof are the CR inversion and the moving plane method, applied to the solution of the lifted problem in the half-space \mathbb H^n\times \mathbbR^+

Risultati di regolarità per insiemi isoperimetrici con densità

In this note, we present some recent regularity results for sets which minimize a weighted notion of perimeter under a weighted volume constraint. We focus on the case of two different densities which are merely alpha-Holder continuous, and describe what are the main issues and techniques used in order to establish the optimal regularity C1, alpha/(2-alpha) for the reduced boundary of such sets.In questa nota, presentiamo alcuni recenti risultati di regolarità per insiemi che minimizzano una nozione pesata di perimetro sotto un vincolo di volume pesato. Ci focalizziamo sul caso di due densità diverse, che siano solo Holderiane di ordine alpha e descriviamo quali sono le maggiori difficoltà e le tecniche usate per provare la regolarità ottimale C1, alpha/(2-alpha) per la frontiera ridotta di tali insiemi