Related papers: Stable $s$-minimal cones in $\mathbb{R}^2$ are fla…

We prove that half spaces are the only stable nonlocal $s$-minimal cones in $\mathbb{R}^3$, for $s\in(0,1)$ sufficiently close to $1$. This is the first classification result of stable $s$-minimal cones in dimension higher than two. Its…

We show that if the rotation set of a homeomorphism of the torus is stable under small perturbations of the dynamics, then it is a convex polygon with rational vertices. We also show that such homeomorphisms are $C^0$-generic and have…

We show that stable cones for the one-phase free boundary problem are hyperplanes in dimension $4$. As a corollary, both one and two-phase energy minimizing hypersurfaces are smooth in dimension $4$.

We investigate stability of (2+1)-dimensional ring solitons of the nonlinear Schrodinger equation with focusing cubic and defocusing quintic nonlinearities. Computing eigenvalues of the linearised equation, we show that rings with spin…

We study stable solutions to the fractional Allen-Cahn equation \linebreak $(-\Delta)^{s/2} u = u-u^3$, $|u|<1$ in $\mathbb{R}^n$. For every $s\in (0,1)$ and dimension $n\geq 2$, we establish sharp energy estimates, density estimates, and…

In this article, we thoroughly investigate the stability inequality for Ricci-flat cones. Perhaps most importantly, we prove that the Ricci-flat cone over CP^2 is stable, showing that the first stable non-flat Ricci-flat cone occurs in the…

We prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter $s$ when $s\rightarrow 1^-$. As a consequence, we obtain that all the nonlocal minimal cones are flat and that all the…

In this work we classify the stable regions (second order minima of perimeter under an area constraint) in tori of revolution with piecewise continuous decreasing Gauss curvature from the longest parallel and with a horizontal symmetry.…

It is well-known that a minimal graph of codimension one is stable, i.e. the second variation of the area functional is non-negative. This is no longer true for higher codimensional minimal graphs. In this note, we prove that a minimal…

We prove that the stable 2-systole is uniformly bounded on the space of Riemannian metrics with scalar curvature at least one for closed spin 2-essential manifolds, which includes $S^2 \times S^2$, $S^2 \times T^n$, and…

Algebraic varieties which are locally isomorphic to open subsets of affine space will be called {\em plain}. Plain varieties are smooth and rational. The converse is true for curves and surfaces, and unknown in general. It is shown that…

We prove that 0 the only classical solution of the Lane-Emden equation in the half-space which is stable outside a compact set. We also consider weak solutions and the case of general cones.

These notes record the lectures for the CIME Summer Course taught by the first author in Cetraro during the week of June 19-23, 2017. The notes contain the proofs of several results on the classification of stable solutions to some…

Let S be a C^2 H-minimal noncharacteristic hypersurface in the first Heisenberg group. We show that if S contains a graphical strip, then it is not a stable minimal surface. Moreover, we show that if S is a C^2 H-minimal noncharacteristic…

We study the stable behaviour of discrete dynamical systems where the map is convex and monotone with respect to the standard positive cone. The notion of tangential stability for fixed points and periodic points is introduced, which is…

The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one-phase free boundary problem. The problem of classifying stable (or minimal) homogeneous solutions in dimensions $n\geq3$ is completely open. In…

We show that a complete, two-sided, stable minimal hypersurface in $\mathbf{R}^5$ is flat.

This article studies a class of semilinear scalar field equations on the real line with variable coefficients in the linear terms. These coefficients are not necessarily small perturbations of a constant. We prove that under suitable…

We establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the $s$-fractional perimeter as a particular case. On the one hand, we establish universal $BV$-estimates in every dimension…

In this article, we systematically investigate the stability properties of certain warped product Einstein manifolds. We characterize stability of these metrics in terms of an eigenvalue condition of the Einstein operator on the base…