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We consider a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions in a bounded domain Ω ⊂ R n whose boundary has an ( n − 2 ) -dimensional singularity. Assuming 1 p n + 2 n − 2 , we prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.

In this paper we show that minima and stable solutions of a general energy functional of the form ∫ Ω F ( ∇ u , ∇ v , u , v , x ) d x enjoy some monotonicity properties, under an assumption on the growth at infinity of the energy. Our results are quite general, and comprise some rigidity results which are known in the literature.

We prove existence, qualitative properties and asymptotic behavior of positive solutions to the doubly critical problem $$\begin{aligned} (-\Delta )^s u=\vartheta \frac{u}{|x|^{2s}}+u^{2_s^*-1}, \quad u\in \dot{H}^s(\mathbb {R}^N), \quad N> 2s,\quad 0

We consider surfaces which minimize a nonlocal perimeter functional and we discuss their interior regularity and rigidity properties, in a quantitative and qualitative way, and their (perhaps rather surprising) boundary behavior. We present at least a sketch of the proofs of these results, in a way that aims to be as elementary and self contained as possible, referring to the papers [CRS10, SV13, CV13, BFV14, FV, DSV15, CSV16] for full details.

In this paper we show that a nonlocal minimal surface which is a graph outside a cylinder is in fact a graph in the whole of the space. As a consequence, in dimension 3, we show that the graph is smooth. The proofs rely on convolution techniques and appropriate integral estimates which show the pointwise validity of an Euler–Lagrange equation related to the nonlocal mean curvature.

We discuss some results related to a phase transition model in which the potential energy induced by a double-well function is balanced by a fractional elastic energy. In particular, we present asymptotic results (such as Gamma-convergence, energy bounds and density estimates for level sets), flatness and rigidity results, and the construction of planelike minimizers in periodic media. Finally, we consider a nonlocal equation with a multiwell potential, motivated by models arising in crystal dislocations, and we construct orbits exhibiting symbolic dynamics, inspired by some classical results by Paul Rabinowitz.

We consider a nonlocal equation set in an unbounded domain with the epigraph property. We prove symmetry, monotonicity and rigidity results. In particular, we deal with halfspaces, coercive epigraphs and epigraphs that are flat at infinity., Final version, accepted for publication on Mathematische Annalen

We provide a series of rigidity results for a nonlocal phase transition equation. The prototype equation that we consider is of the form $$ (-\Delta)^{s/2} u=u-u^3,$$ with~$s\in(0,1)$. More generally, we can take into account equations like $$ L u = f(u),$$ where $f$ is a bistable nonlinearity and $L$ is an integro-differential operator, possibly of anisotropic type. The results that we obtain are an improvement of flatness theorem and a series of theorems concerning the one-dimensional symmetry for monotone and minimal solutions, in the research line dictaded by a classical conjecture of E. De Giorgi. Here, we collect a series of pivotal results, of geometric type, which are exploited in the proofs of the main results in the companion paper.

We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a Levy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case. As ambient space, we can consider: $ \bullet $bounded domains, $ \bullet $periodic environments, $ \bullet $transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one. In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.

This paper contains three types of results: the construction of ground state solutions for a long-range Ising model whose interfaces stay at a bounded distance from any given hyperplane, the construction of nonlocal minimal surfaces which stay at a bounded distance from any given hyperplane, the reciprocal approximation of ground states for long-range Ising models and nonlocal minimal surfaces. In particular, we establish the existence of ground state solutions for long-range Ising models with planelike interfaces, which possess scale invariant properties with respect to the periodicity size of the environment. The range of interaction of the Hamiltonian is not necessarily assumed to be finite and also polynomial tails are taken into account (i.e. particles can interact even if they are very far apart the one from the other). In addition, we provide a rigorous bridge between the theory of long-range Ising models and that of nonlocal minimal surfaces, via some precise limit result.

We consider the equation $-\epsilon^{2}\Delta u + u = u^ {p}$ in a bounded domain $\Omega\subset\R^{3}$ with edges. We impose Neumann boundary conditions, assuming $1, Comment: 24 pages. Second Version, minor changes. To appear in Annales de l'Institut Henri Poincar\'e - Analyse non lin\'eaire

For a smooth, bounded domain [math] , [math] , [math] we consider the nonlocal equation ¶ ϵ2s(−Δ)su + u = up in Ω ¶ with zero Dirichlet datum and a small parameter [math] . We construct a family of solutions that concentrate as [math] at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case [math] , the leading order of the associated reduced energy functional in a variational reduction procedure is of polynomial instead of exponential order on the distance from the boundary, due to the nonlocal effect. Delicate analysis is needed to overcome the lack of localization, in particular establishing the rather unexpected asymptotics for the Green function of [math] in the expanding domain [math] with zero exterior datum.

In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant.

We consider a system of nonlocal equations driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinic, homoclinic and chaotic trajectories are constructed. This is the first attempt to consider a nonlocal version of this type of dynamical systems in a variational setting and the first result regarding symbolic dynamics in a fractional framework.

Given $s,\sigma\in(0,1)$ and a bounded domain $\Omega\subset\mathbb{R}^n$, we consider the following minimization problem of $s$-Dirichlet--plus--$\sigma$-perimeter-type $ [u]_{ H^s(\mathbb{R}^{2n}\setminus(\Omega^c)^2) } + {\rm Per}_\sigma (\{u>0\},\Omega),$ where $[ \cdot]_{H^s}$ is the fractional Gagliardo seminorm and ${\rm Per}_\sigma$ is the fractional perimeter. Among other results, we prove a monotonicity formula for the minimizers, glueing lemmata, uniform energy bounds, convergence results, a regularity theory for the planar cones, and a trivialization result for the flat case. The classical free boundary problems are limit cases of the one that we consider in this paper, as $s\nearrow1$, $\sigma\nearrow1$, or $\sigma\searrow0$.

Given $s\in(0,1)$, we consider the problem of minimizing the fractional Gagliardo seminorm in $H^s$ with prescribed condition outside the ball and under the further constraint of attaining zero value in a given set $K$. &nbsp We investigate how the energy changes in dependence of such set. In particular, under mild regularity conditions, we show that adding a set $A$ to $K$ increases the energy of at most the measure of $A$ (this may be seen as a perturbation result for small sets $A$). &nbsp Also, we point out a monotonicity feature of the energy with respect to the prescribed sets and the boundary conditions.

We consider the equation $$v_t=L_s v-W'(v)+\sigma_\epsilon(t,x) \quad {\mbox{ in }} (0,+\infty)\times\R,$$ where $L_s$ is an integro-differential operator of order $2s$, with $s\in(0,1)$, $W$ is a periodic potential, and $\sigma_\epsilon$ is a small external stress. The solution $v$ represents the atomic dislocation in the Peierls--Nabarro model for crystals, and we specifically consider the case $s\in(0,1/2)$, which takes into account a strongly nonlocal elastic term. We study the evolution of such dislocation function for macroscopic space and time scales, namely we introduce the function $$ v_{\epsilon}(t,x):=v\left(\frac{t}{\epsilon^{1+2s}},\frac{x}{\epsilon}\right). $$ We show that, for small $\epsilon$, the function $v_\epsilon$ approaches the sum of step functions. From the physical point of view, this shows that the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. We also show that the motion of these dislocation points is governed by an interior repulsive potential that is superposed to an elastic reaction to the external stress.

We study the symmetry properties for solutions of elliptic systems of the type [Formula: see text] where x ∈ ℝm with 1 ≤ m < N, X = (x, y) ∈ ℝm × ℝN-m, and F1,…,Fn are the derivatives with respect to ξ1,…,ξn of some F = F(x,ξ1,…,ξn) such that for any i = 1,…,n and any fixed (x,ξ1,…,ξi-1,ξi+1,…,ξn) ∈ ℝm × ℝn-1 the map ξi → F(x,ξ1,…,ξi,…,ξn) belongs to C2(ℝ). We obtain a Poincaré-type formula for the solutions of the system and we use it to prove a symmetry result both for stable and for monotone solutions.

We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $\Omega$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $\Omega$. Moreover, we construct examples of sets for which the limit does not exist.

International audience; We present some basic facts about the controllability of nonlinear finite dimensional systems. We introduce the concepts of Lie bracket and of Lie algebra generated by a family of vector fields. We then prove the Krener theorem on local accessibility and the Chow-Rashevskii theorem on controllability of symmetric systems. We then introduce the theory of compatible vector fields and we apply it to study control-affine systems with a recurrent drift or satisfying the strong Lie bracket generating assumption. We conclude with a general discussion about the orbit theorem by Sussmann and Nagano.