114 research outputs found
Saddle-shaped solutions for the fractional Allen-Cahn equation
No full textWe establish existence and qualitative properties of solutions to the fractional Allen-Cahn equation, which vanish on the Simons cone and are even with respect to the coordinate axes. These solutions are called saddle-shaped solutions. More precisely, we prove monotonicity properties, asymptotic behaviour, and instability in dimensions 2m = 4, 6. We extend to any fractional power s of the Laplacian, some results obtained for the case s = 1/2 in [19]. The interest in the study of saddle-shaped solutions comes in connection with a celebrated De Giorgi conjecture on the one-dimensional symmetry of monotone solutions and of minimizers for the Allen-Cahn equation. Saddle-shaped solutions are candidates to be (not one-dimensional) minimizers in high dimension, a property which is not known to hold yet
Existence and non-existence results for a semilinear fractional Neumann problem
No full textWe establish a priori -estimates for non-negative solutions of a semilinear nonlocal Neumann problem. As a consequence of these estimates, we get non-existence of non-constant solutions under suitable assumptions on the diffusion coefficient and on the nonlinearity. Moreover, we prove an existence result for radial, radially non-decreasing solutions in the case of a possible supercritical nonlinearity, extending to the case $
On fractional Hardy-type inequalities in general open sets
No full textWe show that, when sp > N, the sharp Hardy constant hs,p of the punctured space ℝN \ {0} in the Sobolev–Slobodeckiĭ space provides an optimal lower bound for the Hardy constant hs,p(Ω) of an open set Ω ⊂ ℝN. The proof exploits the characterization of Hardy’s inequality in the fractional setting in terms of positive local weak supersolutions of the relevant Euler–Lagrange equation and relies on the construction of suitable supersolutions by means of the distance function from the boundary of Ω. Moreover, we compute the limit of hs,p as s ↗ 1, as well as the limit when p ↗ ∞. Finally, we apply our results to establish a lower bound for the non-local eigenvalue λs,p(Ω) in terms of hs,p when sp > N, which, in turn, gives an improved Cheeger inequality whose constant does not vanish as p ↗ ∞
GEOMETRIC INEQUALITIES FOR FRACTIONAL LAPLACE OPERATORS AND APPLICATIONS
No full textAbstract. We prove a weighted fractional inequality involving the solution u of a nonlocal semilinear problem in Rn. Such in-equality bounds a weighted L2-norm of a compactly supported function φ by a weighted Hs-norm of φ. In this inequality a geo-metric quantity related to the level sets of u will appear. As a consequence we derive some relations between the stability of u and the validity of fractional Hardy inequalities. 1. introduction In this paper, following the ideas contained in [17], we prove a weighted Poincare ́ inequality that gives us useful informations con-cerning the geometry of the level surfaces of stable solutions of the fractional semi-linear equation (−∆)su = f(u) in Rn, (1.1) where s ∈ (0, 1) and f is C1 in the range of u. For every locally integrable function u: Rn → R such that
Quantitative flatness results and -estimates for stable nonlocal minimal surfaces
No full textWe 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 n >= 2 for stable sets. Namely, we prove that any stable set in B-1 has finite classical perimeter in B-1/2, with a universal bound. This nonlocal result is new even in the case of s-perimeters and its local counterpart (for classical stable minimal surfaces) was known only for simply connected two-dimensional surfaces immersed in R-3.On the other hand, we prove quantitative flatness estimates for minimizers and stable sets in low dimensions n = 2, 3. More precisely, we show that a stable set in B-R, with R large, is very close in measure to being a half space in B-1 - with a quantitative estimate on the measure of the symmetric difference. As a byproduct, we obtain new classification results for stable sets in the whole plane
Risultati di regolarità per insiemi isoperimetrici con densità
No full textIn 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
Flatness results for nonlocal phase transitions
No full textWe consider a nonlocal version of the Allen-Cahn equation, which models phase transitions problems. In the classical setting, the connection between the Allen-Cahn equation and the classification of entire minimal surfaces is well known and motivates a celebrated conjecture by E. De Giorgi on the one-dimensional symmetry of bounded monotone solutions to the (classical) Allen-Cahn equation up to dimension 8. In this work, we present some recent results in the study of the nonlocal analogue of this phase transition problem. In particular we describe the results obtained in several contributions where the classification of certain entire bounded solutions to the fractional Allen-Cahn equation has been obtained. Moreover we describe the connection between the fractional Allen-Cahn equation and the fractional perimeter functional, and we present also some results in the classifications of nonlocal minimal surfaces
The fractional mean curvature flow
No full textIn this note, we present some recent results in the study of the fractional mean curvature flow, that is a geometric evolution of the boundary of a set whose speed is given by the fractional mean curvature. The flow under consideration is of nonlocal type and presents several interesting difference with respect to the classical mean curvature flow. We will describe the main contributions in this field, with particular emphasis on some tipically nonlocal behaviors which are in contrast with the classical local case.In questa nota, presentiamo alcuni risultati recenti riguardanti lo studio del moto per curvatura media frazionaria, che descrive l'evoluzione del bordo di un insieme la cui velocita è data dalla curvatura media frazionaria. Tale flusso ha natura nonlocale e presenta alcune interessanti differenze rispetto al flusso per curvatura media classica. Descriviamo i principali contributi in questo ambito, con particolare enfasi ai comportamente tipicamente nonlocali che sono in contrasto col caso classico
Neckpinch singularities in fractional mean curvature flows
No full textIn this paper we consider the evolution of boundaries of sets by a fractional mean curvature flow. We show that for any dimension n >=2, there exist embedded hypersurfaces in R^n which develop a singularity without shrinking to a point. Such examples are well known for the classical mean curvature flow for n >=3. Interestingly, when n = 2, our result provides instead a counterexample in the nonlocal framework to the well-known Grayson’s Theorem, which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point. The essential step in our construction is an estimate which ensures that a suitably small perturbation of a thin strip has positive fractional curvature at every boundary point
The ε− εβ property, the boundedness of isoperimetric sets in Rn with density, and some applications
No full textAbstract. We show that every isoperimetric set in RN with density is bounded if the density is continuous and bounded by above and below. This improves the previously known bound-edness results, which basically needed a Lipschitz assumption; on the other hand, the present assumption is sharp, as we show with an explicit example. To obtain our result, we observe that the main tool which is often used, namely a classical “ε − ε ” property already discussed by Allard, Almgren and Bombieri, admits a weaker counterpart which is still sufficient for the boundedness, namely, an “ε − εβ ” version of the property. And in turn, while for the validity of the first property the Lipschitz assumption is essential, for the latter the sole continuity is enough. We conclude by deriving some consequences of our result about the existence and regularity of isoperimetric sets. 1
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