1,721,011 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
Quantitative flatness results and -estimates for stable nonlocal minimal surfaces
Full text linkWe 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
GEOMETRIC INEQUALITIES FOR FRACTIONAL LAPLACE OPERATORS AND APPLICATIONS
Full text linkAbstract. 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
-Convergence of variational functionals with boundary terms in Stein manifolds
Full text linkLet be an open subset of a Stein manifold and let be its boundary. It is well known that M inherits a natural contact structure. In this paper we consider a family of variational functionals defined by the sum of two terms: a Dirichlet-type energy associated with a sub-Riemannian structure in and a potential term on the boundary M. We prove that the functionals -converge to the intrinsic perimeter in M associated with its contact structure. Similar results have been obtained in the Euclidean space by Alberti, Bouchitté, Seppecher. We stress that already in the Euclidean setting the situation is not covered by the classical Modica–Mortola theorem because of the presence of the boundary term. We recall also that Modica–Mortola type results (without a boundary term) have been proved in the Euclidean space for sub-Riemannian energies by Monti and Serra Cassano
Sharp energy estimates for nonlinear fractional diffusion equations
Full text linkWe study the nonlinear fractional equation (−Δ)su=f(u) in Rn, for all fractions 0<s<1 and all nonlinearities f . For every fractional power s∈(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≤s<1 . This result is the analogue of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation −Δu=f(u) in Rn . It remains open for n=3 and s<1/2 , and also for n≥4 and all s .Peer ReviewedPostprint (published version
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
A quantitative stability estimate for the fractional Faber-Krahn inequality
Full text linkWe prove a quantitative version of the Faber-Krahn inequality for the first eigenvalue of the fractional Dirichlet-Laplacian of order s. This is done by using the so-called Caffarelli-Silvestre extension and adapting to the nonlocal setting a trick by Hansen and Nadirashvili. The relevant stability estimate comes with an explicit constant, which is stable as the fractional order of differentiability goes to 1. (C) 2020 Elsevier Inc. All rights reserved
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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