1,721,001 research outputs found
Regularity of nonlocal minimal cones in dimension 2
No full textWe show that the only nonlocal s-minimal cones in R2 are the trivial ones for all S ∈ (0, 1). As a consequence we obtain that the singular set of a nonlocal minimal surface has at most n - 3 Hausdorff dimension
<i>Γ</i>-convergence for nonlocal phase transitions
No full textWe discuss the Γ-convergence, under the appropriate scaling, of the energy functional∥u∥ Hs(Ω) 2+∫ ΩW(u)dx, with s∈(0,1), where ∥u∥H s (Ω) denotes the total contribution from Ω in the H s norm of u, and W is a double-well potential. When s∈[1/2,1), we show that the energy Γ-converges to the classical minimal surface functional - while, when s∈(0,1/2), it is easy to see that the functional Γ-converges to the nonlocal minimal surface functional
Density estimates for a variational model driven by the Gagliardo norm
No full textWe prove density estimates for level sets of minimizers of the energyε2s{norm of matrix}u{norm of matrix}Hs(Ω)2+∫ΩW(u)dx, with s∈(0, 1), where {norm of matrix}u{norm of matrix}Hs(Ω) denotes the total contribution from Ω in the Hs norm of u, and W is a double-well potential.As a consequence we obtain, as ε→0+, the uniform convergence of the level sets of u to either an Hs-nonlocal minimal surface if s∈(0,12), or to a classical minimal surface if s∈[12,1)
Density estimates for a nonlocal variational model via the Sobolev inequality
No full textWe consider the minimizers of the energy with , where denotes the total contribution from in the norm of u, and W is a double-well potential. By using a fractional Sobolev inequality, we give a new proof of the fact that the sublevel sets of a minimizer u in a large ball occupy a volume comparable with the volume of
Elliptic PDEs with fibered nonlinearities
No full textIn R m ×R n-m, endowed with coordinates x=(x′,x′′), we consider bounded solutions of the PDE Δ u(x)=f(u(x))χ(x'). We prove a geometric inequality, from which a symmetry result follows
Local and global minimizers for a variational energy involving a fractional norm
No full textWe study existence, uniqueness, and other geometric properties of the minimizers of the energy functional
∥u∥2Hs(Ω)+∫ΩW(u)dx,
where ∥u∥Hs(Ω) denotes the total contribution from Ω in the H s norm of u and W is a double-well potential. We also deal with the solutions of the related fractional elliptic Allen-Cahn equation on the entire space Rn . The results collected here will also be useful for forthcoming papers, where the second and the third author will study the Γ-convergence and the density estimates for level sets of minimizers
Boundary behavior of nonlocal minimal surfaces
Full text linkWe consider the behavior of the nonlocal minimal surfaces in the vicinity of the boundary. By a series of detailed examples, we show that nonlocal minimal surfaces may stick at the boundary of the domain, even when the domain is smooth and convex. This is a purely nonlocal phenomenon, and it is in sharp contrast with the boundary properties of the classical minimal surfaces. In particular, we show stickiness phenomena to half-balls when the datum outside the ball is a small half-ring and to the side of a two-dimensional box when the oscillation between the datum on the right and on the left is large enough. When the fractional parameter is small, the sticking effects may become more and more evident. Moreover, we show that lines in the plane are unstable at the boundary: namely, small compactly supported perturbations of lines cause the minimizers in a slab to stick at the boundary, by a quantity that is proportional to a power of the perturbation. In all the examples, we present concrete estimates on the stickiness phenomena. Also, we construct a family of compactly supported barriers which can have independent interest
Minimization of a fractional perimeter-Dirichlet integral functional
Full text linkWe consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely (Formula presented.), with σ ∈ (0,1). We obtain regularity results for the minimizers and for their free boundaries ∂u>0 using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler-Lagrange equations and extension problems
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
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