1,721,123 research outputs found
Fractional laplacians, perimeters and heat semigroups in carnot groups
No full textWe define and study the fractional Laplacian and the fractional perimeter of a set in Carnot groups and we compare the perimeter with the asymptotic behaviour of the fractional heat semigroup
Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application
No full textLiouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions
No full textWe consider a class of equations in divergence form with a singular/ degenerate weight. Under suitable regularity assumptions for the matrix A and f (resp. F) we prove Holder continuity of solutions which are even in y, and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the C0,alpha and C1,alpha a priori bounds for approximating problems. Finally, we derive C0,alpha and C1,alpha bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems
A fractional isoperimetric problem in the Wiener space
No full textWe introduce a notion of fractional perimeter in an abstract Wiener space and show that half-spaces are the only volume-constrained minimisers
Liouville type theorems and regularity of solutions to degenerate or singular problems part II: Odd solutions
No full textWe consider a class of equations in divergence form with a singular/degenerate weight. Under suitable regularity assumptions for the matrix A, the forcing term f and the field F, we prove Holder continuity of solutions which are odd in y, and possibly of their derivatives. In addition, we show stability of the C0,alpha and C1,alpha a priori bounds for approximating problems. Our method is based upon blow-up and appropriate Liouville type theorems
A fractional Gehring lemma, with applications to nonlocal equations
No full textTo Carlo Sbordone on his 65th birthday.This paper reports the content of a talk given by the second-named author at the Accademia dei Lincei on November 26, 2013.International audienceWe describe a fractional version of the classical Gehring lemma. As a consequence, new self-improving regularity properties of solutions to integrodifferential equations emerge
Nonlocal Equations with Measure Data
No full textInternational audienceWe develop an existence, regularity and potential theory for nonlinear integrodifferential equations involving measure data. The nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional -Laplacean operator with measurable coefficients. We introduce a natural function class where we solve the Dirichlet problem, and prove basic and optimal nonlinear Wolff potential estimates for solutions. These are the exact analogs of the results valid in the case of local quasilinear degenerate equations established by Boccardo & Gallou\"et \cite{BG1, BG2} and Kilpel\"ainen & Mal\'y \cite{KM1, KM2}. As a consequence, we establish a number of results which can be considered as basic building blocks for a nonlocal, nonlinear potential theory: fine properties of solutions, Calder\'on-Zygmund estimates, continuity and boundedness criteria are established via Wolff potentials. %In particular, optimal Lorentz spaces continuity criteria follow. A main tool is the introduction of a global excess functional that allows to prove a nonlocal analog of the classical theory due to Campanato \cite{camp}. Our results cover the case of linear nonlocal equations with measurable coefficients, and the one of the fractional Laplacean, and are new already in such cases
Nonlocal self-improving properties
No full textInternational audienceSolutions to nonlocal equations with measurable coefficients are higher differentiable. Specifically, we consider nonlocal integrodifferential equations with measurable coefficients whose model is given by integral(Rn)integral(Rn)[u(x) - u(y)][eta(x) - eta(y)]K(x, y) dx dy = integral(Rn) f eta dx for all eta is an element of C-c(infinity) (R-n), where the kernel K( . ) is a measurable function and satisfies the bounds 1/Lambda vertical bar x - y vertical bar(n+2 alpha) 1, while f is an element of L-loc(q)(R-n) for some q > 2n/(n + 2 alpha). The main result states that there exists a positive, universal exponent delta equivalent to delta(n, alpha, Lambda, q) such that for every weak solution u the self-improving property u is an element of W-alpha,W-2 (R-n) double right arrow u is an element of W-loc(alpha+delta,2+delta) (R-n) holds. This differentiability improvement is a genuinely nonlocal phenomenon and does not appear in the local case, where solutions to linear equations in divergence form with measurable coefficients are known to be higher integrable but are not, in general, higher differentiable. The result is achieved by proving a new version of the Gehring lemma involving certain families of lifted reverse Holder-type inequalities in R-2n and which is implied by delicate covering and exit-time arguments. In turn, such reverse Holder inequalities are based on the concept of dual pairs, that is, pairs (mu, U) of measures and functions in R-2n which are canonically associated to solutions. We also allow for more general equations involving as a source term an integrodifferential operator whose kernel does not necessarily have to be of order alpha
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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