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A NONLOCAL VECTOR CALCULUS WITH APPLICATION TO NONLOCAL BOUNDARY VALUE PROBLEMS
, 2009
"... Abstract. We develop a calculus for nonlocal operators that mimics Gauss ’ theorem and the Green’s identities of the classical vector calculus for scalar functions. The operators we treat do not involve the gradient of the scalar function. We then apply the nonlocal calculus to define variational fo ..."
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Cited by 21 (8 self)
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Abstract. We develop a calculus for nonlocal operators that mimics Gauss ’ theorem and the Green’s identities of the classical vector calculus for scalar functions. The operators we treat do not involve the gradient of the scalar function. We then apply the nonlocal calculus to define variational formulations of nonlocal “boundary ” value problems that mimic the Dirichlet and Neumann problems for secondorder scalar elliptic partial differential equations. For the nonlocal variational problems, we derive fundamental solutions, show how one can derive existence and uniqueness results, and show how, under appropriate limits, they reduce to their classical analogs.
Decay estimates for nonlocal problems via energy estimates
 Journal de Mathematiques Pures et Applique’es
"... Abstract. In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form ut(x, t) = Rd G(x ..."
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Cited by 18 (6 self)
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Abstract. In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form ut(x, t) = Rd G(x − y)(u(y, t) − u(x, t)) dy. For example, we will consider equations like, ut(x, t) =
The limit as p→∞ in a nonlocal p−Laplacian evolution equation. A nonlocal approximation of a model for sandpiles
"... Abstract. In this paper we study the nonlocal ∞−Laplacian type diffusion equation obtained as the limit as p→ ∞ to the nonlocal analogous to the p−Laplacian evolution, ut(t, x) = RN J(x − y)u(t, y) − u(t, x)p−2(u(t, y) − u(t, x)) dy. We prove existence and uniqueness of a limit solution that verif ..."
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Cited by 17 (8 self)
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Abstract. In this paper we study the nonlocal ∞−Laplacian type diffusion equation obtained as the limit as p→ ∞ to the nonlocal analogous to the p−Laplacian evolution, ut(t, x) = RN J(x − y)u(t, y) − u(t, x)p−2(u(t, y) − u(t, x)) dy. We prove existence and uniqueness of a limit solution that verifies an equation governed by the subdifferetial of a convex energy functional associated to the indicator function of the set K = {u ∈ L2(RN) : u(x)−u(y)  ≤ 1, when x − y ∈ supp(J)}. We also find some explicit examples of solutions to the limit equation. If the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L∞(0, T;L2(Ω)) to the limit solution of the local evolutions of the p−laplacian, vt = ∆pv. This last limit problem has been proposed as a model to describe the formation of a sandpile. Moreover, we also analyze the collapse of the initial condition when it does not belong to K by means of a suitable rescale of the solution that describes the initial layer that appears for p large. Finally, we give an interpretation of the limit problem in terms of MongeKantorovich mass transport theory. 1.
A MongeKantorovich mass transport problem for a discrete distance
 J. Funct. Anal
"... Abstract. This paper is concerned with a MongeKantorovich mass transport problem in which in the transport cost we replace the Euclidean distance with a discrete distance. We fix the length of a step and the distance that measures the cost of the transport depends of the number of steps that is nee ..."
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Cited by 10 (10 self)
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Abstract. This paper is concerned with a MongeKantorovich mass transport problem in which in the transport cost we replace the Euclidean distance with a discrete distance. We fix the length of a step and the distance that measures the cost of the transport depends of the number of steps that is needed to transport the involved mass from its origin to its destination. For this problem we construct special Kantorovich potentials, and optimal transport plans via a nonlocal version of the PDEformulation given by Evans and Gangbo for the classical case with the Euclidean distance. We also study how this problems, when reescaling the step distance, approximate the classical problem. In particular we obtain, taking limits in the reescaled nonlocal formulation, the PDEformulation given by EvansGangbo for the classical problem. 1. Introduction and
Variational Theory and Domain Decomposition for Nonlocal Problems
, 2011
"... In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the wellposedness of associated boundary value problems, proving a nonlocal Poincaré inequality. To determine the cond ..."
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Cited by 6 (2 self)
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In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the wellposedness of associated boundary value problems, proving a nonlocal Poincaré inequality. To determine the conditioning of the discretized operator, we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix. We then introduce a nonlocal twodomain variational formulation utilizing nonlocal transmission conditions, and prove equivalence with the singledomain formulation. A nonlocal Schur complement is introduced. We establish condition number bounds for the nonlocal stiffness and Schur complement matrices. Supporting numerical experiments demonstrating the conditioning of the nonlocal one and twodomain problems are presented.
Partial IntegroDifferential Equation in Granular Matter and its Connection with Stochastic Model
"... Abstract. Our aim is to introduce and study a new partial integrodifferential equation (PIDE) associated with the dynamics of some physical granular structure with arbitrary component sizes, like a sandpile or sea dyke. Our PIDE is closely related to the nonlocal evolution problem introduced in [F. ..."
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Cited by 4 (3 self)
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Abstract. Our aim is to introduce and study a new partial integrodifferential equation (PIDE) associated with the dynamics of some physical granular structure with arbitrary component sizes, like a sandpile or sea dyke. Our PIDE is closely related to the nonlocal evolution problem introduced in [F. Andreu et al., Calc. Var. Partial Differential Equations, 35 (2009), pp. 279–316] by studying the limit, as p → ∞, of the nonlocal pLaplacian equation. We also show the connection between our PIDE and the stochastic model introduced by Evans and Rezakhanlou in [L. C. Evans and F. Rezakhanlou, Comm. Math. Phys., 197 (1998), pp. 325–345.] for modeling the sandpile problem. Key words. partial integrodifferential equation (PIDE), physical granular structure, granular matter, nonlocal evolution problem, nonlocal pLaplacian, growing sandpile, particle system, stochastic model for sandpile
Classical, Nonlocal, and Fractional Diffusion Equations on Bounded Domains
 INTERNATIONAL JOURNAL FOR MULTISCALE COMPUTATIONAL ENGINEERING
, 2010
"... The purpose of this paper is to compare the solutions of onedimensional boundary value problems corresponding to classical, fractional and nonlocal diffusion on bounded domains. The latter two diffusions are viable alternatives for anomalous diffusion, when Fick’s first law is an inaccurate model. ..."
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Cited by 4 (0 self)
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The purpose of this paper is to compare the solutions of onedimensional boundary value problems corresponding to classical, fractional and nonlocal diffusion on bounded domains. The latter two diffusions are viable alternatives for anomalous diffusion, when Fick’s first law is an inaccurate model. In the case of nonlocal diffusion, a generalization of Fick’s first law in terms of a nonlocal flux is demonstrated to hold. A relationship between nonlocal and fractional diffusion is also reviewed, where the order of the fractional Laplacian can lie in the interval (0, 2]. The contribution of this paper is to present boundary value problems for nonlocal diffusion including a variational formulation that leads to a conforming finite element method using piecewise discontinuous shape functions. The nonlocal Dirichlet and Neumann boundary conditions used represent generalizations of the classical boundary conditions. Several examples are given where the effect of nonlocality is studied. The relationship between nonlocal and fractional diffusion explains that the numerical solution of boundary value problems, where the order of the fractional Laplacian can lie in the interval (0, 2], is possible.
Local and nonlocal weighted pLaplacian evolution equations with Neumann boundary conditions, Publ
 Mat
"... Abstract. In this paper we study existence and uniqueness for solutions of the nonlocal diffusion equation with Neumann boundary conditions ut(t, x) = Ω J(x − y)g x+ y 2 u(t, y) − u(t, x)p−2(u(t, y) − u(t, x)) dy in]0, T [×Ω, and for solutions of its local counterpart{ ut = div g∇up−2∇u in]0, T ..."
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Cited by 3 (1 self)
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Abstract. In this paper we study existence and uniqueness for solutions of the nonlocal diffusion equation with Neumann boundary conditions ut(t, x) = Ω J(x − y)g x+ y 2 u(t, y) − u(t, x)p−2(u(t, y) − u(t, x)) dy in]0, T [×Ω, and for solutions of its local counterpart{ ut = div g∇up−2∇u in]0, T [×Ω, g∇up−2∇u · η = 0 on]0, T [×∂Ω. We consider 1 ≤ p < ∞ and g ≥ 0. We pay special attention to the case in which g vanishes somewhere in Ω, even in a set of positive measure. 1.
Nonlocal anisotropic discrete regularization for image, data filtering and clustering
, 2007
"... In this paper, we propose a nonlocal anisotropic discrete regularization on graphs of arbitrary topologies as a framework for image, data filtering and clustering. Inspired by recent works on nonlocal regularization and on the TV digital filter, a family of discrete anisotropic functional regulariza ..."
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Cited by 3 (2 self)
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In this paper, we propose a nonlocal anisotropic discrete regularization on graphs of arbitrary topologies as a framework for image, data filtering and clustering. Inspired by recent works on nonlocal regularization and on the TV digital filter, a family of discrete anisotropic functional regularization on graphs is proposed. This regularization is based on the Lpnorm of the nonlocal gradient and the discrete pLaplacian on graphs. It can be viewed as the discrete analogue on graphs of the continuous pTV anisotropic functionals regularization formulations. After providing definitions and algorithms to resolve such a discrete nonlocal anisotropic regularization, we show its properties for filtering, clustering on different types of data living on different graph topologies (image, data). In particular we investigate the cases of p = 2, p = 1 and p < 1, this latter being very few considered in literature. Key words: anisotropic discrete regularization; nonlocal operators; graph pLaplacian; filtering; clustering 1
Fractional p–Laplacian evolution equations
"... Abstract. In this paper we study the fractional p−Laplacian evolution equation given by ut(t, x) = ..."
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Cited by 1 (1 self)
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Abstract. In this paper we study the fractional p−Laplacian evolution equation given by ut(t, x) =