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ANALYSIS AND APPROXIMATION OF NONLOCAL DIFFUSION PROBLEMS WITH VOLUME CONSTRAINTS
, 2012
"... Abstract. A recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in Rn. The nonlocal vector calculus also enables striking analogies to be drawn between ..."
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Abstract. A recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in Rn. The nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion, including a notion of nonlocal flux. The ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory. In particular, it is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion we consider. The numerous applications elucidate different interpretations of the operator and the associated governing equations. For example, a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process. Sufficient conditions on the kernel of the nonlocal operator and the notion of volume constraints are shown to lead to a wellposed problem. Volume constraints are a proxy for boundary conditions that may not be defined for a given kernel. In particular, we demonstrate for a general class of kernels that the nonlocal operator is a mapping between a constrained subspace of a fractional Sobolev subspace and its dual. We also demonstrate for other particular kernels that the inverse of the operator does not smooth but does correspond to diffusion. The impact of our results is that both a continuum analysis and a numerical method for the modeling of anomalous diffusion on bounded domains in Rn are provided. The analytical framework allows us to consider finitedimensional approximations using both discontinuous or continuous Galerkin methods, both of which are conforming for the nonlocal diffusion equation we consider; error and condition number estimates are derived.
(0.2) ‖∇g‖ ∞ ≤ Cα,
, 2008
"... We correct an inaccuracy in the proof of a result in [Aus1]. 2000 MSC: 42B20, 46E35 Key words: CalderónZygmund decomposition; Sobolev spaces. We recall the lemma. Lemma 0.1. Let n ≥ 1, 1 ≤ p ≤ ∞ and f ∈ D ′ (R n) be such that ‖∇f‖p < ∞. Let α> 0. Then, one can find a collection of cubes (Qi), func ..."
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We correct an inaccuracy in the proof of a result in [Aus1]. 2000 MSC: 42B20, 46E35 Key words: CalderónZygmund decomposition; Sobolev spaces. We recall the lemma. Lemma 0.1. Let n ≥ 1, 1 ≤ p ≤ ∞ and f ∈ D ′ (R n) be such that ‖∇f‖p < ∞. Let α> 0. Then, one can find a collection of cubes (Qi), functions g and bi such that (0.1) f = g + ∑ and the following properties hold:
On the CalderónZygmund lemma for Sobolev functions
, 2008
"... We correct an inaccuracy in the proof of a result in [Aus1]. 2000 MSC: 42B20, 46E35 Key words: CalderónZygmund decomposition; Sobolev spaces. We recall the lemma. Lemma 0.1. Let n ≥ 1, 1 ≤ p ≤ ∞ and f ∈ D ′ (R n) be such that ‖∇f‖p < ∞. Let α> 0. Then, one can find a collection of cubes (Qi), func ..."
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We correct an inaccuracy in the proof of a result in [Aus1]. 2000 MSC: 42B20, 46E35 Key words: CalderónZygmund decomposition; Sobolev spaces. We recall the lemma. Lemma 0.1. Let n ≥ 1, 1 ≤ p ≤ ∞ and f ∈ D ′ (R n) be such that ‖∇f‖p < ∞. Let α> 0. Then, one can find a collection of cubes (Qi), functions g and bi such that (0.1) f = g + ∑ and the following properties hold: (0.2) ‖∇g‖ ∞ ≤ Cα, i bi (0.3) bi ∈ W 1,p
Author manuscript, published in "Journal of Geometric Analysis (2012) 11731210" WEIGHTED NORM INEQUALITIES ON GRAPHS
"... Abstract. Let (Γ, µ) be an infinite graph endowed with a reversible Markov kernel p and let P be the corresponding operator. We also consider the associated discrete gradient ∇. We assume that µ is doubling, a uniform lower bound for p(x, y) when p(x, y)> 0, and gaussian upper estimates for the iter ..."
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Abstract. Let (Γ, µ) be an infinite graph endowed with a reversible Markov kernel p and let P be the corresponding operator. We also consider the associated discrete gradient ∇. We assume that µ is doubling, a uniform lower bound for p(x, y) when p(x, y)> 0, and gaussian upper estimates for the iterates of p. Under these conditions (and in some cases assuming further some Poincaré inequality) we study the comparability of (I −P) 1/2 f and ∇f in Lebesgue spaces with Muckenhoupt weights. Also, we establish weighted norm inequalities for a LittlewoodPaleyStein square function, its formal adjoint, and commutators of the Riesz transform with bounded mean oscillation functions. hal00863448, version 1 19 Sep 2013 1.
A Meyers type regularity result for approximations of second order elliptic operators by Galerkin schemes
, 2012
"... Abstract. We prove a Meyers type regularity estimate for approximate solutions of second order elliptic equations obtained by Galerkin methods. The proofs rely on interpolation results for Sobolev spaces on graphs. Estimates for second order elliptic operators on rather general graphs are also obtai ..."
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Abstract. We prove a Meyers type regularity estimate for approximate solutions of second order elliptic equations obtained by Galerkin methods. The proofs rely on interpolation results for Sobolev spaces on graphs. Estimates for second order elliptic operators on rather general graphs are also obtained.