Periodic Homogenization for Nonlinear Integro-Differential Equations

Schwab, Russell W.

doi:10.1137/080737897

Cited by 48 publications

(60 citation statements)

References 19 publications

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“…In particular, as shown in [29,Section 4.6], when ellipticity occurs with respect to M ± L , then the min-max may be restricted to only utilize linear functionals (or linear operators) that also satisfy the extremal inequality in (6.1). This also appeared in a homogenization result by one of the authors in which they were unable to show that the limit operator had an explicit integrodifferential formula, but rather was only integro-differential and uniformly elliptic in the sense of [9, Definition 3.1] ( see the homogenization in [47]). Example 6.7.…”

Section: Some Examplesmentioning

confidence: 99%

Min–Max formulas for nonlocal elliptic operators on Euclidean Space

Guillen

Schwab

2020

Nonlinear Analysis

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An operator satisfies the Global Comparison Property if anytime a function touches another from above at some point, then the operator preserves the ordering at the point of contact. This is characteristic of degenerate elliptic operators, including nonlocal and nonlinear ones. In previous work, the authors considered such operators in Riemannian manifolds and proved they can be represented by a min-max formula in terms of Lévy operators. In this note we revisit this theory in the context of Euclidean space. With the intricacies of the general Riemannian setting gone, the ideas behind the original proof of the min-max representation become clearer. Moreover, we prove new results regarding operators that commute with translations or which otherwise enjoy some spatial regularity.Date: Monday 18th February, 2019 (This is the revised version per referee suggestions.) 2010 Mathematics Subject Classification. 35J99, 35R09, 45K05, 46T99, 47G20, 49L25, 49N70, 60J75, 93E20 .

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Section: Some Examplesmentioning

confidence: 99%

Min–Max formulas for nonlocal elliptic operators on Euclidean Space

Guillen

Schwab

2020

Nonlinear Analysis

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“…We also can mention homogenization results for singular kernels. We refer to the works of Cazeaux and Grandmont, 15 Schwab, 16 and Schwab 17 emphasizing that those ones deal with the coefficients involved in the equation and not with perforated domains as it is the case here. For random homogenization of an obstacle problem we cite the work of Caffarelli and Mellet.…”

Section: Corollary 1 Under Hypotheses Of Theorem 1 and Conditionmentioning

confidence: 99%

“…First, we pass to the limit in ‖ũ (t, ·)‖ L 2 (Ω) as → 0. From (16), with =ũ , and assuming condition (5), we have, due to (15), (18), and (20), that…”

Section: Proof Of Theorem 1 and Their Corollariesmentioning

confidence: 99%

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Nonlocal evolution equations in perforated domains

Pereira

2018

Math Methods in App Sciences

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In this paper, we study a nonlocal evolution equation posed in perforated domains. We consider problems of the form u t (t,We think about Ω as a fixed set Ω from where we have removed the subset A that we call the holes.Moreover, we take J as a nonsingular kernel. Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of Ω has a weak limit, ⇀  weakly * in L ∞ (Ω) as → 0, we analyze the limit of the solutions proving a nonlocal homogenized evolution equation.KEYWORDS asymptotic analysis, Neumann problem, nonlocal equations, perforated domains 6368

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“…Silvestre [46] assumes the min-max in proving regularity results for critical nonlocal equations, where the nonlocal term is of order 1, the same as the drift. One of the authors in [43] and [44] assumes the nonlocal operators to have a min-max so as to be able to set-up a corrector equation in homogenization for some nonlocal problems. Furthermore, in [43] and [44] a homogenized limit equation is proved to exist, but it is only known as an abstract nonlinear nonlocal operator of a certain ellipticity class, and its precise structure is left as an unresolved question.…”

Section: 4mentioning

confidence: 99%

Min–max formulas for nonlocal elliptic operators

Guillen

Schwab

2019

Calc. Var.

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In this work, we give a characterization of Lipschitz operators on spaces of C 2 (M ) functions (also C 1,1 , C 1,γ , C 1 , C γ ) that obey the global comparison property-i.e. those that preserve the global ordering of input functions at any points where their graphs may touch, often called "elliptic" operators. Here M is a complete Riemannian manifold. In particular, we show that all such operators can be written as a min-max over linear operators that are a combination of drift-diffusion and integro-differential parts. In the linear (and nonlocal) case, these operators had been characterized in the 1960's, and in the local, but nonlinear case-e.g. local Hamilton-Jacobi-Bellman operators-this characterization has also been known and used since approximately since 1960's or 1970s. Our main theorem contains both of these results as special cases. It also shows any nonlinear scalar elliptic equation can be represented as an Isaacs equation for an appropriate differential game. Our approach is to "project" the operator to one acting on functions on large finite graphs that approximate the manifold, use non-smooth analysis to derive a min-max formula on this finite dimensional level, and then pass to the limit in order to lift the formula to the original operator. This is the Director's cut, and it contains extra details for our own sanity.

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Periodic Homogenization for Nonlinear Integro-Differential Equations

Cited by 48 publications

References 19 publications

Min–Max formulas for nonlocal elliptic operators on Euclidean Space

Min–Max formulas for nonlocal elliptic operators on Euclidean Space

Nonlocal evolution equations in perforated domains

Min–max formulas for nonlocal elliptic operators

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