On Nonlocal Variational and Quasi-Variational Inequalities with Fractional Gradient

Rodrigues, José Francisco; Santos, Lisa

doi:10.1007/s00245-019-09610-0

Cited by 12 publications

(16 citation statements)

References 9 publications

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“…As recently addressed by several authors [6,7,13,34,48,49], this fractional gradient seems to be the suitable notion, from a (merely) mathematical perspective, for such a differential object. In particular, it has been proved in [60] that formula (2) determines up to a multiplicative constant the unique object fulfilling some minimal consistency requirements from the physical and mathematical point of view, such as invariance under rotations and translations, s-homogeneity under dilations and some weak continuity properties.…”

Section: Introductionmentioning

confidence: 94%

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A new functional space related to Riesz fractional gradients in bounded domains

Bellido¹,

Cueto²,

Mora-Corral³

2022

Preprint

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We present a new functional space suitable for nonlocal models in Calculus of Variations and partial differential equations. Our inspiration are the Bessel spaces H s,p (R n ), which can be regarded as the completion of C ∞ c (R n ) functions u under the norm u p + D s u p . Here D s u is Riesz' s-fractional gradient (with 0 < s < 1) and 1 ≤ p < ∞ is the integrability exponent. Having in mind models in which it is essential to work in bounded domains Ω of R n , we define a truncation D s δ u of Riesz' fractional gradient, where δ > 0 represents the interaction distance (horizon, in the terminology of Peridynamics), so that D s δ u(x) is determined by the values of u in the ball of centre x ∈ Ω and radius δ. The corresponding functional space is defined as the completion of C ∞ c functions under the natural norm u p + D s δ u p . We prove a nonlocal fundamental theorem of Calculus, according to which u can be expressed as a convolution of D s δ u with a suitable kernel. As a consequence, we show inequalities in the spirit of Poincaré, Morrey, Trudinger and Hardy. Compact embeddings into L q spaces are also proved. As an application of the direct method of Calculus of Variations, we show the existence of minimizers of the associated energy functionals under the assumption of convexity of the integrand, as well as the corresponding Euler-Lagrange equation.

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Section: Introductionmentioning

confidence: 94%

“…If sp < n, the result is a particular case of Theorem 6.1. If sp ≥ n, we take any q satisfying 1 < q ≤ p, sq < n and p ≤ q * s , (48) which is easily seen to exist. Indeed, if n ≥ 2 we can take q = np n+sp , while if n = 1 we choose any q such that 1 < q < 1 s , p 1 + sp ≤ q, which exists because 1 < 1 s and p 1+sp < 1 s .…”

Section: Appendicesmentioning

confidence: 99%

A new functional space related to Riesz fractional gradients in bounded domains

Bellido¹,

Cueto²,

Mora-Corral³

2022

Preprint

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“…Here we are interested in complementing and extending some results of [10] on elliptic fractional equations of second σ-order, subjected to a σ-gradient constraint…”

Section: Introductionmentioning

confidence: 99%

“…which is even continuous if Ω is convex (see [11] for references). Although (2.14) is an open question in the general case of Theorem 2.1, for strictly positive bounded threshold g, it has been shown to hold in the sense of finite additive measures in [10], following the case σ = 1 of [3].…”

Section: Introductionmentioning

confidence: 99%

On a class of nonlocal problems with fractional gradient constraint

Azevedo¹,

Rodrigues²,

L.³

2022

Preprint

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We consider a Hilbertian and a charges approach to fractional gradient constraint problems of the type |D σ u| ≤ g, involving the distributional fractional Riesz gradient D σ , 0 < σ < 1, extending previous results on the existence of solutions and Lagrange multipliers of these nonlocal problems.We also prove their convergence as σ 1 towards their local counterparts with the gradient constraint |Du| ≤ g.

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“…Petrosyan and Pop [17] considered the obstacle problem for the fractional Laplacian with drift in the subcritical regime s ∈ ( 1 2 , 1), and Fernández-Real and Ros-Oton [11] studied the critical case s = 1 2 . There has also been some works on other types of nonlocal free boundary problems, like the work of Rodrigues and Santos [19] on nonlocal linear variational inequalities with constraint on the fractional gradient.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal fully nonlinear double obstacle problems

Safdari

2021

Preprint

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We prove the existence and C 1,α regularity of solutions to nonlocal fully nonlinear elliptic double obstacle problems. We also obtain boundary regularity for these problems. The obstacles are assumed to be Lipschitz semi-concave/semi-convex functions, and we do not require them to be C 1 . Our approach is to adapt a penalization method to be applicable to the setting of nonlocal equations and their viscosity solutions.

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On Nonlocal Variational and Quasi-Variational Inequalities with Fractional Gradient

Cited by 12 publications

References 9 publications

A new functional space related to Riesz fractional gradients in bounded domains

A new functional space related to Riesz fractional gradients in bounded domains

On a class of nonlocal problems with fractional gradient constraint

Nonlocal fully nonlinear double obstacle problems

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