On fractional Poincaré inequalities

Hurri-Syrjänen, Ritva; Vähäkangas, Antti V.

doi:10.1007/s11854-013-0015-0

Cited by 52 publications

(72 citation statements)

References 17 publications

(14 reference statements)

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“…By the fractional Poincaré inequality (cf. [13]), we have |t − ξ| 1+ps dtdξ = cτ ps |v| p W s,p (0,T ;L 2 (Ω)) , which implies the desired assertion.…”

Section: Discussionsupporting

confidence: 52%

Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint

Jin

2018

IMA Journal of Numerical Analysis

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In this work, we present numerical analysis for a distributed optimal control problem, with box constraint on the control, governed by a subdiffusion equation which involves a fractional derivative of order α ∈ (0, 1) in time. The fully discrete scheme is obtained by applying the conforming linear Galerkin finite element method in space, L1 scheme/backward Euler convolution quadrature in time, and the control variable by a variational type discretization. With a space mesh size h and time stepsize τ , we establish the following order of convergence for the numerical solutions of the optimal control problem: O(τ min(1/2+α− ,1) + h 2 ) in the discrete L 2 (0, T ; L 2 (Ω)) norm and O(τ α− + 2 h h 2 ) in the discrete L ∞ (0, T ; L 2 (Ω)) norm, with any small > 0 and h = ln(2+1/h). The analysis relies essentially on the maximal L p -regularity and its discrete analogue for the subdiffusion problem. Numerical experiments are provided to support the theoretical results. optimal control, time-fractional diffusion, L1 scheme, convolution quadrature, pointwise-in-time error estimate, maximal regularity. arXiv:1707.08808v2 [math.NA] 21 Dec 2017 UK.

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“…By the fractional Poincaré inequality (cf. [13]), we have |t − ξ| 1+ps dtdξ = cτ ps |v| p W s,p (0,T ;L 2 (Ω)) , which implies the desired assertion.…”

Section: Discussionsupporting

confidence: 52%

Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint

Jin

2018

IMA Journal of Numerical Analysis

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“…The parameters τ and σ can be taken arbitrarily as long as τ, σ ∈ (0, 1), while 1 < p ≤ q ≤ np n−σp , p < n/σ. In [28] the domain Ω is assumed to belong to the class of John domains (for a definition and properties of this class see for instance [33]); this class is much broader than the one of star-shaped domains. Now we set τ = 1/2 in (4.7).…”

Section: Graded Meshesmentioning

confidence: 99%

A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations

Acosta¹,

Borthagaray²

2017

SIAM J. Numer. Anal.

251

308

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Abstract. This paper deals with the integral version of the Dirichlet homogeneous fractional Laplace equation. For this problem weighted and fractional Sobolev a priori estimates are provided in terms of the Hölder regularity of the data. By relying on these results, optimal order of convergence for the standard linear finite element method is proved for quasi-uniform as well as graded meshes. Some numerical examples are given showing results in agreement with the theoretical predictions.

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“…More recently, some authors have turned their attention to fractional generalizations of Poincaré and Sobolev-Poincaré inequalities, where a fractional seminorm appears instead of the norm in W 1,p (Ω). Indeed, in [13,18] the following inequalities were introduced for John domains:…”

Section: Introductionmentioning

confidence: 99%

Improved Poincaré inequalities in fractional Sobolev spaces

Drelichman¹,

Durán²

2018

Ann. Acad. Sci. Fenn. Math.

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On fractional Poincaré inequalities

Abstract: Weighted fractional Poincaré-type inequalities are proved on John domains whenever the weights defined on the domain depend on the distance to the boundary and to an arbitrary compact set in the boundary of the domain.

Cited by 52 publications

References 17 publications

Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint

Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint

A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations

Improved Poincaré inequalities in fractional Sobolev spaces

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