Optimal rearrangement problem and normalized obstacle problem in the fractional setting

Julián Fernández Bonder; Zhiwei Cheng; Hayk Mikayelyan

doi:10.1515/anona-2020-0067

Optimal rearrangement problem and normalized obstacle problem in the fractional setting

Julián Fernández Bonder^*
, Zhiwei Cheng
, Hayk Mikayelyan

^*Corresponding author for this work

School of Mathematical Sciences

Research output: Journal Publication › Article › peer-review

8 Citations (Scopus)

487 Downloads (Pure)

Abstract

We consider an optimal rearrangement minimization problem involving the fractional Laplace operator (-Δ)^s, 0 < s < 1, and the Gagliardo seminorm jujs. We prove the existence of the unique minimizer, analyze its properties as well as derive the non-local and highly non-linear PDE it satises -(-Δ)^sU - x-(-Δ)^sU+; 1 =U>0g, which happens to be the fractional analogue of the normalized obstacle problem Δu = xu>0.

Original language	English
Pages (from-to)	1592-1606
Number of pages	15
Journal	Advances in Nonlinear Analysis
Volume	9
Issue number	1
DOIs	https://doi.org/10.1515/anona-2020-0067
Publication status	Published - 1 Jan 2020

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Fractional partial differential equations
Obstacle problem
Optimization problems

ASJC Scopus subject areas

Analysis

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10.1515/anona-2020-0067

Optimal rearrangement problem and normalized obstacle problem in the fractional settingFinal published version, 467 KBLicence: CC BY

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title = "Optimal rearrangement problem and normalized obstacle problem in the fractional setting",

abstract = "We consider an optimal rearrangement minimization problem involving the fractional Laplace operator (-Δ)s, 0 < s < 1, and the Gagliardo seminorm jujs. We prove the existence of the unique minimizer, analyze its properties as well as derive the non-local and highly non-linear PDE it satises -(-Δ)sU - x-(-Δ)sU+; 1 =U>0g, which happens to be the fractional analogue of the normalized obstacle problem Δu = xu>0.",

keywords = "Fractional partial differential equations, Obstacle problem, Optimization problems",

author = "Bonder, \{Juli{\'a}n Fern{\'a}ndez\} and Zhiwei Cheng and Hayk Mikayelyan",

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AU - Bonder, Julián Fernández

AU - Cheng, Zhiwei

AU - Mikayelyan, Hayk

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N2 - We consider an optimal rearrangement minimization problem involving the fractional Laplace operator (-Δ)s, 0 < s < 1, and the Gagliardo seminorm jujs. We prove the existence of the unique minimizer, analyze its properties as well as derive the non-local and highly non-linear PDE it satises -(-Δ)sU - x-(-Δ)sU+; 1 =U>0g, which happens to be the fractional analogue of the normalized obstacle problem Δu = xu>0.

AB - We consider an optimal rearrangement minimization problem involving the fractional Laplace operator (-Δ)s, 0 < s < 1, and the Gagliardo seminorm jujs. We prove the existence of the unique minimizer, analyze its properties as well as derive the non-local and highly non-linear PDE it satises -(-Δ)sU - x-(-Δ)sU+; 1 =U>0g, which happens to be the fractional analogue of the normalized obstacle problem Δu = xu>0.

KW - Fractional partial differential equations

KW - Obstacle problem

KW - Optimization problems

UR - https://www.scopus.com/pages/publications/85085874293

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DO - 10.1515/anona-2020-0067

M3 - Article

AN - SCOPUS:85085874293

SN - 2191-9496

VL - 9

SP - 1592

EP - 1606

JO - Advances in Nonlinear Analysis

JF - Advances in Nonlinear Analysis

IS - 1

ER -

Optimal rearrangement problem and normalized obstacle problem in the fractional setting

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