Some evaluations of the fractional p-Laplace operator on radial functions
We face a rigidity problem for the fractional $ p $-Laplace operator to extend to this new framework some tools useful for the linear case. It is known that $ (-\Delta)^s(1-|x|^{2})^s_+ $ and $ -\Delta_p(1-|x|^{\frac{p}{p-1}}) $ are constant functions in $ (-1, 1) $ for fixed $ p $ and $ s $. We eva...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-02-01
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| Series: | Mathematics in Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mine.2023015?viewType=HTML |
| _version_ | 1797825626421329920 |
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| author | Francesca Colasuonno Fausto Ferrari Paola Gervasio Alfio Quarteroni |
| author_facet | Francesca Colasuonno Fausto Ferrari Paola Gervasio Alfio Quarteroni |
| author_sort | Francesca Colasuonno |
| collection | DOAJ |
| description | We face a rigidity problem for the fractional $ p $-Laplace operator to extend to this new framework some tools useful for the linear case. It is known that $ (-\Delta)^s(1-|x|^{2})^s_+ $ and $ -\Delta_p(1-|x|^{\frac{p}{p-1}}) $ are constant functions in $ (-1, 1) $ for fixed $ p $ and $ s $. We evaluated $ (-\Delta_p)^s(1-|x|^{\frac{p}{p-1}})^s_+ $ proving that it is not constant in $ (-1, 1) $ for some $ p\in (1, +\infty) $ and $ s\in (0, 1) $. This conclusion is obtained numerically thanks to the use of very accurate Gaussian numerical quadrature formulas. |
| first_indexed | 2024-03-13T10:56:43Z |
| format | Article |
| id | doaj.art-d76370a300244676bd26fbfd3f6bfd35 |
| institution | Directory Open Access Journal |
| issn | 2640-3501 |
| language | English |
| last_indexed | 2024-03-13T10:56:43Z |
| publishDate | 2023-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematics in Engineering |
| spelling | doaj.art-d76370a300244676bd26fbfd3f6bfd352023-05-17T01:30:49ZengAIMS PressMathematics in Engineering2640-35012023-02-0151123Some evaluations of the fractional p-Laplace operator on radial functionsFrancesca Colasuonno0Fausto Ferrari1Paola Gervasio 2Alfio Quarteroni 31. Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, piazza di Porta S. Donato, 5, 40126 Bologna, Italy1. Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, piazza di Porta S. Donato, 5, 40126 Bologna, Italy2. Dipartimento di Ingegneria Civile, Architettura, Territorio, Ambiente e di Matematica, Università degli Studi di Brescia, via Branze, 43, 25123 Brescia, Italy3. MOX, Dipartimento di Matematica, Politecnico di Milano, via Bonardi, 9, 20133 Milano, Italy 4. EPFL Lausanne, Switzerland (Professor Emeritus)We face a rigidity problem for the fractional $ p $-Laplace operator to extend to this new framework some tools useful for the linear case. It is known that $ (-\Delta)^s(1-|x|^{2})^s_+ $ and $ -\Delta_p(1-|x|^{\frac{p}{p-1}}) $ are constant functions in $ (-1, 1) $ for fixed $ p $ and $ s $. We evaluated $ (-\Delta_p)^s(1-|x|^{\frac{p}{p-1}})^s_+ $ proving that it is not constant in $ (-1, 1) $ for some $ p\in (1, +\infty) $ and $ s\in (0, 1) $. This conclusion is obtained numerically thanks to the use of very accurate Gaussian numerical quadrature formulas.https://www.aimspress.com/article/doi/10.3934/mine.2023015?viewType=HTMLfractional $ p $-laplacianstrong comparison principle$ p $-fractional torsion problemgaussian quadrature formulasnumerical approximation of singular integrals |
| spellingShingle | Francesca Colasuonno Fausto Ferrari Paola Gervasio Alfio Quarteroni Some evaluations of the fractional p-Laplace operator on radial functions Mathematics in Engineering fractional $ p $-laplacian strong comparison principle $ p $-fractional torsion problem gaussian quadrature formulas numerical approximation of singular integrals |
| title | Some evaluations of the fractional p-Laplace operator on radial functions |
| title_full | Some evaluations of the fractional p-Laplace operator on radial functions |
| title_fullStr | Some evaluations of the fractional p-Laplace operator on radial functions |
| title_full_unstemmed | Some evaluations of the fractional p-Laplace operator on radial functions |
| title_short | Some evaluations of the fractional p-Laplace operator on radial functions |
| title_sort | some evaluations of the fractional p laplace operator on radial functions |
| topic | fractional $ p $-laplacian strong comparison principle $ p $-fractional torsion problem gaussian quadrature formulas numerical approximation of singular integrals |
| url | https://www.aimspress.com/article/doi/10.3934/mine.2023015?viewType=HTML |
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